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HOro ooyqeHHH». M., «I1poCBeIIJ.eHHe», 1977.
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06XO.n.HMblX npH3HaKoB MJIeKOnHTalOllI.HX HBJIHeTCH HaJIHqHe CKeJIeTa,
He06xo.n.HMbIM npH3HaKOM KBa)J.paTa HBJIHeTCH paBeHcTBO )J.JIHH ero )J.Ha-
rOHaJIeH H T. ,It.
4eM II1Hpe .l{aHHOe nOH5ITHe, TeM 60JlbWe ero oO'beM, TeM MeHbwe 3anac cBoiicTB,
06I1!HX BceM 06'beKTaM cooTBeTCTBYIOLUero KJlaCca. TaK, nOH!lTHe «napaJlJleJIOrpaMMt
II1Hpe nOHHTH!I «np5IMoyrOJlbHHK» - BC5IKHii np!lMoyroJlbHHK HBJI!leTC!I napaJlJleJIo-
rpaMMoM, HO 06paTHoe YTBep>K.l{eHHe JlO>KHO. n09TOMY JIIOOOH np!lMoyroJlbHHK 06J1a-
.l{aeT BceMH cBoiicTBaMH, npHcYUJ.HMH BceM napaJIJleJIOrp8MMaM (HanpHMep, .l{Haro-
HaJlH JIIoGoro np5IMoyrOJlbHHKa .l{eJI!lT .l{pyr .l{pyra nOnOJlaM). B TO >Ke BpeM5I y npJl
MoyroJlbHHKOB ecTb cBoiicTBa, KOTOpbIMH He oOJIa)l.aIOT OCTaJIbHble napaJIJleJIOrpaMMbI
(HanpHMep, l\JlHHbI .l{HarOHaJleii paBHbI). TO'lHO TaK >Ke nOH!lTHe «MJleKOnHTaroUJ.ee» -
lJaCTHbIii CJlY'laii nOH!lTH!I «n03BOHOlJHoe», H y MJleKOnHTalOUJ.HX eCTb cBOiiCTBa, OT-
CYTCTBYIOLUHe y OCTa.1JbHblX n03BOHOlJHblX (HanpHMep, HaJIHlJHe MOJlOlJHblX >KeJIe3),
a TaK>Ke CBOHCTBa, KOTOpblMH oOJla){aeT JlHIIIb lJaCTb OCTaJIbHblX n03BOHOlJHblX (Ha-
npHMep, TenJlOKpOBHOCTb, KOTOpa!l ecTb y nTHlJ;, HO OTCYTCTByer y npeCMblKalOLUHXCJI
HJlH pblO).
Cpe,lJ.H CYIlLeCTBeHHblX CBOHCTB ,ltaHHOrO nOHHTHH BbI.lteJIHIOTCH ero
xapaKmcpUCmUttCCKUC CBOHCTBa, T. e. CBOHCTBa, npHCYIlLHe 06'beKTaM
COOTBeTCTB)'IOllI.ero KJIaCCa H He npHCYIlLHe HHKaKHM HHbIM o6'beKTaM.
HanpHMep, CBOHCTBO paBeHCTBa MHH ,ltHarOHaJIeH HBJIHeTCH xapaKTe-
pHCTH'IeCKHM ,ltJIH npHMoyrOJIbHHKOB B KJIaCCe napaJIJIeJIOrpaMMOB.
31'0 03HaqaeT, qTO y Bcex npHMoyrOJIbHHKOB ,ltHarOH3JIH HMelOT paBHYIO
,ltJIHHY, npHtJeM JII060H napaJIJIeJIOrpaMM C ,ltHarOHaJIHMH paBHOH ,ltJIHHbI
HBJIHeTCH npHMoyrOJIbHHKOM. B TO )!{e BpeMH B KJIaCCe qeTbipexyrOJlb-
HHKOB 9TO CBOHCTBO He xapaKTepH3YeT npHMoyrOJIbHHKH - CYllI.ecTBY-
lOT qeTbipexyrOJIbHHKH C )J.HarOHaJIHMH paBHOH ,ltJIHHbI, He HBJIHlOllI.HecH
7

npHMoyrOJIbHHKaMH (HanpHMep, paBH060qHaH Tpanell,HH). .D.JIH nOHH-
THH «KBa)J.paT» CBOHCTBO paBeHCTBa )J.JIHH )J.HarOHaJJeH He HBJIHeTCH xa-
paKTepHCTHqeeKHM )J.3)!{e B KJJaCCe napaJJJJeJJorpaMMOB: Bce KBa.n.paTbi
HMelOT )J.HarOHaJJH paBHOH )J.JIHHbI, HO CYlll.eeTBYIOT napaJIJIeJIOrpaMMbI (a
HMeHHO, pa3HOCTopOHHHe npHMoyrOJIbHHKH), Y KOTOpbIX )J.JIHHbI )J.Haro-
HaJIeR paBHbI, XOTH OHH H He HBJIHIOTCH KBa)J.paTaMH.
MO)KeT CJIyqHTbCH, qTO CBORCTBOM a 06JIa.n.aIOT HeKOTopble (He 06H-
3aTeJIbHO Bce) 06'beKTbI )J.aHHoro KJIaCCa, H He 06JJa)J.aeT HH O)J.HH 06'beKT,
He npHH3)l.JIe)!{aIll.HH K 9TOMY KJIaccy. Tor)J.a 3TO CBOHCTBO Ha3bIBalOT
OocmamOttIiblM. npU3liaKOM. COOTBeTCTBYlOllI.erO nOHHTHH. HanpHMep,
paBeHCTBO )J.JIHH CTOpOH - ,ltOCTaTOtJHblR npH3HaK )J.JIH Toro, qT06b1 qe-
TbipexyrOJIbHHK 6bIJI napaJIJIeJIOrpaMMOM, HO 3TOT npH3HaK He HBJIHeTCH
He06xo)J.HMbIM: 3TO 3HaqHT, qTO eeJIH YCTaHOBJJeHO paBeHCTBO )J.JJHH Bcex
CTOpOH, TO '1eTbIpexyrOJJbHHK HaBepHHKa HBJJHeTCH napaJJ,'IeJJOrpaMMOM,
HO 06paTHoe HeBepHO - CYllI.eCTBYIOT napaJJJIeJJOrpaMMbI, CTOpOHbI KO-
TOpbIX HMelOT pa3JIHqHYIO )J.JJHHY.
MbI paCCMOTpeJIH pa3JIHqHbIe CBH3H Me)K.lty nOHHTHHMH H CBOHCTBaMH.
AHaJIOfHqHble CBH3H eCTb H Me)K.lty caMHMH CBOHCTBaMH. ECJIH 3a)J.aHbI
cBoikTBa a H , TO B03MO)!{Hbi TaKHe CJIyqaH:
a) CYlll.eeTBYIOT 06'beKTbI Bcex qeTbipex BH)J.OB, T. e. 06'beKTbI, 06-
JIa,ltalOllI.He 060HMH CBOHCTBaMH, 06'beKTbI, 06JIa)J.alOllI.He JIHIIIb CBORCT-
BOM a, 06'beKTbI, 06JIa.n.alOIlI.He JIHIIIb CBOHCTBOM , H 06'beKTbI, He 06JIa-
,ltalOllI.He HH O)J.HHM H3 3THX CBOHCTB. TaKHe CBOHcTBa Ha3bIB'aIOT IiC301JU-
CUMbLM.U. HanpHMep, CBOHCTBO HaTypaJIbHOrO '1HCJIa )J.e.'IHTbCH Ha 2 He-
3aBHCHMO oT CBOHCTBa )J.eJIHMOCTH Ha 3: eCTb '1HCJJa, )J.eJIHIlI.HeCH H Ha 2,
H Ha 3; )J.e.TJHIlI.HeCH Ha 2, HO He )J.eJIHIlI.HeCH Ha 3; )J.eJJHIlI.HeCH Ha 3, HO
He )J.eJIHIlI.HeCH Ha 2; HaKOHell" He )J.eJJHIlI.HeCH HH Ha 2, HH Ha 3;
6) JII060H 06'beKT, 06JIa)J.aIOIlI.HH CBOHCTBOM ex, 06JIa)J.aeT H CBORCT-
BOM . Tor)J.a rOBopHT, qTO  - cAcocmeuc CBOIICTBa ex. HanpHMep, ,lte-
JIHMOCTb HaTypaJIbHOrO qHCJIa Ha 5 - CJIe)J.CTBne ero )J.eJIHMOCTH Ha 10
(BcHKoe HaTypaJIbHOe qHCJlO, )J.eJIHi.u.eecH Ha 10, )J.eJIHTCH Ha 5). TOqHO
TaK )!{e HaJIHqHe CKeJIeTa y )\(HBOrO cYlll.ecTBa - CJIe)J.CTBHe TenJIOKpOB-
HOCTH, TaK KaK Bce TenJIOKpOBHble cYllI.eeTBa HMelOT CKeJIeT;
6') CBORCTBO ex - CJJe)J.CTBHe CBOHCTBa ;
B) JII060H 06'beKT, 06JIa)J.aIOIlI.HH CBOHCTBOM ex, 06,'Ia)J.ae-r H CBOHCT-
BOM , a JII060H 06'beKT, 06JIa)J.aIOIlI.HH CBOHCTBO:VI , 06JIa)J.aeT H CBORCT-
80M a. B 3TOM CJJyqae CBOHCTBa ex H  paeHOCUAbHbL. HanpHMep, CBOH-
CTBO qeTbIpexyrOJIbHHKOB HMeTb CTOpOHbI O)J.HHaKOBOH )J.JIHHbI paBHO-
CHJIbHO CBOHCTBY «)J.HarOHaJIH qeTbIpexyrOJJbHHKa nepneH)J.HKY JIHpHbI H
eJIHT ,ltpyr )J.Pyra nOnOJIaM»;
r) HH O,ltHH 06'beKT, 06JIa)J.aIOIlI.HH CBOHCTBOM a, He 06JIMaeT CBOH-
CTBOM . Tor,lta CBOHCTBa a H  Ha3b1BaIOT IiCC08M.CCmUM.blM.u. HanpHMep,
CBORCTBO )\(HBOrO CYllI.eeTBa HMeTb )\(a6pbl HeCOBMeCTHMO CO CBORCTBOM
TenJIOKpOBHOCTH;
,It) JII060H 06'beKT 06JIa)J.aeT O)J.HHM H TOJIbKO O)J.HHM H3 CBOHCTB a H .
B 3TOM CJIY'lae CBOHCTBa a H  Ha3bIBalOT npomueonOAO:J/CIiblM.U. HanpH-
Mep, cBoikTBa qeTHOCTH H HeqeTHOCTH HaTypaJJbHbIX '1HCeJI npOTHBono-
JIO)!{Hbi - BCHKoe HaTypaJIbHOe qHCJIO JIH60 '1eTHO, JIH60 HeqeTHO.
B

Y 0 pam B e H I' ,J
I. )].Jl5I nOH5ITH5I «np5IMoyroJlbHHK» YKa>KHTe ero xapaKTepHCTH'IeCKHe cBoACTBS.
2. Haii){HTe oOUJ.He cBoiicTBa TpaneU:HH H pOM08; TpeyroJlbHHKa H napaJlJleJIO-
rpaMMa; np5IMoyroJlbHHKa H Kpyra.
3. HaiiAHTe napbI paBHOCHJlbHbIX cBOiiCTB AJl5I HaTypaJlbHblx lJHCeJI.
4. HesaBHcHMbI JlH paBeHcTBO .II.JIHH CTOpOH H paBeHcTBO .II.JIHH .IIHarOHaJleA qe.
TWpexyroJlbHHKa? '
5. HesaBHcHMbI JlH pSBeHcTBo 1I.JIHH CTOpOH H nepneH)l.HKYJl5IpHOCTb )l.HarOHaJleii
lJpexyroJlbHHKa?
6. KaKYIO CB5I3b Me>K.ll.Y cBOikTB8MH lJeTbipexyroJlbHHKa YCTaHaBJlHBaeT 'N-
opeMa: «npOTHBOnoJiOJKHbIe cropOHbI napaJlJleJIOrpaMMa HMelOT paBHbIe )l.JlHHbI»
7. npHBe)l.HTe npHMepbl paBHOCHJlbHblX cBoiicTB.
8. npHBe)l.HTe npHMepbI npoTHBOnOJlO1KHbIX cBoikTB.
9. liTO MO>KHO CKa3aTb 0 BblnOJlHeHHH cBoiicTBa a 1I.JI!I napaJlJleJIOrpaMMoB, eCJIU:
a) CBoiicTBO a npHCYUJ.e BceM np5IMoyroJlbHHKaM; 0) CBOiicTBO a npHCYUJ.e HeKOTOp\jJM
np5IMoyrOJlbHHKaM; B) CYLUeCTBYIOT np!lMoyroJlbHHKH, He OOJla){aIOLUHe CBoiicTBOM it;
r) HH O)l.HH np!lMoyroJlbHHK He OOJla){aeT CBOiicTBOM a?
nO)l.OepHTe npHMepbI .11.J15I Ka>KllOrO CJlYlJa5I.
10. KaKHe H3BeCTHbIe BaM nOH5ITH5I onpelleJI5IIOTC5I CJlellYIOLUHMH npellJlO>KeHH5IMH:
8) MHoroyroJlbHHK C HaHMeHbIllHM lJHCJlOM cropOH; 0) HaHOOJlbIlla!l xoplla OI("'')KHOCT"i
B) OCb CHMMeTpHH OKPY>KHOCTH; r) TpeyroJlbHHK, HMelOUJ.Hii XOT5I Obi OllHY OCb CHM-
MeTpHH; A) )l.Be np!lMble, )l.eJI5ILUHe nJlOCKOCTb Ha 4 lJaCTH; e) xop)l.a. npoxoll!lIp.a!l
lJepes UeHTp OKPY>KHOCTH; >K) paBHocTOpOHHHii lJeTbipexyroJlbHHK, O)l.HH H3 yrJloB
KOTOporo np5IMoii?
II. YKa>KHTe JlOrHlJeCKHe OillHOKH. onycTHTe JlHIIIHHe CJlOBa HJlH )l.ooaBbTe He-
oOXO)l.HMble CJlOBa B CJle)l.YJOUJ.Hx onpe)l.eJIeHH5Ix: a) JlYlJ eCTb np!lMa5I, OrpaHHlJeHHa5I
C o)l.Hoii CTOpOHbI; 0) oTpe3KoM Ha3bIBaeTC!I np5IMa5I, OrpaHHlJeHHa5I C ooeix
cropOH; B) ceKYLUeii Ha3bIBaeTC5I oecKoHelJHa!l np!lMa5I, npOXO)l.5ILUa!l lJepe3 KaK"e-
HHOY)l.b ){Be TOlJKH OKPY>KHOCTH; r) KBallpaTOM Ha3b1BaeTC!I np!lMoyrOJlbHHK, llHaro-
HaJIH KOTOporo HMelOT paBHYJO 1I.JIHHY H lleJI!lT llpyr ){pyra nOnOJlaM; A) nJlOCKa!l
reoMeTpHlJeCKa!l «pHrypa Ha3bIBaeTC!I OKPY>KHOCTblO, eCJIH Bce ee TOlJKH ollHHaKooo
y.zJ.aJleHbI OT o)l.Hoii H Toii >Ke TOlJKH; e) llBa KOHrpY9HTHbiX yrJIa Ha3bIBaIOTC!I BepTH-
KaJIbHbIMH, eCJIH CTOpOHbI O)l.HOrO H3 HHX !lBJI!lIOTC!I npO)l.OJl>KeHH!lMH CTOpOH ){pyroro;
>K) )l.Ba yrJla Ha3b1BaIOTC!I CMe>KHblMH, eCJlH OHH HMelOT oOLUYIO CTOPOHY; 3) ){Ba yrJl8
H83bIBa1OTC!I CMe>KHbIMH, ecJlH cropoHa O){HOrO H3 HHX 5IBJI!leTC5I npO)l.OJl>KeHHeM
CTOpoHbI )l.pyroro.
12. 5IBJl5IeTC5I JlH B nOCJIellOBareJIbHOCTH '1embtpexyeOJlbHUK, napaJlJleJlOepaMM.
pOM6. npJIMoyeoJlbftUK. K8aiJpam Ka>K)l.oe CJlellylOLUee nOH5ITHe lJaCTHbIM CJlYlJaeM nj>e-
AhI.Ilymero? liTO Ha)l.O C)l.eJIaTb. lJTOObi 9TO HMeJIO MecTO?
13. 5IBJl5IeTC5I JlH nOH5ITHe paBHocTopoHHero TpeyroJlbHHKa lJaCTHbIM CJlYlJ8eM
nOH5ITH5I: a) npaBHJlbHOrO MHoroyroJlbHHKa; 0) np5IMoyroJlbHoro TpeyroJlbHHKa;
B) paBHoyroJlbHoro TpeyroJlbHHKa? .
14. Y nOTpeOJl5I!I CJlOBa «Bce», «HeKoTOpbIii», «Ka>Kllblii», «HH O){HH», YKa>KijTe
OTHoweHHe no OO'beMY Me>KllY CJle){YIOLUHMH nOH5ITH!lMH: a) np!lMoyroJlbHblii Tpe-
yroJlbHHK H paBHOOe)l.peHHbIii TpeyroJlbHHK; 0) paBHocTopOHHHii TpeyroJlbHHK H
p8BH06ellpeHHbIii TpeyroJlbHHK; B) np!lMoyroJlbHHK H pOMO; r) np!lMoyroJlbHHK H
KB8)l.paT; ll) Kpyr H reoMeTpHlJeCKa5I «pHrypa, OrpaHHlJeHHa5I OKPY>KHOCTblO.
15. 5IBJI!leTC5I JlH Ka>KllbIii KBallpaT np5IMoyroJlbHHKOM? 5IBJI!leTC!I JlH Ka>KJJ,blA
KBa)l.paT pOMOOM? 5IBJI!lIOTC!I JlH HeKoTOpbIe pOMObi KB8)l.paTaMH? 5IBJI!llOTc!I JlH Bee
poMObi KB8)l.paTaMH?
16. liTO MO>KHO CKa3aTb 0 npHH8)I.Jle>KHOCTH cBOiicTBa a np!lMoyroJlbHHKY, eCJIIt:
a) CBOHCTBO a npHCYLUe BceM napaJIJleJIOrpaMMaM. 0) CBoiicTBO a npHcyme HeKOTopblM
napaJlJleJIOrpaMMaM, B) CBOHCTBO a He npHCYUJ.e HH O)l.HOMY napaJIJleJIOrpaMMY?
2. JlorH4ecKHe paCCYJK)I.eHHB. JTI060e paCCY)!{)J.eHHe COCTOHT H3 ue-
nOQKH BbICKa3bIBaHHH, BbITeKalOllI.HX )J.pyr H3 )J.pyra no onpe)J.eJIeHHblM
npaBHJIaM. YMeHHe paccYJK.n.aTb, npaBHJIbHO 06OCHoBbIBaTb CBOH BbI-
BO)J.bI He06xo.n.HMO .11O)J.5IM JJI060H nPo<PecCHH. PaCCYJK,ltaTb qeJJOBeK
9

yqHTCH C TOrO MOMeHTa, KOr,ll.a Ha'lHHaeT rOBOpHTb, HO lI.eJIeHaJ1paBJJeH-
Hoe 06Y'IeHHe JIOrHKe paCCY)!{.lI.eHHH HaqHHaeTCH B llIKOJIe. Y)!{e Ha'laJJb-
HhlU KYpC MaTeMaTHKH npe.n.nonaraeT pa3BHTHe Y YtJam,HXCH HaBblKOB
npOBe.n.eHHH CpaBHeHHH, KJJaCCH<pHKall.HH 06'beKTOB, aHanH3a <paKTOB,
,ltOKa3aTeJJbCTBa npOCTei'fIIIHx yrBep)!{,lteHHH. JIOrHqHOCTb paCCY)K)teHHH
Tpe6yeTcH He TOJIbKO )l,lIH pellIeHHH MaTeMaTHqeeKHX 3 a,lta 'I , HO H ,lI.JIH
rpaMMaTHQecKoro aHaJIH3a, YCBoeHHH HaqaJJ npHpO,ltOBe.n.eHHH H T. ,It.
n03TOMY YQHTeJIb HaqaJIbHbIX KJJaCCOB ,lI.OJJ)!{eH 6bITb 3HaKOM C JIOrHKoA:,
T. e. C HaYKoA 0 3aKOHax H <PoPMax MbIllIJIeHHH, 06 OOIll.HX cxeMax pac-
cy)!{.n.eHHH.
OCHOBHbIe THnbl CY)!{.lI.eHHH H YM03aKJII04eHHH paCCMaTpHBalOTCH B
KJIaCCHtJecKOH JJOrHKe, C03,ltaHHOH ,lI.peBHerpeqeeKHM <pHJIOCocpoM ApH-
crOTeJIeM (384-322 rr. ,lto H. 3.).
npHBe,lteM HeKOTopbIe npHMepbI paCCY)!{.lI.eHHH.
a) Bce MJIeKOnHTalOllI.He HMelOT )!{eJIY,ltOK. JIbBbI - MJIeKOnHTalO-
IlI.He; CJJe)¥>BaTeJIbHO, Bce JIbBbI HMelOT )!{eJIY,lI.OK.
6) Y Bcex nTHlJ, ,lI.Ba KpbIJIa. Y OpJIOB ,lI.Ba KpbIJIa; CJIe,ltOBaTeJIbHO,
OpJIbI - nTHlI.bI.
B) Bce KHTbI MorYT JIeTaTb, n03TOMY HeKOTopbIe JIeTalOllI.He npe,ltMeThI
HBJIHIOTCH KUTaMH.
B npHMepax a) H 6) MbI nYTeM paccym,ll.eHHH: npHllIJIH K BepHbIM
bIBo,ll.aM (y JJbBOB eCTb meJIY,lI.OK, a OpJIbi - nTHlI,bI). Ho JIHllIb nepBoe
ll3 3THX paccY)K)teHHH npaBHJIbHO nocTpoeHo. OHO COCTOHT H3 Tpex
pa3J1H'IHblX BblCKa3bIBaHHH. B nepBOM H3 HHX rOBopHTCH 0 TOM, lITO
Bce npe,ll.MeTbI HeKOToporo KJIaCCa (KJIaCCa MJIeKOnHTaIOIlI.HX) 06JI3)l.a-
lOT HeKOTopbIM CBOHCTBOM (HMeIOT meJIY,lI.OK), T. e. lITO 3TO CBOHCTBO -
cyw.eCTBeHHoe CBoiicTBO MJIeKOnHTaIOIlI.HX. BTopoe BbICKa3bIBaHHe CO-
CTOHT B TOM, qTO nOHHTHe «JIeB - qaCTHbIH CJJY'IaH 06ll1.ero nOHHTHH
({MJIeKOnUTaIOIlI.He» (Bce JIbBbI - MJIeKOnHTaIOIlI.He). Ha 3TOM OCHOBaHHH
B TpeTbeM YTBep)!{.n.eHHH ,lI.eJIaeTCH BblBO,ll., qTO OTMeqeHHOe BbIllIe CBOU-
CTBO MJIeKOnHTalOllI.HX npHHa)tJJe)!{HT BceM JIbBaM (Bce JIbBbI HMelOT )!{e-
JiY,lI.OK). 061l1.8H <popMa 3Toro paCCY)!{,lI.eHHH, Ha3bIBaeMoro CUAA02U3M.OM.,
f1:lKOBa:
flYCmb 8ce npeOMembt KAacca a 06AaoalOm c80t1cm80M ex u nycmb 067JCM.
fU/.acca b - ttacmb 06'TJCM.a KAaCca a. T020a 8ce 06'TJCKmbl KAacca b 06-
Jla{)alOm C80Uctn80M. ex.
PaccY>K.lI:eHHe )!{e B npHMepe 6), XOTH H npHBeJIO Hac K npaBHJIbHOMY
bIBO,ll.Y, nOCTpoeHO HenpaBHJIbHO. Ty )!{e <PoPMY HMeeT H CJIe,ltYlOllI.ee
paCCY)!{,lteHHe: «Bce HerpbI qepHOBOJIOCbI; apMHHe qepHOBOJIOCbI; 3Ha-
ftHT, apMHHe- HerpbI». 3,l1.eeb 06a HCXO,ll.HbIX YTBep)K)teHHH HCTHHHbI, a
SbIBO,LI. JIO)!{eH.
cfJopMa paCCY)K)teHHH CtJHTaeTCH npaBHJIbHOH, eCJIH npH HCTHHHOCTH
HCXO,ltHbiX yrBep)!{.lI.eHHH OHa Bcer,lta npHBO,ll.HT K HCTHHHbIM BbIBO,ltaM.
Pa3YMeeTCH, eeJIH HCXO,ll.HbIe YTBep)!{.lI.eHHH JIOmHbI, TO H npaBHJIbHoe
no <PoPMe paCCY)!{.lI.eHHe MO)!{eT npHBeCTH K JIOmHOMY BbIBO,lty. TaKoB
i1pHMep B) - XOTH cpopMa .9TOrO paCCY)K)teHHH HCTHHHa, ero BbiBOA
AO)!{CH, TaK K8K OllIH60'lHO HCXO,ll.HOe YTBep)!()teHHe «Bce KHTbI Moryr
JteTaTb».
to

PaCCMOTpHM ee HeKOTOpbIe npHMepbI.
a) ECAU BCC a fl8AJl1Omcfl b, a BCC b fl8AfllOmcJl C, mo 8CC a !l8AJl1OmcJl,
(TaK, ecJIH Bce KBa,llpaTbI HBJJHIOTCH npHMoyrOJJbHHKaMH, a Bce npHMQ-
yrOJJbHHKH--napaJJJJeJJorpaMMbI,ToBCeKB3.ltPaTbI--napaJJJJeJJorpaM).
6) Ec.4U./iU oOliO a H.C fl8AJ/.CmcJl b, a Bee C MAJlIOmcJl a, mo HU OOfW 0
H.C M/IJU!mcJl b (HanpHMep, eeJJH HH y O,ltHOrO lIeTHOrO qHCJIa ,lteeflTHtJHaSi
sanHCb He KOHqaeTCH UH<PPOH 5 H Bee qHCJIa, .lJ.eJJHHecH Ha 4, tJeTHbI,
TO HH Y o,ltHoro qHCJIa, ,lteJJHIlI.erOCH Ha 4, ,ltecHTHqHaH 3anHCb He KOH-
lIaeTCH uH<PPOH: 5).
Paccy)K)teHHH a) H 6) npaBHJJbHbI. HenpaBHJJbHbIMH HBJJHIOTCH, Ha-
npHMep, TaKHe <PoPMbI paccy)!{)teHHH.
B) ECAU H.U oOliO b H.C fl8AJlemcJl a U H.U oOH.O C IiC fl8AflCmcJl b, mo He-
KOmopblC C H.C fl8AJl1OmcJl a.
r) ECAU 8CC a fl8AfllOmcJl b U H.U oOrw C He fl8AJlCmcJl a, mo H.U OOIiO 0
HC fl8AJlemCJl b.
B HenpaBHJJbHOCTH 3THX <PoPM paccy)!{)teHHH MO)!{HO y6e)tHTbCH, nQ-
.n06paB npHMepbI, KOr,lta C HX nOMOIll.blO 83 npaBHJJbHbIX yrBep)!{,lteHHH
nOJJyqalOTCH JJ02KHbIe BbIBO.ltbl. .lI.JJH <PoPMbI B) paCCMOTpHM TaKOH npH-
yep: «ECJJH HH o,ltH:a pbI6a He HBJJHeTCH MJJeKOnHTalOHM H HH O,ltHQ
Me)tBe,ltb He HBJJHeTCH PbI60H:, TO HeKOTopbIe Me)tBe,ltH He HBJJHIOTCH Mne-
KOnHTaIOIlI.HMH».
A )tJIH <PoPMbI r) onpoBepralOIlI.HM HBJJHeTCfl npHMep «ECJIH Bce JJ1I-
CHIJ.bI HBJJSlIOTCH MJJeKOnHTaIOIlLHMH H HH O,ltHH 3aSlll. He HBJIHeTCSI JJHCHu,eH,
TO HH O,ltHH 3aHU He HBJJfleTCH MJJeKOnHTaIOIIJ.HM».
nOJIyqeHHbIe HBHO HeBepHbIe pe3YJJbTaTbi nOKa3bIBalOT HenpaBHJIb-
HOCTb <pOpM B) H r).
YnpamUeUBJI
I. ]:tJUI CJle.nYX)III.HX paccY>KAeHHH nOCTpOHre HX JlOrH'lecK)'1O ctx>PMY H npo.
nepbTe, nO.ZJ.OHpa!l COOTBeTCTBYIOUJ.He npHMepbI, np8BHJIbHa JlH 3Ta ctx>pMa.
a) ECJlH Bce KBlIJI.paThi !lBJUIIOTC!I np!lMoyroJlbHHKaMH, TO HeKOTOpbIe np!lMoyrOJlb-
HHHH He BBJUIIOTCH KBlIJI.paTaMH;
6) ECJIH HH OAHH KHT He MO>KeT JleTaTb, TO HH O./lHH JleTaJOIII.Hii npe,!lMeT He !lB-
oIUIeTCSI XHTOM;
B) ECJIH Bcex JlbBOB MOJKHO npHpY'IHTb, a JlbBbI -. XHUJ.HHKH, TO Bcex XHIIJ.HHKOB
MO>KHO npHpY'IHTb;
r) ECJlH HeKOTOpbIX XHUJ.HHKOB MO:IKHO npHpy<lHTh, a Bee JlbBbI - XHUJ.HHKH, TO
HeKOTOpbIX JlbBOB MO>KHO npHpY'lHTb;
.n.) ECJlH Bcex XHUJ.HHKOB MO>K!fO npHpY'IHTh, a Beex JlbBOB MOJKHO npHpY'IHTb,
TO Bee JlbBbI - XHIII.HHKH;
e) ECJlH BCeX XHIIlHHKOB MOJKHO npHpy'IHTb, a Bee JlbBbI - XHIIlHHKB, TO scex
JlbBOB MOJKHO npHpy<lHTb;
:IK) ECJIH Bce 3aiiu.bI - XHIII.HHKH H HH OJlHH 3aBU He MO:IKeT JleTlITb, TO HeKOTOphie
XHIII.HHKH He Moryr neTlITb;
3) ECJIH HH OAHH KHT He !lBJI!leTCH pbl60ii H Bee m.YKH - pbIGbI, TO HH OAHa lIlyxa
He BBJI!leTC!I KHTOM;
H) ECJIH HH OAHH JleB He !lBJI!leTC!I pbIooii H Bce JlbBbI :lKHBYT Ha CYIIIe, TO HH O.llHa
pbI6a He 1KH1IeT Ha cyme.
B KaKHX HS 3THX npHMepoB HenpaBHJIbHO nocTpoeHHbIe pacCY:IK.Il.eHH!I npHBeJIH
K ./IO>KHbIM BbIBOAaM? K HCTHHHbIM BbIBOA8M?
2. TIpoaHaJIH3HPyiire CJIel!)'lOIII.He pacc)'JKJI.eHHB:
a) Bee nHCaTeJIH - Ae!lTeJIH HCKycCTBa. HeKOTOpbIe p,e!lTeJIH UCKyccTBa - Ta-
./I8BTJlBBbIe JIIOAH. 3Ha'lHT, HeKoTOpbIe nHCaTeJIH TaJIaHTJlHBbI.
11

0) Bce JlIO.llH CMepTHbi. Bce JlIO,lJ.H - >KHBbIe CYLUeCTBa. 3H8'1HT. BCe >KHBbIe cy-
UJ.eCTBa CMepTHbi.
B) Bce KOlliKH HBJI!lIOTC!l pbloaMH, a Y Bcex pblO - '1eTbipe HorH. 3Ha'lHT, Y KOWKIt
'1eTbipe HorH.
r) HeKOTOpbIe n03BOHO'lHble !lBJI!lIOTC!I MJleKOnHT8IOLUHMH, H HeKOTOpble n0300-
HO'lHble !lBJI!lIOTC!I JI!lrYIIIK8MiI. 3Ha'lHT, HeKOTOpble JI!lrYlliKH - MJleKOnHTaIOLUHe.
KaKHe H3 9THX paCCY>K.!IeHHH HMelOT npaBHJlbHYIO <jJopMY? B KaKHX H3 HHX HC-
TIIJlHbI HCXO){Hbie yTBep>K,lJ.eHH!I? B KaKHX H3 HHX HCTHHHbI BblBO,lJ.bI? MO>KHO JlH H3
.i1O>KHbIX YTBep>K.!IeHHH nyreM npaBHJlbHblX paCCY>K){eHHH nOJlYlJHTb HCTHHHbIH BblBO,lJ.?
3. HCTHHHble H JI01KHble BblCKa3b1BaHHH. B npe.n.bI.n.YIlI,eM nYHKTe
6bIJJO nOKa3aHO,tJTO .n.JIH npOBepKH BbIBo.n.a, nOJIYtJeHHoro paccY)K.lI.eHHeM,
Ha,lto, Bo-nepBbIx, y6e.n.IiTbcH B HCTHHHOCTH HCXO.n.HbIX npe,ltnOJIO)!{eHHii,
a BO-BTOPbIX, BbIHCHHTb, npaBHJIbHa JIIi HCnOJIb30BaHHaH <popMa JIOrH-
tJeeKoro BbIBo.n.a. B 06bI.n.eHHoH mH3HH MbI tJaCTO nOJJb3yeMcH BbIpame-
HHHMH, BbICKa3blBalOIIJ.HMH pa3JIHtJHYlO CTeneHb YBepeHHocTH B HCTHH-
HOCTH BbICKa3b1BaHHH. HanpHMep. rOBopHM: «51 nO'lTH YBepeH, liTO...»,
«B03MQ}KHO, '1TO ...», «He HCKJJIO'IeHO, '1TO ...», «MaJIOBepOHTHO, liTO...»
H T. .11.. I1cnOJIb30BaTb TaKHe BbICKa3bIBaHIiH npH aHaJIH3e paccy)!{.n.eHHH
6bIJIO 6bI 3aTpy.n.HHTeJIbHO, H nOTOMY B JIOrHKe paCCMaTpHBalOT JIHilib
YTBep)!{.n.eHHH, OTHOCHTeJIbHO KOTOpbIX MO)!{HO HaBepHHKa CKa3aTb, HC-
THHHbI OHH HJJH JJO)!{HbI. Bce .n.aJJbHeHiliHe onepaUHH H3)I. 3THMH BbICKa-
3bIBaHHHMH npOH3Bo.n.HTCH 6e3 BbIHCHeHHH HX CMbICJIa, a onHpaHCb JIHllib
Ha TO, HCTHHHbI 3TH BbICK33bIBaHHH HJIH JIOJKHbI. 31'0 CYIIJ.eCTBeHHO no-
TOI\1Y, tJTO CeH'IaC .n.,'IH pellieHHH MHOfHX 3a.n.a'l JIOfHtJeeKOfO aHaJIH3a
npHMeHHIOT 6bICTpo.n.eHCTBYIOIIJ.He BbI'IHCJIHTeJJbHbIe MaIlIHHbI, a OHH
nepepa6aTbIBalOT nOJIy"eHHYIO HH<popMaUIiIO, He HCnOJIb3YH CMbICJIa BbI-
CKa3bIBaHHH, 3 JIHilib nOJIb3YHCb onpe,lteJIeHHbIMH npaBHJIaMH (aJIrOpHT-
MaMH). '
npaBHJI, no KOTOpbIM npe06pa3YIOTCH BbICKa3bIBaHHH, HanOMHHalOT
npaBH.'Ia npe06pa30BaHHH aJIre6paH'IeCKHX BbIpa)!{eHHH. npaBHJIa npe-
q6pa30BaHHH BbICKa3bIBaHHfi COCTaBJIHIOT '1aCTb M.amCM.amUf.{,CCKot1 AO-
ZUKU. B M3TeMaTH'IeCKOH JJOrHKe 8bLCKa3bl8aIiUCM. Ha3bIBalOT JII060e no-
BeeTBOBaTeJIbHOe npe.n.JIO)!{eHHe, 0 KOTOpOM MO)!{HO CKa3aTb, HCTHHHO OHO
HJIU JIO)!{HO. TaKHM 06pa30M, npe.n.JIO)!{eHHH «Bce JIbBbI HMelOT )!{eJIY,ltOK»
H «Bce KHTbI YMelOT JIeTaTb» HBJIHIOTCH BbICKa3bIBaHHHMH. DpH 3TOM nep-
Boe H3 HHX HCTHHHO, a BTopoe JIO)!{HO.
B 06bl){eHHOH >KH3HH npe)I.JIO>KeHHIO «51 B3!1J1 KHHry no MaTe.!ITHKe co CTOJla.
MO>KHO npH;\aTb MHoro pa3J!HQH!?IX OTTeIlKOB CMblCJla, CTaBH no-pa3HoMY JlOrHlJeCKHe
Y,lJ.peHH!I. H!.'np!lMep, MO{KHO nO,lJ.lJepKIIyTb rOJlOCOM, lJTO KHHry B3!1J1 J!, a He KTO-
HHY,lJ.b ){pyroii, JlH60 lJTO KHHra fi8J!ma, a He nOJlO>KeHa Ha CTOJI, JlHOo '1TO B3!1Ta HMeHHO
KHuea, a He, C!i:,a)l{eM, Tp8,lJ.b, JlHOO '1TO 9Ta KHHra 1W MameMamUKe, JlHOo, HaKOHeu,
'1TO KHHra B3Ta co cmO/la, a He, CKa>KeM, H3 II1Ka<f>a. Bce 9TH O'I'TeHKH HeCYUJ.eCTBeHHI,(
AJI)! MaTeMaTHlJeCKOH JlOrHKH - 3,lJ.eCb Ba>KHO JlHIIIb O.llHO, HCTHHHO 9TO BblCKa3b1BaHHe
wiil HeT.
He CtJHTalQTCH BbICK3bIB3HHHMH BonpOCHTeJIbHbIe H BOCKJIHUaTeJ.Ib-
Hble npe.n.JJOJKeHHH (<<KoTOpbIH 'lac?», «LLa 3paBcTBYeT COJIH»..e!»),
n9fKo.1bKY 0 HHX HeJJb3f1 CKa3aTb, HCTHHHbI OHH HJIH JIO)!{HbI. npeA
JIo>i<eHHH, cop.ep)!{aIll.Iie nepeMeHHbIe, KOTopbIe MorYT npHHHMaTb pa3-
JIHtJHbIe 3HaQeHHH, TO)!{e He ClIHTalOT BbICKa3bIBaHH5IMH, TaK KaK npH
12

O,ltHIiX 3HatJeHIiHX nepeMeHHbIX OHIi CTaHOBHTCH HCTIiHHbIMH BbICKa3bIBa-
HIiHMIi, a npli ,ltPyrIix - JJO)!{HbIMIi. HanpHMep, He HBJJHeTCH BbICKa3bI-
BaHIieM npe.n.JJO)!{eHHe «CJJOBO x - rJJarOJl:t. ECJJH x 03HaQaeT «tJIiTaTb:t,
TO CKa3aHHoe HCTHHHO, a eCJJIi x 03HatJaeT «KHHra», TO CKa3aHHoe JJO)!{HO.
no TeM )!{e C006pa)!{eHIiHM He HBJJHeTCH BbICKa3bIBaHHeM YTBep)!{.n.eHHe
«CY!\1Ma. tJIiCeJJ x Ii y paBHa 5» - OHO CTaHOBHTC5.I HCTIiHHbIM, eCJJH
x=2, y=3, H JJO)!{HbIM, eeJJIi x=3, y=4.
BbIcKa3bIBaHIiH MorYT 0603HatJaTbCH He TOJJbKO C nOMOIIJ.blO CJJOB,
HO H C nOMOIll.blO pa3JJHtJHbIX CIiMBOJJOB, HanpHMep:
3+ 2=5 (IicTIiHHoe BbIcKa3bIBaHIie);
-7<-8 (JJO)!{HOe BbIcKa3bIBaHIie); .
22-5:;6:18 (HcTHHHoe BbIcKa3bIBaHHe);
H 2 S0 4 - KIiCJJOTa (IicTHHHoe BbIcKa3bIBaHIie).
npH 3TOM O.n.HO Ii TO )!{e BbICKa3bIBaHHe MO)!{eT 6bITb CCPOPMYJIIiPO;
BaHO MHOfIiMH paBHOCIiJJbHbIMIi cnoc06aMIi. HanpHMep, BbICKa3bIBaHIiH
«ECJJIi K .n.BYM npIi6aBIiTb TPH, nOJIYtJHTCH nHTb», «2+3=5», «l I + II I =
=V», «Two plus three equals five» 03HatJalOT O.n.HO Ii TO )!{e. LLBa
BbICKa3b1BaHIiH 3KBHBaJJeHTHbi no CMbICJJY, eCJJIi OHIi HBJJHIOTCH
o.n.HOBpeMeHHO IiCTHHHbIMIi IiJJIi o.n.HOBpeMeHHO JIO)!{HblMH. HanpIiMep,
Bce paBHO CKa3aTb: «BtJepa 6bIJJ qeTBepr» IiJJIi «3aBTpa 6y.n.eT cy66OTa:t-
eeJJIi cero.n.HH nHTHIiua, TO 06a BbICKa3bIBaHHH HCTHHHbI, a B JJI060A
.n.pyroH: .n.eHb OHIi 06a JJO)!{HbI.
BbIcKa3bIBaHIiH MorYT OTHOCHTbCH KaK K OT.n.eJJbHblM 06'beKTaM H
HX cBoikTBaM, TaK Ii K KJJaCCaM 06'beKTOB H IiX cBoikTBaM. HanpHMep:
«4HCJJo,295 .n.eJJ!iTcH Ha 3», «4eTbIpexyrOJJbHIiK ABCD, H306pa>KeHHbIH
Ha .n.aHHOM pIicYHKe,- napaJJJJeJJOrpaMM», «BCHKHH npHMoyrOJJbHHK
HBJ1HeTCH reOMeTpIitJeeKoH <PIirYPOH», «HeKoTopbIe ytJeHHKIi B IlIKo-
J1e - OTJJIitJHIiKH», «ECJJIi MHKOTb ap6Y3a KpaCHaH, TO ap6Y3 Cne-
JI bUI» .
B .n.aJJbHeHIlIeM Mbl 6y.n.eM 0603HatJaTb BbICKa3bIBaHIiH 60JJbIlIIiMH
6YKBaMH JJaTHHCKoro aJ1<paBIiTa Ii, KaK CKa3aHO BbIIlIe, He HHTepeeYHCb
CMbICJJOM 3TMX BbICKa3hlBaHIiii, 6y.n.eM ytJHTbIBaTb JJIiIlIb HX MCTIiHHOCTb
IiJJH JJO)!{HOCTb. ECJ1H 3a,ltaHbI BbICKa3bIBaHIiH A Ii B, TO H3 HIiX MO>KHO
COCTaBHTb HOBbIe BbICKa3bIBaHIiH, IiCnOJJb3YH CBH3KIi «Ii:t, «IiJJIi» , «ee-
J1Ii ..., TO. ..», «JJIi60..., JJIi60.. .», «B TOM H TOJJbKO B TOM CJJytJae, eeJJIi:t,
a TaK)!{e tJaCTIiuy «He». nYCTb, HanpIiMep, A 03HatJaeT BbICKa3bIBaHIiC
«Ceiiqac COJJHetJHO», a B - BbICKa3bIBaHIie «CeiitJac BeTpeHo». Tor.n.a
BbICKa3bIBaHHe «A Ii B» 03HaqaeT: «CeiitJac COJJHetJHO H BeTpeHO», BbI-
CKa3bIBaHHe «ECJJH He A, TO He B» - «ECJ1Ii ceH:tJac He COJJHetJHO, TO H
He BeTpeHO:t.
TaKIie BbICKa3bIBaHHH Ha3bIBaIOT cocmaelibtM.U, a BXO,ltHIIJ.Me B HIiX
BblCKa3bIBaHIiH A H B - 8ACM.ClimapftbtMU 8blCKa8b16aHWlM.U. LLBa co-
CTaBHblX BbICKa3bIBaHHH A Ii B Ha3bIBalOTCH paeIiOCUAbHblM.U (IiJlH f:1K-
8U8aAClimliblM.U) , ecJJIi OHIi o.n.HoBpeMeHHo IiCTIiHHbI IiJJIi o.n.HoBpeMeHHo
J10>KHbI npli JII06bIX npe.n.nOJJO)!{eHIiHX 06 HCTIiHHOCTIi BXO.n.HIlI.IiX B HIiX
.9J1eMeHTapHbIx BbIcKa3bIBaHIiH. B 3TOM cJJytJae nIilliYT: A =B.
nepBOM 3 a.n.a'I eff, KOTOpylO HaM Ha.n.o peIlIIiTb, HBJJHeTCH YTOtJHeHIie
CMhICJ1a COCTaBHbIX BbICKa3bIBaHHH: He06xoIiMO onpe,lteJJHTb, npH Ka-
13

KHX npe.n.nOJIO)!{eHHHX 06 HCTMHHOCTM A H B 3TH BbICKa3bIBaHMH HMO
CtJMTaTb IiCTMHHbIMIi, a npM KaKMX JIO)!{HbiMM. 3aTeM Mbi BblHCHMM, Ka-
KHe M3 COCTaBHblX BblCKa3bIBaHMM, COCTaBJJeHHblX li3 .n.aHHblX 3JIeMeH-
TapHbIX BblCKa3b1BaHIiH, 3KBMBaJIeHTHbI, T. e. o.n.HOBpeMeHHO MCTMHHbI
HJIM o.n.HOBpeMeBHO JIO)!{Hbi.
YnpaHCReUIiJi
1. Y Ka>KHTe cpe,lJ.H CJle,lJ.YIOUJ.Hx npe)J.JIo>KeHHii BblCKa3b1BaHHII:
a) JIYHa - cnYTHHK 3eMJlH.
0) Bce CTy){eHTb! JlK>6!1T MareMaTHKY.
B) OpHHeCH MHe, nO>KaJIyilcTa, KHHry!
r) HeKOTopbIe JlIO)l.H HMelOT ro.ny6b1e r.naaa.
)I.) BbI ObIJlH B TeaTpe?
2. KaKHe H3 CJle)l.YIOUJ.Hx npe)J.JIO>KeHHii IIBJlIlIOTCII HCTHHHYMH H KaKHe .nO>KHblMH
BblcKa3bIBaHHIIMH?
a) 4HCJlO -2 MeHbIIJe '1HCJla O.
6) 4acTHoe OT )l.eJIeHHII '1HCJla 7 Ha HYJlb paBHo O.
B) CYMMa '1HCJla 5 H '1HCJla x paBHa 10.
r) CYUJ.ecTByeT TaKoe ,lJ.eilcTBHTeJIbHOe '1HCJlO x, '1TO 2x+5= 19.
,lJ.) (13-2.7).4=-4.
e) Bce TpeyrOJlbHHKH paBH06e,lJ.peHHbIe.
3. Cpe,n;h CJle,lJ.YIOUJ.Hx BbICKa3b1BaHHii YKa>KHTe napbl 9KBHB8JJeHTHblX no CMbICJlY:
A - «CerO)l.H!I BOcKpeceHbe»; B - «B'Iepa OblJla cyG6oTa»;
C - «3aBTpa 6Y)l.eT BTOpHHK»; D - «3aBTpa oY,lJ.eT nOHe,lJ.eJIbHHK».
4. JJ:aHbI BbICKa3b1BaHHII - A: «3eMJlII BpaUJ.aeTcH BOKpyr CoJlHua» H B: «3eMJIII
HMeeT <jJopMY II1apa». 06pa3yilre H3 ,lJ.aHHbIX BbICKa3bIBaHHii COCTaBHble H nO.ll'lepKHHTe
enOBa, npH nOMOUJ.H KOTOpblX OHH oGpaaoBaHbI.
5. Cpe,lJ.H CJle)l.YIOUJ.Hx BblcKa3bIBaHHii YKa>KHre COCTaBHbIe; BbI)I.eJIHTe B HHX 9J1e-
MeHTapHble, 6003Ha1JHB Ka>K)l.Oe H3 HHX 6YKBOiI; 3anHIIIHre C nOMOUJ.blO OYKB Ka>KJ:\Oe
rocTaBHoe BblCKa3bIBaHHe.
a) Ha ypoKe MareMaTHKH Y'laUJ.HecII OTBe'laJlH Ha BonpocbI Y'IHTeJIII H nHCaJIH
C3MOCTOllTeJIbHYIO pa60TY.
0) Mbi noil,neM KaTaTbC!I Ha KOHbKax HJlH Ha JlbI>Kax.
B) ECJlH B '1eTbIpexyrOJlbHHKe )l.HafOHaJlH HMelOT paBHYIO )J.JIHHY. TO 9TOT '1eTbI-
pexyrOJlbHHK - pOM6.
r) -17<0.
,n) 4HCJlO 15 )l.eJIHTC!I Ha 3 TOr,lJ.a  TOJlbKO TOr,lJ.a, Kor.u.a CYMMa uH«pP '1HCJla 15
AeJIHTCII Ha 3.
* 2. OIIEPAl(HH HA,n BbICKA3bIBAHHJlMH
1. OTpHaMMe BblCKa3b1BaHHH. ECJJM A - HeKOTopoe BblCKa3b1Ba-
HMe, TO, YTBep)!{.n.aH, 'ITO OHO JIO)!{HO, Mbi nOJIY'IaeM BOBoe BbICKa3bIBa-
HHe, KOTOpoe Ha3bIBalOT ompUlaliUCM. BblCKa3b1BaHMH A H 0603Ha'IaIOT
CIiMBOJJOM A. CHMBOJJ It 'IMTaIOT: «He-A». TaK, eCJJM A - BblCKa3bIBaHIie
«PeKa AHrapa - npMToK EHHceH», TO ero OTpMlI.aHHeM 6y.n.eT BblCKa3bI-
BaHIie A: «HeBepHo, 'ITO peKa AHrapa - npIiTOK EHIiceH». 31'0 BbICKa-
BbIBaHIie MO)!{eT 6bITb c<PoPMYJIMpOBaHo Ii B TaKOM BM,lte: «PeKa AHrapa
He HBJIHeTCH npIiTOKOM EHIiceH». B .n.aHHOM npHMepe BbICKa3bIBaHIie A
BCTIiHHO, a ero OTpHuaHIie, BbICKa3bIBaHMe A ,- JIO)!{HO.
OTpHuaHMeM JIO)!{HOrO BbICKa3bIBaHMH B (<<32=6») HBJIHeTCH MCTMH-
Hoe BblCKa3bIBaHMe B (<<3¥6»).
14

TaKUM. 06pasoM, ec.llU HCKOmopoe 8blCK£13bllJ(1IiUe UCfl1UIiHO, mo eeo
OmpUlfaHUC .Il0J/CHO, U Ha06opom. 3TOT BbIBO.n. MO)!{HO 3anHcaTb "PH
nOMOIll.H Ta6JIHUbI, B KOTOpOH 6YKBa «11» 03HatJaeT HCTHHHoe BbIcKa3i.r-
BaHHe, a 6YKBa «Jb - JIO)!{Hoe. Ta6JIHUbi nO,lt06HOrO BH,lta Ha3bIBaIOT
ma6.1lUl¥lMU ucmUHHocmu.
A I A
" I JI
JI I "
Pa306paHHbIe BbIIlIe npHMepbI HarJIH,ltHO nOKa3bIBa1OT, qTO OTpHuaHHe
HeKOToporo BbICKa3bIBaHHH A MO)!{HO nOJIyqHTb, ecJIH nepe,lt,ltaHHbIM BbI-
CKa3bIBaHHeM nOCTaBHTb CJIOBa «HeBepHO, qTO» HJIH K cKa3yeMOMY H3 BbI-
CKa3bIBaHHH A ,lt06aBHTb qaCTHIl.y «He».
ECJIH BbICKa3bIBaHHe A y)!{e CO.n.ep)!{HT qaCTHIl.Y «He» nepe,lt cKa3ye-
MbIM, TO ,ltJIH 06pa30BaHHH BbICKa3bIBaHHH A ee H3)l.O oT6poCHTb. Ha-
npHMep, ecJIH A - BbICKa3bIBaHHe «I1BaHoB He peIlIHJI 3a,ltaqy», TO BbI-
CKa3bIBaHHe If 6y,lteT TaKHM: «I1BaHoB peIlIHJI 33)1.aqy».
nYCTb A - HeKOTopoe BbICKa3bIBaHHe. Ero OTpHuaHHe it TaK)i{e
HBJIHeTCH BbICKa3bIBaHHeM, H nOTOMY O)!{HO 06pa30BaTb oTpHuaHHe
BbICKa3bIBaHHH .4, T. e. BbICKa3bIBaHHe .4, KOTopoe Ha3bIBaIOT oeOUHb/.Jt
ompUl{aHUCM. BbICKa3bIBaHHH A. TaK, eCJIH A eCTb BbICKa3bIBaHHe «17 -
npOCToe qHCJIO», TO oTpHuaHe A HMeeT BH,lt: «17 He HBJIHeTCH npOCTbIM
qHCJIOM», a BbICKa3bIBaHHe it 6y,lteT TaKHM: «HeBepHO, qTO qHCJIO 17-
HenpocToe». qro BbICKa3bIBaHHe MO)!{HO C<P<>PMYJIHpOBaTb KOpOqe!
«17 - npocToe qHCJJO>>' TaKHM 06pa30M, C,lteJIaB ,ltBoHHOe OTpHuaHHe,
MbI CHOBa BepHYJIHCb K BbICKa3bIBaHHIO A.
Bo6w.e JIIo60e BbICKa3bIBaHHe A paBHOCHJIbHO BbICKa3bIBaHHIO X,
T. e. A=A. .lI.eiiBHTeJIbHO, COCTaBHM TaMHIl.Y HCTHHHOCTH ,ltJIH B
CKa3bIBaHHH A H if. 113 3TOH Ta6JIHUbI BH.n.HO, lITO BbICKa3bIBaHHH A H if
JIH60 O,ltHOBpeMeHHo HCTHHHbI, JIH60 O,ltHoBpeMeHHo JIO)!{HbI H, CJIe.n.OBa-
TeJIbHO, OHH paBHOCHJIbHbI.
A I A I A

JII"IJI
YnpamueuuJI
I. OGpa3yiITe OTpHUaHHe Ka>K)loro H3 CJleAYJOUJ.HX BblcKa3blBaHHii; YKa>KHTe,
UBJI!leTCH HCTHHHbIM AaHHoe BbICKa3b1B8HHe HJlH ero oTpHuaHHe:
A: «4HCJlO 3 - ]:leJIHTeJlb '1HCJla 523»;
B: «3Ha'leHll'e BblpIDKeHHH 18: (7-7) He CYUJ.ecTByen;
C: «JlI060e ]:leiiCTBHTeJIbHOe 1JHCJlO x HBJlHeTCJ1 KopHeld ypaBHeH!!H 5x-12=«nJ
D: «CYUJ.eCTBYIOT napaJlJleJIOrpaMMbI C KOHrpy9H1'iI!JlMH liHarOHaJIJ1MH».
lB

2.)].aHO BbICK83bIBaHHe A: «CYJIJ.eCTBYIOT '1eTHbie rrpOCTbIe '1HCJla». Onpe){eJlHTe
HCTHHHO OHO HJlH JlO>KHO; YKa»WTe Cpe){H CJle){yroUJ.HX BbiCKaSblBaHHH OTpHuaHHe
BblCK83bIBaHHS! A.
B: «CYllI.eCTByroT HelJeTHbie npOCTbIe '1HCJla»;
C: «HeBepHO, '1TO CYllI.eCTB)'IOT '1eTHbie npocTble '1HCJla»;
D: «.TIr060e rrpocroe '1HCJlO He'leTHO».
3. )].JlS! BbiCKaSblBaHHS! A: «.TIr06bIe .lI.Ba TpeyrOJlbHHKa nOJl06HbI1I cqx,pMYJlHpyAre
OTpHuaHHe H JlOOHHOe oTpHuaHHe. KaKHe H3 9THX Tpex BbIcKa3bIBaHHH HCTHHHbI?
2. KOH'bIOHKHH BblCKa3b1BaHHn. TIYCTb A H B - ,ltBa 3JIeMeHTapHbIx
BbICKa3bIBaHHH. COe.n.IiHIiB HX COlO30M «H», nOJIY'IHM HOBoe BblCKa3bIBa-
HHe, KOTopoe Ha3bIBaeTCH KOIi7JIOIiKl{UCU ,ltaHHblX BblcKa3bIBaHHH: H 0603-
Ha'IaeTCH A !\B. 3anHcb A!\B 'IHTaIOT: «A H B».
flo onpeiJcAcliUlO, KOIi7JIOIiKl{UR oeyx 8blCKa3bl8aIiUU ucmUlilW 8 moM.
'! mOAbKO 8 moM. cAyttae, K020a ucmUli1ibl 06a 8blCKa3bl8GJtUfl. ECAU ;JKC
xomfl 6bl oOliO U3 IiUX AO:JfCIiO, mo U KOH7JIOIiKl{Ufl AO:JfCHa.
31'0 onpe,lteJIeHIie KOH'bIOHKI1.HH MO)!{HO 3anHcaTb npH nOMot{ Ta6-
JIHI1.bI HCTHHHOCTH:
A I B I AI\B
H I H I H
H I J1 I J1
J1 I H I J1
J1 I J1 I J1
PaCCMOTpHM BbICKa3bIBaHHe «7-4=3 H 4 - 'IeTHOe 'IHCJIO». OHO
HBJIHeTCH KOH'bIOHKUIieH: ,ltBYX BbIcKa3bIBaHIiH:: «7-4=3» H «4-tJeTHOe
HCJIO». TaK KaK 06a 3TH BbICKa3b1BaHIiH IiCTHHHbI, TO no onpe.n.eJJeHIiIO
HCTHHHa H HX KOH'bIOHKUHH.
06pa3yeM H3 .n.BYX BbIcKa3bIBaHHH «3<8» H «8<11» KOH'bIOHKI1.HIO:
«3<8 H 8<11». OHa HCTHHHa, nOCKOJIbKY IiCTIiHHbI 06a BbICKa3blBaHHH,
BXO,ltHIlI.He B Hee. 06bI'IHO BblCKa3bIBaHIie «3<8 H 8<11» 3ahIicbIBalOT
KOpO'Ie: 3<8<11. TaKHM 06pa30v, iJeOt1HOC ttUCA080C HcpaecHcmeo npeiJ-
cmaeAflcm c06ot1 KOH7JIOHKl{UIO 08YX Hcpaemcme. 3HaH 3TO, MO)f{HO CKa-
iJaTb, tJTO, HanpHMep, HepaBeHcTBo 9<8<11 JIO)f{HO, TaK KaK JIO)f{HO
O,ltHO H3 BbIcKa3bIBaHHH, BXO.n.HIlI.IiX B Hero, a IiMeHHO HepaBeHCTBo 9<8.
ECJIH B KOH'bIOHKUHH A!\B nOMeHHTb MeCTaMIi BbICKa3bIBaHIiH A Ii
B, TO nOJIY'IIiM KOH'bIOHKUHIO BH.n.a B!\ A. 113 Ta6JIHUbI IiCTHHHOCTH
BH,ltHO, 'ITO <PopMYJIbI A!\B H B!\A npH pa3JIIi'IHbIX 3HatJeHHHX BbICKa-
3b1BaHHH: A H B JIH60 O,ltHOBpeMeHHo HCTIiHHbI, JIIiOO o.n.HoBpeMeHHo
JIO)!{HbI:
A I B I AI\B I BI\A
H I H I H I H
H I J1 I J1 I J1
J1 I H I J1 I J1
J1 I J1 I J1 J J1
16

CJIe.n.OBaTeJIbHO, OHM paBHOCMJIbHbI, M ,ltJIH JIlOObiX BbICKa3bIBaHIIll A
H B HMeeM: '
AAB=BAA.
3Ta 3anMCb BbIpa)!{aeT KOM.M.ymamU8lioe C80UcmBO KOIi'bIOIiKtfUU,
n03BOJIHIOIlI.ee MeHHTb MeeTaMH <meHbI KOH'bIOHKIJ.HH. TaK, ecJIM A -
BbICKa3bIBaHHe «4HCJIO 12 KpaTHo 4», a B - BbICKa3bIBaHMe «4MCJIO
12 KpaTHo 3», TO BbICKa3bIBaHIie «4HCJIO 12 KpaTHo 4 M 3» paBHOCMJIbHO
BbICKa3b1BaHHIO «4HCJIO 12 KpaTHo 3 H 4».
CocTaBHB Ta6JJHUbi HCTHHHOCTH .lI.JIH (AAB)Ae H AA (BAC), no-
J1Y"HM, liTO npH JII06bIX 3HatJeHHHX IiCTHHHOCTH BbICKa3bIBaHMH A, B M C
3HatJeHHH HCTHHHOCTH BbICKa3bIBaHHH: (A AB) A e H A A (B /\ C) COBna-
,ltalOT. TaKHM 06pa30M, (A/\B)/\e=A/\ (B/\C).3T0 paBeHCTBO Bblpa-
)!{aeT CBOHCTBO Q£C0tfuamU8liocmu KOIi'bIOIiKtfUU; OHO n03BOJIHeT ony-
CKaTb CK06KH B KOH'bIOHKIJ.HH Tpex M 60JIee BbICKa3bIBaHHH:. TaKaH
KOH'bIOHKIJ.HH HCTHHHa Tor.n.a H TOJIbKO Tor.n.a, Kor.n.a Bce Bxo.n.HIlI,He B Hee
BbICKa3b1BaHHH HCTHHHbI.
06pa3yeM KOH'bIOHKIJ.HIO HeKOToporo BblCKa3bIBaHHH A Hero OTpH-
IJ.aHHH .4, T. e! KOH'bIOHKIJ.HIO A AiL KaKHM 6bI HH 6bIJIO B 3TOM CJIytJae
BbICKa3bIBaHIie A (HCTHHHbIM MJIM JIO)!{HbIM), BbICKa3bIBaHMe A 1\.4 Bcer-
,lta JIO)KHO, TaK KaK npe,ltcTaBJIHeT KOH'bIOHKIJ.HIO .n.BYx BbIcKa3bIBaHHM,
KOTopbIe He MorYT 6b1Tb HCTMHHbIMH o.n.HOBpeMeHHO.
HanpHMep, BbICKa3bIBaHMe «4HCJIO 7 HBJIHeTCH npocTblM H He HBJIHeT-
CH npOCTbIM» 3aBe.n.OMO JIO)!{HO, He3aBHCHMO OT Toro, npOCToe tJHCJIO 7
MJIH HeT - eCJIH OHO npOCToe, TO JIO)!{Ha BTopaH nOJIOBHHa KOH'bIOHKIJ.HH,
a eeJIH OHO He HBJI,HeTCH np9CTblM, TO JIO)!{Ha nepBaH tJaCTb KOH'bIOHIJ.HH.
ECJIH COCTaBMTb Ta6JIMIJ.Y HCTHHHOCTH .n.JIH BbICKa3bIBaHMH A AA, TO
cToJI6eIJ. 3HatJeHMH 3Toro BbICKa3bIBaHMH co.n.ep)!{HT TOJIbKO Jl:
A I A I A/\A
11 I JI I JI
JI I 11 I JI
B 3TOM cJIYtJae rOBopHT, tJTO <popMYJIa A AA mO:JfCOCCmBCIiIiO AO:JfClia,
M 3anHCblBalOT: A /\ if =JI. 3.n.ecb JI 03HatJaeT 3aBe,ltOMO JIO)!{HOe BbI-
cKa3bIBaHHe.
YupamUeUBJI
I. [(aHbI BbICKa3bIBaHHII A: «4.2=8» H B: «(-5) 2=25». OGpa3yiiTe KOH'bIOHKUHIO
.aaHHblx BbIcKa3bIBaHHH H Onpe,lJ.eJIHTe, HCTHHHa OHa HJlH JlO1KHa.
2. Cpe,lJ.H CJle,lJ.YIOUJ.Hx COCTaBHblX BblCKa3b1BaHHii YKa>KHTe KOH'bIOHKUHH H on-
PeJJ.eJIHTe, HCTHHHbI OHH HJlH JI()>KHbI:
a) 4HCJlO 27 KpaTHo 3 H 9.
0) 17<25<23.
B) [(HarOHaJIH JII06oro napaJIJleJIOrpaMMa nepneH,lJ.HKYJlllpHbI H ,lJ.eJIIIT ,lJ.Pyr .apyra
nonOJlaM.
r) ,AaHHbIH TpeyroJlbHHK paBHooe,qpeHHbIii HJlH paOHOCTOpOHHHK.
ro Y I6== -4, HO -4'F (-2) II.
H

3. .il.aHf1I BbICKblBaHHHJ
A: « KynHJI BeJIOCHne,lJ.»; B: « nYTeIIleCTBOB8JI no CCCP» H C: « YlJaCTBOBaJI
B oopeBHOBaHH!lX no BeJlOCnopTY».
Ofx>PMYJlHPyiiTe BblCKa3b1B3HHH, COOTBeTCTBYlOUJ.He CJle)l.YlOUJ.HM «popMy.nay!
8) A/\B; 0) A/\B/\C; B) A/\C; r) A/\B; 11) B/\G.
4. nOJlb3YHCb BblCKa3hlBaHHHMH A, B H C, 3a11aHHbIMH B ynp8>KHeHHH 3, UnH'
WHTe C nOMOUJ.blO CHMBOJlOB JlOrHKH BbIcKa3bIBaHHA ciIe)l.YlOUJ.He BblcKa3bIBaHHH: .
8)  He nYTeIIlecTBOBaJI no CCCP.
(I)  KYnBn BeJIOCHne)l. H Y'IaCTBOBaJI B oopeBHOBaHH!lX.
B)  He nyreIIlecTBOBaJI no CCCP H He KynHJI BeJIoCHnel1.
r)  KynHJI BeJIOCHne)l., HO He Y'IaCT80BaJI B oopeBHOBaHHHX.
 HeBepHO, '!TO !I Y'laCTBOBaJI B oopeBHOBaHHHX H nYTeIIlecT80BaJI no CCCP.
5. .il.OK8JKHTe, '1TO A/\A=A.
8. (;oCTaBbTe Ta6J1HUbi HCTHHHOCTH 11J1J1 CJle)l.YlOUJ.Hx «POPMYJI:
a) .4/\B; 6) /t/\B; B) A/\B .
3. ltH3'bIOHKHH BblCK83b1BaHHii. COe,ltHHHB ,ltBa 3JJeMeHTapHbIX BbI-
CKa3bIBaHHH A H B COI030M «HJJH», nOJIyqHM HOBoe BbICKa3bIBaHHe, Ha-
3b1BaeMOe OU3'OIOIiKl{UCt1 .n.aHHblX BbICKa3bIBaHHH. .lI.H3'bIOHKUHIO BbiCKa-
3bIBaHHH A H B 0603HatJaIOT A VB H qHTaIOT: «A HJJH B». J(U3'OIOIiK4UJ1
AO:JfClia mOAbKO 8 mOM. CAyttae, K020a 06a 6bICKa3b16aIiUJl, us KOmopblX
OHa 06pa308alia, AO:JfCIibl; 60 8CCX OCmaAbliblX CAyttaflX OU3'OIOIiKl{UJl UC-
mUlilia. Ta6JIHUa HCTHHHOCTH .n.H3'bIOHKIl.HH HMeeT BH.n.:
A I B I AvB
H I H I H
H I JI I H
JI I H I H
JI I JI I JI
CJIe)l.yer 06paTHTb BHHMaHHe Ha HeKOTOpylO oc06eHHOCTb ynoTpeUJleHH!I COlO3a
«HJlH» B JlOrHKe. B 06b1,lJ.eHHoH pe'lH COIOS «HJlB» ynoTpe6J1!1eTcH B )l.BYX CMblCJlax: KaK
Hep83)1.eJIHTeJlbHblH H KaK pa311eJlHTej!bHbli\. PaCCMoTpHM, HanpHMep, BbICKa3b1BaHHe
cOrnpaBJleHHe nOe3)1.a 3anpeUJ.aeTC!I, eCJIH M8WHHHCTY He nepel1aJIH >Ke3J1 HJlH ,lJ.e-
>KYPHblH no CTaHUHH o)l.er He no <jJopMe». CHO, lJTO OOe3){ HeJlb3!1 OrnpaBJI!lTb H B TOM
CJlYlJae, KOr){a Y MaIIlHHHCTa HeT >Ke3JJa, a )l.e>KypHbIH no CTaHUHH O,lJ.eT He no «PoPMe.
3)1.eCb COlO3 «HJlH» !lBJlJlerc!I Hepa3)1.eJIHTeJlbHbIM, OH lie npoTHBonOCTaBJl5IeT ooe){HH!I-
eYbie HM '1aCTH. HHYlO pOJlb HrpaeT 9TOT oolO3 B npe,lJ.JIO>KeHHH «3aBTpa B 12 '1aCOB
AHH H 6Y11Y B HHCTII1)'1'e BJIH Ha 3aBOl1e». T8K KaK HeB03MO>KHO 6bITb O)l.HOBpeMeHHO
H B HHCTHTyre, H Ha 3aBOl1e, TO 311eCb COlO3 «HJlH» nOHHMaeTC!I KaK pa3){eJlHTeJlbHblii -
OH B3aHMHO HCKJllO'IaeT ooel1HH!leMbie HM '1aCTH.
H3 Ta6J1HUbi BH,lJ.HO, '1TO B MaTeMaTH'IecKoH JlOrHKe COlO3 «HJlH» nOHHMaJOT KaK
Hepa3)1.eJIHTeJlbHblH - eCJIH 06a Coe,lJ.HH!leMbiX HM BblCKa3bIBaHH!I HCTHHHbI, TO H Bce
BblCKa3b1BaHHe HCTHHHO. ECJlH XOT!lT IIO)I.'1epKHYTb pa3,lJ.e.1HTeJIbHbIH CMbIC.1, TO BMecTO
«HJlH» rOBOpHT «.1IH60..., JlH60...» (HanpHMep, «3aBTpa B 12 '1acOB ,!1H!I !I 6y){y JlHOO
B HHCTHTyre, JlHOO Ha 3aBOl1e»).
06pa3yeM H3 .n.BYX 3JJeMeHTapHbIX BbIcKa3bIBaHHA dO>7», «10=7:.
,ltH3'bIOHKIl.HIO «10>7 HJJH 10=7». HeTpy,ltHo y6e.n.HTbcH B TOM, qro 3Ta
.n.H3'bIOHKIl.HH HCTHHHa, TaK KaK HCTHHHO BXO.n.51IIJ.ee B Hee BbICKa3b1BaHHe
«10>7» B..w' aaWfe «JQ("wui..l.D.=7» 06bIqHO 3anHCbIBaIOT B BH,lte:
1O7. t aKHM 8p!f30M\:.HeCf!lp020C ttUtM060C IiCpaeeliCm60 npCOCmaBAR.em
18

cOOoU OU3'OIOIiKtfUIO Cmp02020 liepaeeliCmBa U pQ8CH,CmBa. 3HaH 3TO, JIerKO
onpe.n.eJJIiTb, HCTIiHHbI IiJIIi JIO)KHbI, HanpHMep, TaKHe HepaBeHCTBa:
22, 23. HepaBeHCTBO 2 IiCTIiHHO,TaK KaK HBJIHeTCH .n.Ii3'bIOHKUMeii
BbIcKa3bIBaHHH 2=2 Ii 2<2, nepBoe Ii3 KOTOpbIX IiCTIiHHO. HepaBeHcTBO
23 JIO)KHO, nOCKOJJbKY HBJIHeTCH .n.H3'bIOHKUHeii .n.BYX JIO)KHbIX BbI-
CKa3b1BaHMii: 2>3 H 2=3.
.lJ.JIH .n.Ii3'bIOHKUIiH, TaK )Ke KaK Ii .n.JIH KOH'bIOHKUIiH, MO)KHO YKa3aTb
pH,lt paBHOCIiJIbHOcTeH. .lJ.JIH JIl06bIX BbICKa3bIBaHIiii A, B Ii C HMeeM:
A V B=BV A (KoMMYTaTHBHOCTb ,ltH3'bIOHKlLMM);
(A V B)VC=A V (BVC) (acCOll.MaTHBHoCTb ,ltH3'bIOHKUIiIi).
TaK, ecJJH A - BbICKa3b1BaHHe «CeiitJac COJIHetJHO», a B - BbiCKa-
SblBaHHe «CeHqaC .u.O)K.ltJIHBO», TO BbICKa3bIBaHIie «CeiitJac CO.'IHetJHO HJIH
,nO)K)tJIHBO» paBHOCIiJIbHO BbICKa3bIBaHHIO «Ceiiqac .n.O)K.n.JIIiBO IiJIH COJI-
HeqHO».
CBOHCTBO aCCOUHaTHBHOCTH .n.H3'bIOHKUIiH n03BOJIHeT onYCKaTb CKOO-
KM H nIicaTb A V B V C BMeeTO (A V B) V C.
npli nOMOIi Ta6JIIiU HCTHHHOCTH HeTPy.n.HO YCTaHoBIiTb, lITO
(A V B);\C=(A;\C)V(B;\C);
(A ;\B)VC=(A VC);\(BVC).
nepBoe paBeHcTBo BbIpa)KaeT oucmpu6ymu8liblu 3aKOli KOIi7JIOIiKl{UU
OmH.OCUmCAbIiO OU37JIOIiKl{UU, a BTopoe - oucmpu6ymU8/iblt13aKOIi OU37J-
IOIiKtfUU omliocumeAbliO KOIi7JIOIiKl{UU. ,
06pa3yeM .n.H3'bIOHKIJ.IiIO HeKoToporo BbICKa3bIBaHIiH A Ii ern OTPH-
uaHIiH .4, T. e. BbICKa3bIBaHIie A V It. TaK KaK H3 .n.BYx BbICKa3bIBaHIiH
A H It Bcer.n.a O.u.HO HCTHHHO, a .n.pyroe JIO)KHO, TO .n.Ii3'bIOHKUHH A V It
oy,lteT lICTIiHHa ,ltJIH JIl000ro BbICKa3bIBaHHH A. ECJJIi COCTaBIiTb Ta6JIH-
uy IiCTIiHHOCTIi .n.JIH BbICKa3bIBaHHH A V It, TO MbI YBIi.n.HM, lITO CToJI6eu
3HatJeHHH :1Toro BbICKa3bIBaHIiH co.n.eP)KIiT TOJIbKO 11.
A I II lAVA
H I J1 I H
J1 I H I H
B 3TOM CJJYtJae rOB.?pHT, tJTO <PoPMy.ha A V A mO:JfCOccmeClilio ucmU/ilia
H 3anIiCbIBalOT: A V A=I1.
PaccMoTpIiM TaKoe BbICKa36IBaHIie: «YpaBHeHHe x 2 +5=0 MMeeT
,neikTBIiTeJJbHbIe KOpHIi IiJIII He HMeeT HX». 0603HaqHM BbICKa3blBaHlIe
«YpaBHeHHe x 2 +5=O IiMeeT ,lteHCTBIiTeJJbHble KOpHH» tJepe3 A. Tor.n.a
BbICKa3bIBaHIie «YpaBHeHHe x 2 +5=0 He HMeeT .n.eM:CTBIiTeJIbHbIX KOp-
HeM:» IiMeeT BH.n. A. TaKHM 06pa30M, ,ltaHHOMY COCTaBHOMY BbICKa3bIBa-
HHID COOTBeTCTByeT <PoPMYJIa A V A, TO)!{.n.eCTBeHHO HCTMHHaH npM JIIO-
OblX 3HatJeHHHX A. n03TOMY H .u.aHHoe BblCKa3bIBaHHe TO)K.n.eCTBeHHO
HCTHHHO.
OnepaUIiIi KOH'bIOHKUIiIi, .u.Ii3'bIOHKUIiIi H oTpIiuaHIiH CBH3aHbI CJJe-
19

,LI.YIOIlI.IiMIi COOTHOllieHIiHMIi, c:;npaBe,LI.JIIiBOCTb KOTOpbIX MO)KHO YCTaHO-
BHTb npli rrOMOIll.Ii Ta6JJIiIl. IiCTIiHHOCTH:
a) AAB= AVB;
6) A v B =A AD.
3TH COOTHOIlleHHH HaSblBalOT cpopM.YAaM.U oc MopeaHa 1) .
TaK, ecJIM A - BbICKaSblBaHIie «51 IirpalO B IllaXMaTbI», a B - B-
CKaSbiBaHMe «51 IirpalO B IllaIIIKIi», TO BbICKaSblBaHIiIO «HeBepHO, 'ITO 51
IirpalO B IllaXMaTbI IiJJIi B IllaIIIKIi» COOTBeTCTBYeT <popMYJJa A VB . Ho
nOCKOJJbKY <PoPMYJJa A VB paBHOCIiJIbHa <PoPMYJJe A AB, TO BbICKa3bI-
BaHIie «HeBepHO, 'ITO 51 IirpalO B lliaXMaTbI IiJIIi IllaIIIKIi» MO)!{HO saMe-
HIiTb BbICKa3bIBaHIieM «51 He IirpalO B IllaXMaTbi Ii He IirpalO B IllaIIIKM».
YnpamBCBDJI
1. CpeAH CJleAYIOLUHX BbIcKa3bIBaHHH YK8>KHTe 1\H3'bIOHKUHH 1\BYX 8J1eMeHTapHblX
BbIcKa3bIBaHHH H onpe1\eJIHTe, HCTHHHbI OHH HJlH JlO>KHbI:
a) KB8)l.paTHbIiI KopeHb H3 '1HCJla 36 paBeH 6 HJlH -6;
6) '1HCJlO 29 npocroe HJlH COCTaBHoe; B) 15<:15;
r) 0-7.
2. OjJopMYJlHpyilTe oTpHuaHHe BbICKa3bIBaHHII A: «53». BepHo JlH, '1TO BbiCKa.
3bIBaHHe if TaKoao: «5<3»?
3. )].aHo BbICKa3bIBaHHe «-4<-9». YK8>KHTe cpe1\H CJle1\YIOIII;HX BbIcKa3bIsaHHii
ero oTpHuaHHe:
a) -4>-9; 0) -4*-9; B) -4-9;
r) HeBt!pHO, '1TO -4<:-9; 1\) HeBepHo, '1TO -4<-9.
4. )].oKaJKHTe, '1TO If.JIII JlI06b1x BbICKa3bIBaHHH A, B H C:
a) A V A=A; 6) A 1\ (BVC)= (A I\B)V (A 1\ C);
B) A V (BI\C)= (A v B) 1\ (A vC); r) AV (AI\B)=A;
1\) AI\(AvB)=A; e) AvB= AI\B;
>K) AI\B= AvB; s) AI\A=A.
4. HMnJJHKaIl.HH BblCK83b1BI\HHn. PaccMoTpIiM COCTaBHoe BbiCKaShI-
naHIie, KOTopoe 06pa30BaHO Ii3 .l!.BYX 3JJeMeHTapHbIX npli nOMOIIJ.Ii CJIOB
«eCJIIi..., TO...».
nYCTb, HanpIiMep, .n.aHbI BbiCKaSbiBaHMH A: «B'Iepa 6b1JJO BOCKpe-
ceHbe» Ii B: «51 He 6bIJI Ha pa60Te». Tor.n.a COCTaBHoe BbiCKaSblBaHIie
«ECJJM B'Iepa 6bIJJO BOCKpeeeHbe, TO H He 6bIJJ Ha pa60Te» IiMeeT <PoPMY
«ECJJM A, TO B».
BbICKa3bIBaHIie «ECJJIi A, TO B» HaSblBalOT UM.nAUKauea 8blCKa3blIJa-
HUt1 A, B M npM nOMOM CMMBOJJOB sanIiCbIBaIOT TaK: A*B. BblCKa3bI-
BaHIie A, BXo.n.Hee B MMnJJIiKaIl.IiIO A::;.-B, Ha3bIBaIOT YCA08ueM. MMnJIM-
KaIl.IiM, a BbiCKaSbiBaHIie B - ee 3aKAlOtteHueM..
YCJIOBMJIMCb C'IIiTaTb, 'ITO UM.nAUKaltUfl «ECAU A, mo B» AO:JfCHa
AUtub 8 oOliOM. CAyttae: 8blCKa3blJJaH.UC A ucmUHHO, a BblCKa3bl8aH.ue B
AO:JfCHO; 80 8cex opyeux CAyttaRX UM.nAUKaltUfl ucmUHlia.
n03TOMY Ta6JiHIl.a IiCTIiHHOCTIi IiMnJJIiKaUIiIi «ECJIM A, TO B» IiMeeT
BIi,lt:
1) )].e MopraH (1806-1871) - IIIOTJlaH1\cKHii MareMaTHK H JlOrHH.
20

A I B I A  B
11 I 11 I 11
H I JI I JI
JI I 11 I 11
JIIJII 11
COrJIaCHO onpe.n.eJJeHHIO IiMnJJIiKaUIUi BbicKa3b1BaHHe «ECJIIi Blfe-
pa 6b1JJO BOCKpeeeHbe, TO fI He 6bIJJ Ha pa60Te» JIO)!{HO JIIillib B O.n.HOM
CJIYlfae: Blfepa .n.eiicTBIiTeJJbHO 6b1JJO BOCKpeceHbe (T. e. YCJIOBIie «BtJepa
6b1JIO BOCKpeeeHbe» HCTIiHHO), a H 6bJJI Ha pa60Te (T. e. 3aKJIIOlfeHIie
«51 He 6bJJJ Ha pa60Te» JJOmHO). Bo Bcex ,ltpymx CJIYtJaHx .n.aHHoe COCTaB-
Hoe BblCKa3b1BaHIie HCTIiHHO: H Tor.n.a, Kor.n.a BblCKa3b1BaHIiH «BtJepa
6bIJIO BOCKpeceHbe» Ii «51 He 6bIJJ Ha pa60Te» HCTIiHHbI, H B Tex CJIYlfaflX,
Kor.n.a BblCKa3bIBaHIie «B'Iepa 6bJJJO BOCKpeceHbe» JJOmHO. (ECJJH BbJ-
CKa3bIBaHIie A JJO)KHO, TO no onpe.n.eJJeHHIO IiMnJJIiKaUIiH A*B Bcer.n.a
IiCTIiHHa, He3aRUCHMO OT TOro, IiCTHHHO IiJJH JJOmHO BblCKa3b1BaHIie B.)
YnoTpe6J1eHlie CJlOB «eCJIIi..., TO ...» B JlOrliKe OTJlIi'laeTC!! OT ynoTpeGJleHIi!! HX
B OObI)l.eHHOii pelJli, r){e MbJ, KaK npaBIiJlO, ClJliTaeM, '1TO eCJlIi BbICKa3bIBaHlie A JlO>KIIO,
TO BblCKa3b1BaHHe «ECJlIi A, TO B» BoooLUe He HMeeT CMbICJla.
KpoMe TOro, CTPO!! npe){JlO)!{eHIi!! BIi){a «ECJlH A, TO B» B oObI)l.ellHOH pe'lli, MbI
nO'ITH BCer){a nO){pa3YMeBaeM, '1TO npe){JlO>KeIlHe B CJle)l.yeT (BbiTeKaer) H3 npe){JlO-
>KeIlHSI A. YnoTpeOJleHlie CJlOB «eCJlH..., TO ...» B JlOrHKe lie TpeGyeT 9Toro, nOCKOJlbKY.
KaK Mbi Y>Ke OTMelJaJlIi paHbIIle, CMbICJI BblCKa3b1BaIlHii B JlOrHKe lie paccMaTpIiBaeTC!!.
n09TOMY C TO'IKH 3pellHSI JlOrliKIi ){OnYCTHMbJ, lIanpHMep, TaKHe HMnJlHKaUIiIi: «EcJlH
2.2=5, TO BO){a B peKe COJlelIa!!» IiJlIi <illCJIIi CJlOBO «)l.OM» SlBJI!!eTC!! r.'1arOJlOM, TO ropoJl.
BapIIlaBa - CTOJlHua nOJlbIIlH». B npHBe){elllIbiX npHMepax HMnJlHKaUIiH YCJlOBlie H
3aKJlIO'IeIlHe IIIiKaKIiM oopa30M He CB!!3aHbI ){pyr C )l.pyroM, 110 no onpe){eJleHIiIO ooe
HMnJlHKaUHH HCTHIIHbI, nOTOMY '1TO B nepBOM npHMepe YCJIOBHe H 3aKJlIO'IeHlie JlO>KlIbI.
a BO BTOpOM YCJlOBlie JlO>KHO, a 3aKJlIOlJellHe IiCTHIIIIO.
OnepaUIiH IiMnJJIiKaUHH .n.BYX BblCKa3bIBaHIilf MomeT 6b1Tb BbJpa)KeHa
tJepe3 onepau.HH OTpIiuaHIifl H .n.Ii3'bIOHKUIiIi; .n.JJH JJI06bIX BblCKa3bIBaHIiR
A Ii B HMeeM:
(A*B)=(AV B).
PaBHOCIiJIbHOCTb <POPMYJI A*B H A VB JJerKO YCTaHaBJJHBaeTCSI
npH nOMOIll.Ii Ta6JJIiU IiCTIiHHOCTIi:
A! B! A I A  B !AVB
11 I 11 I JI I 11 1 H
111 JII JI I JI I JI
JII I1 I I1 I H I H
JII JII I1 I 11 1 11
nYCTb IiMeeTCH IiMnJIHKaUIiH A*B. nepeeTaBHB MeCTaMH ee YCJIOBHe
H 3aKJIlOlfeHIie, nOJIYlfIiM HOBYIO HMnJIIiKall.HIO B*A, KOTOPYIO Ha3bJBa-
21

1:. U,llIl1![!CCliUC:'I, OJjJL::il!iO:l U.iflflOU. 1 JK, .;J.,IH I1MII.1HKall.hH (<1::'C.11i
CYMMa lI.H<pP tJHCJIa 138 KpaTHa 3, TO H CaMO tJHCJJO 138 KpaTHo 3» 06-
paTHoH 6y.n.eT TaKaH HMnJIHKall.HH: «ECJIH tJHCJIO 138 KpaTHo 3, TO CYMMa
lI.H<pP IfHCJIa 138 KpaTHa 3». B .n.aHHOM cJIYlfae H .n.aHHaH HMnJJHKall.HH,
HeM 06paTHaH HCTHHHbI. Ho TaK 6b1BaeT He Bcer.n.a, HanpHMep, HMnJIH-
KaUHH «ECJIH 5>2, TO 5 - tJHCJIO tJeTHoe» JIO)!{Ha, a 06paTHaH eM «ECJIH
5 - tJHCJIO tJeTHOe, TO 5>2» - HCTHHHa, nOCKOJIbKY JIOJKHO ee YCJJOBHe.
, lI.PyraH HMnJJHKall.HH nOJJYtJaeTCH H3 HMnJJHKaUHH A=:>B nYTeM
3aMeHbI BblCKa3b1BaHHH A H B HX OTpHlI.aHHHMH. OHa HMeeT BH.n. A=:>B H
Ha3b1BaeTCH npomu60nOAOJICli0t1 HMnJIHKall.HH A=:>B, HaKOHell., ecJJH
3aMeHHTb A H B HX OTpHuaHHHMH H o,ltHoBpeMeHHo nepeCTaBHTb MeeTa-
MH YCJIOBHe H CJIe,ltCTBHe, TO nOJIYtJHM HMnJIHKalI.HIO B=>A. C nOMOIlLbIO
Ta6Jmu HCTHHHOCTH .JIerKO npOBepHeTCH, 'ITO BblCKa3b1BaHHH A=>-B H
B=>-A paBHOCHJIbHbI (SaJWIi KOH.mpanOSUl{Uu):
A=>-B=kA.
31'0 n03BOJIHeT no Ka)K)tOH HMnJIHKalI.HH CTpOHTb paBHOCHJIbHYIO eH
HMnJIHKalI.HIO. HanpHMep, HMnJIHKalI.HH «ECJJH llOCJJe,ltHeH lI.H<pPOH B
,neeHTHtJHOH 3anHCH tJHCJJa 140 HBJJHeTCH 0, TO 3TO tJHCJJO .n.eJIHTCH Ha 5»
paBHOCHJIbHa HMllJJHKaUHH «ECJJH tJHCJIO 140 He .n.eJJHTCH Ha 5, TO no-
CJJe.n.HHH lI.H<ppa ee .n.eCHTHtJHOH 3anHCH OTJJHtJHa OT 0». 06e 3TH HMnJJH-
KaUHH HCTHHHbI. nepBaH HCTHHHa nOTOMY, tJTO HCTHHHO H YTBepJK,lteHHe,
'ITO 140 KOHtJaeTCH lI.H<pPOH 0, H YTBepJK,!I.eHHe, 'ITO 140 ,lteJJHTCH Ha 5, a
Tor.n.a HCTHHHa H HMnJJHKall.HH A=:>B; BTopaH )!{e HMnJIHKall.HH HCTHUUS
nOTOMY, tJTO B HeH H YCJIOBHe, H CJIe,ltCTBHe JIO)!{Hbi (HeBepHO H TO, tJTO
140 He .n.eJJHTCH Ha 5, H TO, tJTO llOCJIe,ltHHH lI.H<ppa 3Toro tJHCJIa OTJJHtJHa
OT 0). A B 3TOM CJIYtJae '1MnJJHKalI.HH TO)!{e _HCTHHa.
3aMeTHM, tJTO HMnJIHKall.HH B=:>A H A=:>B TOJKe paBHOCHJJbHbI:
B=>-A=A=>-B.
BbIHCHHM B 3aKJJlOtJeHHe, KaK COCTaBHTb OTpHuaHHe HMllJJHKaIJ.HH
A=>-B. Mbi y )!{e YCT aHOBH;H, tJTO A=>-B=A VB. n03TO MY MO JKH 3anH-
caTb A=>-B =A VB. Ho no cpopMYJIe.n.e MopraHa HMeeM A V B=A I\B=
=A 1\13. 3HatJHT, A=>-B= A 1\13.31'0 03HatJaeT CJIe.n.YIOIII.ee: Iim06blooKa-
8llJnb, Ilmo us A He cAeoyem B, liaiJo nOKa3atnb, Ilmo A ucmuH.H.O U B AO:JfCH.O.
YopamUeUBJI
I. B CJleyJOUJ.HX COCTaBHblX BblCK83b1BaHHHX BbIJ].eJIHTe COCTaBJI!lIOUJ.He HX Me.
lIIeHT.apHble BblCKa3b1BaHH!I; YKIDKHTe cpeH HHX HCTHHHble HMnJlHKaUHH:
s) ECJlH '1HCJlO 48 KpaTHo 8, TO OHO KpaTHO 4;
6) ECJlH -3<-1, TO 3 2 =6;
B) ECJlH 23= 10, TO Y c06aKH '1eTb1pe HorH;
r) ECJlH 2.2=5, TO cYLUecTBYIOT BebMbl.
2. B KIDK)l.Oi!: H3 CJle)l.YIOLUHx HMnJlHK8UHi!: BbIJ].eJIHTe YCJlOBHe H SSKJllOlJeHHe H
oJx>pMYJlHPyi!:Te HMnJlHKaUHH, npOTHBOnOJlO1KHyJO JlSHHOi!: H 06paTHyio hpOTHBODO-
JI01KHOH; onpe.D;eJIHTe, HCTHHHbI OHH HJlH JlO>KHbI:
s) ECJlH H.D;eT .D;O>K)l.b, TO MOCTOBS!I MOKpa!l;
6) ECJlH H On03aJl Ha pa60TY, TO MHe 06'b!lB!lT 8b1rOBOp;
22

B) ECJII Y'IeIlHKa nepeBeJIH B CJle){YIOLUHR KJlaCC, TO OH nOJlY'IHIl no BceM open-
MeTaM OTJlH'IHble OTMeTKH.
3. L{aHbI BblCKa3b1BaHHSI A: «qeTbipexyroJlbHHK MN PQ - napaJIJleJlOrpaMM»
H B: «L{HarOHaJIH '1eTbipexyroJlbHHKa MNPQ B TO'IKe nepeCe'leHHSI .a;eJI!!TC!! nOnOJl8M».
C<poPMYJlHPyiiTe BblCKa3b1BaHHSI, cooTBeTcTBYIOLUHe cJle.a;YIOLUHM <lx>pMYJlaM:
a) AB; 6) B  A; B) A; r) If;
It) AB; e) l b A.
4. CoCTaBbTe Ta6J1HUbI HCTHHHOCTH )I.JI!! CJle.n;YIOUJ.Hx «popMYJf:
a) A ::::;> (BvC); 6) (A ::::;> B)V (A ::::;> C).
5. L{oKa>KHTe, 4TO eCJlH A, B H C - HeKoTopble 9J1eMeHTapHble BblCK33bIBaHHSI, TO
a) A -;> (B ::::;> C)= (A A B) ::::;> C;
0) (A::::;>B) A (A -;> C)=A ::::;> (BAC).
5. 3KBHBaJleHHH BblCKa3b1BaHHn. TaBTOJIOrHH. 113 .n.BYX BbICKa3b1-
BaHHH A H B MOJKHO COCTaBHTb HOBoe BblCKa3bIBaHHe, KOTopoe tJHTaeTCH
TaK: «A B TOM H TOJlbKO B TOM CJJYtJae, eCJlH B». 31'0 BblCKa3bIBaHHe Ha-
3b1BalOT aK8U8aACliltUCU BblCKa3b1BaHHH A H B H 0603HatJalOT: AB. CtJH-
TalOT, tJTO 6blCKa3bl8aliUe AB ucmUIiIiO, CCAU 06a 6blCKO.3bt8aflUfl A U B
ucmUlilibl UAU 06a 8btCKa3bt8a1iUfl A U B AOJICIibl. B OCmaAbliblX CAyttaJlX
(m. e. CCAU oaliO UJ amux 8blCKa3bl8aliut1 ucmUIiIiO, a apY20C AO:JfC1W)
9K8U8aAeHltUIO Cl1.umalOm AO:JfCliot1. TaKHM 06pa30M, Ta6J1HlI.a HCTHHHOCTH
.n.JIH 3KBHBaJIeHUHH A H B HMeeT BH.n.:
AI BIAB
I1 I I1 I I1
I1IJII . JI
JI I I1 I JI
JI I JI I I1
HanpHMep, BbICKa3bIBaHHe «4HCJJO 129 .n.eJJHTCH Ha 3 B TOM H TOJIbKO
B TOM CJJYtJae, Kor.n.a CYMMa ero UH<pP .n.eJJHTCH Ha 3», HMeeT BH.n. AB,
r,lte A - BbICKa3bIBaHHe «4HCJJO 129 .n.eJJHTC5I Ha 3», a B - BbICKa3W-
BaHHe «CYMMa lI.H<pP tJHCJJa 129 .n.eJIHTCH Ha 3». TaK KaK 06a BbICKa3b1-
BaHH5I HCTHHHbI, TO H BblCKa3b1BaHHe AB HCTHHHO. I1cTHHHO H BbiCKa-
3b1BaHHe «lJHCJIO 127 .n.eJIHTC5I Ha 3 B TOM H TOJIbKO B TOM CJJYtJae, KOr,lta
CYMMa ero lI.H<pP .n.eJlHTCH Ha 3» - B 3TOM CJJYtJae H A H B JlO)!(Hbi (Ha
caMOM .n.eJJe \fHCJlO 127 He .n.eJIHTCH Ha 3, a cYMMa ero lI.H<pp 1 +2+ 7= 10
TO)!{e He .n.eJIHTCH Ha 3). BblcKa3bIBaHHe )!{e «lJHCJIO 195 .n.eJJHTCH Ha 3
B TOM H TOJIbKO TOM CJJYtJae, Kor.n.a CYMMa ero lI.H<pP .n.eJIHTCH Ha 4» JIO)!{-
HO, TaK KaK CaMO tJHC.no 195 .n.eJJHTCH Ha 3, a cYMMa ero uH<pP paBHa 15
H Ha 4 He .n.eJJHTCH.
B npe.n.bI,1J.YIIJ.HX nYHKTax Mbi nOKa3aJIH, KaK C nOMOlIJ.blO OTpHuaHH5I
Ii Onepall.HH KOH'bIOHKlI.HH, ,ltH3'bIOHKlI.HH, HMnJIHKall.HH MO)!{HO H3 3a.n.aH-
HbiX BbICKa3b1BaHHH A, B, e CTpOHTb HOBwe BblCKa3b1BaHHH. DpH 3TOM
OblJIO nOKa3aHO, HanpHMep, tJTO He3aBHCHMO OT TOro, HCTHHHbI HJIH JIO)!{-
Hbi BbICKa3bIBaHIiH A H B, COCTaBHbIe BbICKa3b1BaHH5I A!\ B H B!\ A HJlH
o.n.HoBpeMeHHo HCTHHHbI, HJIH o.n.HoBpeMeHHo JIO)!{HbI. Tenepb Mbi MO)!{eM
TO )!{e caMoe CKa3aTb HHatJe: 3KBHBaJIeHHH A !\BB!\A Bcer.n.a HCTHH-
23

H3, T. e. A;\BB;\A-H, r,lte H 03HatJaeT 3aBe.n.OMO MCTRHHOe BbI-
CKa3bIBaHHe. Boo6III.e,ecJJH BblCKa3b1BaHH5I A H B 3KBHBaJIeHTHbI, TO
9KBHBaJIeHl{H5I AB HCTHHHa. CnpaBe.n.JIHBO H 06paTHoe, eCJIR AB=
=H, TO A=B.
CoCTaBHble BblCKa3b1BaHHH, HCTHHHble npH JII06b1X npe.n.JIO)!{eHH5IX 0
BXO.n.51III.HX B HHX 9JIeMeHTapHblX BblCKa3b1BaHH5IX,Ha3b1BaIOT maemoAO-
2UflM.U. 3HatJHT, A ;\BB;\A H A V BBV A - TaBTOJIOrHR. DpHBe-
.n.eM eIII.e o.n.HH Ba)!{HblH npHMep TaBTOJIOrHH:
«A=>B);\A)=>B. 3Ta TaBTOJIOrH5I 03HatJaeT, tJTO ecJJH HCTHHHa HM-
nJIHKal{H5I A=>-B H HCTHHHO YCJIOBHe A, TO HCTHHHO H CJIe.n.CTBHe B.
'uJI5I .n.OKa3aTeJJbCTBa Ha.n.o COCTaBHTb Ta6JIHl{Y HCTHHHOCTH.
YopamneUHH
I. )].aHbI BblCKa3b1BaHH!! A: «qHCJlO 729 KpaTHo 9» H B: «CYMMa uH«pP '1HCJla 729
KpaTHa 9». C«PoPMYJlHPYHTe BblCKa3b1BaHH!!:
a) A:::;>B; 0) B ::;> A; B) AB.
2. )].aHbI BblCKa3b1BaHH!! M: «TpeyrOJlbHHK ABC - paBHooe.a;peHHblH» H N:
«)].Be BbiCOTbi TpeyroJlbHHKa ABC KOHrpY9HTHbI». OjJOpMyJlHpYHTe BblCKa3b1BaHH!!:
a) M ::;> N; 0) N  M; B) MN.
l(aKHe H3 HHX HCTHHHbI, a KaKHe JlO>KHbI?
3. )].aHbI BblCKa3b1BaHH!! A: «Cero.lJ;HH HCHO», B: «Cero)l.H!! H.D;eT CHer», C: «51 oY)l.y
nHcaTb nHCbMa» H D: «Cero.lJ;H!! nOHe)l.eJIbHHK».
npOlJHTaHTe, nOJlb3Y!!Cb oObllJHblM !!3b1KOM, COCTaBHble BblCKa3b1BaHH!!:
a) A I\D; 0) A 1\8; B) D I\A I\C; r) A V B;
A) B::;>C; e) CD; >K) (A 1\ B)D.
4. 113 3J1eMeHTapHblX BblCKa3b1BaHHH A: «3aBTpa oY)l.eT .D;O>K)l.b», B: «MbI noiiD.eM
B reaTp», C: «3aBTpa oY.D;eT COJlHe'lHO» H D: «38BTpa SaH!!TH!! HalJHYTC!! paHbI1Ie OObllJ-
HOro» 06pa30BaHbI CJle)l.)'IOLUHe COCTaBHble.
a) ECJlH 3aBTpa oY.D;eT )l.OJK,ll.b, TO 3aH!!TH!! Ha'lHyrc!! paHbI1Ie OObllJHOro H Mbi
noH.D;eM B TeaTp.
6) 3aBTpa oy)l.eT COJlHe'tHO HJlH oY.D;eT .D;OJK,lJ.b; H 3aMTH!! Ha'lHYTC!! paHbI1Ie oObI'I.
HOro. /'
B) 3aBTpa 3aH!!TH!! Ha'lHyrc!! paHbIIle 06b1'1HOfO H Mbi nOH.D;eM B TeaTp B TOM H
'lUJIbKO 'roM CJlyqae, eCJIH  6YJl.eT Jl,OJK,ll.H H 6y.D; COJiHe'lHO_
3anHIIIHTe .D;aHHble COCT8BHble BbiCKaSblB8HHH, HCnOJlb3Y!! CHMBOJlbI JlOrHKH BJ,I-
CKa3b1BaHHii. O.D;H03Ha'lHO JlH MO>KHO nOH!!Tb BblCKa3b1BaHHe B)?
5. KIDKJI.oe H3 3J1eMeHTapHblx BblCKa3b1BaHHii, BXO.D;!!mee B CJle.D;YlOmHe HH>Ke co-
CTaBHble BblCKa3b1BaHH!!, 0603Ha'lbTe 6YKBOH: H 3anHIIIHTe 9TH COCTaBHble BblCKa3b1-
BaHH!! B CHMBOJlH'IeCKoi!: <lx>PMe.
a) B ropo.D; !! noe.n.y Ha aBT06yce, HJlH Ha TpaMBae, HJlH Ha TaKCH.
6) ECJlH np!!Ma!! AB nepneHJI.HKYJI!!pHa npHMoii CD H np!!Moi!: KL, TO npHMhIe
CD H KL napaJIJleJIbHbI.
B) qeTblpexyrOJIbHHK HBJI!!eTC!! napaJlJleJIOfpaMMOM TOf.D;a H TOJlbKO TOf.D;a, KOfJl,a
ero npoTHBOnOJlO>KHble CTOpoHbI nonapHO nap8JIJIeJIbHbI.
6. CoCTaBbTe T86J1HUY HCTHHHOCTH JVl!! <lx>PMYJlbI (AB) 1\ (BA) H Cp8BHHTe
ee C Ta6J1Huei!: HCTHHHOCTH 9KBHBaJIeHUHH A  B.
PaBHOCHJlbHbI JlH BblCKa3b1BaHH!! (AB)I\ (BA) H A B?
7. )].oKa>KHTe CJle.D;)'IOmHe T8BTOJIOfHH: a) A V B 4:;)A I\B; 0) «AB)A)A.

rJlaBa II
MHOmEA,KOYmmU,KOMBUHA1OPUKA
 1. OnEPAIHI HA.II: MHOmECTBAMH
t.' MH01KeCTBa. nOHHTHe MHO)!{eeTBa HBJIHeTCH O.n.HHM H3 OCHOBHbIX
nOHHTHH MaTeMaTHKH. nOHCHHM ero npHMepaMH. TaK, MO)!{HO rOBopHTb
o MHO)!{eCTBe rJIaCHbiX 6YKB pYCCKoro aJI<paBHTa, 0 MHO)!{eCTBe yqa-
III.HXCH HeKOTopOH IllKOJIbI, 0 MHO)!{eCTBe peIlleHHR HepaBeHCTBa x+5>
>12x, 0 MHO)!{eeTBe nl:)pT B .n.aHHOH ay.n.HTOpHH H T. .11.. B nOBCe,ltHeBHOH
)!{H3HH BMeeTO CJ10Ba «MHO)!{eeTBO» ynOTpe6JIHIOT CJIOBa «Ha60p», «coO-
paHHe», «KOJIJIeKlI.H:H», «CT3)l.O», «Ta6YH» H T. .11..
06'beKTbI JII060H npHpo.n.bI (JIIO.n.H, .n.OMa, KHHrH, reOMeTpHtJeCKHe
<PHrYPbl, tJHCJ1a H T. .11..), COCTaBJIHIOIII.He MHO)!{eeTBO, Ha3bIBalOT ero
9ACM.CIimaM.U. HanpHMep, tJHCJ10 3 HBJIHeTCH 3JIeMeHTOM MHO)!{eCTBa Ha-
TypaJIbHblX IIHCeJI, MaR - 3JIeMeHTOM MHO)!{eeTBa MeeHueB B ro.n.y.
OTHOIlleHHe Me)!{.n.y MHO)!{eCTBOM H ero 3JIeMeHTaMH, KOTopoe MbI BbI-
pa3HJIH CJIOBaMH fl8AflCmCfl 8ACM.ClimOM., BblP.a)!{alOT H npH nOMoIIl.H CJlQ-
Ba npuliaOAC:JfCUm. TaK, MO)!{HO CKa3aTb, liTO tJHCJIO 3 npHHa.n.JIe)!{HT
MHO)!{eeTBY HaTypaJIbHblX tJHCeJI.
lI.JIH COKpaIIl.eHHH 3anHCH pa3J1HtJHblx BblCKa3b1BaHHH 0 MHO)!{eCTB3X
H HX 3JJeMeHTax npHHHTa cJIe.n.YIOIII.aH CHMBOJIHKa: MHO)!{eeTBa OObillHO
0003HatJalOT 60JIbIllHMH 6YKBaMH JIaTHHCKoro aJI<paBHTa, a HX 3JIeMeH-
Tbi - MaJJblMH, CJ10BO «npHHa)tJJe)!{HT» 3aMeHHIOT CHMBOJIOM E:-
BblCKa3b1BaHHe «06neKT a npHHa)tJJe)!{HT MHO)!{eCTBY A» 3anHCbIB3-
lOT TaK: a E: A. 3anHcb a E A MO)!{HO npOqHTaTb HHatJe: «06'beKT a
eeTb 3JIeMeHT MHO)!{eeTBa A».
BblCKa3b1BaHHe «3JIeMeHT a He npHH3)l.JIe)!{HT MHO)!{eCTBY A» 3anHCbI-
BalOT TaK: a  A, npHqeM 3Ta 3anHCb MO)!{eT ObITb npotJHTaHa H HHaqe:
«06'beKT a He HBJIHeTCH 3JIeMeHTOM MHO)!{eCTBa A».
HanpHMep, ecJJH A - MHO)!{eCTBO lIeTHbiX HaTypaJIbHbIX tJHCM,
TO CJIe.n.YIOIII.He BblCKa3b1BaHHH 06 3JIeMeHTax MHO)!{eeTBa A HCTHHHbI:
2 .
]6 E A, 328 E A, 17 A, 13 A.
.lI.JIH HeKOTopblX qHCJIOBbiX MHO)!{eCTB HMelOTCH cneUHaJJbHbIe 0603-
HatJeHHH. TaK, MHO)!{eeTBO Bcex HaTypaJIbHblX tJHCeJI 0603HatJalOT OYK-
BOH N, MHO)!{TBO lI.eJIbiX HeOTpHlI.aTeJIbHblX tJHCeJI - 6YKBOH Zo,
MHO)KeeTBO Bcex UeJIbiX tJHCeJI - 6YKBOH Z, MHO)!{eeTBO Bcex pall.HQ-
HaJIbHblX IIHCeJI - 6YKBOH Q H MHO)!{eeTBO Bcex .n.eHCTBHTeJIbHblX IIH-
CeJI - 6YKBOH R.
25

MHO}!{eCTBa MOrYT CO,ltep)!{aTb KaK KOHetJHOe tJHCJIO 3JIeMeHTOB, TaK
H 6ecKOHetJHOe. TaK, MHO)!{eeTBO npe.n.MeTOB, H3ytJaeMbiX B IllKdJIe, KO-
HetJHO, a MHO)!{eeTBO TOtJeK npHMOH 6ecKOHetJHO.
MHO)!{eCTBO MO}!{eT cO,ltep}!{aTb H O,ltHH 3JIeMeHT. HanpHMep, MHO-
}!{eCTBO rJIaCHbIX 6YKB B CJIOBe «Illap» COCTOHT H3 o.n.Horo 3JIeMeHTa -
6YKBbl «a».
PaCCMaTpHBalOT B MaTeMaTHKe H MHO)!{eCTBO, He co.n.ep}!{alIJ.ee HH
O,ltHoro 3JIeMeHTa; ero Ha3bIBalOT nYCmblM. M.IiO:JfCCcmeoM. H 0603HatJalOT
CHMBOJIOM 0. CYlIJ.eeTByeT JIHIllb O,ltHO nYCTOe MHO)!{eeTBO. npHMepaMH
rrYCTOro MHO}!{ecTBa MorYT CJIy}!{HTb: MHO}!{ecTBO JIIO,lteH Ha CoJiHue,
MHO)!{eCTBO HaTypaJIbHbIX tJHCeJI, pacnOJIO}!{eHHbIX Ha tJHCJIOBOH npHMoH
JIeBee e)I.l1H H UbI , MHO)!{eCTBO ,lteHCTBHTeJIbHbIX KopHeii ypaBHeHHH
4x+5=4 (x-7).
3.aeMeHTaMH MHO}!{eeTBa MorYT 6bITb MHO)!{eCTBa. HanpHMep, MO}!{HO
rOBOpHTb 0 MHO)!{eCTBe KJIaCCOB HeKOTopoii IllKOJIW. 3JIeMeHTW 3TOro
MHO)!{eCTBa - KJJaCCbI, HBJIHlOIlI.HecH B CBOIO OtJepe,ltb MHO)!{eCTBaMH
YtJarn:HXCH. Ho YtJarn:HeeH y)!{e He HBJIHIOTCH 3JIeMeHTaMH MHo)!{ec:rBa
KJIaCCOB lI1KOJIbI.
YnpamnenDJI
1. KaK Ha3bIBa1OT: -a) MHO>KeCTBO Bcex nHoHepoB o)l.Hoii llIKOJlbl; 0) MHO>KeCTBO
Bcex nHoHepoB O)l.HOrO KJlaCCa; B) MHO>KeCTBO apTHCTOB, pa60TalOUJ.HX B O)l.HOM TeaTpe?
2. npHBe,llHTe npHMepbl MHO>KecTB, COCTaBJleHHblX H3: a) '1HCM; 0) reoMeTpH'Ie-
CKHX «pHryp. '
3. A - MHO>KecTBO napaJlJlMOrpaMMoB. npHHaJVIe>KHT JlH 9TOMY MHO>KecTBY:
a) pOMO; 0) TpaneUH!I; B) ,llHaroHaJlb napaJIJlMOrpaMMa; r) np!lMoyroJlbHHK; ,ll) na-
p aJIJleJle nH ne,ll?
4. Zo - MHO>KecTBO UMbiX HOOTpHuaTeJlbHblX '1HCM. Ha30BHTe HeCKOJlbKO '1HCM,
npHHa,llJle>KaUJ.Hx 9TOMY MHO>KeCTBY. KaKHe H3 CJle,llyIOUJ.Hx BblCKa3b1BaHHii HCTHHHbI:
a) 136EZo; 0) OEZo; B) -18EZo; r) -3,5EZo?
5. .LI.aHbI MHO>KeCTBa: M - MHO>KeCTBO HaTypaJIbHblX '1HCM, OOJlblliHX 8 H MeHb-
llIHX 18; P - MHO>KeCTBO HaTypaJlbHbIx '1HCM, OKaH'IHBaIOUJ.HXC!I uH«ppoii: 7.
a) YKa>KHTe, KaKHM H3 9THX "V!0>KecTB npHH8)l.Jle>KaT '1HCJla 12, 17, 0, 3, 7. 3a-
nHIIIHTe 9TO C nOMO[IJ.blO CHMBOJla E.
0) YKa>KHTe, KaKHM H3 9THX MHO>KecTB He npHH8)l.Jle>KaT Ha3BallHble '1HCJla.
3anHIIIHTe 9TO C noMOIII,bIO CHMBO.na. -
6. X - MHO>KeCTBO OTp!l,llOB HeKOTOpofl nHOHepCKoii )l.PY>KHHbI. KaKHe H3 CJle-
ltyIOUJ.HX BbICKa3b1BaHHii HCTHHHbI:
a) 9J1eMeHTaMH MHO>KeCTBa X !lBJI!lIOTC!I OTp!l,llbi nHOHepCKOii ,llPY>KHHbI;
tJ) 9J1eMeHTaMH MHO>KecTBa X !lBJI!lIOTC!I nHOHepCKHe 3BeHb!l 3TOii: ,llPY>KHHbI;
B) 9J1eMeHTaMH MHO>KeCTBa X He !lBJI!lIOTC!I nHOHpbl 9TOii ,llPY>KHHbI?
7. )].oKa>KHTe, '1TO CJle,llyIOllI.He MHO>KeCTBa nYCTbI:
a) MHO>KeCTBO HarypaJlbHblX '1HCM, MeHbIIlHX HyJl!l;
6) MIIO>KecTBO HaTypaJIbHblX '1HCM, OOJlbllIHX 8 H MeHbIIlHX 9;
B) MHO>KeCTBO HaTypaJlbHblX '1HCM, KB8)l.paT KOTOpbIX paBeH -I.
8. J-!3, BeCTHO '1TO xER. Haii)l.HTe MHO>KecTBO peIIleHHii: Ka>K)l.oro H3 ypaBHeHHii.
KaKHe H3 HHX rryCTbI?
a) 7x+5=7(x+12); 6) 2 (x-5)=3x; B) x2+3=2.
9. npO'lHTaiiTe c.lIe,llylOllI.He BblCKa3b1BaHH!I; YKa>KHTe Cpe,llH HHX HCTHHHble:
a) -7E Q; 0) -7EZ; -a) -7ER;
r) OEZo; ,ll) OZo; e) OER;
3 3 Li
>K) 4 E Qi 3) "4 'toR.
26

.) Ci.G('C<'U :.&;wmI:'1 r.:!Dt::eCTD. Fi:mF;C m;r;;::CCEQ.. .\lEGKeCTEO
C411TalOT Sa.n.aHHbIM, eeJIH 0 JII060M 06'beKTe MOJKHO CKa3aTb, npHHa,lt-
JIe)!{HT OH 3TOMY MHO)!{eeTBY HJIH He npHHa.n.JIe)!{HT. MHO)!{eCTBO MO)!{HO
3a,ltaTb, nepetJHCJIHB Bce ero 3JJeMeHTbI. TaK, eCJIH a, b, c, d - 060SHa-
lIeHHH pa3JIHIIHblX o6'beKToB, TO MHO)!{eCTBO A 3THX o6'beKTOB 3anHCbJ-
BalOT:
A= { a' b' C' d }
, , ,
H tJHTaIOT: «A - MHO)!{eeTBO, 3JIeMeHTbI KOToporo a, b, c, d».
YKa3aHHblH: cnoc06 3a.n.aHH5I MHO)!{eeTB npHMeHHM TOJIbKO .n.JIH Ko-
HellHblX MHOJKeCTB, ,lta H TO npli YCJIOBHH, tJTO IIHCJIO 3JIeMeHTOB MHO-
)!{eeTBa HeBeJJHKo.lI.pyrOH: cnoc06 3a.n.aHIiH MHO)!{eCTB COCTOHT B CJIe.n.y-
lOIlI.eM: cpopMYJJHPYIOT xapaKmepuCmUttCCKOC C60ucm60 3JIeMeHTOB MHO-
)!{eCTBa, T. e. CBOH:CTBO, KOTOpblM 06JJa.n.alOT Bce 3JJeMeHTbI 3Toro
MHO)!{eCTBa Ii TOJIbKO OHH. HanpHMep, MHO)!{eCTBO M HaTypaJIbHblX
IIHCeJI, MeHblllHX 6. 3T0MHO)!{eCTBO 3a.n.aHO BTOpblM cnoco60M: YKa3a-
HO xapaKTepHCTIilleeKoe cBoikTBO Bcex 3JIeMeHTOB MHOJKeCTBa M, a
HMeHHO CBOH:CTBO 6bITb HaTypaJIbHbIM IIHCJIOM H MeHblliHM IIHCJIa
6. B .n.aHHOM cJJYtJae 3JJeMeHTbI MHOJKeCTBa M MOJKHO nepe'lHcJ1HTb:
M={I; 2; 3; 4; 5}.
MIiO)!{eCTBO, .n.JIH 3JIeMeHTOB KOToporo YKa3aHo HeKOTopoe xapaK-
TepHCTHlleeKoe C'BOH:CTBO, YCJIOBIiMCH 0603HallaTb TaK: B <PHrypHbIX CK06-
Kax nHllleTCH 0603HalleJ:IHe 3JJeMeHTa, 3aTeM npoBo.n.IiTC'H BepTIiKaJJbHaH
lIepTa, nOCJIe KOTOpOH:' nHllleTCH CBOHCTBO, KOTOpbIM 06JIa.n.alOT 3JIe-
MeHTbI .n.3HHoro MHO)!{eeTBa H TOJIbKO OHIi. HanpHp, MHOJKeCTBO M
HaTypaJibHbIX IIHCeJI, MeHblliHX 6, 3anHllleTc!I TaK:
M={x/x€:Nl\x<6} HJIH M = {xix EN, x<6}.
TaKHM 06pa30M, MH TOro IIT06bI 3a.n.aTb HeKOTopoe MHOJKeCTBO, Ha-
.no JIH60 nepellHCJIHTb ero 3JIeMeHTbI, JIH60 YKa3aTb xapaKTepIiCTH-
tJeeKoe CBOHCTBO ero 3JIeMeHTOB. OTMeTHM, liTO BTOpOH cnoc06 npHMe-
HHM MH 3a.n.aHHH MHO)!{eeTB, co,ltep)!{aIII.HX KaK KOHellHoe IIHCJIO 3JIe-
MeHToB, TaK H 6ecKoHe l lHo MHoro 3JIeMeHTOB. HanpHMep, MHOJKeCTBO A
TOlleK M, JIe)!{aIII.HX Ha OKPY)!{HOCTH pa.n.Hyca r C ueHTpOM B TOIIKe 0,
3anHllieTCH CJIe.n.YIOIII.HM 06pa30M:
A={MI IOM/=r}.
MHO)!{eeTBa A H B CtJHTalOT pa8H.blM.U, ecJIR OHH COCTOHT H3 O.lI,HHXII
II Tex )Ke 3JJeMeHT08. B 3TOM CJIyqae nHlnyr A=B. HanpHMep, MHO-
)!{eCTBa A={P; 2 2 ; 3 2 ; 4 2 } H B={Vf; VT6; V S1 ; V 256} paBHbI,
TaK KaK 06a MHO)!{eCTBa COCTOHT H3 tJHCeJI 1, 4, 9 H 16. MHOJKeCTBO
,ltBY3HatJHblX llIiCeJI, Y KOTOpblX UH<ppa e,ltIiHHU: COBna.n.aeT C U:WppOH .n.e-
CHTKOB, paBHO MHO)!{eeTBY .n.BY3HSqHblX tJHCeJI, ,lteJIHIII.HXCH Ha 11-
06a MHO)!{eeTB8 COCTOHT HS tJHcen 11, 22, 33, 44, 55, 66, 77, 88, 99.
27

YopamUeUHJI
I. nepe'lHCJlHTe 9J1eMeHTbI CJle,ll;ytOUJ.HX MHO>KeCTB:
A - MHO>KeCTBO pa3J1H'IHblX 6YKB B CJlOBe «oapaoaH»;
B - MHO>KeCTBO CTOJlHLL CCCP;
C - MHO>KecTBO pa3J1H'IHblX uH«PP '1HCJla 214425;
D= {xl x - )l.eHb He)l.eJlH, H a3BaHHe KOTOporo H a'lHH aeTC!I C OYKBbI «II»}.
2. 1'!3oopa3HTe Ha '1HCJlOBOii np!lMoii CJle){yIOUJ.He MHO>KeCTBa:
A= { X1XER. x>2}; B= {xix EZo. x<2};
c= xlxEZ, -3<x<7}; D={xlxER, --3<x<7};
E= xlxEN, x<2}.
3. OJ>opMYJlHpyiiTe xapaKTepHCTH'IeCKHe CBOiiCTBa MHO>KecTBa: a) napaJlJleJIo-
rpaMMoB; 0) TynoyroJlbllblx TpeyroJlbHHKOB; B) TO'IeK OKPY>KHOCTH.
4. BbioepHTe lIa JlHCTe oYMarH TO'IKy 0 H OTMeTbTe II1TpHXOBKOii MHO>KecTBO TO'leK
K TaKHX, '1TO:
a) IOKI=5cM; 0) !OKI>5cM; B) IOKI<5cM.
5. nocTpoihe oTpe30K AB. r){e Jle>KaT TaKHe TO'IKHM,.D.JI!I KOTOpblX I AMI=IBMI?
6. OJ>opMyJlHpyiiTe XapaKTepHCTH'IeCKOe cBoiicTBO MHO>KecTBa TO'IeK oliCCeKTpl1CbI
yrJla.
7. YKa>KHTe Cpe){H CJle.n.yIOLUHX MHO>KeCTB paBHble:
A - MHO>KeCTBO pOMOOB C np!lMbIMH yrJlaMH;
B - MHO>KeCTBO KBa.n.paToB;
C - MIIO>KeCTBO np!lMoyroJlbHHKoB C KonrpY311THblMH cTOpollaMH,
D - MIIO>KeCTBO '1eTbIpexyroJlbHHKOB C np!lMblMH yrJlaMH;
E - MHO>KeCTBO np!lMoyroJlbHHKOB.
3. nO)J.MHoMeCTBa. )J.HarpawMbI 3nJlepa - BeHHa. nYCTb A - MHO-
)!{eeTBO peK B EBpone, a B= {BoJJra; lI.Henp; CeHa; OKa}. MHO)!(eCTBO B
HBJJHeTCH 'IaCTblO MHO)KeeTBa A, nOCKOJJbKY Ka)!{.n.blM 3JIeMeHT B HB-
JIHeTCH peKoH, npOTeKalOlIJ.eH B EBpone. rOBOpHT, 'ITO B HBJIHeTCH
noOM.liOJICCCm80M. MHO)KeCTBa A.
Bo061IJ.e, MHO)KeCTBO B Ha3bIBalOT nOaM.liw/CccmeoM. MHO)!{eCTBa A
Tor.n.a H TOJJbKO Tor.n.a, Kor.n.a Ka)K.n.bIH 3JJeMeHT B npHHa.n.JJe)!{HT MHO-
)KeeTBY A.
BbICKa3b1BaHHe «MHO)KeCTBO B HBJIHeTCH no.n.MHO)!{eCTBOM MHO)KeCT-
Ba A» 3anHCblBaeTCH npH nOMOIIJ.H CHMBOJJOB TaK: BeA. rOBOpHT TaK)!{e
«B BKJJIO'IeHO B MHO)KeCTBO A». o BbICKa3bIBaHHe 3KBHBaJJeHTHO CJIe-
,ltYIOIlI.eMY: «B MHO)!{eCTBO A BKJJIO'IeHO MHO)!{eCTBO B», TO eeTb A =>B.
COrJJaCHO .n.aHHOMY onpe.n.eJJeHHIO no.n.MHO)KeeTBa, Ka)K.n.oe MHO)KeCT-
BO HBJIHeTCH no.n.MHomeCTBOM caMoro ce6H: A eA. KpoMe TOro, C'IHTa-
lOT, 'ITO nYCToe MHO)!{eCTBO eCTb no.n.MHO)!{eeTBO JII060ro MHO)KeCTBa
A : 0eA.
Pa3JIH'IaIOT .n.Ba BH.n.a no.n.MHO)!{eCTB MHO)!{eeTBa A: caMO A H 0
Ha3b1BaIOT IiCC06Cm8CliliblMU noOM.lioJICccmeaM.u; Bce OCTaJIbHbIe no.n.MHO-
)!{eCTBa MHO)KeeTBa A, eeJIH OHH CYlIJ.ecTBYIOT, - c06cm8Cliliblc noOM.lio-
:JfCccmea A.
HanpHMep, MHO)KeCTBO A = {m; n; p} HMeeT llleeTb co6cTBeHHbIX
no.n.MHO)KeeTB: {m}, {n}, {p}, {m; n}, {m; p}, {n; p} H .n.Ba HeeOOCTBeH-
HblX: {m; n; p}, 0.
BbI.n.eJJHTb no.n.MHO)!{eeTBO B H3 MHO)!{eCTBa A MO)!{HO, YKa3aB CBOM-
CTBO, KOTOpbIM 06JJa.n.alOT HeKOTopbIe 3JIeMeHTbI MHO)KeCTBa A. TaK,
ecJIH A - MHO)KeCTBO 6YKB PYCCKoro aJJ<paBHTa, TO, 3a.n.aB CBOMCTBO
robITb rJIacHoib, Mbi BbI.n.eJJHM H3 A no.n.MHO)!{ecTBO B Bcex rJIaCHbIX 6YKB.
28

OqeBH.n.HO, liTO eCJIH BeA H A eB, TO A=B. 113 3TOrO YTBep)K.n.eHHH
BbiTeKaeT O.n.HH H3 cnoco60B .n.OKa3aTeJJbCTBa paBeHCTBa .n.BYX MHO)!{eCTB:
ecJJH .n.OKa3aHO, liTO JII060H 3JJeMeH'J H3 B HBJJHeTCH 3JIeMeHTOM MHO-
meCTBa A H, B CBOIO otJepe.n.b, JJl060H 3JJeMeHT H3 A HBJIHeTCH 3JJeMeH-
TOM MHO)KeCTBa B, TO .n.eJJaIOT BbIBO.n., qTO A=B.
KpoMe Toro, eCJIH A e B H Be C, TO A e C. lI.eHcTBHTeJJbHO, eCJ1H
Ka)!{.lI.blH: 3JIeMeHT MHO)KeCTBa A npHH3.lI.JIe)!{HT B, a Ka)!{.lI.blH 3JIeMeHT
MHO)!{eCTBa B HBJJHeTCH, B CBOIO otJepe,lt!>, meMeHTOM C, TO Ka)K)tbIH: 3JIe-
MeHT H3 A npHHa)tJJe)KHT MHO)!{eCTBY C.
MbI HCnOJlb30BaJIIi pa3J1H'IHble 0003HatteHH!I )I.JI!I OTHOIIleHHii «3J1eMeHT a "PH-
Ha){Jle>KHT MHO>KecTBY A» (aE A) H «MHO>KeCTBO B BKJlIO'IeHO B MHO>KeCTBO A» (BC::,A).
3m BbI3BaHO TeM, '1TO 9J1eMe1lTbI MHO>KecTBa caMH MorYT OblTb MIIO>KeCTBaMH. Ha-
npHMep, ecJIH A - MHO>KeCTBO OTp!l){OB nHoHepcKoii ){PY>KHHbI, a - O){HH H3 3THX
OTp!l){OB, TO 3anHCb aEA BepHa, a 3anHCb aeA HeBepHa. ECJlH Callia I1BaHoB-
nHOHep OTp!l){a a, TO Callia I1BaHoB Ea. n09TOMY eCJIH ObI MbI HanHCaJIIi aeA, TO
nOJlY'IIiJlH Obi, '1TO Callia I1BaHoB EA. Ho MHO>KeCTBO A COCTOHT H3 OTp!l){OB )l.aHHoil:
){PY>KHHbI, a He H3 OT){eJIbHbIX nHOHepOB. TaKHM oopa30M, 3anHCb aeA npHBeJIa K
OlliHOKe.
MHO)KeCTBO, COCTO!lLUee H3 O){IIOrO 9J1eMeHTa a, 0003HalJalOT {a}. CJIe){yeT pa3Jja-
'1aTb 3anliCH a H {a} - nepBa!l H3 HHX 03Ha'iaeT 9J1eMeHT a, a BTOpa!l - MHO>KeCTBO.
COCTO!lLUee H3 O)l.HOrO 9JIeMeHTa a. MO>KHO HanHcaTb aEA H {a}eA. HO HeJlb3!1 nHcaTb
aeA Ii, BooOrn.e rOBOp!l, HeJlb3!1 nHcaTb {a}EA.
DOHHTHH MHO)KeCTBa H nO.n.MHO)KeCTBa HCnOJJb3YIOTCH npH onpe,l1.e-
JIeHHH MHOrHX nOHHTHH MaTeMaTHKH, B tJaCTHOCTH npH onpe.n.eeHHH
reOMeTpHqeCKOH <PHrYPbI.
FCOMcmpUttCCKOii ifJU2ypoii Ha3b1BaeTCH BCHKOe MHO)!{eCTBO TOlleK.
T aKHM 06pa30M, OTpe30K, JIY'l, npHMaH, TpeyrOJIbHHK, map, Ky6 H T. .11..-
Bce 3TO reoMeTpH'leCKHe <PHrYPbI. TaK KaK MHO)!{eCTBO MO)!{eT COCTOHTb
H3 o.n.Horo 3JJeMeHTa, TO OT.n.eJJbHO B3HTaH TOtJKa (TOqHee rOBopH, MHO-
)!{eeTBO, COCTOHlIJ.ee H3 O.n.HOH TOqKH) HBJIHeTCH reOMeTpH'leCKOH: <pHry-
pOH, TaK )!{e KaK H JII060e KOHeqHOe MHO)!{eeTBO TO'leK. ECJJH <pHrypa
P l HBJIHeTCH c06cTBeHHbIM no.n.MHO)!{eeTBOM <PHrYPbI F 2, TO rOBopHT
TaK)!{e, tJTO F l - ttacmb <PHrypbI F 2 . HanpHMep, OTpe30K AB - tJaCTb
npHMOH AB.
lh06b1 HarJIlI,ltHO H306pa)!{aTb MHO)KeCTBa H OTHOJ.lieHHH Me)K.lI.Y HHMH,
PHCYIOT reoMeTpHtJeCKHe <PHrypbI, KOTopbIe Haxo.n.HTCH Me)K.n.y co60A
B 3THX OTHomeHHHX. HanpHMep, eCJIH MbI XOTHM HarJJH.n.HO H306pa3HTb,
tJTO MHO)KeeTBO A HBJJHeTCH COOCTBeHHbIM no.n.MHO)KeCTBOM MHO)!{eeTB8
B, TO pHcyeM 3TH MHO)!{eCTBa KaK Ha pHcYHKe 1. ECJIH )!{e H3.lI.O nOKa3aTl>,
tJTO MHO)!{eCTBa A H B He HMelOT 06IIJ.HX 3JIeMeHTOB, TO 3TH MHO)!{eeTB8
H306pa)!{alOT TaK, KaK Ha pHcYHKe 2. TaKHe H306pa)!{eHHH' MHO)!{ecTB
Ha3b1BalOT ouazpaM.MaMU .9iiAcpa - BC/ilia 1).
lI.HarpaMMbI 3HJJepa - BeHHa .n.eJJalOT HarJJH.n.HblMH pa3JJHtJItwe
YTBep)K)teHHH, KaCalOIlI,HeeH MHO)!{eCTB. HanpHMep, .n.HarpaMMa, ItQ-
Ka3aHHI;1H H8 pHcYIJKe 3, .n.eJJaeT otJeBH.n.HblM YTBep)K.n.eHHH: eCJIH A cB
H BeC, TO A eC.
I) 3iiJlep (/707-1783) - II1BeiiuapcKHii MaTeaTHK, lJJIeH neTepoyprcKoii AKa-
Jl,eMHH lIaYK, ,n*OH BeRR (/834-1923) - alirJlHHCKHH MaTeMaTHK.


PHC. I
80
PIiC. 2
Hepe,ltKO 6bIBaeT TaK, 'ITO paCCMaTpHBa-
lOT TOJIbKO no,ltMHO)!{eCTBa o.n.Horo H Toro )!{e
MHO)!{eCTBa I. TaKoe MHO)!{ecTBO I Ha3bIBalOT
YH.ueepcaAbliblM. M.IiO:JfCccmeoM.. TaK, eCJIH A-
MHO)!{eCTBO CTy,lteHToB nepBoro Kypca HeKO-
Toporo HHCTHTYTa, B - MHO)!{eCTBO cTy.n.eH-
TOK B 3TOM )!{e HHCTHTYTe, C - MHO)!{eCTBO
cnopTCMeHoB 3Toro )!{e HHCTHTYTa, TO B Ka-
lIeeTBe YHHBepCaJlbHOrO MHO)!{eeTBa I MO)!{HO
B3HTb MHO)!{eCTBO BCeX CTy.n.eHTOB ,ltaHHOrO
HHCTHTYTa, nOTOMY lITO Tor,lta A c I, B c I,
ec:.l.
Ha .n.HarpaMMax 3H:JIepa - BeHHa yHH-
BepCaJlbHOe MHO)!{eeTBO I qaCTO H306pa)!{alOT
B BH.n.e npHMoyrOJlbHHKa, a ero nO,ltMHomeCT-
Ba - KpyraMH.
YopamueuuJI
I. Ll.aHbI MHO>KeCTBa: A= {I; 21; 121; IS; 52; 512};
B= {16; 2; 5}; C= {i; 21; IS}; D= {21; IS; 52};
E= {121' 512}' F= {12' 15' 25}' K= {21' 121' 512'
52;15};'Q={512}. ' ., ,..
PIiC. 3 YKa>KIiTe, KaKlie H3 .ltaHHblX MHO>KeCTB !lBJI!lIOTC!I
no,n:MHO>KeCTBaMIi MHOJKeCTBa A. BbI!lcHIiTe. KaKHe H3
CJle)l.YIOUJ.llx BbIcKa3bIBallliH IiCTHHHbI, 11 KaKlie JlO>KHbI: a) BcA; 0) CcA; B) Dc:.
c:A; r) EQ; M FE; e) QcA.
2. 113BecTHo, '1TO MHO>KeCTBO M COCTOIiT H3 .ltBY3Ha'lHblX 'I HCeJl, KpaTHLIx 9.
06pa3YHTe no,n:MHOJKeCTBa MHO>KeCTBa M, COCTO!lLUlie H3 '1HCeJI, KOTOpble: a) KpaTHbi
itll6JIy 4; 0) KpaTHbi '1HCJlY 18; B) He ,n:eJI!lTC!I Ha '1HCJlO 3; r) He !lBJI!lIOTC!I '1eTHbiMIi.
3. P - MHOJKeCTBO CJlOB B npej{JIO>KeHHIi: «11 ,n:oJlro oy,n:y TeM JlIOOe3eH !I Hapo,n:y,
QTO '1YBCTBa ,n:oopble !I Jllipoi!: npooY>K.llaJI...J)
Ha30BIiTe Ii 3anliWHTC 9J1eMeHTbI MHO>KeCTBa P; BbI)I.eJIliTe H3 Hero nO,lJ.MHO>KeCTBa,
9JleMeHTbI KOTOpblX o6.a,'\aIOT CBOi'iCTBOM: .a) «ObITb HMeHeM cYLUecTBHTeJlbHblM»;
6) «OblTb MeCTOIiMeHHeM»; B) «ObITb nOiVIe>KaLUIiM».
4. 113BeCTHo, '1TO A cB Ii a EA. KaKlie 1i3 CJle,n:yIOLUHX BblCKa3b1BaHHi!: HCTHHHbI:
a) aEB; 6) acB; B) aB?
5. nYCTb A - MHO>KeCTBO BblnYKJlbIX MlioroyrOJlbliHKOB, B - MHO)f{eCTBO '1eTbI-
pexyrOJlbHHKOB, C - MHO>KecTBO TpanellliH. D - MHO>KecTBO napaJIJleJIOrpaMMoB,
E..,- MHO>KeCTBO p0M60B. YKa>KIiTe, KaKlie H3 ,n:aHHblX MHO>KecTB !lBJI!lIOTC!I no,n:MHO-
ecTBaMn ){Pyrlix. 3anHWHTe 9TO npH nOMOLUIi CHMBOJla c.
6. Ll.aHo MHO>KecTBO K= {a; b; {e; d}; Ie; f; m}}. Onpe,n:eJIHTe. KaKHe 1i3 CJle,n:ylO-
III.HX BblCKa3b1BaHHii HCTHIIHbI H nO'leM)':
a) {a; {e; d}}cK; 0) {e; d}cK; B) {{e; d}}cK;
r) {e; d}EK; ,n:) Ie; f; m}K; e) aEK.
7. 1130opa3liTe C nOMOil\bIO ,n:HarpaMM 9iiJlepa - BeHHa CJle,n:YIOUJ.ne napbl MHO-
JKecrs:
a) A - MHO>KecTBO cTy){eHTOB HeKoTOporo HHcTHTyra H B - MHO>KecTBO cTy){eH-
TOB ne,n:arorli'lecKoro «PaKYJlbTeTa 9TOro IiHcTHTYTa;
6) M - MHO>KeCTBO ){eBO'IeK B HeKoTOpoii I1IKOJle H P - MHO>KeCTBO nlioHepoB
B 3TOii I1IKOJle;
B) E - MHO>KeCTBO KBa,n:paTOB Ii D - MHO>KeCTBO TpeyroJlbHHKOB.
J) I13 CTHXOTBopeHlIH A. C. nYIIIKHHa «naM!lTHIiIO. .
30

8. nOCTpoJ\Te Jl,lIarp8MMbI 3liJlepa - BeHHa )I.JIH MHO>KeCTB, Jl,aHHblX B ynpIDK-
HeHIIH 5 9TOro naparpacpa.
9. PaccMoTpllM YHHBepcaJlbHoe MHO>KeCTBO 1 KHllr B 6H6J1110TeKe HHcTHTYTa H
ero nO)l.MHO>KeCTBa: M - KHllr no MaTeMaTHKe, A - KHHr no aJIreGpe, F - Kl\jJr
no «p1I3I1Ke, P - MHOJKecTBO KHllr Ha anrJlHHCKOM H3b1Ke. 11306pa311Te 9TH MHO>Ke-
eTBa C nOMOIIl,bIO Jl,Harp8MM 3J\Jlepa - BeHHa.
4. nepece4eHHe MH01KeCTB. nYCTb ,LI.aHbI ,LI.Ba MHO)!{eCTBa: A =
={a; b; c; d} Ii B={c; d; c}. 06pa3yeM HOBoe MHO)!{eCTBO P, COCTOHIlI.ee
H3 Bcex 3JJeMeHTOB, npIiHa,LI.JJe)!{aIll.IiX o.n.HOBpeMeHHO Ii A, Ii B: P= {c; d}.
rOBOpHT, '1TO MHO)!{eeTBO P HBJJHeTCH nepeee'leIjHeM MHO)!{eeTB A Ii B.
Boo6e, ncpccctteliUCM. M.IiOJICccme A Ii B Ha3bIBalOT MHO)!{eCTBO, B
KOTopoe BXO.n.HT Te Ii TOJJbKO Te 3JJeMeHTbI, KOTopbIe O,LI.HOBpeMeHHO
npIiHa,LI.JJe)!{aT MHO)l{eeTBaM A H B. nepeCe'leHHe MHO)!{eCTB A Ii B 000-
3Ha'laeTCH lIepe3 An B, r.n.e CIiMBOJJ n - 3Ha nepeCe'leIfIiH MHo)!{ecTB.
TaK KaK JJI060H 3JJeMeHT X Ii3 MHO)l{eeTBa A n B 06JJa.n.aeT CBOHCTBOM
«x E A Ii X E B», TO .n.aHHoe onpe,LI.eJJeHIie nepeCe'leHIiH .n.BYX MHO)!{eeTB
MO)!{HO 3anIicaTb B T aKOM BIi,LI.e:
An B= {xix E A/\xE B}.
ECJJH MHO)!{eCTBa A H B He IiMelOT 06Il1.IiX 3JJeMeHTOB, roBOpHT, liTO
3TIi MHO)!{eeTBa IiC nCpCCCKalomcfl, Ii nmllYT A n B= 0. HanpHMep, He
nepeceKalOTCH MHO)!{eeTBa o.n.H03Ha'lHblX HaTypaJJbHbIX llIiCeJJ H ,LI.BY-
3Ha'lHbIX HaTypaJIbHbIX llIiCeJJ.
ECJJIi )!{e MHO)!{eeTBa A Ii B IiMelOT XOTH 6bI O,LI.IiH 06IiH 3JJeMeHT, TO
rOBopHT, '1TO MHO)!{eeTBa A Ii B nCpCCCKalOmcfl IiJJIi liTO nepeee'leHHe
MHO)!{eeTB A Ii B Iicnycmo, Ii nHlliYT A n B=I=0 .
Ha pIicYHKe 4 MHO)!{eeTBO A n B Ii306pa)l{eHO 3alliTpHxoBaHHoH
06JJ aCTblO.
nYCTb a - xapaKTepllCTlIlJeCKOe CBOHCTBO 9J1eMeHTOB MHOJKeCTBa A, a  - xa-
paKTepHCTHlJeCKOe CBOHCTBO 9J1eMeHTOB MHO>KeCTBa B, mr)l.a B MHO>KeCTBO An B BOH-
.nYT TOJlbKO Te 9J1eMeHTbI )l.aHHblX MHO>KeCTB, KOTopbIe 06J1a)l.a1OT CBoJ\CTBaMIl a H p
O)l.HOBpereHHO. HanpHMep, ecJIlI A - MHO>KeCTBO TynoyroJlbHblX TpeyroJlbHHKOB
(3.necb XapaKTepHCTHlJecKoe CBOHCTBO a - «6b1Tb TynoyroJlbHblM TpeyrOJlbHIIKOM»),
a B - MHO>KecTBO paBHoOe.npeHHblx TpeyroJlbllllKOB (xapaKTepllCTlIlJecKoe CBOHCTBO p
3J1eMeHTOB 3TOro MHO>KeCTBa - «6b1Tb paBHoOeJl,peHHblM TpeyrOJlbHHKOM»), TO MHOJKe-
CTBY An B npllHa)l.Jle>KaT TpeyrOJlbHHKII, KOTOpble HBJlHIOTC!I II ryIIOyroJlbHblMII, H
paBll06e.npenHblMII, T. e. ryIIoyroJlbHble paBH06e,l\peHHble TpeyroJlbHIIKIl.
Onepau.IiH nepeCelleHIiH MHO)!{eeTB 06JJa.n.aeT pH,LI.OM CBoilcTB.
1. flcpccc!f.CliUC M.Ii0:JfCCCm8 KOM.-
M.ymamu81i0: .n.JJH JJI06bIX MHO-
)!{eeTB A Ii B HMeeM
AnB=BnA. (I)
31'0 CBOHCTBO BbiTeKaeT Ii3 on-
pe.n.eJJeHHH onepau.IiIi nepeee'le-
HHB MHO)!{eeTB A H B.
PHC. 4
31

A
c
a)
PHC. 5
: )
--
/
6)
2. flcpccc4,CIiUC M.IiOJICccme acco/{uamU8IiO: ,LI.,'!H mo6bIX MHOiKeCTB
A, B, C HMeeM
(A n B) n C=A n (B n C). .
(2)
31'0 n03BOJJHeT 3anHCbIBaTb BbIpa)!{eHIie A n B n C 6e3 CKo60K H Ha-
XO.n.HTb nepeee'leHHe J1Io60ro tJHCJJa MHO)!{eCTB.
Har.IJH.n.Ho CBoikTBO (2) HJJJJIOCTpHpyeTcH npH nOMOlll.H ,LI.HarpaMM
3HJJepa - BeHHa, H306pa)!{eHHbIX Ha pHCYHKe 5. J1eBaH '1aCTb paBeH-
CTBa (2) H306pa)!{eHa Ha pHCYHKe 5, a) (BepTHKaJJbHOH IlITPHXOBKOH OTMe
qeHO MHO)!{eCTBO A n B, a ropH30HTaJIbHOH - MHO)!{eCTBO C); 06JJaCTb,
3alliTpHxoBaHHaH ,LI.B, a)!{,LI.bI H306pa)!{aeT MHO)!{eCTBO (A n B) n C. Ha
pHcYHKe 5, 6) BepTHKaJJbHOH lliTpHXOBKOH OTMeqeHO MHO)!{eeTBO A, a
ropH30HTaJJbHOH - MHO)!{eCTBO B n C; a 06JJaCTb, 3alliTpHxoBaHHaH
,ltBa)!{,LI.bI, H306pa)!{aeT MHO)!{eCTBO A n (B n C). CpaBHHBaH 06JJaCTH, 3a-
lliTpHxoBaHHbIe ,LI.Ba)!{,LI.bI Ha pHCYHKax 5, a) H 5, 6), npHxo.n.HM K BbIBO,LI.Y,
liTO MHO)!{eCTBa (A n B) n C H A n (B n C) paBHbI.
OTMeTHM, HaKoHeu, eIIJ.e O.n.HO CBOHCTBO:
3. EC,1U AcB; mo A nB=A.
lI.eHcTBHTeJJbHO, eCJJH A - nO,LI.IHO)!{eCTBO MHO)!{eCTBa B, TO 3TH
MHO)!{eCTBa H306pa)!{aIOTCH TaK, KaK nOKa3aHO Ha pHcYHKe I. 3JIeMeH-
TallIH, npHHa,LI.JJe)!{aIIJ.HMH O,LI.HOBpeMeHHo H A H B, HBJJHIOTCH 3JleMeHTbi
MHO)!{eeTBa A, T. e. An B=A.
B qaCTHOCTH, ,LI.JJH JII060ro MHO)!{eCTBa A HMeeM:
AnA=A, An0=0, An/=A.
YnpamaeHHJI
J. H aH){IITe nepecelJell He MIIO>KeCTB:
a) A= j a; b; c; d; e; f}, B= U; d; a; m};
) A= x!xEN, x<IO}, B={xlxEN, x<:7}; .
B) A= xlxEN, x- ){BY3HalJHOe lJIICJlO};
B= xlxEN, x OKaH'IHBaeTC!I UH«PPOH 7};
r) A={xlxEN}, B= {xlxEZo}'
2. )].aHbJ MHO>KeCTBa E H F. y1306pa3HTe 9TII MHO>KeCTBa C nOMoLUbIO )l.HarpaMM
32

3HJlepa - BeHHa H OTMeTbTC II1TpIlXOBKOH MHO>KeCTBO En F, eCJIH: a) HaF;
0) FeE; B) EnF=0; r) E=F.
3. HcnoJlb3Y!l lJHCJlOBYIO rrp!lMYIO, HaH,D.HTe rrepeCeqeHlle CJle)l.YIOLUHx MHO>KecTB:
a) A= {x/xEZ, x>-2} II B= {xlxEZ. x.,.:8};
0) A= {xlxER. x-2} II B= {xlxER, x.,.:8};
I
B) A={xlxER, 0<x<3 2 } II B={xlxER, -3<x<lh
r) A={xlxER, x<-1,7} H B={xlxER, x>-I,2}.
4. nYCTb A - MHO)KeCTBO cTy){eHTOK B lIeKoTOpoM IIHCTHTYTe, B - MIIO>KecTBO
yqaLUllxc!I B 9TOM HHCTHTYTC, KOTopble cTapIIle 20 JleT. OnllIIlIlTe MHO>KeCTBO An 8
H H300pa3HTe MHO)KeCTBa A, B H An B rpa«plIlJeCKIl.
5. P - MHO)KeCTBO Bcex np!lMoyrOJlbHIIKOB. R - MHO>KeCTBO Bcex pOM008.
KaKHe «PHrypbl npHlla)lJIeiKaT MHO>KeCTBY P n R?
6. OjJopMYJlHpyihe CBoHcTBa 3J1eMeHTOB MHO>KeCTBa P n Q n S II rrepelJlICJlIlTe ero
3J1eMeHTbi, CCJlH
P= {xlxEN, x<20},
Q= {xlxEN, x<20. x - lJeTHOe lJIICJlO}.
S= {xlxEN, x<20, x KpaTllo 3}.
7. Ha pHcYHKe 6 11300pa>KeHbI MHO>KecTBa M, P II Q. nepepllcyiiTe IIX 8 TeTPIWt
H OTMeTbTe WTpllXOBKOH Ka>K){Oe H3 MHO>KecTB:
a) Mnp; 0) MnQ; B) pnQ; r) MnpnQ;
A) (Mnp)n (pnQ).
8. PacnOJlO>KHTe ){Ba yrJla TaK, lJTO IIX nepecelJeHlle ecTb: a) TpeyroJlbllHK;
0) JlY'I; B) TOlJKa; r) OTpe30K; .l!) lJeTbipexyroJlbHHK.
9. Haii){HTe nepeCelJeHHe: a) CMe>KHblX yr JlOB; 6) BepTIIKaJIbHblX yrJloB.
10. KaK Ha3b1BaeTC!I nepeCelJellHe: a) Kpyra II np!lMOH; 0) Kpyra II rroJlyrrJlocKOCTH;
B) Kpyra II yrJla C BepIIlHHoii B ero u.eHTpe?
5. 06'be.IJ.HHeHHe MHO)KeCTB. PaccMOTpHM eIII.e O,LI.HH H3 cnocoOOu
nOJIyqeHHH HOBoro MHomeCTBa H3 .n.BYX .n.aHHbIX.
06uOUlicliUCM. 06YX M.IiOJICccme A H B Ha3bIBaeTCH MHomecTBO.
COCTOHIII.ee H3 3JIeMeHTOB, KOTopbIe npHHa.n.JIe)!{aT XOTH 6bI O.n.HOMY H:I
3THX MHO)!{eeTB. OO"be,LI.IlHeHHe MHomeeTB A H B 0603HaqalOT A U B, r,lte
CHMBOJI U - 3HaK 06"be.n.IiHeHHH MHomeCTB.
HanpHMep, OO"be.n.lIHeHHeM MHomeCTB A = {m; n; p; k; 1} H B=
={p; r; s; n} HBJJHeTCH MHomecTBO A UB={m; n; p; k; 1; r; s}.
Ha pHcYHKe 7 MHomeeTBO A U B H306pameHO 3alliTpHxOBaHHoi
06JJaCTblO.
no onpe.n.eJJeHHIO B 06"be.n.HHeHHe .n.BYx MHomeeTB A H B MorYT BXo-
.n.HTb 3JIeMeHTbI Ii3 A, He npHH3.lI.JJe)!{aIII.He MHomeCTBY B, 3JIeMeHTbI H3
B, He npHHa.n.JIemaIII.He A, H 3JIeMeHTbI, npHHa.n.JIemaIII.He MHomeeTBaM
A H B O,LI.HOBpeMeHHO.
PaccMoTpHM eIII.e O.n.HH npHMep. nYCTb A - MHomeCTBO yqaIII.HXCH
HeKOToporo KJIaCCa, 3aHHMalOIUHXCH B Bo.TJei16oJIbHOH ceKUHH, a B -
PHC. 6
PHC. 7
! Nt 702
33

MlIO?i5ecTBO Y'Iaw.HXCH TOrO )!{e KJIaCCa, 3aHIiMalOIIJ.IiXCH B MaTeMaTH-
lJ.eeKOM KpymKe. Tor.n.a B MHomeCTBO A U B BOH.n.YT Te YIJ.aIIJ.HeeH,
KOTopbIe 38HIiMaIOTCH B BOJJeH60JJbHOH ceKUHH IiJJIi B r.,aTeMaTIitIeCKOM
KpY)!{Ke. Cpe.n.H HIiX MorYr 6bITb TaKIie, KOTopble 3aHiMaIOTcH TOJJbKO
B BOJJeH60JJbHOH CeKlI.IiIi, JJIi60 TOJJbKQ B MaTeMaTHtIecKoM KpymKe,
H MorYT 6bITb TaKHe, KOTopbIe 38HIiMalOTCH Ii B BOJJeHoo.nbHoH CeKlI.IiIi
H B MaTeMaTHIJ.eCKOM KpymKe.
TaK KaK JJI060H 3JJeMeHT x Ii3 MHomeCTBa A U B 06JIa.n.aeT CBOHCTBOM
ex E A HJIH X E B», TO onpe,lteJJeHIie ooo.e,ltIiHeHIiH .n.BYX MHomeCTB MQ)K-
no 3anIicaTb TaK:
A UB={xlxEAVxEB}.
ECJlH 0: - xapaKTepHCTH'Iecxoe CBOHCTBO 9JIeMeHTOB MHO>KecTBa A,  - xa-
J>8.KrepHCTII'IeCKOe cBoiiCTBO 9JIeMeHTOB MHO>KeCTBa B, TO B MHO>KeCTBO A U B BOH.D,YT
T3KHe 3J1eMeHTbI, KOTOpbIe 06J1a)l.8JOT cBoiiCTBOM 0: IIJlII cBoiicTBOM . (Co103 «HJIII» 3)1.ecb
JlOHIlMaeTC!I TaK, KaK ero IlDHHMak>T B MareM3THlJecKOH JlOrllKe, T. e. KaK Hepa3.l\eJIH-
TeJlbHbIii. )
Onepan.HH 06'be,ltHHeHIiH MHO)!{ecTB 06JIa,ltaeT TaKIiMH CBOHCTBaMIi:
1. I1.Afl A106blX M.IiO:JfCccme A U B UMCCM.
AU B=B U A (KOMMYTaTIiBHocTb).
2. I1.AR A106blX M.IiO[JICccme A, B u C UM.CCM.
(A U B) U C=A U (B U C) (aCCOII.IiaTIiBHOCTb).
STo CBOHCTBO n03BOJIHeT nIicaTb Bblpa)!{eHIie (A U B) U C 6e3 CKO-
60K Ii rOBopIiTb npo 06'be,ltIiHeHIie JJI060ro tIIiCJJa MHomeeTB.
3. ECAU BeA, mo A UB=A.
B qaCTHOCTIi, JJ)IH JII060ro MHO)l{ecTBa A HMeeM:
AUA=A; AU0=A; AU/=/.
3TH CBOHCTBa ,110Ka3bIBaIOTCH aHaJrOrIitJHO TOMY, KaK .n.OKa3b1Ba-
JIIiCb CBOHCTBa nepeeetIeHIiH MHO)!{eCTB.
CBH3b Me)!{.n.y OnepaII.HHMH nepeeetIeHIiH Ii 06"be.n.IiHeHIiH MHO)!{eeTB
OTpa)l{aIOT CBOHCTBa oucmpu6J!mu8Hocmu;
4. II.Afl AmlX M.IiOOlCecmB A;I B U C cnpa6eoAu6bl paeeHcmea:
a). A n (B U C)=(A n B)U,(A n C);
6) A U (B n C)=(A U B) n (A U C).
a)
34
8
c
6)
PIIC. 8

CBoiicTBa .n.HCTpH6ynmHocTH (KaK, BrIpOtJeM, H npe.n.bI.n.YIIJ.He TPH
cBoikTBa) HJJJJIOCTPHPYlOTCH Ha .n.HarpaMMax 3iiJIepa - BeHHa. TaK.
Ha pHcYHKe 8 npHBe.n.eHbI .n.HarpaMMbI, COOTBeTCTBYIOIIJ.He JJeBOH H npa-
Bon tJacTH COOTHoweHHSI 46). Ha .n.HarpaMMe a) BepTHKaJJbHOH IIITPH-
XOBKOH OTMetreHo MHO)!{ecTBO A, rOpH30HTaJJbHOii - MHO)l{eeTBO B n C.
BCH 3aIlITpHXOBaHHaH 06JIacTb npe.n.cTaBJIHeT c060ii MHO)l{eeTB()
AU (B n C). Ha .n.HarpaMMe 6) BepTHKaJIbHOii lliTpHXOBKOii OTMetJeHO
MHO)l{eCTBO AU B, ropH30HTaJJbHOH - MHO)!{eCTBO AU C. 06JIaCTb,
aaIlITpHXOBaHHaH .n.B, a)!{.n.bI H306pa)!{aeT MHO)!{eeTBO (A U B) n (A U C).
PaccMaTpHBaH nOJIY'IeHHbIe 06JIaCTII, npHXO,ltHM K BbIBO.n.y, l[TO MHO)l{e-
CTBa A U (B n C) H (A U B) n (A U C). paBHbI.
YopamReR.R
1. Ha!i.1(HTe OO'be){HHeHHe MHO>Kecra A II B, ecJIH:
a) A= b; 0; M}, B= {M; 0; p; e};
6) A - MHO>KecTBO UH«}Jp '1RCJla 482 528. B - MHO>KecTBO uR«PP qRCJla 5 283 824;
B) A={xlxEN. x.,.:9}, B={xlxEN, x>lO};
I 
r) A={xlxER. 0<xo;;;;;42"}' B={xlxER, -7<x<2}.
2. GpoPMYJlHpyihe XapaKTepHCTII'IeCKOe CBOHCTBO 9JleMeHTOB MHO>KecTBa K=
= M Up. eCJIII M - MHO>KeCTBO qeTHbIX HarypaJrbHblx qHCeJl, a P - MHO>KecTBO
)l.BY3RaqHbIX HarypaJIbHblX qHCM. YKa>KHTe, KaKHe R3 CJIe.ll.yiOLURX BblCKa3b1BaHHA
RCTHHHbI, a KaKHe JlO}/{HbI:
a) 12EK; 0) 8K; B) 13EK; r) OK;
.n:) -24EK; e) 139K.
3. P - MIIO>KeCTBO np!lMoyroJlbllHKOB, R - MIIO>KecTBO pOM6oB. KaKHe «PHrYPli
npHH8)l.JJe>KaT MHO>KeCTBY PUR?
4. Ll.aHbI MHO>KeCTBa: A={xlxER. -I <x<2}, B={xlxER. 0<x<4}. C=
={xlxER, -2.,.:x.,.:0}. .
I1cnoJlb3Y!l '1HCJlOBYIO np!lMyiO. YKa>iCIITe xapaKTepllCTRqecKoe CBOHCTBO 9JleMeH-
1'OB Ka>K)l.OrO 113 CJle){YIOLUHX MHOCTB: .
a) AUB; 0) AUC; B) BUC; r) BnC;
.n:) AnB; e) AnC; >K) (AUB)nC; 3) (AnC)U(AnC);
H) AUBUC; K) AnBnC.
I. KaKIrn: «pHrypa HBJI!leTC!I o61.e,zufHeHHeM )l.BY:X CMe>KHblX yrJlOB?
6. npe){CTaBbTe KBa){paT B BH){e OO'be){HHeHII!I qeTbipex TpeyrOJlbHIlKOB.
6. ,lJ,onOJIHeHHe K nO)l.MHO)KeCTBY. PaSllOCTb MHO)l(eCTB. nyCTh B-
no.n.MHO)!{eCTBO MHO)!{ecTB3 A. MHO)KeCTBO BCeX 9JIeMeHTOB H3 A, He
npHH3.lI.JJe)!{aIIJ.HX MHO)l{ecTBY B, Ha3b1BaIOT oonOAliCliUCM. K nOOM.Ii03lCe-
cmey B H 0603Hattf1IOT B A . TaK, eCJIH A - MHO)!{ecTBO Y'laIIJ.HXCH B He-
KOTOpOM K.'1aCCe, a B - MHO»<ecTBO ,lteBOtJeK B HeM, TO B A eeTb MHO)!{e-
CTBO MaJIb'lHK B, 3TQtd. KJIacce.
ECJJH Be A, 1'0 Ha ,ltHarpaMMe 3HJIepa - BeHHa
npHH3.lI.JIe)!{aT Bce TO'lK1i MHO)!{eeTBa A, He
npHH3.lI.JIe)!{aIIJ.He B. Ha pncYHKe 9 MHO)l{eCT-
BO B A 3aurrpHxOBaHo.
.u.onOJIHeHHe K MHO:H<ecTBy B B YHHBep-
CaJIbHOM MHO)!{eCTBe I 0603Ha'lalOT B ' . Han-
pHMep, eeJIH I - MHO)!{eeTBO BCex TpeyrOJIb-
HHKOB, a B - MHO)!{eCTBO npHMoyrOJIbHhlX
TpeyrOJIbHHKOB, TO .n.onOJIHeHHeM K nO)l.MHo-
2*
MHO)KecrBy. B A
'A
Pac. 9
85

A
B
PHC. 10
JKeeTBY B 6Y.LI.eT MHO)!{eeTBO B', COCTOHIlI.ee
H3 BceX TpeyrOJJbHHKOB, He 51BJJ5IIOIlI.HXCH
np5lMoyrOJJbHbIMH.
D.JJ5I JJI06bIX nO.LI.MHO)!{eCTB A H B YHH-
BepCaJJbHOrO MHO)!{eCTBa / HMelOT MeCTO CJJe-
,ltYIOIlJ,He paBeHcTBa:
1) (AnB)'=A I UB ' ;
2) (A U B)' =A / n B ' .
B caMOM .n.eJJe, B CHJJY onpe.n.eJJeHHti onepaUI1H Ha.LI. MHO)!{eCTBaMH
HCTHHHbI CJJe.n.YIOIlI.He 9KBHBaJJeHlI.HH:
x E (A n B)' x(A n B)xA V xB
(xE A ' )V(x E B')X E (if' UB').
3HaqHT, MHO)!{eCTBa (A n B)' HA' UB' COCT05lT H3 O.LI.HHX H Tex )!{e 3JJe-
MeHTOB, a nOToMY (A n B)' =A' U B'. BTOpoe paBeHCTBO .n.OKa3b1BaeTC5I
TOqHO TaK )!{e.
Pa8IiOCmblO d8YX M.IiOJICccme A H B Ha3bIBaeTC5I TaKoe MHO)!{ecTBO,
B KOTopoe BXO.n.51T Bce Te 3JJeMeHTbI, KOTopble npHHa.n.JJe)!{aT A H He
qpHHa.n.JJe)!{aT B. Pa3HOCTb MHO)!{ecTB A H B 0603HaqalOT CHMBQJJOM
A"B. HanpHMep, eCJJH A={aj bj Cj d; e}, a B={b; dj c; kj f; n},To
A"B={aj c}. TaKHM 06pa30M, A"B=A,,(A n B). OTMeTHM, tJTO
A'=/"A. ,.
TaK KaK JJl060H 3JJeMeHT x H3 MHO)!{eCTBa A'.B 06JJa.n.aeT CBOHCTBOM
«x npHHa.n.JJe)!{HT A H He npHHa.n.JJe)!{HT B», TO onpe.LI.eJJeHHe pa3HOCTH
MHO)!{eCTB A H B MO)!{HO 3anHcaTb B TaKOM BH.LI.e:
A"B={xlxE A/\xB}.
Ha pHcYHKe 10 MHO)!{ecTBO A"B 3aIIITpHxoBaHo.
.lI.JIH JI106b1x MHO)!{ecTB A, Dol H t cnpaBe.n.JJHBbI CJJe.n.YIOIlJ,He paDeH-
CTBa, CBJI3b1BaIOIlI.He BbltJHT8HHe MHOJKecTB c ,ltPyrHMH Onepal1HJIMH
H8.lI. MHo1keCTBaMH: '
a) A,,(B n C)=(A"B) U (A"C)j
6) A,,(B U C)-=(A"B) n (A"C)=(A"B)"C.
YopSIHHeHHJI
J. 8AAHTe JlOIIO,JlHeH)le K MHOJKecTBY B JlO MHO>KecTB8 A, ecJlH:
 A'If.{1I; 12; 4ai 54; 1}, B={7; 12};
d) Aa{xlxEN, x<:IO}, B={xlxEN, x<5};
B) A - OJKecrBO Y'!ai!J.HxCfI HeKoroporo uacca, B - MHOJKecTBO OTJlH'IHHKOB
B 9TOM ¥8c&.
2. 0&98'1HM 'Iepea N' )l.QIIOJlHeHHe K MHOJKeCTBY Bcex HaTyp8JIbHbIX 'IHCM B
MHOJKect9 g Bcek Ue.fIbIX imce.n. YKaJKHTe CpeJlH CJleJlyiOIUHX Bl,iCK89h1BaHHft HCTHH-
HbIe:
8) -3eN'; 6) OeN'; B) ON'; r) -3,5EN';
JI.) 2EN'; e) 12N".
3. J.I3BecTHO, 1JTO Pel, QCi.1 H pnC2*.0(p»c. II). OrMeTbTe 06JIaCTH, HsoO-
pa>K8IOIUHe MHO>KeCTBa:
36

PIIC. II
J
@8
a) 5)
PIIC. 13
PIIC. 12
a) p'; 0) Q'; B) (P n Q)'; r) (P U Q)'; A) P' U Q'; e) P' n Q'. YKa>KIlTe cpeAH 9'I'HX
MHO}!{ecTB napbI paBHblx.
4. D - MHO>KeCTBO ){BY3Ha'lHblX HaTypaJIbHblX '1I1CeJI, P - MHO>KeCTBO Bcex
He'leTHbiX HaTypaJlbHblx '1I1CeJI. QpopMYJIIIPYHTe XapaKTepKCTH'IecKoe CBOHCTBO 9Jle-
MeHTOB MHO>KeCTB D"-.P II P"'-.D. BepHo JIH, 1JTO D"'-.P= P"-.D?
5. Haiii\HTe pa3HocTb MHO>KeCTB P II S, eCJIII:
a) P= {xlxEZ, -4.o;;:x.o;;:6}, S= {xlxEN. 3.o;;:x.o;;:10};
6) P={xlxER, -7.o;;:x.o;;:O}; S={xlxER, -3,5.o;;:x.o;;:3}.
6. Ha PHCY!lKe 12 11306pa>KeHbI MHO>KecTBa T II S. nepepllcYHTe IIX B rerp3J1&
H OTMeTbTe WTPIIXOBKOH CJleAYIOLUlle MHO>KecTBa:
a) T"'-.S; 6) S"-.T; B) (TUS)"'-.S; r) T"'-.(TnS);
A) (TUS)",T; e) (TUS)".(TnS).
7. ,AHarpaMMbI 3ftJlepa - BeHHa H cjJopMbI JlOrH'IeCKHX paccYJKJJ.eHHft. lI.Har.
p8MMaMIl 3iiJlepa - BeHlla YA06HO nOJIb30BaTbC!! JVIfi lIarJlfIAHoro H306paJKeHHH
cooTHollieHIIH Me>KAY pa3JIII'IlIbiMH nOH!!TRfIMIl. nYCTb )l.aH1>I nOHfITllfi a H b. H<J<?6pa31111
a)
8@cQ)
PHC. 14
@oo
5) 6)
PIIC. 15
Q)
o
0)
PHC. 16
37

HX 061>eMbi A H B ){HarpaMMaMH 3ftJlepa - BeHHa.
ECJlH Jli060e b !!BJI!!eTC!! a, 1'0 06'beM nOH!!TH!! b-
nO){MHO>KeCTBO 061>eMa nOHRTH!! a, BC:A. B 9TOM
CJlYlJae )l.HarpaMMa 3llJlepa - BeHHa HMeeT BHII, H30-
Opa>KeHHbIH Ha pHcYHKe 13, a HJlH 13, 6. ECJlH >Ke
HH O)l.HO a He !!BJI!!eTC!! b, TO MHO>KeCTBi.\ A H B He ne-
peCeKaIOTC!!, A nB=0', H HarpaMMa 8llJlepa - BeH-
Ha HMeeT BH, H306pa>KeHHbIH Ha pHcYHKe 2 (c.30).
ECJlH HeKoTOpble a !!B,'!!!IOTC!! b, TO MHO>KeCTBa A H B
PHC. 17 HMelOT 06LUHe 3J1eMeHTbI. a 9TOM cJlYlJae B03MO)KHbI CH-
Tyal\HH, YKa3aHHble Ha pHcYHKe 14. ECJlH HeKOTO.
pble a He !!BJI!!IOTC!! b, TO B03MO)!{Hbi CJlY'IaH, H300pa>KeHHble Ha pHcYHKe 15.
PaccY>K){eHHe «ECJlH Bce a !!BJI!!IOTC!! b, a Bce b !!BJI!!IOTC!! C, TO Bce a !!BJI!!JOTC!! C»
B nepeBO)l.e Ha !!3bIK TeopHH MHO>KecTB 03HalJaeT, lJTO eCJlH Ac::B H BC::C, TO AcC.
A 9TO YTBep>K)l.eHHe O'leBH)l.HO (pHC. 3, C. 30). TO'lHO TaK >Ke H3 pHcYHKa 16 BH){HO, '1TO
ecJIH Bce a !!BJI!!IOTC!! b H HH OHO C He !!BJI!!eTC!! b, TO HH O)I.HO C He !!BJI!!eTC!! a: eCJlH
ACB H CnB=0, TO CnA=0.
B TO >Ke BpeM!! paCCY>K)I.eHHe no «p<>pMe ECJlH Bce a !!BJI!!IOTC!! b H HH 0)1.110 C He
"BJI!!eTC!! a, TO HH O,lJ,II0 C He !!BJI!!eTC!! b» HenpaBHJlbHO nOCTpoeHO. Ha !!3b1Ke TeopHH
MHO>KeCTB OHO 0603Ha'laT, '1TO eCJIH AcB H CnA=0', TO H CnB=0'. Ho PRCYHOK
J7 nOKa3b1BaeT, '1TO 9TO, Bo06LUe rOBOp!!, He TaK.
YopamUeUDJI
I. J1YCTb I - MHO>KeCTBO lJeTbipexyrOJlbHHKOB nJlOCKOCTH, A - nO)l.MHO>KeCTBO
Tpanel\Hll, B - napaJlJleJIOrpaMMOB, C - pOM60B, D - np!!MoyrOJlbIlHKOB, E-
ICB8)l.paTOB, F - lJeTblpexyrOJlbHHKOB C nepneHHKYJI!!PHbIMH )l.HarOHaJl!!MH, G-
'leTbipexyrOJlbHHKOB, ){HarOHaJIH KOTOpblX )l.eJ\iIT ){Pyr pyra nOnOJlaM. l1306pa3HTe
3TH MHO)l{eCTBa C nOMOUJ.b1O Harp8MMbI 3ii:.nepa - BeHHa.
2. l1306pa3HTe C nOMOLUblO ){HarpaMM 3iiJlepa - BeHHa CJleYIOLUHe cjx>pMbl
.IIOrHtleCKHX paccY)K)l.eHHii.
a) Ec.'1H HH O)l.HO a He !!BJI!!eTC!! b H HeKOTOpble C RBJI!!IOTC!! a, TO HeKOTopbIe C
He !!BJI!!IOTC!! b;
6) ECJlH Bce a !!BJI!!IOTC!! b H HeKoTOpble C He !!BJI!!IOTC!! b, TO HeKOTOpble C He
JlBJlJIIOTC!! a;
B) ECJlH HeKOTopbIe a !!BJI!!IOTC!! b, a HeKOTOpble b !!BJI!!IOTC!! C, TO IIeKOTOpbIe a
RBJI!!IOTC!! c.
KaKHe H3 9THX <jJOPM paCCY>K,!I.eHHH npaBHJlbHbI, a KaKHe HenpaBHJlbHbI? no-
crpoHTe npHMep, nOKa3bIBalOLUHH HenpaBHJlbHOCTb <jJOpMbI B).
3. nOCTpOHTe HarpaMMbI 3HJlepa - BeHHa )l.JI!! ynpa>KHeHHll n. 2,  I.
 2. 3JIEMEHTbI KOM6HHATOPHKH
l. l{eKapToBO npOH3BeJl,eHHe Jl,BYX MHO)KeCTB. Ha'lHeM C npuMepa.
qIiCJIO 27 3anIiCbIBaeTCH C nOMOIIJ.blO .n.BYx UH<pP 2 Ii 7. 3TH UH<PPbI CJIe-
,ltyeT 3anIiCbIBaTb B onpe.n.eJIeHHOM nOpH.n.Ke: CHa'laJIa 2, a IIOTOM 7.
ECJIIi HX nepecTaBIiTb, nOJIY'lIiTCH .n.pyroe '1HCJIO 72. fOBOpHT, '1TO
(2; 7) - ynopfliJOttCliliafl napa '1IiCeJJ. YnopH.n.o'leHHYIO napy '1IiCeJI x
H y 6y.n.eM 3anIiCbIBaTb cJIe.n.YIOIIJ.IiM 06pa30M: (x; y). B '1IiCJIO 44 BXO.n.HT
,ltBe o.n.IiHaKOBbie UIi<PPbI. OHIi 06pa3YIOT ynopH.n.o'leHHYIO napy (4; 4).
TaKHM 06pa30M, B ynopH.n.o'leHHbIx napax 'I IiCJIa MorYT nOBTO-
pHTbCH.
YnopH.n.O'leHHbIe napbI MO)!{HO COCTaBJIHTb He TOJJbKO Ii3 quceJI,
HO Ii H3 3JIeMeHTOB JIl06bIX MHO)KeeTB. nYCTb 3a.n.aHO MHO)KeCTBO X
88

H nYCTb x H y - 3JJeMeHTbI 3TOrO MHO>KecTBa (npH 9TOM MO>KeT CJIytJHTb-
CH, tITO X=y). Ha30BeM (x; y) ynoPWOttCIi1WU napot1, a x H y - KOM.nD-
HelimaM.U HJIH KOOpOUliatna.Mu 3TOH naphI. naphI (Xl; Yl) H (Xz; yz) CtJH-
TaI9TCH c06naOalOl4UMu B TOM H TOJIbKO TOM CJIytJae, Kor.n.a Xl =XI R
Yi=Y2' n03TOMY ecJIH X=FY, TO naphI (x; y) H (y; x) pa3JIHtIHbI.
HanpHMep, H3 6YKB MHO)!{ecTBa X = {a; 6; B} MO)!{HO COCTaBHTb,lte-
BHTb ynopH.n.OtIeHHbIX nap: (a; a), (a; 6), (a; B), (6; a), (6; 6), (6; B),
(B; a), (B; 6), (B; B).
npHMepOM ynopH.n.OtJeHHOH napbI HaTypaJIbHbIX tJHCeJI MO)!{eT CJIY-
)!{HTb napa, COCTaBJJeHHaH H3 tJHCJIHTeJIH H 3HaMeHaTeJJH .n.p06H - BMe-
CTO Toro '1T06bI nHcaTb 3/6' MomHO 3anHcaTb (3; 5). npH nepecTaHoBKe
tJHCeJJ 3 n 5 nOJJY'laeTCH HHaH .n.p06b Y3'
EIIJ.e 60JIee 06IIJ.ee nOHHTHe ynopH.n.OtIeHHOH napbl nOJJytIaeTCH, eeJIII
6paTb ee KOMnOHeHTbI H3 pa3JJHtIHbIX MHO>KecTB, HanpHMep KOMnOHeH-
TY X H3 MHO)!{eeTBa X, a y - H3 MHO)!{ecTBa Y. nYCTb, HanpHMep, 3a-
,ltaHbI .n.Ba MHO)!{eeTBa X={a; b; c} H Y={4; 5}. 06pa3yeM H3 3JIeMeH-
TOB 3THX MHO)!{eeTB napbI TaK, tIT06bI nepBaH KOMnOHeHTa napbl npH-
Ha,ltJIe)!{aJia MHO)!{eeTBY X, a BTopaH MHO)!{ecTBY Y. Bce 3TH napbI COC-
TaBJJHIOT MHO)!{eeTBO: {(a; 4); (a; 5); (b; 4); (b; 5); (c; 4); (c; 5)}, KQ-
Topoe Ha3bIBalOT .LI.eKapTOBbiM npOH3Be.LI.eHHeM MHO)!{eeTB X H Y H 0603-
HatialOT X X Y.
Bo06IIJ.e, OCKapm06blM. np0U36COCIiUCM. MHO)!{eCTB X H Y Ha3bIBalOT
MHO)!{eCTBO X X Y, 3JJeMeHTaMH KOToporo HBJJHIOTCH Bce napbI (x; y)
TaKHe, tITO x EX, YE Y, T. e.
X XY={(x; y}lx E Xl\yE V}.
ECJIH MHO)!{ecTBa X H Y COBnaJ(alOT, T. e. X = y, TO MHO>KecTBO
X X X COCTOHT H3 Bcex nap (x; y) TaKHX, tITO x EX, y EX. HanpHMep,
ecJIH X = {m; n; p}, TO X X X = {m; m); (m; n); (m; p); (n; m); (n; n);
(n; p); (p; m); (p; n); (p; p)}.
nOJIaralOT, tJTO Xx0=0XX=0.AJIS'l JIIOOOrO MHomecTBa X.
.lJ.eKapToBO npOH3Be.n.eHHe MHO>KecTB, Boo6IIJ.e rOBopH, He OOJIa.n.aer
HII CBOHCTBOM KOMMyr.aTHBHOCTH, HH CBOHCTBOM aCCOI1.HaTHBHOCTH,
T. e.:
1) CCAU X=#=y, mo XxY=#=YxX;
2) CCAU IiU oOliO U3 MIiO:JfCeCme X, Y, Z He nycmo, mo
Xx (YxZ)=#=(XxY)xZ.
.lJ.eHCTBHTeJIbHO, 3JJeMeHTaMH MHO)!{ecTBa X X Y HBJIHIOTCH naphl
(x; y) TaKHe, tJTO x E X H Y E Y, a 3JIeMeHTaMH MHO)!{ecTBa Y X X -
napbI (y; x), r.n.e y E Y H X EX. Ho npH X=FY naphI (x; y) H (y; x) pa3-
JIR'IHbI, CJIe,!lOBaTeJIbHO, ecJIH X=#=Y, TO MHO>KecTBa X X Y H Y X X
pa3JIHtIHbI.
3JIeMeHTbI ,lteKapTOBa npOH3Be,!leHHH ,ltBYX KOHetJHhIX MHO)!{ecT8
y.n.06HO pacnOJIaraTb B BH.n.e Ta6JIHUbI, r.n.e no BepTHKaJJH pacnOJIaralOT
!lJIeMeHTbI MHO)!{eeTBa X, no ropH30HTaJiH - 3JIeMeHTbI MHO)!{ecTBa Y,
39

I 4 I 5
a I (a; 4) I (a; 5)
b I (b; 4) I (b; 5)
c I (c; 4) I (c; 5)
3 3JJeMeHTbI MHOmeCTBa X X Y nIiIllYT Ha
nepeCe'leHIiHX COOTBeTCTBYlOllI.IiX CTpOK H
CTOJJ6l1.0B. TaK, Ha Ta6JJIiue, npIiBe.n.eHHOii
HIime, Ii306pameHbI 3JJeMeHTbI .n.eKapTOBa
npOH3Be.n.eHIiH MHO)!{eCTB X = {a; b; c} Ii
Y={4; 5}.
YnpamUeUBJI
I. Ha30BHTe n!lTb ynop!l.a;o'leHHblx nap .a;eiicTBHTeJIbHbIX '1I1CeJI, !lBJIJIIOLUIIXC!I
peweHHeM ypaBHeHH!I 2x-3y=7.
2. )].aHbI .a;Ba MHO>KeCTBa: X= {6; r; .a;} II Y= {a; II; oJ.
3anHwHTe 9JIeMeHTbI MHO>KeCTBa X X Y B BH)l.e Ta6JIHltbI.
3. BbinHIIIHTe Bce ){BY3Ha'lHble 1JHCJla, B KOTOpbIX 1JHCJIO ){eC!lTKOB npHHa.a;Jle>KIIT
loIIIO>KecTBY {8; 6; 2}, a '1HCJlO eHHHlt - MHO>KeCTBY {3; 5; OJ.
4. CocTaBbTe Bce ){P06H, '1HCJlHTeJlb II 3HaMeHaTeJIb KOTOpbIX - 0.a;H03Ha'lHOe
'IIICJlO H3 MHO>KeCTBa {3; 4; 5; 7}. CKOJIbKO .a;p06eii nOJIY'IHJIOCb?
5. A - MHO>KeCTBO rJIaCHblX, B - MHO>KeCTBO rJIyxHx COrJIaCHbiX. BbinHIIIIlTe
9JleMeHTbI MHO>KeCTB A X B H BX A. .
6. )].aHbI MHO>KeCTBa A= {a; b; c}, B= {I; 2} H C= {2; 3; 4}.
a) 3anHIIIHTe MHO>KeCTBa AXB, AXC H BXC.
6) BbI!lCHHTe, KaKHe 9JIeMeHTbI npHH8)l.Jle>KaT MHO>KeCTBaM (A X B) n (A X C) II
AX (BnC). Bepno JIH, 1JTO (AXB)n (AXC)=AX (BnC)?
7. )].OKa>KHTe, '1TO M!I JII06b1x MIIO>KeCTB A, B, C CnpaBe){JIHBO cJle.a;YIOLUee pa-
BeIICTBO: AX (BUC)= (AXB)U (AX C).
2. KOpTe)l{Jf. ECJJIi 3a.n.aHO MHomeCTBO X, TO Ii3 ero 3JJeMeHTOB MO)KHO
COCTaBJJHTb He TOJJbKO ynopH.n.O'leHHbIe napbI, HO Ii ynopH.n.O'leHHble
TpoiiKH 3JJeMeHTOB, '1eTBepKH 3JJeMeHTOB Ii T. .11.. HanpIiMep, 6YKBbI
CJIOBa «TeJJecpoH» 06pa3YlOT ynopH.n.O'leHHYIO ceMepKY. BBe.n.eM 06Il1.ee
MaTeMaTIi'leCKOe nOHHTIie, '1aCTHbIMH CJJY'laHMIi KOToporo HBJJHIOTCH H
ynopH.n.O'leHHbIe napbI, Ii ynopH.n.O'leHHble TpoiiKIi, Ii ynopH.n.O'leHHble
lJeTBepKH.
nYCTb .n.aHbI MHO)l{eeTBa Xl' X 2 , . . ., Xn. B03bMeM KaKoii-HIi6y.lI.b
9JJeMeHT ai Ii3 MHO)l{eCTBa Xi, nOTQM 3JJeMeHT a 2 Ii3 MHO)!{eCTBa X 2J . . .,
9JJeMeHT an Ii3 MHomeCTBa Xn. Bbl6paHHble 3JJeMeHTbI paCnOJIO)l{IiM no
1J0pH.n.KY: (a l ; a2; . . .; an). Mbi nOJJY'laeM ynopflOOtteliliYIO n-KY 3JJeMeH-
TOB ('1HTaeTCH: «3HKa»), Bbl6paHHblX Ii3 MHOmeCTB Xi, X 2, . . ., Xn.
BMeCTO CJIOB «ynopH.n.O'leHHaH n-Ka» rOBOpHT KOpO'le - «KOpTe)l{»
(paHlI.Y3CKoe CJJOBO «KopTe)K» 03HatJaeT TOp)l{eCTBeHHOe WeCTBIie, H8-
npHMep, rOBOpHT «CB3.lI.OOHblH KOpTe)K» IiJJH «KOpTe)l{ aBTOMaWIiH»).
1JHCJIO n Ha3b1BaIOT OAUIiOt1 KOpme:JfCa, 3JIeMeHTbI ai, a2' . . ., an- ero
KOM.noliclimaM.U.
MHomeeTBa Xi, X 2J . . ., X n MorYT IiMeTb 061l1.Iie 3JJeMeHTbI IiJJIi
.u.a)l{eCOBna.n.aTb .n.pyr c .n.pyrOM. HanpHMep, CJJOBO «TeJJe<P<>H» - KOp-
Te)l{ .n.JJIiHbI 7, COCTaBJJeHHbIH Ii3 3JJeMeHTOB MHO)l{eeTBa X = {a; 6; B; . . .;
10; 51} (npli 3TOM B CJJOBO «TeJIe<P<>H» BXO.n.HT He Bce 6YKBbI 3Toro MHO)l{e-
CTBa, a JJIiWb '1aCTb 3TIiX 6YKB). npe.n.JJOmeHIie «Y MeHH 3a3BOHIiJJ Te-
JIe<P<>H» - KopTem .LI.JJHHbI 4, KOMnOHeHTaMH KOToporo HB.TJHIOTCH CJJOBa
pyccKoro H3bIKa. K:am.n.oe Ii3 3TIiX CJJOB - KOpTe)!{, COCTaBJJeHHblH H3
6YKB. TaKIiM 06pa30M, KOMnOHeHTaMH KopTema MorYT 6b1Tb Ii KOpTe)l{H.
40

MO)KHO COCTaBJIHTb H KOpTe)KH, KOMnOHeHTaMH KOTOpbIX HBJISIIOTCSI MHO-
)!{eCTBa, HanpHMep
({a; b}; {c; d}; {c; t}).
B MaTeMaTHKe npHMepOM KOpTe)!{a MO)KeT CJIY)KHTb Ha60p lI.H<PP,
BXO,ltHIlI.HX B .n.eCHTH'IHYIO 3anHCh KaKoro-HH6y.n.h '1HCJIa. 31'0T KOpTe)l(
COCTaBJIeH H3 UH<pP 0, 1, 2, 3, 4, 5, 6, 7,8,9, npHtJeM UH<PPbI MorYT
nOBTopHTbCSI, a npH nepecTaHoBKe uH<pP MO)KeT nOJIYtJHTbCSI HHoe tJHC-
JIO. T aK, KOpTe)!{ uH<PP tJHCJIa 112 231 HMeeT BH.n. (1; 1; 2; 2; 3; 1).
J1.Ba KOpTe)Ka (al; a 2 ; . . .; an) H (b l ; b 2 ; . . .; b m ) Ha3bIBalOT PM-
IiblM.U, eCJJH OHH HMelOT o.n.HHaKoBYIO .n..rIHHY, T. e. n=m, H Ka)!{.n.aH
KOMnOHeHTa nepBoro KOpTe)Ka paBHa KOMnOHeHTe BToporo. KOpTe)!{a C
TeM)!{e HOMepOM, T. e. al=b h a 2 =b 2 , . . ., an=b m .
HanpHMep, KOpTe)!{H (a; b; c) H (a; b; c) paBHbI, a KOpTe)KH (a; b; c)
H (b; a; c) He paBHbI, TaK )!{e He paBHbI H KOpTe)KH (a; b; c) H
(a; b; c; d).
I1cnoJIb3YH nOHSITHe KOpTe)!{a, MO)!{HO onpe,lteJIHTb nOHHTHe .n.eKap-
TOBa npOH3Be.n.eHHSI Tpex, tJeTbIpeX H, Bo061IJ.e, n MHO)KeeTB.
nYCTb 3a,ltaHbI n MHO)KeCTB: Ah A2' . . ., An (MHO)!{eCTBa MorYT
liMeTb 06Il1.He 3JJeMeHTbI). :113 JIeMeHTOB 3THX MHO)!{eCTB 06pa3yeM KOp-
Te)!{H .n.JIHHbI n, nepBaH KOMnOHeHTa KOTOpblX npHHa.n.JIe)!{HT MHomeeTBy
Ai, BTopaH - MHO)!{eeTBY A2' . . ., n-SI - MHO)!{eCTBY An. MHO)!{eeTBO
TaKHX KOpTe)!{eH Ha3b1BaIOT OCKapm08blM. np0U38COCIiUCM. M.IiO:JfCCCm8
Ai, A2' . . ., An H 0603HatJalOT A 1 xA 2 x. . . xAn.
HanpHMep, ,lteKapTOBO npOH3Be.n.eHHe MHO)!{eeTB A 1 ={I; 2}, A 2 =
={3; 4}, A3={5; 6; 7} liMeeT BH.n.:
A 1 XA 2 xA 3 ={(I; 3; 5); (1; 3; 6); (1; 3; 7); (1; 4; 5); (1; 4; 6); (1; 4; 7);
(2; 3; 5); (2; 3; 6); (2; 3; 7); (2; 4; 5); (2; 4; 6); (2; 4; 7)}.
¥opamUeUBJI
1. CpaBHBTe nOHHTBH KOpTeJKa B MHO>KecTBa. YKa>KKTe, B '1eM HX CXO.D.CTBO H B
'leu pa3JlH'IHe.
2. 3anHIIIHTe MHO>KeCTBO pa3JlH'IHblX 6YKB CJlOBa «napaJIJlMOrpaMM». 3anHIIIHTe
KOpTe>K 6YKB B 3TOM CJlOBe. KaKoBa JJ,JIHHa 9TOro KOpTe>Ka?
3. CKOJlbKO UH<PP B sanBCB '1HCJla 235 535? CKOJlbKO pa3J1H'IHblX uH«PP B sanBCR
3TOro '1HCJla? .
4. QIJoPMYJlHpyiiTe 3aJla'lY 3, HcnoJlb3YH nOHHTHH MHO>KecTBa H KOpTe>Ka.
. X e
5. Ha MHO>KeCTBe N Bcex HarypaJIbHblX '1HcM 3aJ(aHO ypaBHeHHe y = 7'
Ha30BHTe HecKOJlbKO '1eTBepOK '1HCeR, npHHaJ(JleJKmnHX MHO>KecTBY peIIleHHii T JlaB-
Horo ypaBHeHHH. BepHO JlH, '1TO (3; I; 9; 3) E 17
6. 06pa3yiiTe BceB03MO>KHble KOpTe>KH .D.JlHHbI 3 H3 9JleMeHTOB MHO>KecTBa
A= {a; b; c; d}.
7. IfcnoJlb3YH UH<PPbl 2, 7, 0, 4, SanHIIIHTe aceB03MO>KHbIe TpeX3Ha'lHble '1HCJla
(UH«PPbl B 3anHcB '1HCJla He nOBTOpHIOTCH).
8. )].aHbI MHO>KeCTBa: A= {a; b; c}, B= {m; n}, C= {x; 1/; z}. 3anHIIIHTe MHO>Ke-
I CTBa: A X BX C H BX A X C. BblHCHHTe, KaKBe B3 CJIJlYIOInIfX BLickablBaHHii HCTBIt-
'. HbI, a KaKHe JlO>KHbI: '
a) (b; m; x)EAXBXC; 6) (bi mj x) BXAXCI
B) AXBXC=BXAXC.
41

3. KOM6HHaTopHKa. npaBHJlO CYMMbI. Ha npaKTHKe tJaCTO npHxo-
,ltHTCH BhI6HpaTh H3 HeKOToporo MHO)!{eeTBa 06'beKTOB ero no.n.MHO)!{e-
CTBa, pacnOJIaraTh 3JIeMeHTbI KaKorO-TO MHO)!{eeTBa B TOM HJIH HHOM
nopH.n.Ke H T. .11.. TaK, MacTepy npHxo.n.HTCH pacnpe.n.eJISITh pa3JIHtJHble
BH)tbI pa60T Me)K)ty pa6oqHMH, o<PHuepy - Bbl6HpaTh HapH,lt H3 C(>JI,lJ,aT
B3Bo.n.a, lllaXMaTHCTY - H3 HeeKOJIhKHX cepHH XO.n.OB Bbl6HpaTh HaHJlyq-
rnylO. nOCKOJIbKY B TaKHX 3MatJax pe'lh H.n.eT 0 Tex HJIH HHblX KOM6HHa-
II.HHX paoOT, COJI.n.aT, XO,ltOB H T. .11.., HX Ha3b1BaIOT KOM.6uHamOpliblMU.
06JIaCTb MaTeMaTHKH, B KOTOpOH H3YtJalOT KOMOHHaTopHble 3a.n.atJu,
Ha3bIBalOT KOM.6uliamopUKOCi. nO CYTH .n.eJIa, B KOM6HHaTopHKe H3YtJaIOT
KOHetJHhIe MHO)!{eeTBa, HX no.n.MHO)!{eeTBa, OT06pa)!{eHHH, a TaK)!{e KOp-
Te)!{H, COCTaBJleHHble H3 3JIeMeHTOB KOHetJHblX MHO)!{eeTB. n03TOMY KOM-
6HHaTopHKY MO)!{HO paCCMaTpHBaTh KaK tJaCTh TeopHH KOHeqHblX MHO-
)KeeTH.
PellleHHe 60JIhlllHHCTBa KOM6HHaTopHhIX 3a.n.atJ OCHOBaHO Ha .lI.Byx
npOCThIX npaBHJIaX, KOTopbIe Ha3hIBalOT npaBHJIaMH CYMMhI H npOH3Be-
,lteHHH. npaBHJIO CYMMbI n03BOJIHeT HaHTH tJHCJIO 3JIeMeHTOB B OO'be.n.H-
HeHHH .n.BYX KOHetJHbIX MHO)!{eeTB, a npaBHJIO npOH3Be.n.eHHH - '1HCJIO
3JIeMeHTOB HX .n.eKapTOBa npOH3Be.n.eHHH.
0603HatJHM tJHCJIO 3JIeMeHTOB KOHetJHoro MHomeeTBa X qepe3 n(X),
NHO)!{eCTBO, COCTOHIIJ.ee H3 n 3JIeMeHTOB, Ha30BeM n-M.lio:J/CCCtn80M.. Ha-
npHMep, eeJIH X={a; b; c; d; C; f}, TO n (X)=6, H n03TOMY X HBJIHeTCSI
6-MHO)!{eeTBOM.
nYCTh X co.n.ep)!{HT m 3JIeMeHTOB, a Y co.n.ep)!{HT n 3JIeMeHTOB.
aMeM, CKOJIhKO 3JIeMeHTOB co.n.ep)!{HT OO'be.n.HHeHHe Xu Y. O.n.HOOHaq-
HblH OTBeT Ha 3TOT Bonpoc MO)!{HO .n.aTb JIHllib B CJIyqae, Kor.n.a MHO)!{ecTBa
X H Y He nepeceKalOTC1. B 3TOM CJIYtJae MHO)!{eeTBO X U Y CO,ltep)!{HT
m+n 3JJeMeHTOB. HanpHMep, eCJIH X={a; b; c; d}, Y={c; f; g}, TO
Xu Y ={a; b; c; d; C; f; g} co.n.ep)!{HT 4+3=7 3JIeMeHTOB. TaKHM 06pa-
30M, cnpaBe.n.JIHBO CJIe.n.YIOIIJ.ee YTBep)!{.n.eHHe.
ECAU M.1i0:J/Cccm80 X coOcp:J/Cum m 3ACM.ClimOe, a M.lio:J/Cccmeo Y -
n 3/leM.Clim08, npUtteM. 3tnU M.lio:»cecmea He nepCCCKaJOnu:fl, mo X U Y co-
iJep:JfCum m+n fJAeM.Clim08.
I1HblMH CJJOBaMH, U3 X n Y = 0 cACoyem
n(XU Y)=n(X)+n(Y). (1)
3T0 OtJeBH)tHoe YTBep)!{.lI.eHHe Ha3b1BaIOT_ B KOMoHHaTopHKe npMUAOM.
CYM.M.bl.
B cJIyqae, Kor.n.a nepeeeqeHRe MHO)KeeTB X H Y HenycTo, ,lteJIO 06-
C1'OHT CJIO)KHee. HanpHMep, OO'be,ltHHeHHe MHO)!{eeTB X={a; b; C; d; e}
u Y = (d; c; f; g} COCTOHT JJHllih H3 cm 9JIeMeHTOB: Xu Y ={a; b; c;
dj c; f; g}, a He H3 5+4=9 3JIeMeHTOB. 31'0 OO'bHCHHeTCH TeM, qTO 3JIeMeH-
TH d Ii e npHHa;J.JIe)!{aT H X H Y, a B OO'be,ltHHeHRe Xu Y 9TH 9J1eMeHTbI
BXO.n.HT JIHlllh O.n.HH pa3 (MH MHO)!{eeTH He HMeeT CMblCJJa rOBopHTh, 1ffO
HeKOTopbIH 3JIeMeHT BXO.n.HT B HHX HeeKOJIbKO pa3). n03TOM)' H3 CYMMbI
5+4 Ha.n.o BbItJeeTb 2, T. e. tJHCJIO 3JIeMeHTOB nepeeeqeHHH X n Y.
BOOOlIJ.e, OM A106blX oeyx M.1i03ICeCme X u Y cnpaeeoAU80 paseH£fll6O
n(X U Y)=n(X)+n(Y)-n(Xn V).
4

I1TaK, ttUCAO 8ACMelim08 067JCOUIiCIiUfl 08YX MIiO:J/CCCm8 paelio CYMMe
ttuceJl aACMClim08 8 Ka:J/COOM U3 H.UX! YMCliblUCliliOfi Iia ttUCJlO aAeMClim08
nCpCCetteliufl amux MIi0:J/Cccm8.
3aMeTHM, qTO B HatJaJJbHOH lllKOJJe npH H3YtJeHHH CJJO)!{eHHH TaK)!{e
HCXQ,ltHT H3 OO'be.n.HHeHHH MHO)!{eCTB, HO npH OM, KOHetJHO, CtJHTalOT,
tJTO OHH He HMelOT 06IIJ.HX 3JJeMeHTOB.
npHBe.n.eM oe3 BbIBo.n.a <POPMYJJY ,ltJJH tJHCJJa JIeMeHTOB B OO'be.n.H-
HeHHH Tpex MHO)!{eCTB:
n(XU YUZ)=n(X)+n(Y)+n(Z)-n(Xn Y)-
-n (X n Z)-n(Y n Z)+n(X n Y n Z).
YupamueUHJI
I. 113 40 cTY)l.eHTOB rpynnbl 35 '1eJIOBeK ycnelliHo C)l.aJlH 9K3aMeH no MaTeMaTHKe,
a 37 - no PYCCKOMY H3b1KY. )].aoe CTY)l.eHTOB nOJlY'IHJlH HeY)l.OBJleTBOpHTeJlbHble
OTMeTKH no 060HM npe,nMeTaM. CKOJlbKO CTY)l.eHTOB HMeIOT aK8)l.eMH'IeCKYIO 3a)l.OJl>KeH-
HOCTb?
2. 11380 II1KOJlbHHKOB 40 HrpalOT B «pyrooJl, a 50 - B BOJleii60Jl, npH'IeM 27 II1K()Jlb-
HHKOB HrpalOT H B «pyrooJl, H B BOJleiiGoJl. CKOJlbKO II1KOJlbHHKOB HrpaeT XOTI> B OJ.lHY
H3 9THX Hrp? CKOJlbKO II1KOJlbHHKOB HrpaeT- JlHIIIb B O)l.HY H3 9THX Hrp?
3. 113 100 cTYiJ.eHTOB aHrJlHi\cKHH !!3b1K H3Y'laIOT 28 '1eJIOBeK, HeMeUKHij -- 30,
«ppaHUY3cKHH - 42, aHrJlHHCKHH H HeMeuKHH - 8, aHrJlHHCKHH H «ppaHUY3cKHH -
10, HeMeUKHH H «ppaHUY3cKHH - 5, Bce TPH !!3b1Ka H3Y'laIOT Tpoe CTYiJ.eHTOB. OCTaJIb-
Hble cTy){eHTbI H3Y'lmoT HcnaHCKHH !!3b1K. CKOJlbKO trY)l.eHTOB H3Y'IalOT HcnaHCKHA
113b1K? CKOJlbKO CTY)l.eHTOB R3Y'lalOT TOJlbKO O)l.HH !!3b1K
4. npaBHno npoH3BeAeHHH. BTopoe OCHOBHoe npaBHJJO KOM6HHaT
pHKH KaCaeTCH no.n.ctJeTa tJHCJJa KOpTe)!{efi, KOTopbIe MO)!{HO COCTaBHTb
H3 3JIeMeHTOB .n.aHHbIX KOHetJHbIX MHO)!{eeTB. PaccMOTpHM CHatJaJJa Ta.
KYIO 3MatJY.
CKOAbKO nap 8uoa (XII; Yl) MO:J/CIiO cocmaeumb U3 8ACMeIitn08 MIiOJICeCfIUJ
X={Xl; X2; . . .; xm} H Y={Yl; Y2; . . .; Yn}?
3anHilleM BCe 3TH napbI B BH.n.e CJJe.n.YIOIIJ.efi Ta6JIHII.bI:
(Xl; Yl), (Xl; Y), . . ., (Xl; Yn),
(x; Yl), (X 2 ; Y2), . . ., (X; Yn),
(Xm; Yl), (Xm; yz), . . ., (Xm; Yn)'
Mbi BH.n.HM, tJTO OHa COCTOHT H3 m CTpOK, B K3)!{.n.ofi H3 KOTOpbIX C
n.ep)!{HTCSI n 3JJeMeHTOB. 3HatJHT, OOIll.ee tJHCJJO nap paBHO mn.
I1TaK, ttUCAO ynopMotteliliblX nap, KOmopblC MOJICHO cocmaeumb U3
iACM.elim08 m-MIiO:J/Cccmea X U n-MIiO:J/Cccmea Y, paelio mn, m. e. npoUaee-
H.UIO ttucAa 8ACMClim08 8 X Iia ttUCAO 8ACMelim08 8 Y.
MHO)!{eeTBO ynopH.n.OtJeHHbIX nap, COCTaBJJeHHb!X H3 3JJeMeHTOB MHO-
)KeeTB X H Y, MbI Ha3BaJJH B n. 1 ,lteKapTOBbIM npoH3Be.n.eHHeM 3THX MHO-
)KeeTB H oo03HatJHJJH X X Y. n03TOMY ,ltOKa3aHHoe YfBep)l{)teHHe MO)l(HO
KOpOTKO 3anHcaTb TaK:
n (X X Y)=n (X)'n (Y).
(11


MomHo .n.OKa3aTb, tJTO CnpaBe.n.JIHBO 60JIee
06Il1.ee YTlep)!{.n.eHHe, Ha3b1BaeI\lOe npa8UAOM
np0U38COCIiUfl:
n(X l xX 2 X. . .xX n )=
=n(X 1 ).n(X 2 )... ..n(X n ). (2)
B KOM6HHaTOpHKe paBeHcTBo (1) 06bItJHO <POPMYJIHPYIOT CJIe.n.YlOlIJ.HM
06pa30M.
ECAU fJACMClim x MO:J/CIiO 8bl6pamb m cnoc06aMu, a fJACMClim y MO:J/CIiO
8bl6pamb n cnoc06aMu, mo ynopflOO'iCIiIiYIO napy (x; y) MO:J/CIiO 8bl6pamb
mn cnoc06aMu.
n p Ii M e p 1. 113 .n.epeBHIi A B .n.epeBHIO B Be.n.YT TPH .n.Op OrH , a
H3 B B C Be.n.YT .n.Be .n.oporH. CKOJIbKHMH cnoc06aMH MO)!{HO npoHTH H3
ABC tJepe3 B?
4T06bI pelllHTb 3 a.n.atJY , 0603HatJHM .n.oporH H3 A B B tJHCJIaMH 1, 2
H 3, a H3 B B C - 6YKBaMH a H b. Tor.n.a K3)!(.n.bIH BapHaHT nYTH H3
ABC 3a.n.aeTCH napoH, COCTOHIIJ.efi H3 tJHCJIa H 6YKBbI. HanpHMep, nYTb,
BbI;\eMHHbIH Ha pHcYHKe 18, 3a.n.aeTCH napoH (2; a). Ho tJHCJIO nap Ta-
Koro BH.n.a no npaBHJIY npoH3Be.n.eHHH paBHo 3 -2=6. BOT 3TH BapHaHTbI:
(I; a), (2; a), (3; a), (1; b), (2; b), (3; b).
HHor.n.a .n.JIH pellieHHH 3a.n.atJ npHxo.n.HTCH nOJIb30BaTbCH 0606U{e1i1iblM
npMUAOM npoU38cocIiUfl. 5b1BaeT, tJTO XOTH pa3JIHtJHble BapHaHTbI Bbl60pa
8JIeMeHTa y onpe.n.eJIHIOTCH y)!{e c.n.eJIaHHblM Bbl60pOM 3JIeMeHTa x, tJHCJIO
cnoc06oB BhI6paTb y npH JlI060M BbI60pe x O.n.HO H TO )!{e. B 3TOM CJIYtJae
napy (x; y) TO)!{e MomHO Bbl6paTb mn cnoc06aMH, r,lte m - tJHCJIO cno-
co60B BbI6paTb 3JIeMeHT x, an - tJHCJIO cnoc06oB Bbl6paTb y nOCJJe Toro,
KaK 3JIeMeHT x y)!{e Bbl6paH.
n p H M e p 2. HaH.n.eM tJHCJIO CJIOB, co.n.ep)!{alIJ.HX 4 6YKBbI, B Ko-
TOpblX JI106b1e ,ltBe coce.n.HHe 6YKBbI pa3JIHtJHbI ('1HCJIO 6YKB B aJI<paBHTe
paBHo 33; npH 3TOM .n.onycKalOTcH H CJJOBa, JJHllleHHble CMblCJIa, Ha-
npHMep «Baxa»).
nepBYIO 6YKBY CJIOBa MO)!{HO BbI6paTb 33 cnoc06aMH. nOCJIe Toro
KaK OHa BbI6paHa, CJIe.n.ylOlIJ.YIO 6YKBY MO)!{HO Bbl6paTb JIHlllb 32 cnoco-
6aMH, TaK Kal\ nOBTopHTb BbI6paHHYlO 6YKBY HeJIb3H. TpeTbH 6YKBa
OTJIH'IHa OT BTOpOH, XafH H MO)l(eT COBna,B:aTb C nepBoH, a nOToMY ee
MO)!{HO BbI6paTb 32 cnoc06aMH, paBHO KaK H tJeTBepTYIO. n03TOMY 06IIJ.ee
QHCJJO cnoc060B BbI60pa paBHo 33 -32 .32 '32= 1 081 344.

A 2 C
J B b
PHC. 18
YupamUeUHJI
!. I1Mee'f(;,!I n!lTb BH){OB KOHBepTOB oe3 MapoK H '1eTbipe BH){a MapOK. CKOJlbKHMH
cnocooaMH MO>KHO BbiopaTb KOHBepT H MapKY ){JI!I nOCblJlKH nHcbMa?
2. CKOJlbKHMH cnocooaMH MO>KHO H3 CJlOBa «3){aHHe» BbiopaTb )l.Be OYKBbI, O){Ha
H3 KOTOpblX rJlacH1!.!I, )l.Pyra!l CorJlaCHa!l?
3. CKOJlbKHMH cnocooaMH MO>KHO H3 CJlOBa «KOCMOHaBT» BbiopaTb ABe 0YKBbI,
oJl.Ha H3 KOTOpblX rJlaCHa!l, a )l.pyra!l COrJlaCHa!l?
4. CKOJlbKM.H cnoc06aMH MO>KHO YKa3aTb Ha II1axMaTHoii )l.OCKe ABa KBa)l.paTa -
6eJIblH H '1epHbJA?
5. CKOJlbKHMH cnoco6aMH MO>KHO Bbl6paTb Ha II1aXMaTHOH ){OCKe OeJlblH H '1epHblH
KBBApaTbI, He Jle>KaLUHe HH Ha O)l.HOH ropH30HTaJlH, HH Ha O)l.HOH BepTHKaJIH?
44

6. 113 12 CJlOB MY)KCKOrO pO)l.a, 9 >KelICKoro H 10 Cpe)l.HerQ W)l.a H8)l.O BbI6paTb
no olQloy CJlOBY Ka1kJ1oro pO)l.a. CKOJlbKHMH cnoc06a!d!l 3TO MO»<i!O C)l.eJlaTb?
7. CKOJlbKO CYIUecTBYeT CJlOrOB, Y KOTOpbIX nepB811 6YKBa rJlaCHa!l, a BTOpaa
cor JlaCH a!l?
5. Pa3MeeHHH  nOBTOpeHHHMH. 113 4-MHO)!{eCTBa X = {a; b; e; d}
MO)!{HO COCTaBHTb 16 KOpTe)!{efi ,ltJIHHbI 2:
(a; a), (a; b), (a; e), (a; d)
(b; a), (b; b), (b; e), (b; d),
(e; a), (e; b), (e; e), (e; d),
(d; a), (d; b), (d; e), (d; d).
PemHM 06IIJ.YlO 3a)tatJY.
Haiimu ttUeAO KOpmeJICeii OAUlibl k, KOmopble MOJICIiO eocmaeumb U3
:JACMelim08 m - MIiOJICeemea X.
4 T06bI pemHTb 3TY 3a.lI.atJy, Ha.lI.O HaiiTH tJHC.'IO KOpTe)!{efi B .lI.eKap-
TOBOM npOH3Be.lI.eHHH X X X X. . . X X, cO.lI.ep)!{aIll.eM k MHomHTeneH
(3TO .lI.eKapTOBO npoH3Be,lteHHe KaK pa3 H COCTOHT H3 TaKHX KOpTe)!{eiI).
Ho no npaBHny npoH3Be.n,eHHH tJHCJIO 3JIeMeHTOB B .lI.eKapTOBOM npoH3Be-
,lteHHH xxX x.. .XX paBHO n(X).n(X)... ,'n(X). TaK KaK no YC-
'---..---' '---,,----'
k pS3 k pa3
JIOBHIO n(X)=m (MHO)!{eeTBO X CO,ltep)!{HT m 3neMeHTOB), TO
n(XXXX. . .XX)=m.m.. . . .m=m k .
'- ....,.,- --"
k pS3
HTaK, t{UCAO KopmeJICeii OAUlibl k, eocma8ACliliblX U3 :JACMCIif1W6
m-MIiOJICCemea X, paelio m k .
HanpHMep, H3 33 6YKB pyccKoro aJI<paBHTa MO)!{HO COCTaBHTb 33-
CJIOB ,ltJIHHbI 2 (aa, a6, aB, . . ., aH, 6a, 66, . . ., HIO, HH), 33 3 CJIOB MHHbI
3,33. CJIOB ,ltJIHHbI 4 H T.,lt. TOtJHO TaK)!{e H3 10 lI.H<PP 0, 1, 2, 3, 4, 5, 6,
7,8,9 MO)!{HO COCTaBHTb 10 1 ,ltBY3Ha'lHbIx HOMepoB (00,01, . . ., . . .,99),
loa Tpex3HatJHbIx H9MepOB H T. ,It.
KopTe)!{ ,ltJIHHbI k, cocraBJIeHHbIH H3 3JIeMeHTOB m-MHO)!{eeTB8, Ha31i-
BalOT pa3M.CU{eliueM e n08mopcliUflMU U3 m 9ACMCIitn06 no k, a '1HCJIO T8-
KHX KopTe)!{eH 0603Ha'lalOT A (OT <ppaHlI.Y3cKoro CJIOBa arrangement -
pa3MelIJ.eHHe). TaKHM 06pa30M, .
A=mi.
$opMYJIa n03BOJIHeT pemHTb TaKYIO 3a,lta'lY.
Hatlmu ttUeAO noOMIiOJICCCme m-MIiO:JfCeemea X.
nepeHYMepyeM 3JIeMeHTbI MHO)!{eeTBa X: X= {Xl; XI; ...; X m }.
Ka)K)toe no,ltMHO)!{eeTBO A c: X MO)!{HO «3amHcppOBaTb» C nOMOIll.blO KOP-
Te)!{a ,ltJIHHbI m H3 Hynefi H e.n,HHHU; MbI nHmeM Ha HeKOTopOM MeeTe 1,
eeJIH 3JIeMeHT C ,ltaHHbIM HOMepOM BXO.lI.HT B nO.lI.MHO)!{eeTBO, H 0, eeJIH
OH He BXO,ltHT. HanpHMep, eenH X={Xl; X 2 ; Xs; X4; X6}, TO KO}>Te)!{
(0; 1; 1; 0; l)mH<ppYeT nO.lI.MHO)!{eeTBO{X2; X3; X6}, KOpTe}K (0; 0; 0; 0; 0)-
rrycToe nO,ltMHO)!{eeTBO, a KOpTe)!{ (1; 1; 1; 1; 1) - Bce X.
n03TOMY HaHTH '1HCJIO no,ltMHOlKeeTB m-MHO)!{eCTBa X - 3T0 Bce
paBHO, tJTO HafiTH '1HCJIO KOpTe)!{efi ,ltJIHHbI m, COCTaBJIeHHbIX H3 3JIe-
45

MeHTOB 2-MHOJKeCTBa {a; I}. no <p<>pMYJIe (1) qHCJJO TaKHX KopTe-
>KeH: paBHO 2 m . 3HaqHT, 'tUCAO noOMIi03fCccme m-MIiOOICecmoa X pa8-
No 2 m .
HanpHMep, MHO)!{eeTBO X = {a; b; e} HMeeT 2 3 =8 n0.lI.MHO)!{eeTB:
0, {a}, {b}, {e}, {a; b}, {a; e}, {b; e}, {a; b; e}.
)"opamHeHHJI
1. CKOJlbKO '1eTbIpeX3H2'IHbiX '1l1CeJI MO>KHO COCTaBHTb H3 UH«PP I, 2, 3, 4, 5, 6?
2. CKOJlbKHMH cnoc06aMH H3 KOJlO)l.bI, COAep>Karn.eii 36 KapT, MO>KHO BbI6paTb
no o)l.Hoii KapTe Ka>K){oii MacTH?
3. B HeKoTopoM rOCY)l.apCTBe He 6bIJlO )l.Byx >KHTeJleiJ: C o.a;HHaKOBhIM HaoopoM
ay60B. KaKOBa MO>KeT 6b1Tb HaHOOJlbIIla!! '1HCJleHHOCTb HaCeJIeHH!! rOCY)l.apCTBa (y
oaeKa He ooJlee 32 3y60B)?
4. n!!Tb pa3J1H'IHbIX npe)l.MeTOB pa3)1.alOT 8 JIJO){!!II4, npHQeM MO>KeT CJI)"IHTbC!!,
IJTO HeKoropble nOJlY'lalOT no HecIroJIbKY npe){MeroB. CKOJIbKHMH cnocOOaMH MO-
1it4:!T 6bITb npOH3Be){eH pa3){eJI?
6. YnopH.II,04eHHbIe MHo.»(eCTBa. I(oHeqHOe MHO}l(eeTBO X Ha3bIBaeTC5J
ynopfli)ottCliliblM, eCJIH ero 3JIeMeHTbI nepeHYMepoBaHbI HeKOTopbIM 06pa-
30M: X = (Xi; X 2 ; . . .; X m ). OoH5ITHe ynopH.lI.OqeHHOrO MHO}I(eeTBa-
IJaCTHbIH CJIytJaH nOHHTHH KOpTe)!{a. OHO BbI.lI.eJIHeTCH H3 06m.ero nOHH-
THH KOpTe)!{a YCJJOBHeM, qTO B ynopH.lI.OqeHHOM MHO}l(eeTBe Bce 3JJeMeHTW
pa3JIHqHbI. HanpHMep, KOpTe)!{ (a; b; a; e; a; d) He HBJIHeTCH ynopH.lI.O-
IJeHHbIM MHO)!{eCTBOM, a (a; b; c; d; c; f) - ynopH.lI.OqeHHoe MHO)!{eeTBO.
O.lI.HO H TO )!{e MHO)!{eeTBO MOJKHO ynopH,lto<IHTb pa3JIlf1IHblMH eno-
c06aMH. HanpHMep, MHO)!{eeTBO lllKOJJbHHKOB B KJIaCCe MOJKHO ynOPMo-
1JHTb no B03paCTY, POCTY, Beey, aJnpaBHTY H T. .lI.. PemHM CJIe,lt)'lOll{ylO
S3,Zlaqy.
CKOAbfCUMU enoc06aMu M03tCIiO ynopflOOttUmb m-MIiO:JICectn80 X?
Ka)K)toe ynopH.lI.oqeHHe 3aKJIIOqaeTCH B TOM, lITO KaKOH-TO 3JIeMeHT
nOJIyqaeT HOMep 1, KaKoH-TO - HOMep 2, . . ., KaKoH-To - HOMep m.
HOMep 1 MO)!{eT nOJIyqHTb mo6oH: H3 3JJeMeHTOB MHO)!{ecTBa X. 3HatJHT,
BbI60p nepBoro 3.'IeMeHTa MO)l(HO C.lI.e.rlaTb m cnoc06aMH. ECJIH nepBbIH
3JIeMeHT Bbl6paH, TO Ha BTOpoe MeeTO OCTaeTCH JIHllib (m-l) KaH)I.H.lI.aT,
TaK KaK nOBTopHTb C.lI.eJIaHHbIH BbI60p HeJIb3H. 3HaqHT, HMeeM m-l
cnoc06 BbI60pa BToporo 3JIeMeHTa. TpeTHH 3JIeMeHT MO)!{HO BbI6paTb
(m-2) cnoc06aMH H T. ,It. nOCJIe.nHHH: 3JIeMeHT MO)!{HO BbI6paTb JIHllib
O,ltHHM cnoc060M - Bce OCTaJlbHhle 3JJeMeHTbI nOJJyqHJI8 caOH MeeTa, 8
OCTaJICH JJHlllb O.lI.HH 3JJeMeHT, KOTOpblH H 3aHHMaeT m-e MeeTO. no npa-
BMJIY npOH3Be.n.eHHH nOJJyqaeM, tJTO 06m.ee qHCJIO cnoc06OB ynopMOqe-
HH paBHO m(m-l).. . .'1.
TIpoH3Be,lteHHe nepBbIX m HaTypaJIbHblX qHCM B MaTeMaTHKe Ha3h1-
salOT «m-tjJaKtrWpuaA» H 0603HaQalOT m!. HanpHMep, 41 = 1 .2.3.4=24.
TaKHM 06pa30M, qHCJIO pa3JIHqHhlX ynopH.lI.oqeHHH m-MH01KeeTBa X
paBHO m!. Pa3JIHqHbIe ynopMotJeHHH m-MHO)!{eeTBa COCT05JT 83 O.lI.HHX
H Tex )!{e £meMeHTOB, a OTJIH'laIOTCH .lI.pyr OT .lI.pyra JIHllib nOpSUI.KOM 3THX
3JleMeHToB. npH 3TOM 3JIeMeHTbI B HHX He nOBTopHIOTCH. n09TOMY HX
H33blBalOT nepecmali08KaMU 6C3 noemopcliuii ti3 m 3AeM.eIif1WfJ. 4HCJlO
46

TaKHX nepeeTaHOBOK 0603HatJalOT Pm (OT<ppaHlI.Y3CKOrO CJIOBa permuta-
tion - «nepeeTaHOBKa»). HTaK, Mbi ,ltOKa3aJIH, tJTO
Pm=m!. (1)
HanpHMep, H3 tJeTbipex 6YKB a, b, c, d MO)l{HO C.lI.eJIaTb 4!=24 ne-
peeTaHOBKH:
abcd adbc bcad cabd cdab dblW
abdc adcb bcda cadb cdba dbca
acbd blWd bda1; cbad dabc dcab
lWdb bade bdca cbda dlWb deba
TO'lHO TaK )!{e pernaeTCH 60JIee OOIIJ.aH 3a.lI.atJa.
CKOAbKO ynopWOtteliliblX k- MIi03fCccme M03fCIiO cocma8umb U3 8ACMCIi-
moe m- MIi03fCccm8a X?
OTJIHtJHe 3TOH 3a,ltatJH OT npe.n.bI.lI.ym.eH 3aKJIIO'IaeTCH B TOM, tJTO COC-
TaBJIeHHe ynopH.lI.OtJeHHoro k-MHO)!{eCTBa 3aKaHtJHBaeTCH, KOr.lI.a I\lhI BbI-
6epeM k 3JIeMeHTOB. n03TOMY, tJTo6b1 HaHTH tJHCJIO TaKHX ynop5I.lI.O'leH-
HhIX nO.lI.MHO)!{eCTB, Ha.lI.O nepeMHO)!{HTh k '1HCeJI: m, m-l, m-2 H T. .lI..
nOCJIe.n.Hee H3 HHX paBHO m-k+ 1 (JIerKO npoBepHTh, tJTO B MHO)!{eCTBe
{m; m-l; . . .; m-k+ I} CO.lI.ep)!{HTCH k tJHCeJI). 003TOMY '1HC.'10 yno-
pH.lI.O'leHHblX k-MHO)!{eeTB, COCTaBJIeHHblX H3 3JIeMeHTOB m-MHO)!{eeTB8
X, paBHO m(m-l)... ,'(m-k+l).
3TH ynopH,ltOtJeHHbIe k-MHO)!{eeTBa H83b1BaIOT pa3MCU{CIiUflMU 6C3
n08mopcliuii U3 m 8ACMClim08 no k, a HX tJHCJIO 0003Ha'laIOT A. MbI
.lI.OKa3aJIH, '1TO
A=m(m-l) (m-2)... ,'(m-k+l). (2)
<f>OPMYJIY .lI.JIH A MO)!{HO 3anHcaTb HHaqe, YMHO)!{HB H pa3.lI.eJIHiJ
nepBYIO tJaCTb cpopMYJIhI (2) Ha 1.2.. . .' (m-k). Mbi nOJJY'IHM:
k m (m-l)(m-2).... .(m-k+ I)(m-k).... .2.1 ml
Am=
1.2.... .(m-k) (m-k)! .
HTaK, A k _ m! (2 ' )
m - (m-k)1 .
npH 3TOM A:;: = Pm = ml, TaK KaK CqHTaIOT, tJTO 01 = 1.
YnpamueUHH
J. CKOJlbKHMH cnoc06aMH MO>KHO COCT8BHTb TpeXUBeTHbIH «pJlar C TpeMIi ropH<
30HTaJlbHblMH nOJlOCaMH O)l.HOH H TOR >Ke II1HPHHI>I, ecJ[H ecTb MaTepHIi nliTH pa3J1HtJHblJ(
uBeTOB? PeIIlHTe TY >Ke 38)1.a'lY, eCJIH O)l.Ha H3 nOJloc )l.OJl>KHa 6b1Tb KpacHoiL
2. CKOJlbKO CJlOBapeH HY>KHO H3)1.aTb, 'IT06b1 MO>KHO 6b1J1o AenOCpe)l.CTBeHHo
BbInOJlH!lTb nepeBO)l.bl C moOoro H3 nliTH !l3bIKOB (PyccKoro, aHr JlHHCKOro, HeMeriKoro,
«ppaHUY3cKoro H HTaJlbIlHCKoro) Ha mo6oii )l.Pyroii H3 HHX?
3. CItOJlbKO MO>KHO COCTaBHTb TpeX3Ha'lHblX tJHCM H3 uH<jJp 1, 2, 3, 4, 5, 6, 7.
ecJ[H HH O)l.HO H3 9THX '1HCeJI He CO)l.ep>KHT )l.BYX nOBTOp!lIOLUHXC!I UH«pp?
4. CKOJlbKHMH cnoc06aMH MO>KHO BblHYTb H3 BOJlHOii KOJIO)l.b! '1eTbipe KapTbi pas-
.IIHtJHOR M!lCTH, 1JT()('jbl cpe.n.H HHX He 6b1J10 nap (ABYx BaJleTOB, )l.BYX ceMepoK H T. .II.)?
5. CKOJlbKHMH cnoc06aMH MO>KHO noca)l.HTb Ha cKaMeHKY 9 '1eJIOBeK?
47

7. C04eTaHHH 6e3 nOBTopeHHH. MbI paccMoTpHM cefitJac cJJe.lI.YIOIlI.ym
3a.lI.atJy KOM6HHaTopHKH.
CKOAbKO nOOMIiO:JfCCCme, coocpJICaw,ux no k BACMClim08 Ka:JfCOOC,
MO:JfCIiO cocmaeumb U3 BACMClim08 oalilio2o m-AtliOJICCCmea X?
TaKHe n0.lI.MHO)!{eeTBa Ha3bIBalOT CottCmaliUflMU 6C3 n08mopcliuii U3
m BACJHClim08 no k, a HX tJHCJJO o603HatJalOT C, (OT <ppaHUY3cKoro CJIOBa
combinaison - «cotJeTaHHe»).
BbIBe.lI.eM <P0PMYJIY, BbIpa)!{alOIlI.YIO C tJepe3 m H k. B03bMeM Ka-
Koe-HH6Y.lI.b k-nO.lI.MHo)j{ecTBo A B m-MHomeCTBe X. TaK KaK A CO.lI.ep)!{HT
k 3JJeMeHTOB, TO ero MO)!{HO ynopH.lI.OtJHTb k! cnoco6ar-lH. OpH 3TOM Kam-
,ltoe ynopH.lI.OtJeHHoe k-MHO)j{eCTBO, COCTOHIlI.ee H3 3JleMeOB MHO)!{eCTBa
X, MO)!{eT 6bITb nOJJYtJeHO TaKHM nYTeM. 3HatJHT, tJHCJJO ynopH.lI.OtJeHHblX
k-MHO)!{eCTB, COCTaBJleHHbIX H3 3JJeMeHT()B MHomeCTBa X, B k! pa36oJJb-
me tJHCJJa HeynopH.lI.OtJeHHbIX k-nO.lI.MHO)j{ecTB B X.
HanpHMep, H3 3JJeMeHTOB MHomeCTBa A = {a; b; c; d} MomHO co-
CTaBHTb tJeTbIpe 3-nO.lI.MHO)!{eCTBa:
{a; b; c;}, {a; b; d}, {a; c; d}, {b; c; d}.
QI1CJJO )He ynopH.lI.otJeHHbIx 3-nO.lI.MHO)!{eeTB B 3! = 6 pa3 oo.'Ibllie:
(a; b; c) (a; b; d) (a; c; d) (b; c; d)
(a; c; b) (a; d; b) (a; d; c) (b; d; c)
(b; a; c) (b; a; d) (c; a; d) (c; b; d)
(b; c; a) (b; d; a) (c; d; a) (c; d; b)
(c; a; b) (d; a; b) (d; a; c) (d; b; c)
(c; b; a) (d; b; a) (d; c; a) (d; c; b)
Ho tJHCJlO ynopH,ltotJeHHblx k-MHO)!{eCTB paBHO A, a tJl1CJJO k-nO.ltMHO-
)KeeTB Mbl 0603HatJI1JJH tJepe3 C. n03TOMY A=k! 'C. TaK KaK A =
ml
>= (m-k)I' TO
C k m!
m = (m-k)! k! .
::na <p<>pMYJJa BbIpa)!{aeT tJHCJJO k-nO.lI.MHO)!{eeTB B m-MHOmeeTBe X.
n p H M e p. CKOJJbKHMH cnOC06aMH MOmHO BbIOpaTb ,lteJJeraUHIO B
5 tJeJJOBeK 113 rpynnbI, cO.lI.ep)!{aIll.eH 12 tJeJIOBeK?
nOCKOJJbKY nopH.lI.OK tJJJeHOB .lI.eJIeraUHH pOJJH He HrpaeT, TO HaM
HMO Y3HaTb, CKOJJbKO MO)!{HO BbI6paTb 5-nO.lI.MHO)!{eeTB 113 12-MHo)!{e-
CTBa. ::no tJl1CJJO TaKOBO:
C5 == 8.9.1O.11.12_ 792
12 7! 5! 1.2.3.4.5 .
YopamUeUHJI
I. CKOJlbKHMH cnoc06aMH MO>KHO BbI6paTb '1eTbipe KpacKH H3 II1eCTR pa3J1H'IHblX
KpacoK?
2. Y O)l.HOrO '1MOBeKa ecTb 8 KHHr no MaTeMaTHKe, a y )l.pyroro - 6 KHHr. CKOJlb-
KHMH cnoc06aMH OHH MorYT 06MeH!lTb TPH KHRrH O)l.Horo '1e.OBeKa Ha TPH KHHrH )l.PY-
raro '1MOBeKa? -
48

3. Y MaMbI 2 !lOJlOKa H 3 rpYlliH. Ka>K)l.blli )l.eHb B Te'leHHe "!lTH )l.He", no.a;p!l.a;
OHa Bbl){aeT CbIHY no O){HOMY «PPYKTY. CKOJlbKHMH cnocooaMH OHa MO>KeT 9TO
C){M aTb?
4. POTa COCTOHT H3 Tpex o«PHUepOB, 6 Cep>KaHTOB H 60 p!I)l.OBbiX. CKOJlbKHMH
cnocooaMH MO>KHO BbI)l.eJlHTb H3 HHX OTP!l){, COCTO!lUJ.HIi H3 O){HOrO o«PHuepa, ){BYX cep-
>KaIlTOB H 20 p!l){OBbIX?
- 5. Ha II1KOJlbHOM BelJepe npHcYTcTBYIOT 12 ){eBYllieK H 15 IOHOllieli. CKOJlbKHMH
cnocooaMH MO>KHO Bbl6paTb H3 HHX 4 napbl ){JI!I TaHua?
6. CKOJlbKHMH cnoc06aMH MO>KHO BbiopaTb 12 lJeJlOBeK H3 17, eCJlH ){aHHbIe ){BOe
qeJlOBeK H3 9THX 17 lie MorYT OblTb BbiopaHbI BMecTe?
7. KaKoe HaHOO.1bWee '1HCJlO TOlJeK nepeCelJeHH!I MorYT HMeTb ){HarOHaJIH n-
yroJlbHHKa?
8. CBoi1cTBa 4HCeJI C::,. 4HCJJa C, BbIpa)!{alOIIJ.He KOJJHtJecTBO k-
nO.lI.MHO}KeCTB B flH,IHomeCTBe X, 06JJa.lI.a1OT ueJJbIM pH.lI.OM 3aMetJaTeJIb-
HbIX CBOllcTB. 3TH CBOHCTBa BbipamalOT pa3JJHtJHbIe COOTlIOllieHHH Me)K)ty
nO.lI.MHOmeCTBaMH MHomeCTBa X.
1) EC,1U Okm, mo ecplio paeclicmeo
C::'=C;;;-k.
(1)
C nOMOIIJ.blO <POPMYJJbI (1) n. 7 3TO YTBepm.lI.eHHe .lI.OKa3bIBaeTCH cpa-
3Y:
C m - k ml
m = (m-k)1 [m-(m-k)JI
_ ml _ C k
(m-k)I kl - m'
2) II.Afl AlO6blx k u m, maKUX, ttmo km, 8CpH.O pMCH.Cfll6O
C,=C\+C!h_l' (2)
11 3TO paBeHCTBO HeTpY.lI.HO nOJIYtJHTb C nOMOIIJ.blO <P<>pMYJIhI (1) n. 7.
B caMOM ,lteJIe, TaK KaK
Ck-l _ (m-I)I (m-I)I k .
m-l - (k-I)I (m-k)1 kl (m-k)! '
k (tn-I)! (m-I)! (m-k)
C m - 1 = kl (m-k-I)I = k! (m-k)1
TO, nO.lI.CTaBJIHH 3TH 3HatJeHHH B npaBYIO tJaCTb <POPMYJJhI (2), nOJIYtJaeM:
Ck-l + Ck = (m-I)! k + (m- I)! (m- k) =
m-l m-l kl (m-k)! kl (m-k)!
_ (m-I)I (k+m-k) m! _ C k
kr(m-k)! kl (m-k)1 - m'
PaBeHCTBO (2) .lI.OKa3aHO. OTMeTHM, '1TO npH k=O nOJJYtJaeM paBeH-
CTBO C:h=C';;:l +C'I-l' TaK KaK Cfh = C:h-l = 1, TO CJJe.lI.YeT nOJJOmHTb
Cri1:1=O. AHaJIOrHtJHO CJJe,ltyeT nOJJO)!{HTb C::'=O npH k>m. Tor.lI.a pa-
BeHCTBO (2) OKa3bIBaeTCH BepHbIM H npH k=m.
TOm.lI.eCTBO (2) n03BOJIHeT BbItJHCJJHTb 3HatJeHHH C, 3HaH C-l
H C::'-:\. I1HbIMH CJJOBaMH, C nOMOIIJ.blO 3Toro TO)K)tecTBa MomHO nOCJIe-
,ltOBaTeJIbHO BbItJHCJJHTb C::, CHatJaJJa npH m=O, 3aTeM npH m= 1, npH
49

m=2 H T. ,It. BbItIHCJIeHHH Y,lt06HO 3anHCbIBaTh B BH,lte TpeyrOJIbHOH Ta6-
JIHUW: .


2
3 3
I 4 6 4 I
5 10 10 5
. . . . .


B (m+ l)-n CTpOKe KOTOpOll no nopH,ltKY CTOHT '1HCna
" Cfn, . . ., C:.
npH 3TOM C:h = C;;; = 1, a OCTaJIbHbIe '1RCna BbI'IIiCJIHIOTCH no cpopMYJIe
(2). nOCKOJIbKY C::'-.!l H C::'-l pacnOJIaraIOTcH B 3TOH Ta6JIHlI.e CTpoKon
BbIIIIe, qeM C::' , H H8XO,ltHTCH B 3TOll CTpoKe CJJeBa H cnpaBa OT Hero,
TO ,ltJIH nOJJY'leHIiH C
 HMO CJIO)!{HTb HaXO.LI.HIIJ.IieCH cneBa Ii cnpaBa OT
Hero tJHcna npe.lI.bI.lI.ym.en CTpOKH. HanpHMep, 3HatJeHHe 10 B lliecTOll
crpoKe MbI nOJIY'IHnH, CJIO)!{HB '1HCJIa 4 H 6 miToH CTpOKH.
31'y TpeyronbHYIO Ta6JIHUY Ha3bIBalOT mpeY2oAb1iUKOM. flaCKaAfl, no
liMeHH <ppaHUY3cKoro MaTeMaTHKa BJI33a OaCKaJJH (1623-1662), B TPY-
,ltax KOToporo OHa BCTpe'laeTCH. 31'0 Ha3BaHHe IiCTOpH'IeCKH HeTO'lHO,
TaK KaK TaKYIO Ta6JIHUY 3HaJJ y)!{e apaOCKHH MaTeMaTHK OMap XaHHM,
)KHBlliHI1 B XIII B.


YopamUeUHJI
.D.oKa
HTe CJle)l.YIOLUHe CBOHCTBa '1Ece..! c::,.
m m-l k-m+ I ,.,m
a) Cm+k-Cm+k - k+ I vm+k;
6) C;;; + C
+l + C;;;+2 = C;;;!i;
B) C;;;+
+1 +C;;;+2+C;;;+3=c:t
;
r) C::,+ctm+...+
+...+
=2ml
)I.) C:h+C
+C
+... =2m-l;
e) cfn+C
+C
+... =
-t.




rJlaBa III
1IPE,lJ;IIKATbI II TEOPEMbl


S 1. IIPE}:{IIKATLI II OIIEPAIum HAJJ: HilMI!


1. O,JI.HOMeCTllhle npeJJ.HKaThI. npe.n..nOlKeHHe «n09T X HanHCaJI
l103MY «nMTaBa»» He HBJIHeTCH BbICKa3blBaHHeM, nOCKMbKY He YKa-
33HO, KaKOH n09T HMeeTCH B BH.l\Y. ECJIH 3aMeHHTb B 3TOM np
01KeHHH
OYKBY x CJIOBOM «nYllIKHH», nOJIY'IHTCH HCTHHHoe BhICKa3hIBaHHe «n03T
nYlllKHH HanHCaJI nmMY «nMTaBa»». ECJIH 1Ke 3aMeHHTb x CJIOBOM
«HeKpacOB», nOJIYtlHTCH J10iKHoe BblCKa3bIBaHHe «n03T HeKpacoo Ha-
nHCaJI n09MY «nOJITaBa»» - CPe.D.H HanHcaHHblX HeKpaCOBblM npOH3-
Be,D.eHHH HeT n09MbI C TaKHM Ha3BaHHeM. 3aMeHHH x B TOM 1Ke npe,l.(Jlo-
}KeIlHH CJIOBOM «,lI.OM», nOJIYtJHM OecCMbICJIeHHoe npe.AJI01KeHHe, TaK KaK
CJIOBOCOtleTaHHe «n09T .nOM» CMbICJla He HMeeT. .D:JIH npocrOThl MbI H
TaKHe npe.nJI01KeHHH oY.l\eM C'IHTaTb JlO1KHbIMH BblCKa3bIBaHHHMH.
nYCTb npe.AJ1O)KeHHe CO.l\ep)KHT nepeMeHHYIO, KOTopaH MO:>Ker npH-
HHM.3Tb pa3JIH'IHble 3HatleHHH, npHtleM nO.l\CT8HOBKa JIJ06oro H3 3Ha-
tJeHHH nepeMeHHoH npeBpaIIJ.aeT npe.D.JI01KeHHe B HCTHHHoe HJIB JIOlKHoe
BWCKa3blBaHHe. Tor.l\a 3TO np
mKeHHe Ha3bIBaIOT odHoMecmHblM npe-
dUtcamOM. .lJ.JlSl KaJK,L{Ol'O O.l\HOMeCTHoro npe.nHKaTa HMO YKa3aTh MHO-
)KecTBO 3HaQeHHH, KOTopble M01KeT npHHHMaTh nepeMeHHaH x. Ero
Ha3hIBaJOT 06.1WfmblO onpeae.AeHWl. npe.D.HKaTa. B paCCMOTpeHHOM BbI-
me npHMepe sa 06JIacTb onpe.neJJeHHSl npe,AHKaTa ecreCTseHHO 6wlO
ObI IIPHHHTh MH01KeCTBO Bcex n09TOB, HO 9TO MHO:>KecTBO He,AOCTaTOqHO
'1eTKO onpe.neJIeHO (CtlHTaTb JIH, HanpHMep, n09TOM 'IeJIOBeKa, CO'lHHHB-
llIero JIHllib CTHXH )l,JIH ,lLOMaWHero TOp1KecTBa?). MH01KeCTBO X .l\OJI1KHO
ObITb onpe.neJIetw O.1lH03Ha'lHO (H
mpHMep, MHp1KecTBO n09TOB, 0 KOTOpbIX
eCTb CTaTbH B «lIHTepaTypHOH 9HII,HKJIOne.nHH» nOCJle)l,Hero H3.l\aHHH).
KaJKJJ.bIH npe.D.HKaT A (x), xE; X, Onpe,l\eJIHeT nO,AMH01KecTBO Tc::.X,
COCTOSlIIJ.ee H3 3J1eMeHTOB, upH nO,ll,CTaHOBKe KOTOpbIX IS A (x) BMec-
CTO x nOJI)"IaeTCH MCTHHHoe BliCKa3bIBaHHe. 31'0 nO.D.MHOJKecTBO
H33bIBalOT MH03lCeCtnOOM ucmuHHocmu npe,AHKaTa. B paCCMOTpeHHOM
BbIme npHMepe MHOEecrBO HCTHHHOCTB COCTOHT H3 .QlUloro 9JIeMeHTa T =
={nyuIKHH} - Y ,APyrHx n03TOB Ber n09MbI C H33B3HHeM «nOJIT3Ba».
HO)I.JIH npe,AHKaTa «n09T nymKHH HaIIHC3JI n03MY x» MHOJKecTBO HCTHH-
BOCTO CQJJ.ep)KHT MHoro 9JIeMeHTOB: T={«PYCJI3H H JIIO.l\MHJIa»; «KaB-
Ka3cKHH nJIeHHHK»; «faBpHHJIHa,Aa»; «BUHM); «BpaThSI pa36oHHHKID;


61




«BaxqIlCapallCKIIll <j:JOHTaH»; «lI.blraHbI»; «rpa<j:J HYJIlfH»; «EBreHIIll OHe-
fHH»; «I10.nT3B3»; «Ta3I1T»; «.lJ.OMHK B KOJIOMHe»; «E3epCKIIll»; «AH,A)!{e-
JIO»; «Me,lLHbIH BCa,lLHHK»}. O,lLHOMeCTHblll npe,lLHKaT HBJIHeTCH He qeM
HHbIM, KaK X3paKTepHCTIIqeCKHM CBOllCTBOM, n03BOJIHIOIIIHM BbI,lLeJIHTb
JJ3 oo.naCTII onpe,lLeJIeHIIH 9Toro npe,lLHK3T3 nO,l\MHOlKeCTBO I1CTHHHOCTH T.
B MaTeMaTIIKe npIlMep3MII O,lLHOMeCTHbIX npe,lLIIK3TOB MorYT CJIYlKIITb
npe,lLJIOlKeHHH «HaTypaJIbHOe qHCJIO X - npOCToe» (3,lLeCb MHOlKecTBO
onpe,lLeJIeHHH X COCTOIlT H3 Bcex H3Typ3JIbHblX qIlceJI, a MHOlKeCTBO
HCTHHHOCTH T - I13 npOCTblX qIlCeJI). «.lJ.HarOH3JIII n3paJIJIe.norpal\1Ma x
nepneH,lLHKYJIHpHbl» (3,lLeCb X - MHOlKeCTBO Bcex n3paJIJIeJIOrpaM-
MOB, a T - MHOlKecTBO n3paJIJIeJIOrpaMMOB C nepneH,lLIIKy.nHpHbIMH
,lLH3rOHa.nHMH, T. e. pOMOOB). O,lLHOMeCTHbIM npe,lLIIKaTOM HBJIHeTCH H
ypaBHeHHe x2-7x+ 12=0. MHolKecTBoM I1CTIIHHOCTII 9Toro npe,lLHKaTa
CJIymIlT MHomecTBo T={3; 4}, COCTOHIlI.ee I13 KopHell ypaBHeHHH.
MHO)\{eCTBO I1CTIIHHOCTII HepaBeHcTB3 3x-2<7, x E N, COCTOIlT H3
tlHCeJI 1 H 2, a MHomecTBo HCTHHHOCTH Hep3BeHcTBa 3x-2<7, x E R,
CO,lLeplKHT oeCKOHeqHO MHoro qIlceJI - Bce tlHCJIa x TaKHe, qTO x<3.
O,lLHoMecTHbIe npe,lLIIK3TbI, 3a,lL3HHbIe H3 MHolKecTBe X, OY,lLeM 0603-
HaqaTb CHMBOJI3MII A (x), x EX; B (x), x EX H T.,lL. 3anHcb A (x), x E X,
tlHT310T TaK: «H3 MHolKecTBe X 33,lLaH npe,lLHKaT A (x)>>. BbIcKa3blBaHHe,
KOTopoe nOJIyqHTCH, eCJIH B npe,lLHKaT A (x) nO,lLCTaBHTb KOHKpeTHOe
3HaqeHHe nepeMeHHoH x=a, oY,lLeM 0003H3t1aTb A (a). H3npHMep, nYCTb
X =N, 3 CHMBOJI A (x) 0603HaqaeT I'lpe,lLIIKaT: «x - qeTHOe qHCJIO».
3TOT npe,lLHK3T 06paTHTcH B BbIcKa3bIBaHHe, eCJIH nepeMeHHYIO x 3aMe.
HHTb HaTypaJIbHbIM 'IHCJIOM. TaK, ecJIH x=6, TO MbI nOJIyqHM BbICK33bI'
BaHHe A (6): «6 - qeTHOe 'IHCJIO»..
npe,n,HKaTbI, 3M3HHbIe Ha KOHetlHbIX MHomeCTBax, MomHO 33,lLaBaT.
TaOJIHII:aMH, B nepBoH CTpOKe KOTOpbIX YKa3bIB3eTCH 9JIeMeHT MHome.
CTB8, a BO BTOpOll - HCTHHHO HJIH JIOmHO BbICKa3bIB8HHe, nOJIytlae.
Moe H3 npe,lLHK3Ta, eCJIH 3aMeHHTb nepeMeHHylO 9THM 9JIeMeHTOM. Ha.
npHMep, nyCTb 33,lL3H npe,lLHKaT A (x): «x - qeTHOe tlHCJIO» Ha MHome.
CTBe X={I; 2; 3; 4; 5; 6}. T3K K3K BbICKa3bIBaHHe «1 - tleTHoe qHCJI01
JIOmHO, TO tlHCJIY 1 COOTBeTCTBYeT 3HaqeHHe npe,lLHKaTa «11» (JIOmb),
qHCJIY me 2 COOTBeTCTBYeT HCTHHHoe BbICKa3bIBaHHe «2 - tleTHOe 'IHC'
JlO». nOJIyqaeM TaKYIO TaoJIHU:Y:

A (x) I JI
MomeT CJIyqHTbCH, 'ITO ,lLBa npe,lLHK8Ta A (x) H B (x), 3a,lLaHHbIe H8
O,lLHOM H TOM me MHomecTBe X, HMelOT O,AHHaKOBbIe MHomecTBa HCTHH-
HOCTII. TaKIIe npe,l\HKaTbI Ha3bIBalOT a1C8U8aJleHmHblMU. HanpIIMep,
9KBHB8JIeHTHbI npe,l\HKaTbI «HaTypaJIbHOe qHCJIO X ,l\eJIHTCH Ha 3» H
.cYMMa I1H<j:Jp ,l\ecHTHqHOll 3anHCH HaTypaJIbHOrO qHCJI3 x ,lLeJIHTCH H8
3». Ooa npe,n,HK3Ta 3a,lLaHbI H3 MHomecTBe HaTypaJIbHbIX tlHCe.n. H O,lLHO-
BpeMeHHo HCTHHHbI HJIH JIOlKHbI. ECJIH npe,lLHK3TbI A (x) H B (x) $KBHB3-
JIeHTHbI, TO nHllIYT A (x)-B (x).
2
3
4
5
6
11
JI
11
JI
11
52

YpaBHeHfUI 3x-5=7 H 3x= 12 HBJIHIOTCH 9KBHBaJIeHTHbIMH npe,l\u-
KaTaMH Ha MHomeCTBe R, TaK KaK JII060e 'IHCJIO x, Y,l\OBJIeTBOpHIOIlI.ee
ypaBHeHHIO.3x-5=7, Y,lLOBJIeTBOpHeT H ypaBHeHHIO 3x= 12. TotIHO TaK
me 9KBHBaJIeHTHbI HepaBeHcTBa 4x<16 H 2x<8.
YupamHeHBJI
I. 113 CJle)(YIOlI1.HX npe)(JIOmeHHH Bbl6epHTe O)(HOMeCTHble npe,l\IIKaTI.I, YKamHTe
RX 06J1aCTI. onpe,l\eJIeHHH H MHomeCTBO RCTHHHOCTH:
a) x+5= 1;
6) npH x=2 BbmOJIHHeTCH paBeHcTBO x2-1=0;
B) )(JIH Bcex 'IHCeJI x BblnOJIHHeTCH paBeHCTBO x+2=2+x;
r) CYlI1.eCTByeT TaKoe nOJIOmHTeJIbHOe 'IIICJlO x, 'ITO x2+ 1 =0;
)() nJIaHeTa x cOBepwaeT BpalI1.eHHe BOKpyr CoJIHua 3a O,l\HH ro,l\;
e) x+3<3x-5.
2. .lI,aHbI npe)(HKaTbl A (x) : 9x2-4=0 H B (x) : 3x-2< -11. HaH,l\HTe MHomeCTBa
RCTHHHOCTH 3THX npe)(HKaTOB, eCJIH HX 06JIaCTI. onpe,l\eJIeHHH eCTb: a) R; 6) R +;
B) N.
3. Ha MHomeCTBe X= {I; 2; 3; 4; 5; 6; 7; 8} 3B)(aH npe)(HKaT A (x) «x - He'leTHOe
'IHCJlO:t. nepenHWHTe Ta6JIHLI,y B TeTpB)(b H 3anOJIHHTe ee, OTMe'laH HCTHHHblC BblCKa-
SblBaHHH 6YKBOH «11», a JIOmHble - 6YKBOH «/1»:
x I
A (x) I
2
3
4
5
6
7
8
2. I(BaHTopbI. nYCTb Ha MHomecTBe X npOCTbIX tIHCeJI 3MaH npe-
.l(uKaT P (x): «npocToe tIHCJIO X - HetIeTHO). nOCTaBHM nepe.A 9THM npe-
,lLUKaTOM CJIOBO «BCHKOe». nOJIYtIHM JIOmHOe BbICKa3bIBaHlfe: «BCHK08
npocroe tIUCJIO X HetIeTHO» (9TO BbICKa3bIBaHHe JIOmHO, TaK KaK 2 -npo-
CToe 'IeTHOe tIHCJIO).
nOCTaBHB nepe,lL ,lLaHHbIM npe,l\HKaTOM P (x) CJIOBO «CYIlI.ecTByen,
nonytIUM UCTHHHoe BbICKa3bIBaHHe: <<CYIlI.ecTBYeT npocToe tIHCJIO x,
HBJIHIOIlI.eecH He'IeTHbIM» (HanpHMep, x=3).
TaKHM 00pa30M, npeBpaTHTb npe,l\UKaT B BblCKa3bIBaHHe MomHO He
TOJIbKO, nO,lLCTaBHB BMecTO lIepeMeHHOH ee 3Ha<JeHHe, HO H nOCTaBHB
nepe.A npe,l\HKaTOM CJIOBa: «Bce», «CYIlI.ecTBYeT» U ,lLp., Ha3bIBaeMbIe B
JIOfUKe K8aHmOpaMU.
Pa3JIHtIaiOT ,lLBa OCHOBHbIX BH,lLa KBaHTopOB: K8aHmOp 06U{Hocmu u
K8aHmOp CYU¥!CmB08aHUfl.
nYCTb P (x) - HeKOTopbIH npe,l\HKaT, 3a.n.aHHbIH Ha MHomecTBe X.
nOCTaBHB nepe,lL HHM KBaHTOp OOIll.HOCTH, nOJIY<JHM BbICKa3bIBaHHe:
«.lJ.JIH Bcex x E X BbInOJIHHeTCH npe.n.UKaT P (x)>>. OHO HCTHHHO B TOM
H TOJIbKO TOM CJIY<Jae, eCJIH ,lLJIH Bcex 9JIeMeHTOB a H3 MHO)\{ecTBa X
BbICKa3bIBaHHH P (a) HCTHHHbI.
PaCCMaTpHBaeMoe BbICKa3bIBaHHe YCJIOBHJIHCb 3anHCbIBaTb npH
nOMOIll.U CHMBOJIOB B TaKOM BH,lLe:
(V x E X) P (x)
('I UTalOT: «.lJ.JIH Bcex x H3 X CnpaBe,lLJIHBO P OT x»).
53

CHMBOJI V - 3TO IIepeBepHYTaH IIepBaH OYKBa aHr.nHHCKOrO CJIOBa
All - Bce. lJaCTo BMeCTO c.nOBa «Bce» MOR<HO BC1'peTHTb c.nOBa «KaR<-
JJ.bIH», «JIIOOOll», «BCHKHll».
Ec.nIl IIOCTaBHTb IIepe.A P (x) KBaHTOp CYIlI.ecTBoBaHHH:I, IIO.rIY'IHM
BbICKa3bIBaHHe: «CYIlI.eCTBYeT TaKoe x EX, tITO BbIIlO.nHHeTCH IIpe,lLIIKaT
P (x)>>. 3T0 BbICKa3bIBaHHe HCTIIHHO B TOM H TO.nbKO B TOM c.nY'Iae, KOr,l1,a
XOTH ObI ,lLJIH O,l1,HOrO 9.neMeHTa a H3 MHOR<eCTBa X BbICKa3bIBaHue P (a) HC-
rHHHO.
BbICKa3bIBaHHe «CYIlI.eCTBYeT TaKoe x, tITO BbIIIOJIHHeTCH npe,lLHKaT
P (x)>> 3alIIICbIBalOT IIPH nOMOIll.I1 CIIMBO.nOB TaK:
(3x E X) P (x)
(tIIITaIOT: «CYIlI.eCTByeT TaKoe x li3 X, tITO P OT x»). CIIMBO.n 3 - 9TO
nepeBepHYTaH IIepBaH oYKBa aHrJIHHCKOro CJIOBa Exist - CyIlI.ecTBYeT.
qaCTo BMeCTO CJIOBa «CYIlI.ecTBYeT» YIiOTpeOJIHIOT OOOpOTbI «H, all,lLeTCH»
«XOTH ObI O,l1,I1H».
npIlBe,lLeM IIpHMep YIiOTpeOJIeHHH KBailTopOB.
nYCTb Ha MHOR<ecTBe N HaTypa.nbHblX tIHCeJI 3a,lLaH IIpe)l.HKaT P (x):
«lJHCJIO x KpaTHO 5». I1cnoJIb3YH KBaHTopbI, H3 ,l1,aHHoro IIpe,lLIIKaTa
MOR<HO IIO.n, ytIHTb HalIpHMep, CJIe,lLYIOIlI.I1e BbICKa3bIBaHHH: 1) m060e
HaTypa.nbHoe tIIIC.nO KpaTHo 5; 2) KalK,lLOe HaTypaJIbHOe tIIICJIO KpaTHO 5;
3) Bce H8Typa.nbHbIe tIIIc.na KpaTHbI 5; 4) CYIll.ecTBYIOT HaTypa.nbHble tIHC-
na, KpaTHbIe 5; 5) HaH.n.eTCH HaTypa.nbHoe tIIIC.nO, KpaTHoe 5; 6) XOTH 6bI
O,lLHO HaTypa.nbHoe tIHCJIO KpaTHo 5.
nepBbIe TPII BblCKa3bIBaHIIH JIOR<HbI II I1MeIOT O,lLHH II TOT R<e CMbIC.n.
npH IIOMOIll.I1 CIIMBO.rIOB OHII 3alIIIllIYTCH TaK: f'tx E N)P (x).
IlOCJIe,lLHIIe TPII BblCKa3bIBaHIIH HCTIIHHbI II I1MeIOT <POPMY:
(3x E N)P(x).
YnpamOeOHJI
1. OjJopMYJIHPYHTe CJle.D.YJOlI1.He BblCKa3b1BaHHH, nOJIb3YHCb 06b!'lHblM H3blXOM,
YCTaHOBHTe IIX IICTHHHOCTb HJlII JlOJKIIOCTb:
a) (VxER) x-l=x; 6) (VxER)x2=4;
B) (VxER)x-l*x; r) (3xE)x2=4. .
2. Ha MHOJKecTBe N Bcex H8TypaJlbHblx '1HCeJI 3B)(aHbI npC)(HK8Tb! P (x): «l!HCJlO x
'IeTHO» H Q(x): «l!HCJlO x KpaTHo 4».
GpopMYJlHpyi!Te CJle)(YIOlI1.He BblCKa3b1BaHHH, nOJIb3YHCb 06bl'lHblM H3bIKOM, H
YKaJKHTe cpe,D,1I HHX IICTHHHble: a) (VxEN)P(x); 6) (3xEN)Q(x); B) (3xEN)P(x);
1') (vxE N)Q (x).
3. BBe)(HTe 0603Ha'leHHH npe)(HKaTOB H 3anHWHTe npH nOMOlI1.H KBaliTOpOB CJle-
AYJOlI1.He BblCKa3b1BaHHH.
a) CYlI1.ecTBYeT AeliCTBHTeJII.HOe '1HCJlO x TaKoe, '1TO x2= I.
6) JlIo60e HaTypaJIbHOe '1HCJlO '1eTHO.
B) Haii'.neTCH AelicTBHTeJIbHOe '1HCJlO x TaKoe, '1TO JVlH KaJK.ZJ;orO AelkTBHTeJII.IlOr<>
'lHCJla a HMeeT MecTO paBeHCTBO ax=a.
r) 1J,JIH JII06b1X Aei!cTBHTeJII.HblX a H b CYlI1.ecTByeT '1HCJlO x TaKoe, '1TO a+b=x.
.n) He cYlI1.eCTBYeT pallHOHaJIbHOrO '1HCJla x TaKor<>, '1TO x2=2.
3. Onepal\HH HaJJ. npeJJ.HKaTaMH. Ilpe,lLHKaTbI, TaK )\{e KaK II BbICKa-
SbIBaHHH, ObIBalOT 9.neMeHTapHbIMH H COCTaBHbIMH. COCTaBHble IIPeAH-
54

KaTbI 06pa3YlOTCH H3 9JIeMeHTapHbIX nplf no-
MOIll.H JIOrH'IecKHx CBH30K: «If», «H.nH», «He-
BepHO, qTO», «ecJllf..., TO...» If ,lLp., CMbICJI KOo
TOpbIX TOT >Ke, LJTO If B JIOrlfKe BbICKa3bIBa-
HHR.
HanplfMep, COCTaBHblMIf HB.nHIOTCH c.ne-
,lLYIOIlI.lfe npe,lLHKaTbI Ha MHO>KecTBe R ,lLeHCT-
BRTeJIbHbIX qlfce.n: «lJIfCJIO X qeTHO H KpaTHo
3.; «x>2 If x=2»; «x>3 IfJIIf x<-2..
nplf oneplfpoBaHlf1f C COCTaBHbIMH npeAHKaTaMIf Ha,lLO HaXO,lLIfTb
If X MHO)\{eCTBa I1CTHHHOCTIf. Y CTaHOBIfM npaBIf.na, KOTopbIe n03BOJIHIOT
HaHTIf MHO)\{eCTBO HCTIfHHOCTIf COCTaBHoro npe,lLIfKaTa, ecJIIf 1f3BecTHbI
MHO)\{eCTBa IfCTHHHOCTIf COCTaBJIHIOIlI.IfX ero 9.neMeHTapHbIx npe,lLIfKaToB.
nYCTb Ha MHO)\{ecTBe X 3a,lLaH npe,lLIfKaT A (x). Ero ompU[fGHueM
Ha3bIBalOT npe,lLIfKaT A (x) , onpe,lLe.neHHbIH Ha TOM )\(e MHO)\{ecTBe X,
nplfqeM npe,lLHKaT A (x) IfCTIfHeH 1) npH Tex 3Ha'IeHIfHX X 1f3 MHO)\{eCTBa
X, nplf KOTOpbIX npe,lLIfKaT A (x) .nO)\{eH, If HaooopoT.
PaccMoTplfM, HanplfMep, Ha MHO)\{ecTBe X={IO; 15; 20; 25; 30}
npeAHKaT A (x): «lJHC.'10 x OKaHqlfBaeTCH I1H<j:JpOH 5». Ero OTplfIlaHHeM
oY,lLeT TaKOH npe,lLHKaT A (x) : «HeBepHo, LJTO qlfCJIO x OKaH'IHBaeTCH I11f<p-
pOR 5», If.nlf, qTO TO )\{e caMoe, «lJHCJIO X He OKaH'IIfBaeTCH I11f<j:JpOH 5..
JIerKo BH,AeTb, 'ITO MHO)\{eCTBOM IfCTIfHHOCTH npe,lLHKaTa A (x) B
,lLaHHOM nplfMepe HB.nHeTCH {15; 25}, a MHO)\{ecTBOM IfCTIfHHOCTH npeAlf-
KaTa A (x) - {1O; 20; 30}, T. e. BTOpoe MHOiKecTBO - ,lLonOJIHeHHe K
nepBoMY B X.
BoooIll.e, eC/lU T - M1iOJlCeCmOO ucmUHHocmu npeiJuKanu:l A (x),
x EX, mo MHOJ/CecmeOM ucmUHHocmu npeiJuKanu:l A (x) , x EX, fl8/lflen1CJI
iJOM/lHeflue T ' K MH.OJ/Cecmey T e M.1iOJ/Cecmee X (pHC. 19).
nYCTb Tenepb Ha MHO)\{ecTBe X 3a,lLaHbI ,lLBa npe,lLHKaTa: A (x) H
B (x). Tor,lLa HX KOH.7JIOHKlJ,UeU oY,lLeT npe,lLIfKaT A (x) 1\ B (x). OH IfCTIfHeH
nplf Tex 3Ha'IeHIfHX x 1f3 MHO)\{eCTBa X, nplf KOTOpbIX IfCTIfHHbI 06a
npeAHKaTa A (x) If B (x).
TaK, ecJIIf Ha MHO>KecTBe X={IO; 15; 16; 20; 35} 3MaHbI npeAH-
KaTbI: A (x): «lJIfCJIO x 'IeTHO., B (x): «lJIfCJIO x KpaTHo 5», TO HX KOH'bIOH-
KlI.lfei'f HBJIHeTCH npe,lLHKaT A (x) 1\ B (x): «lJIfCJIO x 'IeTHO If KpaTHo 5..
MHO)\{eCTBOM HCTHHHOCTH npe,lLHKaTa A (x) B ,lLaHHOM CJIyqae HBJISI-
eTCH {1O; 16; 20}, a MHO)\{ecTBOM IfCTHHHOCTH npe,lLIfKaTa B (x) - {lO;
15; 20; 35}. npe.lLHKaT «lJIfCJIO x LJeTHO H Kparno 5. 06pam:aeTcH B HC-
TIfHHoe BbICKa3bIBaHHe TO.nbKO TOr,lLa, Kor,lLa x=10 If x=2O. MHO)\{ecrno
HCTHHHOCTif {1O; 20} 9Toro npe,lLIfKaTa - nepeceLJeHlfe MHO)\{ecTB HC-
THHHOCTIf npe,lLHKaTOB A (x) If B (x).
BoooIll.e, ec.llU Tl - M.HOJtCeCTn80 ucmUHHocmu npeiJUKanu:l A (x),
xE X, a T 2 - M.HOJlCeCmeO ucmUHHocmu npeiJuKanu:l B(x), xE X, rno
M.HOJ/Cecmeo ucmUH.H.ocmu T npeiJuKanu:l A (x) 1\ B (x), x EX, - 11£pe-
PIIC. 19
II MbI: rOBopHM, '1TO npe./UIKaT A (x) HCTHHeH npH x=a. eCJlH A (0) - HCTHHHoe
BblCKa3b1BaHHe.
56

Ce'ieHUe .M.HOJ/CeCme UCmUHHOCmU npeiJuKam06
A (x) U B (x), m. e. T=TI n T 2 (pHC. 20).
npe,lLHKaT A (x) V B (x), x EX, Ha3bIBaeTCH
iJU37JIOHK4Ueu npeiJUKam08 A (x), B (x), x eX.
011 HCTHHeH rrpH Tex 3Ha'IeHHHX H3 MHOiKecT-
Ba X, rrpH KOTOpbIX HCTHHeH XOTb O,lLHH H3
npe,lLHKaTOB A (x), B (x).
M HOJ/CecmsO.M. T ucmUHHocmu npeiJuKanW
A (x)VB(x), xEX, fl8/lflemCfl T 1 UT 2 , ziJe
T l- .M.HOJ/Cecmso ucmUHHocmu npeiJuKanw
A (x), a T 2- MHOJ/Cecmeo ucmUHHocmu npe-
iJuKanw B(x), T=T1U T 2 (lla pHcYHKe 21
T - BCH 3allITpHXOBaHHaH oo.nacTb).
PaccMoTpHM, HarrpHMep, Ha r.IHOiKeCTBe X
CTY,lLeHTOB HeKOToporo HHcTHTYTa ,lLBa npe,lLH-
KaTa -- A (x): «CTY,lLeHT X - crropTCMeH» H
B (x): «CTY,lLeHT X - rrepBOKypCHHK». .lJ.H3'b-
IOHKIIHeH 9THX npe,lLHKaTOB HB.nHeTCH npe,lLHKaT
A (x) V B (x): «CTY,lLeHT X - crropTCMeH HJIH
nepBOKypCHHK». MHOiKeCTBO HCTHHHOCTH rrpe,lLHKaTa A (x) COCTOHT H3
Bcex CTY,lLeHTOB-cnopTCMeHOB ,lLaHHoro HHCTHTYTa, MHO)\{eCTBO HCTHHHO-
CTH npe,lLHKaTa B(x)-- H3 Bcex CTY,lLeHTOB-nepBOKypCHHKOB 9Toro HH-
CTHTYTa, a MHOiKecTBO HCTHHHOCTH ,lLH3'bIOHKIIHH ,lLaHHbIX npe,lLHKaTOB
npen.craBJIHeT CoDOH oD'be,lLHHeHHe MHOiKecTB HCTHHHOCTH npe,lLHKaTOB
A (x) H B (x), T. e. MHOiKecTBO, xapaKTepHCTH'IeCKOe CBOHCTBO 9JIeMeH-
TOB KOToporo MOiKeT OblTb C<j:JOPMYJIHpOBaHo TaK: «ObITb cnopTCMeHOM
JlJIH nepBoKypCHHKOM ,lLaHHoro HHCTHTYTa».
)(
PIIC. 20
x
Tz
PIIC. 21
YupamoeooJl
1. Ha MHOIKeCTBe X={I; 2; 3; 4; 5} SB)(aHbl npe.l(HKaTbl A (x): «qHCJlO X npo-
CTOe», B(x): «x<3» II C(x): «(x-I)' (x+2)=0».
a) HaAllHTe MHOJKeCTBO HCTHHHOCTH KaJK)(OrO H3 llaHHblX npe.l(HKaTOB.
6) ,[lJIH KaJK)(oro H3 npellHK8TOB c4x>PMYJIHPYHTe ero OTpHlJ,aHHe H HaAllHTe ero
MHOJKecTBO HCTHHHOCTH.
2. PaccMoTpHM npe.ll.HK8Tbl A (x), B (x) H C (x), YKa38HHble B ynpaJKHeHHH 1.
I1oJJb3YHCI. 06blqHblM H3blKOM, npoQHTakre npe,ZJ.HK8Tbl A (x) A B (x); A (x) A C (x);
B(x)VA(x); B (x) AC(x); A (x)VC(x); A (x) AB(x); B(x)V C(x) H H8H,lJ.HTe MHOJKe.
CTBO HCTHHHOCTH KaJK)(OrO H3 HHX. BepHo JIH, qTO MHOJKecTBO HCTHHHOCTH npe.ll.HK8Ta
A (x) A B (x) nycTO?
3. Ha MHOJKecTBe yqaI.l{HXcH HeKoTOporo KJI8CC8 SB)(aHbl npellHKaTbi A (x): «Y'l8'
II1HAcH x HrpaeT B WaXMaTb1» H B (x): «YqaI.l{HHcH x YMeeT nJIaBaTI.:t.
IIOJII.3YHCI. 06b1'1HblM H3blKOM, npo'lHT8ATe CJlellYIOI.I{He npellHK8TbI H YK8>KHTe
Xap8KTepHCTHqecKoe cBoiicTBO MHOJKecTB HCTHHHOCTH KaJK)(OrO H3 HHX:
8) A (x) A B (x);
r) A (x) A B (x);
6) A (x) V B (x);
.Zl) A (x) V B (x);
B) A (x) A B(x);
e) A (x) V B (x).
4. IIpellHKaTbl P (x), Q (x) H R(x) 3B)(aHbl Ha HeKoTOpoM MHOJKeCTBe X. H3BeCTHO,
'ITO Tl - MHOJKeCTBO HCTIIHHOCTH npe.ll.HK8Ta P (x), T 2 - MHOJKeCTBO HCTHHHOCTH npe-
AHKaT8 Q(x), Ta - MHOJKeCTBO HCTHHHOCTH npellHKaTa R (x), npH'IeM Tin t 2 nT #0.
66

11306pa3HTe npH nOMOlI1.H ,ZJ;HarpaMM 3HJIepa - BeHHa MHOJKeCTBa HCTHHHOCTH CJJe.
,ZJ;YJOlLI,HX npe,ZJ;HKaTOB:
8) P (X) 1\ Q (X) I\R (X);
B) P (X) 1\ (Q (X) V R (X»;
,ZJ;) (P(X) I\Q(x»VR(x).
6) P(x)vQ(x)VR(x);
r) (P(x)I\Q(x»vR(x);
5. Ha MHOJKeCTBe N HaTypaJII.HblX 'IHCeJI 3B)(aHbI npe,ZJ;HKaTbi P (x): «4HCJIO Jt
'1eTHO» H Q (x): «4HCJIO X IJ,CJIHTCH Ha 4».
OjJoPMYJIHpyihc CJICIJ,YIOIl\He BbICKa3blBaHHH, n0J1I.3YHCI. 06bl'lHblM H3blKOM, H
YKaiKHTc CpelJ,H HHX HCTHHHbIC:
a) (Vx)P(x)I\Q(x); 6) (3x) P (x) V Q(x) ;
B) ('1 x) P (x) 1\ Q (x); r) (3X)P (X) 1\ Q (x) ;
.n) (3x) P (x) 1\ Q (X); e) (3X )P (x) V Q (x).
4. HMnJIHKall.HH H 3KBHBaJIeHlI.HH npe,ll,HKaTOB. 113 npe,lLHKaTOB
A (x) H B (x), 3a,lLaHHbIX Ha O,lLHOM H TOM )\{e MHO)\{eCTBe X, MO)\{HO COC-
TaBHTb npe,lLHKaT A (x)=?B (x). Ero Ha3bIBalOT UhtnAUKau,ueu 9THX npe-
,lLHKaTOB H 'IHTaIOT: «EcJIH A (x), TO B (x)>>. HanpHMep, H3 npe,lLHKaTOB
«HaTypa.nbHoe 'IHC.nO X ,lLe.nHTCH Ha 3» H «HaTypa.nbHoe 'IHC.nO X ,lLeJIHTCH
Ha 4» MO)\{HO COCTaBHTb npe,lLHKaT «Ec.nH HaTypa.nbHoe 'IHC.nO X ,lLe.nHTCH
'Ha 3, TO OHO ,lLeJIHTCH H Ha 4». .3TOT npe,lLHKaT HCTHHeH npH HeKoTopbIX
HaTypa.nbHbIx 3Ha'IeHHHX X H .nO)\{eH npH ,lLpyrHx. HanpHMep, npH x=
= 12 9TOT npe,lLHKaT npHHHMaeT BH,lL «EcJIH 'IHC.nO 12 ,lLe.nHTCH Ha 3, TO
OHO ,lLe.nHTCH H Ha 4». 3,lLecb HCTHHHbI H YCJIOBHe «<4HC.nO 12 ,lLe.nHTCH Ha 3»)
H c.ne,lLCTBHe «<4HCJIO 12 ,lLe.nHTCH Ha 4»). A Tor,lLa, KaK MbI 3I1aeM (CM.
n. 4  2 r.n. I), HCTHHHa H HMn.nHKaIlHH 9THX BbIcKa3bIBaHHH. I1CTHHHoe
BbICKa3bIBaHHe no.nY'IaeTCH H npH 3aMeHe X Ha 14 «Ec.nH HaTypa.nbHoe
'IHC.nO 14 ,lLe.nHTCH Ha 3, TO OHO ,lLe.nHTCH H Ha 4». .lJ.e.no B TOM, 'ITO 14 He
,lLeJIHTCH Ha 3, a nOToMY B 9TOM c.nY'Iae He BbIno.nHHeTCH yc.nOBHe H HM-
nJIHKaIlHH HCTHHHa (CM. c. 20). Ho npH x= 15 no.nY'IaeTCH .nO)\{HOe BbI-
CKa3bIBaHHe «Ec.nH 15 ,lLe.nHTCH Ha 3, TO OHO ,lLe.nHTCH H Ha 4» - Be,ll,b
B 9TOM c.nY'Iae yc.noBHe BbIno.nHeHO (15 ,lLeJIHTCH Ha 3), a c.ne,lLCTBHe He
BbInOJIHHeTCH (15 He ,lLe.nH1;cH Ha 4).
113 9TOro npHMepa BH,lLHO, tITO npe,l1,HKaT A (x)=>B (x) npeBpaIll.aeTCH
B JIO)\{Hoe BbICKa3bIBaHHe .nHllib npH nO,lLCTaHOBKe BMeCTO x TaKHX 3Ha-
'IeHHH a, ,lI,JIH KOTOpbIX A (a) HCTHHHO, a B (a) .nO)\{HO. I1HbIMH c.nOBaMH,
eCJIH 0603HatIHTb tIepe3 TI MHO)\{eCTBO HCTHHHOCTH npe,lLHKaTa A (x),
a 'Iepe3 T.- MHO)\{ecTBO HCTHHHOCTH npe,AHKaTa B (x), TO A (x)=>B (x)
.nO)\{HO Ha MHOR<ecTBe TI n T; (HMeHHo Ha 9TOM MHO)\{ecTBe A (x) HCTHH-
HO, a B (x) JIO)\{HO). Ha ,lLono.nHeHHH K MHO)\{ecTBY TI n T npe,lLHKaT
A (x)=>B (x) HCTHHeH. Ho no <j:JopMy.ne (1) n. 6  1 r.n. II HMeeM:
(T I n T;)' =T U (T;)' =T U T..
3Ha'IHT, MHOJICeCI1Z8OM ucmU1iHocmu npeiJUKanu:l A (x)=>B (x) fl8AfletnC!I.
067JeiJuHeHue MfWJtCecm T 2 ucmUHHocmu npeiJUKanu:l B (x) U iJono/l-
HeHUfl K MHoJICecmey Ti ucmUHHocmu npeiJuKama A (x) (pHC. 22).
4acTo BCTpe'IaIOTCH TaKHe npe.AHKaTbI, 'ITO H3 HCTHHHOCTH O,lLHorO
H3 HHX cJIe,lLYeT HCTHHHOCTb ,lI,pyroro. HanpHMep, H3 «x ,lI,eJIHTCH Ha 4»
57

CJIe,nYeT: «x ,l\eJIRTCH Ha . Tenepb MbI MO)\{eM
CKa3aTb, tJTO H3 llCTllHHOCTH npe,lLllKaTa A (x)
CJIe,nYeT llCTllHHOCTb npe,lLllKaTa B (x) B TOM H
TOJIbKO B TOM CJIytJae, KOr,lLa nYCTO MHO)\{eCT-
BO TIn T;, Ha KOTOpOM JIO)!{eH npe,lLHKaT
A (x)=?B (x). Ho ll3 pllcYHKa 23 Bll,lLHO, tITO
MHo)\{ecTBa Ti R T llMelOT nYCToe nepece-
tJeHlle JIllllib B CJIY'Iae, KOr,l\a TI - nO,AMHa-
)\{ecTBO T2' T l cT 2 .
I1TaK, npeiJuKam A (x)=>-B (x) ucmUHeH iJ/LJl
ecex x E X e moM. U mOAbKO e mOM CAY'iae,
IWCiJa MHOJlCeCmsO Ti UCmUHHocmu npeiJUKa-
nw A (x) coiJepJ/Cumcfl e MHO:JfCecmse T 2 ucmUH-
Hocmu npeiJUKanw B(x), m. e. KOciJa TiCT2'
B TOM CJIytJae, KOr,lLa HMnJIHKaIlHH
A (x)=?B (x), x EX, I'ICTllHHa npll Bcex 3Ha-
'IeHHHX H3 MHO)\{ecTBa X, rOBopHT, tJTO npe-
iJUKam B (x) AOZUlieCKU cAeiJyem U3 npeiJuKa-
ma A (x), U npeiJuKam B (x) Hll3bI8Cl1Ont
Heo6xoiJUM.btM YCAoeueM iJAfl npeiJuKanw A (x), npeiJuKam A (x) - iJocma-
mO'iHblM YCA08ueM iJ/LJl B (x).
TaK, B llMnJIllKaIlllll «EcJIH x - tJllCJIO HaTypaJIbHOe, TO OHO I1eJIoe»
npe,l\HKaT B (x): «x - tJHCJIO I1eJIoe» JIOrutJecKll CJIe,nYeT ll3 npe,nllKaTa
A (x): «x - tJHCJIO HaTypaJIbHoe». CJIe,noBaTeJIbHO, npe,lLllKaT «x - 'IRC-
JIO I1eJIoe» HBJIHeTCH He06xo,lLHMbIM yc.nOBHeM .lI.JIH npe,lLHKaTa «x-
'IHCJIO HaTypaJIbHoe», a npe,lLHRaT «x - tJHCJIO HaTypaJIbHoe» HB-
JIHeTCH ,lLOCTaTotJHbIM YCJIOBlleM ,lLJIH npe,nllKaTa «x - tJllCJIO lie-
JIoe» .
I1cnoJIb3YH 9Tll TepMllHbI, llMnJIHKaIllllO «ECJIR tJllCJIO HaTypaJIbHoe,
TO OHO I1eJIoe» MOlKHO BbIpa3llTb TaKllMll CJIOBaMll.
1. .lJ.JIH Toro tIT06bI tJllCJIO x 6bIJIO HaTypaJIbHbIM, Heo6XO,lLllMO, tIT06bI
OHO 6bIJIO I1eJIbIM.
2. .lJ.JIH Toro tJT06bI 'IllCJIO x 6bIJIO I1eJIbIM, ,lLOCTaTotJHO, 'IT06bI OHO
6bIJIO HaTypaJIbHbIM.
ECJIH npe,nHKaTbI A (x) II B (x), 3a,lLaHHbIe Ha MHO)\{ecTBe X, 9KBll-
BaJIeHTHbI, T. e. ec.nll MHO)\{eCTBa HX llCTllHHOCTll TIll T 2 CQBIH1.lI,<31OT,
TI =T 2, TO .lI.JIH Bcex x EX llCTllHHa 9KBllBaJIeHIIllH A (x)B (x). Ha-
npllMep, B CJIY'Iae npe,nllKaTOB A (x): «HaTypaJIbHOe tJllCJIO x ,lLeJIllTCH Ha
10. R B (x): «.lJ.ecHTH'IHaH 3anllCb HaTypaJIbHOrO 'IllCJIa x KOH'IaeTCH
II.R<j:JpOH 0» ,l\JIH Bcex HaTypaJIbHbIX 'IllCeJI llCTllHHa 9KBllBaJIeHIIllH A (x)
B (x). HanpRMep, ,lLJIH x= 140 OHa RCTHHHa, nOTOMY 'ITO 140,l\eJIHTCH
Ha 10, a nOCJIe,nHHH I1ll<j:Jpa 9Toro tJllCJIa paBHa HYJIIO (06a BbICKa3bIBaHllH
HCTllHHbI), a npR x= 12 9Ta 9KBllBaJIeHIIllH llCTllHHa, nOCKOJIbKY 06a
BbICKa3bIBaHRH JIO)!{HbI (12 He ,lLeJIllTCH Ha 10, II nOCJIe,nHHH I1ll<j:Jpa 9TOro
'IRCJIa He paBHa 0).
ECJIll npe,nllKaTbI A (x) II B (x) Ha MHO)!{eCTBe X 9KBllBaJIeHTHbI, TO
KaJKJ{bIH ll3 HllX Ha3bIBalOT He06xoiJuMblM II iJocnwmO'iHbtM YCA08ueM.
JIH BToporo. HanpllMep, ,l\JIH TOro 'ITOObI HaTypaJIbHoe 'IRCJIO x ,l\e-
;:x

T I
""--
T'/
Z
PRC. 22
PHC. 23
58

JIHJIOCb Ha 10, HeOOXO,l1,HMO H ,l1,OCTaTO'IHO, t.ITOQ,bI nOCJIe,l1,HHH I1H<j:Jpa B ,l1,e-
CHTHqHOfi3anHCH 3TOrO qHCJIa ObIJIa paBHa O.
OrMCTIIM, '1TO 113 3KBIIB8JIeHTHOCTli )(BYX npC)\IIKaTOB A (x) II B (x) Ha HCKOTOpOM
MHOJKCCTBe X He CJlelJ,yeT, '1TO npe)\HKaTbI, BbIpaJKeHllbie TeMIi JKC CJlOBaMH, OKaJKYTCSl
3KBliBaJIeHTHbiMli Ha IJ,PyroM MHOJKCCTBC Y. HanpliMep, Ha MHOJKCCTBC napa.Me.10rpaM-
MOB 3KBIIBMeHTHbi npCIJ,IIK8TbI «Bcc CTOpOHbI HMCIOT OIJ,HHaKOBYIO IJ,JIHHY» H «,l],Ha-
rollaJIH nepnCHIJ,IIKYJ1HpHbI»: IJ,JIH TOro '1T06bl Bce CTOpOHbI napaJ1JIeJIOrpaMMa IIMeJIH
O)I.HHaKOBYIO IJ,JIlIHY. Heo6XOlJ,IIMO II IJ,OCTaTO'lHO, '1T06b1 ero IJ,lIarOHaJIIi 6bIJ1H nepnCH-
AHKYJIHpII bI. A Ha MHOJKeCTBe '1CTblpexyrOJIbHIIKOB TH npelJ,lIKaTbl He3KBIIBaJ1CHTHbI:
CynJ,eCTBYCT '1CTblpexyrOJIbHHK, IJ,HarOHaJIH KOToporo nepneHIJ,IIKYJIHpHbI, HO CTOpOHbI
HMelOT pa3JIII'IHYIO )(JIIiHY.
YnpamBeBBJI
I. Ha MHOJKCCTBe X={-3; -2; -I; 0; I; 2; 3; 4} 3a)(aHbI npelJ,lIKaTbi A (x):
«411CJ1o x KpaTHo 3» II B(x): «x-I>O».
Onpe)\eJIliTe MHOJKCCTBO IICTIIHHOCTII KaJKlJ,OrO 113 CJ1CIJ,YlOlllHX npCIJ,HKaTOB H
H306pa311Te ero npll nOMollllI IJ,lIarpaMM 3rwcpa - BCHHa:
a) A (x) ===') B (x); 6) A (x) ===') B (x) .
2. HMnJIIiKallllH «ECJlII '1HCJlO x KpaTHo 4, TO OHO '1eTHO» 3alJ,aHa Ha MIIOJKeCTBe
X={l; 2; 3; 4; 5; 6}. .
a) HaHlJ,lITe MHOJKCCTBO HCTHHHOCTH ».8HHOH HMnJIIiKallllll; H306pa3l1Te ero Ha
jl,HarpaMMe 3HJIepa - BeHlla.
6) P.oKaJKIITC, '1TO npe)(HKaT «4HCJlO x KpaTHo 4» - »,OCTaTO'IHOe YCJIOBlie jl,JIH
npelJ,lIKaTa «411CJ1o x '1eTHO».
3. npelJ,lIKaTbl P (x): «,l],HarOHaJIH '1eTbipexyrOJIbllHK8 x KOHrpY9I1THbl» II Q (x):
«4CTblpexyroJIbHIIK x - npHMoyrOJ1bHHK» 3a.naHbI Ha MHOJKeCTBe Bcex '1CTblpexyrOJIb-
HHKOB.
a) apopMYJIHPYHTe IIMnJ1I1KallIlH P (x)===') (x) II Q (x) ===') P (x).
6) ,l],oKaJKHTe, '1TO npCIJ,IIKaT Q (x) HBJ1HCTCH 1J,0CTaTO'IHbIM yc.10BlieM )\JIH P (x).
B) BepHO JIH, '1TO npe)\IIKaT Q (x) - Hc06xolJ,HMOe YCJ10BHC t(J1H P (x)?
4. X - MHOJKeCTBO '1eTblpexyrOJIbHHKOB. Ha 3TOM MHOJKeCTBe 3alJ,aHbl npCIJ,II'
KaTbi
C (x): «4eTblpcxyrOJII.HIIK x - napaJIJ1eJIOrpaMM» II
D (x): «,l],Be npOTIiBOnOJIOJKHble CTOpOHbl '1CTblpexyrOJIbHIIKa KOHrpy3HTHbI».
HaH)(HTe, eCJIH 3TO B03MOJKHO, TaKlie '1CTblpcxyrOJII.HIIKII, IJ,JIH KOTOpbIX: a) npe.
AHKaTbi C (x) II D (x) HB.IlHIOTCH HCTIIHllblMIi BblCKa3b1BaHIIHMII; 6) npelJ,lIKaTbi C (x)
H D (x) HB.IlHIOTCH JIOJKHblMH BblCKa3blBaHIIHMII; B) npelJ,IIK8T C (x) 06pB111aeTCH B J10JK-
Hoe BblCKB3b1BaHHe, a npe)(HKaT D (x) - B IICTHHHoe; r) npelJ,HKaT C (x) 06palllaeTCH
B HCTHHHOe BblCKa3b1BaHlle, a npelJ,IIK8T D (x) - B JIOJKHoe.
5. l(aKoe H3 CJleIJ,YIOlI1.HX BblCKa3b1BallllH IICTIIHHO?
a) 4T06b1 nocTynHTb B BY3, 1J,0CTaTO'IHO OKOH'IIITb CpelJ,HlO1O WKOJIY
6) 4T06b1 nocTynHTb B By3, Heo6xolJ,HMO OKOH'IIITb CpelJ,HIOIQ WKOJ1Y.
6. B KaJKJ\OM 113 CJ1eIJ,YlOlllIIX CJly'lacB YKaJKIITe, KaKoe H3 IJ,BYX YTBCpJKJ\eHHH
JIOrll'leCKIi CJlelJ,yeT 113 )(pyroro II pa3HblMli cnoc06aMli C<jJoPMYJ1I1PYHTe HMnJ1I1KallHIO,
IICnOJIb3YH TepMIIHbl «Heo6xo1J,HMOe YCJIOBHe» 11 «IJ,OCT8TO'IHOe YCJIOBlle»:
a) A (x): «x>O»; B (x): «2x>0», xE R.
6) A (x): «x2+3x=0»; B (x): «x=0», xE R.
7. H3BecTHo, '1TO npe)(HKaT C (x) HB.IlHeTCH Heo6xo)(HMblM YCJIOBHeM IJ,JIH npe)(H-
KaTa B (x), KOTOpblH, B CBOIO O'lepelJ,b, HB.IlHeTCH Heo6xo)(HMblM YCJIOBHeM IJ,JIH npe».H-
KaTa A (x). l(aKIiM YCJIOBHeM HB.IlHCTCH IJ,JIH A (x) npelJ,lIKaT C (x)?
5. MHorOMeCTHbie npe}J,HKaTbi. npe,lLJIO)\{eHHe «n09T X HanHCaJI
n09MY y» CO,l1,ep)\{HT ,l1,Be nepeMeHHbIe X H y. B 9TOM c.nyqae Ha,lLO 3a,lLaTb
,l1,Ba MHO)\{eCTBa X H Y - O,lLHO H3 HHX COCTOHT H3 <j:JaMHJIHH, a BTopoe-
H3 Ha3BaHHH n03M. 113 9.neMeHTOB 9THX MHO)\{eCTB OY,lLeM COCTaBJIHTb
ynopH,lLO'IeHHbIe napbI, HanpHMep: (nYllIKHHj «.lJ.BeH<w..I,aTb»), (<<Dyw-
59

KHH; «LLbIraHbl»), (EceHlW; «XOPOllIO!»), (MaHKOBCKHH; «Bo Becb ro.noc»)
H T. ,lL. HeKOTOpbIM H3 9THX nap (HanpHMep, (TIYllIKHH; «LLbIraHbI»),
(MaHKOBCKHH; «Bo Becb rOJIOc»» COOTBeTCTBYIOT HCTHHHbIe BbICKa3bIBa-
HHH (<<Il09T nYllIKHH HanHca.n n09IY «LLblraHbl»», «TI09T MaHKOBCKHH
HanHca.n n09MY «Bo BeCb ro.noc»»), ,lLpyrHM napaM - .nO)\{HbIe BbICKa-
3bIBaHHH (<<TI09T TIYllIKHH HanHca.n n09MY «LlBeHMIIaTb»»).
TIYCTb, Bo06Il1.e, HeKOTopoe npe,lL.nO)\{eIlHe P (x, y) CO,lLep)\{HT ,lLBe
nepeMeHHble, npH'IeM nepeMeHHa51 x npHHHMaeT 3Ha'IeHHH H3 MHO)\{eCTBa
X, a nepeMeHHaH Y - H3 MHO)\{eCTBa Y (9TH MHOlKeCTBa MorYT H COBna-
.lI.aTb). TIapbI (x; y), x EX, y E Y, oopa3YIOT, KaK MbI 3HaeM, ,lLeKapTOBO
npOH3Be,lLeHHe X X Y MHO)\{eCTB X H Y. TIYCTb ,lL.nH .nIOOOH napbI (a; b)
H3 X X Y npH 3aMeHe B npe,lL.nOlKeHHH P (x, y) nepeMeHHoH x ee 3Ha'Ie-
HHeM a, a nepeMeHHoH y 3HalJeHHeM b nO,1)'lJaeTCH BblCKa3bIBaHHe P (a,b).
Tor,lLa rOBopHT, 'ITO P (x,y) - o8YUteCmHbiu npeowwm, 3a,lLaHHbIH Ha
X X Y. COBoKynHocTb T nap (a; b), npH nO,lLCTaHOBKe KOTOpbIX B ,lLByX-
MecTHblH npe,lLHKaT P (x, y) no.nYlJaeTCH HCTHHHoe BblCKa3bIBaHHe, Ha-
3bIBalOT MHOJ/CeCflU3OM ucmUHHocmu 3Toro npe,lLHKaTa. 3To mo)\{eCTBO
HB.nHeTCH nO,lLMHO)\{eCTBOM ,lLeKapTOBa npoH3Be,lLeHHH X X Y.
TO'IHO TaK )\{e onpe,lLeJJHIOTcH TpexMecTHbIe, 'IeTbIpeXMeCTHbIe H T. ).\.
npe,lLHKaTbI. HanpHMep, npe,ZJ.JIOmeHHe «MaTeMaTHK x pO,lLHJICH B ro,lLY
y, a ,lLHCCepTaIlHIO 3aIll.HTHJI B ro,lLY 2» HB.nHeTCH TpexMecTHbIM npe,lLHKa-
TOM, a npe,lLJIO)\{eHHe «CYMMa LJHCeJI x H y paBHa npOH3Be,lLeHHIO LJHCeJJ
U H U» - LJeTbIpexMecTHbIM npe,lLHKaTOM. n - M£cmHblU npeiJUKam onpe-
,lLeJIeH Ha ,lLeKapTOBOM npOH3Be,ZJ.eHHH X I X X 2 X. . . X Xn. Ero M.HOJtCeCm-
(J() UCmUHHOCmU T HBJIHeTCH nO,lLMHO)\{ecTBOM 9TOro ,lLeKapTOBa npOH3Be-
).\eHHH.
KaK H B CJIYtIae O,lLHOMecTHbIX npe,ZJ.HKaTOB, MHoroMecTHbIe npe).\HKa-
TbI Ha3bIBalOTCH aK8U8a/leHmHbtMU, eCJIH OOJIaCTH Onpe,lLeJJeHHH H MHO-
mecTBa HCTHHHOCTH 9THX npe,ZJ.HKaTOB cOBnMalOT. HanpHMep, npe,ZJ.HKaT
cTpeyroJIbHHK x nO,lL06eH TpeyrO.nbHHKY y» 9KBHBaJIeHTeH npe,ZJ.HKaTY
cYrJIbI TpeyrOJIbHHKa x HMelOT TY )\{e BeJJHLJHHY, LJTO H COOTBeTCTBYIOm:He
yrJIbI TpeyrOJIbHHKa y». A npe,lLHKaT «.lJ.OM x HaXO,lLHTCH Ha nepece-
tleHHH YJIHII y H Z» 9KBHBaJIeHTeH npe,ZJ.HKaTY «.lJ.OM x HaXO,lLHTCH Ha YJIH-
lie y H ,lLOM x HaXO,lLHTCH Ha YJIHIIe z». MI,I oY,lLeM 0603HaLJaTb 9KBHBa.neHT-
HOCTb n pe,lLHKaTOB 3HaKOM ,..,:
A (x, y)"" B (x, y) H T. ,lL.
YpaBHeHHH x+2y=5 H 2x+4y= 10 HBJIHIOTCH 9KBHBaJIeHTHblMH npe-
,lLHKaTaMH. HanpHMep, napa (1; 2) Y,lLOBJIeTBOpHeT ypaBHeHHIO x+2y=5
H 9Ta me napa Y,lLOBJIeTBOpHeT ypaBHeHHIO 2x+4y= 10: 1 +2.2=5 H
2.1 +4.2= 10. T aKHe ypaBHeHHH Ha3bIBalOT aK8U8aAeHmHbtMu. 3KBH-
BaJIeHTHbI H HepaBeHcTBa 2x+4y>1O H x+2y>5.
YupamHeHHR
I. CpC,l\H CJle,l\YIOlI1.HX npe)(JlOJlwHHH YKIDKHTe )(BYXMecTHbiC npeD,HKaTbi.
11) CnpaBKa N2 ... .llaHa TOB. ....
6) rOpO.ll C C1'OHT Ha 6epcry Mopa D.
B) KaKaa pCKa Bna.naer B KacnHiicKoc MOpC?
60

r) HeKOTOpble cTy)(eHTbI )(8HHqA rpynmi 38HHM81OTCH CnOpTOM.
)() 4HCJlO b paBHo 1JHCJiy li.
e) 4HCJlO x HBJIHeTCH J(eJIHTeJIeM 1JHCJ1a 2.
)I{) 3x+4>x.
3) 4HCJ10 2 - KOpCHb ypaBHcHHH x-7=1\x.
2. npe.ll.HK8T «4HCJlO x .HBJIHeTCH )(eJIHTeJICM '1HCJia y» 3a)(aH Ha MHO)l{CCTBe BCeX
He<JTpHlJ,aTe.1lbHblX IJ,eJIbiX '1HCCJI Zoo HB30BHTe )(Be iIaflbl f{HCeJI_HS MHO)l{CCTBa Zo, npH
nO)(CTaHOBKe KOTOpblX B )(aHHblH npC)(HKaT nOJlY'l8ercH JIO)l{HOC BbICK83b1BaHHe. H8-
30BHTe )(BC napbl '1HCeJI H3 MHO)l{eCTB Zo, npHHlIJlJIe)i{8lI1. H C MHO)l{CTBY HCTUHHOCTH l'
)(aHHOrO npe)(HKaTa. BCpHO JIH, '1TO a) (0; 5) E T; 6) (5: 0) E T; B) (3; 24) E T; r) (7; 5) a T?
3. Ha MHO)l{eCTBC A= {3; 4; 5; 6} 3a;J.aH npe)(HKaT «x>y».
a) 3anHl11HTC MHO)l{eCTBO P Bcex nap, COCTaBJIeHHblX H3 9JICMCHTOB MHO)l{CCTB A.
6) BbI)(eJlHTe H3 MHO)l{CCTBa P nO)(MHO)l{CCTBO T TCX nap, )(JIH KOTOpblX )laHhblH
npC)(HKaT 06palI1.aCTCH B HCTHHHoe BblCKa3b1B8HHe.
4. npC)(HK8T «y=x+2» 3a;J.aH Ha MHO)l{CCTBC X={xlxfN, x<:IO}. 3anH111HTe
MHO)l{ecTBO nap, npHHa,zJ.JIC)I{alI1.HX MHO)l{CCTBY HCTHHHOCTH T 9Toro npC)l.HKaTa.
5. Ha MHO)l{eCTBC X={CCCP, 4CCP, nHP, HPB, CPP, C<1>PIO, f,LlP, BHP}
3a,naH npC)(HKaT «CTpaHbI x H y HMelOT 06lI1.YIO rpaHHIJ,Y». 3anHwHTc MHO)l{CCTBO He-
THHHOCTH )(aHHOrO npcJ(HKaTa.
6. Ha MHO)l{CCTBe X={O; 1; 2; 3; -1; -2; -3} 3a)(aH npC)(HKaT x2+y2=4.
CocTaBI.Tc MHO)l{CCTBO nap, npH nO)(CTaHOBKC KOTOpblX )(aHHblH npC)(HKaT 06palI1.aeTCH
B BblCKa3b1BaHHe. KaKoBo MH9)1{CCTBO HCTHHHOCTH 3TOro npC)(HKaTa?
7. Ha f4H0)l{CCTBC A= lxlxEN, x<8} 3a)(SH npe)(HKaT <<x+y npHHa)(J1C)I{HT
MHO)l{CCTBY A». HaH)(HTe M;I10)l{eCTBO HCTHHHOc1'H 3TOro npC)(HKaTa.
8. npC)(HKaT F (x, y): «4HCJ10 x HBJIHCTCH )leJIHTeJICM '1HCJIa y» 3a;J.aH Ha MHO)l{e-
mc M HaTypaJIbHblX 0)(H03Ha'lHI.IX '1HCeJI.
a) HaH.nHTe MHO)l{CCTBO HCTHHHOCTH 3TOro npC)(HKaTa.
6) C<jJopMYJlHPYHTe npc.l(HKaT F (y, x). - .
B) I1MeIOTCH JlH TaKHe napbl '1HCeJI (a; b), aEM, bEM, )(J1H KOTOpI.IX O)(HO HS
BblCKa3bJBaHHA F (a, b), F (b, a) HCTHHHO, a )(pyroe JIO)l{HO? .
r) I1MelOTcH JlH TaKHC napbl (a; b), aEM, bEM,)(JIH KOTOpblX 06a BblCKa3b1BaHHH
F (a, b) H F (b, a) HCTHHHbI?
AI I1MH JlH TaKHe napbl (a; b), aEM, bEM,.l(JIH KOTOpblX 068 BblCK83bIB8HH.II
F (a, b) H F (b, a) JlO>KHbI? _
. 9. Ha MHO)l{ecTBC nJtOCKHX reoMcTpH'IeCKHX c1JHryp 3a.naHbI npeJ(HKaTbi A (x, y):
«<PHrypbl x H y paBHOBeJIHKH) H B (x, y): «<PHrYPbl x H y KOHrp)'9uTHbI». KaKHe liS
c.iIe)(YIOHx BblCKa3b1B8HHA HCTHHHbI?
a) 4ro(!bI c1mrypbl 6b1JIH paBHOBeJlHKHMH, He06xo)(HMO, '1T06b1 OHH 6b1JIH KOH-
rpysHTHblM H .
6) 4r06b1 cl>HrYPbl 6b1J1H p8BHOBeJIHKHMH, .l(OCT8TO'lHO, 'lr06b1 OHH 6b1JIH HOH'
rpy9HTHblMH.
B) 4r06b1 cl>HrYPbl 6b1JIH KOHrpy3HTHblMH, He06xoJ(HMO, '1T06b1 OHH 6b1JIH p8BHO-
BeJlHKHMH.
r) 4r06b1 cl>Hrypbl (SblJIH KOHrpY3HTHblMH, AOCT8ro'lHO, '1r06b1 OHH6b1JIH p8BHO-
BeJIHKHMH.
I 2. TEOPEMbI
J. CTpoeHHe TeOpeMbi. HSY'IeHHe JII060ro paSJl.eJI8 MaTeM8THKpt
conpoBoaeTCH, K8K npaBHJIO, paCCMOTpeHHeM npeJl.JIO>KeHHH, Ha3bI-
B8eMbIX meopeMaMU. B llIKOJIbHOM Kypce MaTeMaTHKH HsyqalOT TeopeMid,
HMelOIII.He pa3Ho06pa3Hble cpOpMbI 3anHCH, cpopMYJIHPYIOT H ,lLOKa3bI81,i.
lOT pa3JIH'IIU:>le BHJl.bI TeOpeM. QT06b1 ycnewHo BblnOJIHSTb TaKYIO pa6oiy,
He0'f5xOJl.HMbI 3HaHHH 0 CTpOeHHH TeOpeMbl, OCHOBaHHble Ha nOHHTHH
BbICKa3bIBaHHH, npe,lLHKaTa H KBaHTOpa.
PaCCMOTpHM CJIe,lLYIOIlI.YIO TeOpeMY: «ECJIH TOqKa JIe>KHT Ha OHCCQ.
TpHCe y.rJIa, TO OHa paBHOYJl.8JIeHa or CTOpOH iToro )lMa». YCJIOBHeM
6i

9TOH TeOpeMbI HBJIHeTCH npeJlJIO}KeHHe «To'IKa JIeJKHT Ha 6HCCeKTpHCe
yrJIa», a 3aK.nI0'IeIfHel\! - npe,lL.nO)\{eHHe «To'lKa paBHOY,lLaJIeHa OT CTO-
pOH yr.na». BH,lLHM, 'ITO H YCJ1OBHe, H 3aK.nlOtJeHHe aHHoii TeopeMbI
npe,lLCTaB.nHIOT c06or1 npe,lLHKaTbI, 3a,lL3HHble Ha MHOJKeCTBe P BceX TO-
qeK nJIOCKOCTH. 1I.eikTBHTeJIbHO, npe.ll.JIO)\{eHHe «To'IKa JIeJKHT Ha OHC-
ceKTpHce yrJIa» OOpaIll.aeTCS! B IiCTHHHOe BbICKa3bIBaHHe', eCJIH roBOpHTb
o roqKe H3 MH01KeCTBa P, JIe1Kam:eH Ha OHCCeKTpHce aHHoro yrJIa, H
6YeT JI01KHbIM ,lJ,JIS! BCex TOtJeK nJIOCKOCTH, KOTopble Ha OHCCeKTpHCe
9TOrO yrJIa He JIe)\{aT. To lKe caMoe MO)l{HO CKa3aTb H 0 npfJ.VI())l{eHHH
«TOqKa paBHoYaJIeHa OT CTopoH yrJla». 0603Ha'lHM 3TH npe;J.HKaTbi CO-
C1I'BeTCTBeHHO LJepe3 A (x) H B (x), r)l,e x 0603HaqaeT Jlfooyro TMKY H3
MHOJKecTBa P. Tor.ll.a pacCMaTpHBaeMaS! TeopeMa COCTOHT B TOM, tlTO
HMnJIHKaIlHH 3THX npe,lLHKaTOB A (x}==?-B (x), TO ecTb npe,lLHKaT «ECJIH
T()qKa x JIeJKHT Ha 6HcceKTpHce yrJIa, TO OHa paBHOYaJIeHa OT CTOpOH
9T0I'0 yrJIa», oopaIll.aeTCH B HCTlfHHOe BblCKa3bIBaHHe npH seex x H3 MHO-
1KecTBa P. npH nOMOIll.H KBaHTopa 06Il1.HOCTH 3TO MOJKHO 3anHcaTb TaK:
(VxE P) (A (xf*B(x».
TaKHM 06pa30M, rOBopS! 0 CTpoeHHH TeopeMbI «EcJIH TO'IKa JIeJKHT
Ha oHcceKTpHce yrJIa, TO OHa paBHOY.ll.aJIeHa OT CTOpOH 9Toro yrJIa»,
Mbi BbmMHJIH B HeM TPH qaCTII:
1. J!cAoBue meopeMbl: npe,AHKaT A (x), 3a,lLaHHblH Ha MHO)l{eCTBe P
Bcex TO'IeK nJIOCKOCTH.
2. 3aKAIO'leHUe meopeMbl: npe.ll.IrKaT B (x), 3a,lLaHHblH Ha MH<»KeCTBe P
Bcex TO'IeK nJIOCKOCTH.
3. Pa37JflCHUmeAbftafl '«lCmb: B HeH onHCbIBaIOTCS! MHroKeCTBa OO'beK-
TOB, 0 KOTOpblX H)l,eT peLJh B TeopeMe. B CHMBOJIHLJecKoH 3aIIHCH TeOpe-
MbI K pa3'bHCHHTMbHOM 'laCTH TeopeMbi CJIe,lLYeT OTHecTH sanHC1» V x E P.
3aMeTHM, 'ITO B CJIOBecHOii: <PoPMYJIHpOBKe TeopeM pa3'bHCHHTMb-
HaS! qaCTb oObI'IHO He ct>OPMYJIHPYeTCS! HBHO, HO OHa BcerAa IIOpa3Y-
MeBaeTCS!, H np" pa60Te c TeOpeMaMH ee He06xOJI.HMO BbI)I,eJ1H.Tb.
O'IeBH.lI.HO, 'ITO CTpoeHHe JIIOOOH TeopeMbI, CJIOBecHaH q,opMYJIlipOB-
K3 KOTOpOH BKJIIOtJaeT CJIOBa «eCJIH..., TO...», 6y TaKHM 1Ke, KaK Ii
CTpoeHHe paCCMOTpeHHOH TeopeMbI, T. e. ee MOJKHO 3anI:!caTb B BH)I,e:
C'lx E X) (A (xf*B (x», (1)
re X - MH01KecTBO, Ha KOTOpOM 3a,lLaHbI npe,lLHKaTbi A (x) Ii B (x).
TeopeMbI, He CO.ll.epJK3IlI.He CJIOB «eCJIH..., TO...», TalOKe MorYT OblTb
sanHcaHbI B BH)I,e (1). HanpHMep, paccMOTpHM TeopeMY «B pOMoe .lI.H-
arOHaJIH B3aHMHO nepneH.IUlKYJIS!pHbI». B HeH YTBep)l{)l,aeTCH, qTO ecJIH
cpe.int: pa3JIHqHblX 'leTbIpexyrOJIbHHKOB BbI6paTb moOoH pow, TO AHaro-
HaJJI:! 8 HeM OKaJKyTCH B3aHMHO IIepneH,ll.HKYJIHpHblMH. n09TOM)' aHHYIO
Te6peMy MOJKHO $pMyJilipOBaTb TaK: «ECJIH 'leTbIpexyroJIbmrK-
p0M6, TO ero .lI.HafOHaJIH ssaHMHO rrepneH.nHK}'JIS!pHbD. 0603HatJHB MHO-
>KecTBO Bcex LJeTbIpexyrOJlbHHKOB Ha nJIOCKOCTH OYKBOH X, a moOoH tIe-
mpexyrOJIbHHK H3 9TOro MHOJKecTBa OYKBOH x, nOJIyqaeM 3anHCb Teope-
MY, 0 KOTOpOH H)l,eT pe1Jb:
C'I x E X) (A (x)==>B (x»,
6'

r)l.e A (x) - npe)l.HKaT «4eTbipexyrOJIbHHK x - pOMO», a B (x) - npe-
,lLHKaT «.lJ.HarOHaJIH qeTbIpexyrOJIbHHKa x B3aHMHO nepneH,lLHKYJIHpHID,
3a,lLaHHbIe Ha MHO)l{ecTBe X.
nYCTb (V x  X) (A (x)=>-B (x» - 3annCb HCTHHHOH TeOpeMbI. Tor-
,lLa ee YCJIOBHe H 3aKJIIO'IeHHe oopa3YIOT HMn.nHKaIlHIO, HCTHHH)'10 npH
Bcex x H3 MHOJKeCTBa X, H, CJIe,lLOBaTeJIbHO, npe)l.HKaT B (x) .norH'IeCKH
CJIe,lLYeT H3 npe)l.HKaTa A (x). n09TOMY 3aKJIIO'IeHHe TeopeMbI B (x) HB-
JIHeTC}l 1ie06xoouMbtM YCA08ueM )l.JIH YCJIOBHH A (x) TeopeMbI, a YCJIOBHe
A (x) - iJocmamO'iHbtM YCA08ueM .MH 3aKJIlO'IeHHH TeopeMbI B (x).
nOJIb3YHCb 9TOH TepMHHOJIorueH, TeOpeMY «.lJ.HaroHa.nH pOMoa B3a-
HMHO nepneH,lLHKYJIHpHbI» MOJKHO npOLJHTaTb CJIe)l.YIOIlI.HM 00pa30M:
1. .lJ.JIH Toro 'IT06bI 'IeTblpexyrOJIbHHK ObiJI pOMOOM, He06xo,lLHMo,
'IT06bI ero )l.HarOHaJIH ObiJIH B3aHMHO nepneH,lLHKy.nHpHbI.
2. .lJ.JIH Toro 'ITOObI )l.HarOHaJIH 'IeTbIpexyrOJIbHHKa ObIJIH B3aHMHO
nepneH,lLHKYJIHpHbI, )l.OCTaTO'IHO, qTOObI OH ObiJI pOMOOM.
I1HOr,lLa BMecTO CJIOB «He06xo,lLHMoe yCJIOBHe», «)l.OCTaTOqHOe yCJIo-
BHe» ynOTpeo.nHIOTcH CJIOBa «HeoOXO,lLHMblH npH3HaK», «,lLOCTaTO'IHbIH
"pH3HaK». TaK, TeopeMa «EC.nH B 'IeTbIpexyrOJIbHHKe npOTHBOnOJIOJKHble
CTOpOHbI nQnapHo KOHrpY9HTHbI, TO 9TOT qeTbIpexyrOJIbHHK - napaJI-
JIeJIorpaMM» HB.nHeTCH lI.OCTaTO'IHbIM npH3HaKOM napaJIJIeJIOrpaMMa.
YnpalKHeHBJI
J. BbI)(eJIHTe B CJIe)(YIOlIJ,HX reopeMax pa3'bHCHHTeJIbHYIO '1acTb, YCJIOBHC H 3a-
KJIIO'IeHHe. 3anHlIlHTe 9TH TeopCMbl, HCnOJIb3YH CHMBOJIbl:
8) ECJIH TO'IKa JICJKHT Ha CepC)(HHHOM nepnCH)(HKYJIHpC K OTPC3KY, TO OHa paB-
HOY)(aJICHa OT cro KOHUOB.
6) ECJIH H8TypaJIbHOe '1HCJIO OKaH'IHBaereH )(BYMH HYJIHMH, TO OHO KpaTHO 4.
B) B paBH06e)(pCHHOM TpeyrOJIbHHKe yrJl&l npH OCHOBaHHH KOHrpY9HTHI.I.
r) ECJIH )(Be npHMblC napaJIJICJII.Hbl TpCTI.CH, TO OHH napaJIJICJII.Hbl MCJK.!!Y co6oH.
)() TIpOTHB 60JII.lIlCH CTOpOHI.I B TpcyroJlI.HHKe JIeJKHT 6oJII.lIlHH yroJI.
c) CepC)(HHa )(HarOHaJIH napaJIJIeJIOrpaMMa HBJIHeTCH cro UCHTpOM cHMMeTpHH.
2. Qfx>PMYJIHpyiiTe reopCMbl a) - c), OCnOJIb3YH TCpMHHbI «Heo6xo)(HMOC YCJIo-
BHC», «CTaTO'IHOC YCJIOBHC».
3. Ha MHOJKCCTBC BCCX TpcyrOJIbHHKOB nnOCKOCTH 3a,D.aHI.I npC,!!HKaTbI A (.x):
«TpcyrOJII.HHK x paBH06c)(pcHHbLib, B (x): «.IJ:Ba BHyrpCHHHX yrJIa TpcyrOJIbHHKa x
KOHrpY9HTHbl MCJK.1\y co6oih, C (x); CTPH BHyrpeHHIlX yrJIa TpcyrOJIbHHKa x KOHrpy-
9HTIIbI MeJK,!{y co60H:t, D (x): «Llae BblCOTbi TpeyroJIbHHKa x KOHrpY9HTHbl MCiKlJ,y co6oii:t.
a) l(aKHe H3 9THX npC)(HKaTOB JIorH'IecKH CJIC,!!)'IQT H3 APyrHx nepe'lHCJIeHBbIX?
COCTaBI.Te H3 COOTBCTCTBYIOlIJ,HX nap npcAHKaTOB TeopeMbI.
6) UPOPM)'JIHPYHTe 9TH reopeMbl, HCItOJII.3YH TepMHHbl ({Heo6XO)(HMOC YCJIOBHC:t,
«AOC'!'8TOQHOe YCJIOBHe».
4. B lIlKOJII.HOM Kypce reoMCTpHH ecTb teop.eMbI, HB3b1o.aellUo!e npH3HaKBMH 118-
paJIJleJIOrpaMMa. BhlHCHHTe, 0 KaKHX npH3H8Kax 8 9THX reopel\lax HAeT pe'lb - )(0-
CTaTO'IHblX HJIH Heo6XO)(HMblX.
2. TeopeMa, 06paTHaH )J.aHHoii. TeopeMa «ECJIH cYMMa I1H<pP HaTY-
paJIbHOrO qMCJIa KpaTHa 9, TO H caMO qHCJIO KpaTHo 911 MOJKeT ObITb 3a-
nHcaHa B TaKOM BH)l.e:
\"f x E N) (A (x)=>-B (x»,
r)l.e 3anHCb V x E N npe)l.CTaBJIHeT pa3'bHCHHTeJIbHYIO tlaCTb TeopeMbl:
TeopeMa cnpaBe.ll..nHBa )l.JIH Bcex HaTypaJIbHblX qHCeJI x; npe,lLHKaT A (x):
63

«CYMMa I1HCPP 'IHCJIa x KpaTHa 9» HB.'IHeTCH
YCJIOBHeM TeopeMbl, a npe,lLHKaT B (x): «4HCJIO
KpaTHo 9» - ee 3aKJIIO'IeHHeM.
nOMeHHeM MeCTar.m YCJIOBHe H 3aKJIIO'Ie-
HHe 9TOH TeopeMbl, OCTaBHB 0e3 H3MeHeHHH
pa3'bHCHHTe.nbHYIO 'IaCTb. nOJIY'IHM HOBYIO
TeopeMY Blf,ll.a ("Ix E N) (B (x)*A (x», CJIO-
BeCHaH cpOPMYJIHpOBKa KOTOpOH TaKOBa: «EcJIH
naTypaJIbHOe 'IHCJIO KpaTHo 9, TO H CYMMa ero
I1Hcpp KpaTHa 9». nOnY'IeHHYIO TeopeMY Ha3bIBalOT o6pamHou oaHHou:
Boo6w.e, ec.nH A (x) H B (x) - ,lLBa npe,ll.HKaTa, 3a,ll.aHHbIe Ha MHO'
}KecTBC X, TO Teopelbl (V x E X) (A (x)*B (x» H ("Ix E X) (B (x)*A (x»
Ha3bIBalOT o6pamHblA/u ,lLpyr ,lLPyry. Pa3'bHCHHTeJIbHaH 'IaCTb Y HHX O,ll.H-
HaKOBa: ("Ix EX).
Teopela «ECJIIf CYMMa I1Hcpp HaTypaJIbHOrO 'IHc.na KpaTHa 9, TO
H carvlO 'IHC.nO KpaTllo 9» H eH oopaTHaH HCTHHHbl. O,lLHaKO TaK 6bIBaeT
He BCer,ll.a. PaCCMOTpHM Ha MHO)\{eCTBe X Bcex 'IeTbIpexyro.nbHHKOB nJIO-
CKOCTH lI.Ba npe,ll.HKaTa A (x): «4eTblpexyrOJIbHHK x - npHMoyrOJIb-
HHK» H B (x): «.lJ.narOHaJIH 'IeTblpexyro.nbHHKa x KOHrpY9HTHbI».
C.oBecHaSl cpOpMy.nHpoBKa TeopeMbI (V x E X) (A (x)*B (x» B 3TOM
cJIY'Iae TaKOBa: «Ec.nH 'IeTbIpexyro.nbHHK x - npm,lOyro.bHHK, TO ero
JJ.HarOHa.nn KOHrpY3HTHbI». 3Ta TeopeMa HCTHHHa. nOCTpOHM TeopeMY,
06paTHYIO ,lLaHHoi'f: «ECJIH ,lLHaroHa.nH 'IeTblpexyrOJIbHHKa KOHrpy-
3HTHbl, TO 3TOT 'IeTblpexyrOJIbHHK - npSlMoyrOJIbHHK». 3T0 BbICKa3bI-
BaHHe JIO)\{HO. (CywecTBYeT 'IeTblpexyro.nbHHK C KOHrpY3HTHbIMII ,lLHarO-
H8JISIMII, He HBJISIIOWHHCH npHMoyrOJIbHHKOM.)
3aMeTHM, 'ITO ecJIH npe,lLHKaTbl A (x): «x - npHMoyro.nbHHK» H
B (x): «.lJ.HarOHaJIH x KOHrpY3HTHbl» paCCMOTpeTb Ha MHO)\{ecTBe P na-
paJIJIe.orpaMMoB n.nOCKOCTH, TO TeopeMbI ("Ix E P) (A (x)*B (x»,
("Ix E P) (B (x)=?-A (x» 0Y,lLYT HCTHHHbI.
nYCTb oDe TeopeMbI ("Ix E X) (A (x)*B (x» H ("Ix EX) (B (x)*A (x»
HCTHHHbl. 3T0 03Ha'IaeT, 'ITO MHO)\{eCTBO HCTHHHOCTH npe,ll.HKaTa A (x)
(MHO)\{ecTBO T I ) HBJIHeTCH nO,lLMHO)\{ecTBOM MHO}KecTBa HCTHHHOCTH npe-
JJ.HKaTa B(x) (MHO}KecTBO T 2 ), H Ha06opoT, T. e. T l c.T 2 H T 2 c.T h a
nOTOMY T I =T 2 (pHC. 24). fOBOpHT, 'ITO Ka}K,lLbIH H3 npe,lLHKaTOB A (x)
H B (x) HBJIHeTCH He06xoouMblM H ooCmamO'iHblM YCA06ueM ,lLJIH ,lLpyroro.
HanpuMep, npe,lLHKaT «CYMMa I1HCPP HaTypaJIbHOrO 'IHCJIa KpaTHa 9»
HBJISleTCH HeOOXO,lLHMbIM H ,lLOCTaTO'IHbIM YCJIOBHeM ,lI..nH npe,lLHKaTa «Ha-
TypaJIbHOe 'IHCJIO KpaTHo 9», H Hao60poT.
ECJIH HCTHHHbI 06e TeopeMbI ("Ix E X) (A (x)=?-B (x» H ("Ix EX)
(B (x)*A (x», TO HX MO)\{HO OO'be,ll.HHHTb B O,lLHY TeopeMY BH,lLa
("IxE X) (A (x)B(x». npHMepOM MO)\{eT CJIY}KHTb CJIe,lLYIOIlI.aH Teope-
Ma: «.lJ..nSl Toro 'ITOObI HaTypaJIbHOe 'IHCJIO OblJIO KpaTHo 9, HeooXo,lLHMO
H ,lLOCT, aTO'IHO 'ITOObI cYMMa ero I1Hcpp obl.na KpaTHa 9».
BMecTo C.'10B «He06xo,lLHMo H ,lLOCTaTO'IHO» 'IaCTO ynOTpe6JIHIOT c.no-
Ba «TOr,ll.a H TO.nbKO TOr,lLa», «Te H TO.bKO Te». HanpHMep, «4eTbIpex-
yrOJIbHHK HBJIHeTCH napaJI.ne,lorpaMMoM Tor,lLa H TOJIbKO TOr,lLa, Korad
ero ,lLHarOHaJIH, nepeceKaHcb, ,lLeJIHTCH nOnOJIaM».
G
x
Puc. 24
64

YopallCsesss
1. B Ka>KOM H3 CJleAYIOIUHX CJlY'IaeB c<jJoPMYJIHpyATe reopeM}' ('IxE X) (A (x)
B (x» H 06paTHYIO eA. YKaJKHTe, B KaKHX CJlY'laHx 06paTHaH reopeMa HCTHHHa.
a) A (x): e4HCJlO x OKaH'IHBaercH HYJIeW, B (x): e4HCJIO x eJIHTCH Ha 5:t, rD.e
xEZ..
6) A (x): eTpeyrOJII.HUK x paBHocropoHHHii:t, B (x): cMeHauhI TpeyroJlbHHlta x
ltoHrpY9HTHhi MeJK,l1y c06oH», xE X, r,l1e X - MHOJKeCTBO TpeyroJIbHHKOB.
B) A (x): ex - pOM6», B (x): e.LI.uarOHaJIH ,l1eJIHT yrJIhi nonOJlaM:t, xE X, rD.e X -
MHOJKeCT80 napaJIJIeJIOrpaMMoB.
2. .LI.JIH KIDK,lloA U3 CJle)(YIOIUHx reopeM c<jJopMY JIHpyATe 06paTHYIO. B CJlY'lHe
ecJIH 06paTHaH reopeMa JIOJKHa, nOKIDKUTe nO'leMY,
a) ECJlH TPH CTOpOHI.I O)(HOrO TpeyrOJII.HHKa KOHrpY9HTHhi TpeM cropoHaM ,l1py.
roro TpeyrOJIbHHKa, TO TaKIlC TpeyrOJII.HUKH KOHrpY9HTHhi.
6) ECJlH yrJIhi CMCJKHbIe, TO cYMMa HX BeJIH'IHH paBHa ]80°.
B) ECJlH KIDK,l10e U3 ,l1BYX CJlaraCMhlX ,l1eJIHTCH Ha ,l1aHHOe '1HCJlO, TO H CYMMa HX
,l1eJIHTCH Ha 9TO '1HCJlO.
r) KBB,D.paT rHnOTeHY3h1 npHMoyrOJII.HOrO TpeyrOJIbHHKa paBeH C}'MMe KBB,D.p8TOB
cro KaTeTOB.
3. Ha MHOJKecTBe X Bcex Tpaneui'l 3B,D.aHbl npe,l1HKaThI:
A (x): «TpaneHH x paBH06e,l1pCHHaH».
B (x): «.LI.uarOHaJIU TpanCUH x KOHrpY9HTHbl:t. nOKaJKHTe, '1TO Ka>K)(hlA H3 npe-
,l1UKaTOB HBJIHeTCH lIeo6XOlJ,HMhlM U llOCTaTO'lHbIM YCJIOBueM ,l1JIH ,l1pyroro. CcpoPMY.
JIHpyiiTc TeopeMY
(\fxE X) (A (x)B (x».
3. TcopcMa, npOTltBOnOJlO1KHaH ,!I,aHHoit ECJIH B TeopeMe
('<Ix E X) (A (x>=*B (x»
(1)
YCJIOBHe H 3aKJIIO'IeHHe 3aMeHHTb HX OTpHIIaHHHMH, TO nOJIY'IHTCSI
HOBaH TeOpeMa BH,lLa
(Vx EX) (A (x) * B (x» .
(2)
3Ty TeopeMY Ha3bIBalOT npomuBon(JAoJ/CHoU llaHHOH TeopeMe (1). nYCTb
HanpHMep, Ha MHO)\{eCTBe N Bcex HaTypa.nbHbIx 'IHCM 3allaHbI npe,lLHKa-
Tbi A (x): «1I.ecHTH'IHaH 3anHCb 'IIICJIa X OKaH'IIlBaeTCH HYJIeM» H B (x):
«4HCJIO x ,lLeJIHTCH Ha 5». Tor,lLa CJIOBecHaH <j:JOPMYJIHpOBKa TeopeMll
BHlla (1) np03BY'IHT TaK: «EcJIH ,lLeCHTll'IHaH 3anHCb HaTypa.nbHoro 'IHC-
JIa OKaH'IHBaeTCH HYJIeM, TO OHO ,lLMHTCH Ha 5». TeopeMa, npOTHBono-
JIO)\{HaH 3TOH TeopeMe, 6Y,lLeT CJIe,lLYlOlllei'I: «EcJIH ,lLecHTH'IHaH 3anHCb Ha-
TypaJIbHOrO 'IHCJIa He OKaH'IHBaeTCH HYJIeM, TO OHO He lleJIHTCH Ha 5».
B llaHHOM npHMepe TeopeMa BH,lLa (1) HCTHHHa, a npOTHBOnOJIO)\{HaH
eH TeopeMa .nO)\{Ha (.nerKo Ha3BaTb HaTypa.nbHoe 'IHCJIO, 3anIlCb KOToporo
He OKaH'IHBaeTCH HYJIeM, HO KOTopoe ,lLMHTCH Ha 5, HanpHMep 5,15, 25
H T. ll.). CJIY'IaeTCH, 'ITO H ,lLaHHaH TeopeMa, H eH npOTHBOnOJIO)\{HaH
HCTHHHbI.
TeopeMY BH,lLa
(Vx EX) (B (x>=*A (x»
(3)
Ha3bIBalOT TeopeIOH, npomU8onoAoJ/CHoU o6pamHou. TaK, TeopeMa «ECJI"
HaTypa.nbHoe 'IHC.no He lle.nHTCH Ha 5, TO ero lleCHTH'IHaH 3anHCb He
OKaH'IHBaeTCH Hy.neM» npOTHBOnOJI01KHa TeopeMe «ECJI" HaTypaJIbHoe
3 ;" 702
65

'mCJIO )l,eJIHTCSl Ha 5, TO ero .neCHTH'IHaH 3anHCb OKaH'IHBaeTCH HYJleM:»,
KOTopaH, B CBOIO OIJep.e.ll.b, 06paTHa TeopeMe «ECJIH ,D.ec.HTHtJHaH 3al1HCb
'1HCJ1a OKaHLJHB8eTCSI HYJIeM, TO OHO ,lLeJIHTCfl Ha 511.
3aMeTHM, tlTO TeopeMhI BO)l.a (1) H (3) p8BHOCHJIbHbI, T. e. TeopeMa
(1) HCTHHHa B TOM H TOJIbKO B TOM CJIYtJae, KOr,lLa HCTHHHa H TeopeMa
(3). 3TOT <PaKT JIHT B OCHOBe ,lLOKa3aTeJIbCTBa MemoooM 1WHmpan03U-
UU, CYTb KOToporo 3aKJIIOtJaeTCH B CJIe)l.y«>IlI.eM: tJT06bl )l.OKa3aTb HC-
THHHOCTb TeopeMbl (Vx E X) (A (x,>*B (x», ,lLOKa3blBaIOT OCTHHHOCTb
TeopeMbl, npOTHBOnOJIO)l{HOH 06paTHoH, T. e. TeopeMbl C'fx E X) (B (x,>*
* A (x» , IlOCKOJIbKY ecJIH reopeMa (3) HCTHHHa, TO H TeopeMa (I) TOlKe
HCTHHHa.
.lJ.oKaJKeM MeTO,lLOM KOHTpan03HIIHH TeopeMY «ECJIH ,ABe npHMble
napaJIJIeJIbHbl TpeTbeH, TO OHH napaJIJIeJIbHbI Me)K)l.Y co6oH». Yc.noBHeM
JJ.3HHOH reopeMbi HBJIHeTCfl npe)J.HKaT «nP"Mble X H Y napaJIJIeJIbHbI npSl.-
MOH C», a 3aKJIIOtJeHHeM - npe,lLHKaT «npflMble X H Y napaJIJIe.nbHbI».
npe.lLnOJIOlKHM, 'ITO 3aKJIIOtJeHHe TeopeMhl JIO)l{HO, T. e. 'ITO npHMbIe x
H Y He napaJIJIeJII.Hbl H nepeceKalOTCH, HanpHMep, B TOtJKe M. Ho TOr,lLa
no 3KCHOMe napaJJJIeJIbHOCTH npHMbIe x H y He MorYT OKa3aTbCH
O,lLHOBpeMeHHO napaJIJleJIbHbIMH npHMOH C. TaKHM 06pa30M, H3 TOro,
'1TO npHMaH x He napaJIJIeJIbHa npHMOH y, CJIe,lLYeT, 'ITO npHMbIe x H y
He napaJIJIeJIbHbI O.ll.HOBpeMeHHO npHMOH C, T. e. HCTHHHa TeopeMa, npo-
THBOnOJIOJKHaH 06paTHoiL CJIe,lLOBaTeJIbHO, H .u.aHHaH TeopeMa HCTHHHa.
¥npaIKHeB8J1
I. .lI.JIH XaJKAOH Jl3 reopeM ynp3JKHeHH!I 2 (n. 2) c<jJopMYJIHpyATe eA npoTHBono-
.IIO)I<HYIO, a 3aTeM o6paTHYlO npoTHBOnOJIo)l<HoA. BblHCHtlTC. HCTHHHbI OHH HJIH
JIOJKHbI.
2. .lI.oKaJKHTe, liTO TeopCMbl ('9'X E X) ( A (x)::::;> B (x» H ('9' x E X) ( B (x) ::::)A (x»
paBHOCRlII.HbI.
3. KaKHe H3 CJIeAYIOIII,HX reopCM HCTHHHbI? K8KHe H3 HHX HBJlHIOTCH no OTHO-
UleHHIO .D.pyr K Apyry 06pBTHblMH. npcTHBOnOJlOJKHbIMH, npOTHBOnOJlmKHblMH 06paT-
HOiI?
a) ECJlH KaJKAQe H3 CJlaraeMbiX ,aeJlHTC!I H8 7, ,10 H cYMMa AeJIHTCR Ha 7.
6) EcnH HH O)(HO H3 CJlaraeMbiX He ,/I,eJIHTCH Ha 7, TO H CYMMa He AeJIHTCH Ha 7.
B) ECJIH CYMMa AeJIHTCH Ha 7, TO H KaJK,ltoe CJlaraeMOe ACJIHTCH Ha 7.
r) ECJlH CYMMa He AeJlHTC!I Ha 7, TO H KaJKJI()e CJlaraeMoe He AeJlHTCH Ha 7.
.II.) ECJlH XOTH ISbi OAHO CJlaracMoe Jl,eJIHTCH Ha 7, TO H cYMMa AihcH Ha 7.
e) ECJIH C)'MMa He Jl.eJIHTCH Ha 7, TO XOTR 6w OJl.HO CJlaraeMOe H AeJIHTCH Ha 7.
4. ueAYlOID,He reopeMbi Jl.OKaJKHTe MeTOJI,OM KOHTpan03HIJ,HH.
a) ECJlH npHMaH nepeceKaeT OAHY H3 napaJIJIeJlbHblX npHMblx, TO OHa nepeceKae-r
B ..pyrylO.
«S) Bo BCJIKOM TpeyroJIbHHKe npOTHB 6oJlbWero yrJla JleJKHT 60JlbWaH CTOpcO.

r JI 1\ B a IV
COOTBETCTBIUI, OTBOmEHHH,
OTOBP AmEBIIH
i 1. COOTBETCTBDH H OIIEP ADD HAA HDMD
1. 6HHapHwe COOTBeTCTBIfSI. Teopll51 MHOJKeCTB H3Y'IaeT cBOHCTBa
MHOJKecTB H onepaIlHH H3.l{ HHMH, OTBJleKa51Cb OT npHpoJJ,bl 9J1eMeHTOB
3'fHX MHO)l{ecTB H OT crroco6a HX 3HHSI; .lI.Ba MHO)l{.ecTBa, 33.l{aHHble
pa3JIHIIHblMH xapaKTepHcTHqecKHMH cBOHcTBaMH, CIIHTaJOTCfl paBHblMH,
ecJIH OHH COCTOHT H3 O.ll.HHX H Tex lKe 9JIeMeHTOB. 4T06b1 MOJKHO ObIJIO
npH.naraTb reopHio MHOlKecTB K pellIeHlllO npaKTH'IeCKHX 3a.ll.all, a TaK-
JKe )I)IH nOCTpOeHHH MaTeMaTlltlecKtlX TeopHH, Ha.ll.O paCCMaTpllBaTb
MHolKecTBa, MeJK.lI.Y 9JIeMeH'l'aMH KOTOpblX onpe,lLeJIeHbI Te HJlH HHbIe OT-
HOilleHHH. He .lI.aBafl noKa TOIIHIdX OIlpe.lI.eJIeHHH, npHBe,lLeM npHMephl.
B MHOJKecTBe ocpHII.epOB .lI.aHHOrO nOJIKa .lI.JISI HeKoTopblX nap 3J1e-
MeRTOB (a; b) cnpaBe.ll.JIHBO YTBepJoK,lLeHHe «Oq>HII.ep a C.nYJKHT B O.ll.HOH
paTe C ocpllIlepoM b», )I)IH .lI.pyrHx nap CnpaBe,LL.nllBo YTBep)!{)l,eHlle «OcpH-
I1ep a cTapllie no 3BaHHIO a<PuIlepa b», .lI..n51 TpeTbliX - YTBep)K,1l,eHHe
«oqm.II.ep a HMeeT TO JKe 3BaBBe, 'ITa H ocpHIJ,ep b». KaJK)J,oe ll3 HX
YTBepJK,lLeHHH 3a.ll.aeT HeKOTopoe OTHOllIeHHe MeJK,lLY ocpHIIepaMH a H b
(coBMeCTHoH CJIYJKObI. CTapllIHHCTBa, paBeHCTBa 3BaHllll). B 9THX npH-
Mepax pe'Ib llIJIa 00 OTHOllIeHHHX Me}K)l,Y 3JIeMeHTaMll O.ll.HOrO II Toro JKe
MHOJKecTBa. MOJKHO rOBopHTb H 0 COOTBeTCTBHH){ MeJK.lI.Y 9JIeMeHTaMH
pa3JIllQHbIX MHOJKeCTB - HanpllMep, YTBep)\{,lLeHHe «ocpllIlep a C.nYJKHT
B pOTe b» 3a.ll.aeT COOTBeTCTBHe Me:>K.lI.Y MHOJKecTBOM ocpllIlepoB H MHOlKe-
CTBOM pOT.
.lJ.a.ll.HM o6uJ.ee onpe.ll.eJIeHHe 6UHapHOCO coomeemcmeufl Me}K)l,Y 3J1e-
MeHTaMH MHOJKeCTB X H Y (CJIOBO «OllHapHbIll» npOHCXO.ll.HT OT JIaTHH-
CKoro CJIOBa bis, 03Ha'IalOIlI.ero «.lI.BalK,lLbI», H nOKa3bIBaeT, 'ITO peQb H.lI.eT
o .lI.BYX MHOJKecTBax X H V). npllBe.ll.eHHbIe BbIllIe npllMepbI nOKa3bIBalOT,
liTO KaJK.lI.Oe TaKoe COOTBeTCTBHe 3a.ll.aeTCH HeKOTopbIM .lI.BYXMecT'HbIM
npe.ll.llKaTOM R(x, y), B KOTOpOM x npOoeraeT MHO)I{eCTBO X, a y - MHO-
)l{ecTBO Y (HanpHMep, npe.ll.HKaTOM «OcpHIIep X C.nY)l{HT B pOTe y» , r,lLe
x npllHa.ll.JIeJKHT MHOJKecTBY X ocpHIIepOB, a y - MHO)l{eCTBY Y pOT).
ECJIH MHOJKeCTBa X Ii Y COBna.ll.aJOT, X = Y, TO rOBopHT He 0 COOTBeT-
CTBHH, a 00 OmJiOULeHUU Me3JCiJy aAeMeHmaMU MHOJ/Cecmea X. npll 3TOM
MOJKeT C.nYQHTbC51, liTO ,lLBa pa3JIH'IHbIX npe,LLHKaTa R (x, y) H S (x, y),
T8KHe, 'ITO B o60HX x npo6eraeT MHOJKecTBO X, a y - MHOJKeCTBO Y,
HMelOT O.ll.HHaKOBoe MHOJKecTBO HCTHHHOCTH T. HanpHMep, npe,lLHKaThI
«4HCJIO x He npeBOCXO.ll.HT IIHCJIa y» H «4HCJIO x HBJlHeTCH ,lLeJIHTeJIeM
)*
61

tJHCJI8 y» P83JIHtlHbI. Ho eCJIH X={2; 4; 75}, 8 Y={l; 4; 8; 16}, TO
MHO)\{ecTBa HCTHHHOCTH 3THX npe.ll.HKaTOB O.ll.HHaKOBbI: OHH COCTOHT H3
nap (2; 4), (2; 8), (2; 16), (4; 4), (4; 8), (4; 16). B TaKHX CJIYtlaHX MbI
0Y.lI.eM CtJHTaTb, tlTO npe.ll.HKaTbI R (x, y) H S (x, y) 3a.ll.alOT O.ll.HO H TO lKe
COOTBeTCTBHe Mem.ll.Y MHOlKecTB8MH X H Y.
TaKHM 00pa30M, tlT06bI 3a.ll.aTb COOTBeTCTBHe R MelK,lLY MHO)KeCT-
BaMH X H Y, .lI.OCT8TOLJHO YK83aTb nO.ll.MHolKecTBO r .lI.eKapTOBa npOH3-
Be.ll.eHHH X X Y. I1HbIMH CJIOBaMH, COOTBeTCTBHe R - TpOllKa MHO)KecTB
(X, Y, r), r.ll.e rc::x X Y. ECJIH a H b - 3JIeMeHTbI MHOlKeCTB X H
Y, Ha KOTOpbIX 3a,lLaH npe.ll.HK8T R (x, y), TO 3anHCb aRb 03HaLJaeT TO
)Ke, 'ITO H BblCKa3bIBaHHe R (a, b). OHa HBJIHeTCH 0606Il1.eHHeM 3anHCH
MHorHX COOTBeTCTBHll H oTHollIeHHll, ynoTpe6JIHeMbIX B MaTeMaTHKe.
CpaBHHTe, HanpHMep, 3anHCb aRb C 3anHCblO OTHOllIeHHH paBeHCTBa H
HepaBeHCTBa ,lLellCTBHTeJIbHbIX tlHCeJI: a=b, a>b, a<b; HJIH C 3anHCblO
OTHOllIeHHH napa.nJIe.nbHocTH H nepneH,lLHKYJIHpHOCTH npHMblX: allb, a1-b.
<l>opMHpoBaHHe nOLJTH Bcex nOHHTHH: HatlaJIbHOrO Kypca MaTeMaTHKH
conpOBOlK,lLaeTCH paCCMOTpeHHeM Tex COOTBeTCTBHH H oTHollIeHHH, KO-
Topble CYIlI.eCTBYIOT MelK.ll.Y H3ytlaeMblMH 06'beKTaMH. TaK, H3YLJaH no-
HHTHe HaTypaJIbHOrO LJHCJIa, MbI rOBopHM 06 OTHOllIeHHHX «60JIbllIe»,
«MeHbllIe», «paBHo»; H3YLJaH oTpe3KH, paccMaTpHBaeM OTHOllIeHHH «.lI.JIHH-
Hee», «KopOtle», «paBHo». Co MHorHMH COOTBeTCTBHHMH H OTHOllIeHHHMH
YtlaIll.HecH HaLJaJIbHblX KJIaCCOB BCTpetlalOTCH npH pellIeHHH 3a.ll.atl.
91'0 H OTHOllIeHHH «cTapllIe», «MOJIOlKe», «CTapllie B... pa3», «HHlKe»,
«BbIllIe» Ha MHolKecTBe JIIO.lI.eH, H COOTBeTCTBHH «,lLOpOlKe, LJeM», «,lLellIeBJIe
 ... pa3», «.lI.OpOlKe Ha ., .», onepHpOBaTb KOTOpbIMH npHXO,lLHTCH npH
cpaBHeHHH CTOHMOCTH p83JIHtlHbIX npe,lLMeTOB.
.lJ.J1H OHHapHbIX COOTBeTCTBHH: npHMeHHeTCH CBOH TepMHHOJIOrHH,
OTJIHtJSIOIIIaHCH OT TepMHHOJIOrHH, npHMeHHeMOH: .lI.JIH .lI.ByxMecTHbiX
Ta6nHu,a 1

nOBe.u.eJIb- BTOpBHK Cpe.u.8
BDK
cJ>8MBJlBH .
AJICK CCCB I" III I
BO'lKapCBa I II " II
BaCHJIbCB I I IIIII
fpHropbcB 111111 I
.lI.euHcoB I II " II
E$pCMOB I I "" IJ
68

Anel<cee8 ·
6o'll<apeOa. .
BaCUlTbeIJ .
rpuzopbefJ .
AeHucotJ
EtppeMotJ
nOHe8enbHUI<
BmopHUK
Cpeaa
PHC. 25
x
v
PRC. 26
npe.ll.HKaTOB. HanpHMep, MHO}KeCTBO X Ha3blBaIOT 06AacmblO Omnpa8M-
HUfl coom8emcmeUfl R, MHO}KeCTBO Y - 06AacmblO npu6blmufl, a MHO-
}KeCTBO r - cpacjJUKOM coomeemcmeUfl. TaKHM 00pa30M, rpacpHK r
COOTBeTCTBHH R - 9TO cOBoKynHocTb Bcex nap (x; y), r.ll.e x EX, y E Y,
,lLJUI KOTOpbIX HCTHHHO xRy. HanpHMep, ecJIH R - COOTBeTCTBHe
«x ,lLeJIHTCH Ha y» H X={IO; 20; 30; 40}, Y={2; 3; 4}, TO r COCTOHT
H3 nap (10; 2), (20; 2), (20; 4), (30; 2), (30; 3), {40; 2), (40; 4).
ECJIH MHO}KeCTBa X H Y KOHeLJHbI, TO OHHapHbIe COOTBeTCTBHH Me}K.lI.Y
HHMH MO)\{HO 3a.ll.aTb TaOJIHIIaMH. HanpHMep, Tao.nHIIa 1 3a,lLaeT COOTBeT-
CTBHe «IIlKo.nbHHK x ,lLe)\{ypHT B .lI.eHb y». 3,lLeCb X = {A.neKceeB; Boq-
KapeBa; BaCHJIbeB; rpHropbeB; .lJ.eHHcoB; EcppeMoB} H Y = {noHe-
.lI.eJIbHHK; BTOpHHK; cpe,lLa}. 3allITpHxoBaHHbIe KJIeTKH TaOJIHIIbI oopa-
3YIOT rpacpHK paccMaTpHBaeMoro COOTBeTCTBHH.
.lJ.JIH KOHe'IHbIX MHO)\{eCTB COOTBeTCTBHH H300pa)\{a1OT TaK)\{e npH
nOMOIll.H OCOObIX 'IepTe)\{ell, COCTOHIlI.HX H3 TOLJeK H HanpaBJIeHHbIX JIHHHH
(CTpeJIOK), H,lLYIlI.HX H3 O,lLHOll TO'IKH B ,lLPyrylO. TaKHe qepTe)\{H B MaTe-
MaTHKe Ha3bIBalOT opueHmup08aHHbiMU cpacjJaMU (OT rpe'IeCKOrO CJIOBa
«rpacpo» - «nHllIY», cpaBHHTe CJIOBa «TeJIerpacp», «rpacpHK»). 3JIeMeH-
TbI MHO)\{eCTB X H Y o603Ha'IalOT TO'IKaMH, a CTpeJIKH npOBO,lLHT H3
TOLJKH X B TOLJKY y, eCJIH (x; y) E r, T. e. eCJIH xRy. HanpHMep, rpa.
H300pa)\{eHHbIH Ha pHcYHKe 25, 3a,lLaeT TO )\{e COOTBeTCTBHe «IIlKo.nbHHK
x .lI.e}KypHT B ,lLeHb y», 'ITO H TaOJIHIIa 1 (rpacpHK ,lLe)\{ypcTB).
Ha pHcYHKe 26 H300pa}KeH rpacp COOTBeTCTBHH Me}K,lLY MHO}KeCTBa-
MH X H Y. B03bMeM KaKYIO-HHOY,lLb TO'IKY MHO)\{eCTBa X, HanpHMep
TO'IKY a, H Bce cTpe.nKH, BbIXO,lLHIlI.He H3 9TOll TO'IKH. MHO}KeCTBO KOH-
II.OB 9THX CTpeJIOK Ha3bIBalOT 06pa30M aAeMeH-
ma a npH COOTBeTCTBHH R H 0003HaqalOT R(a).
3TOT o6pa3 COCTOHT H3 9.neMeHTOB t, x, y
MHO}KeCTBa Y, R(a)={t;x;y}. B 06Il1.eM
CJIYLJae oopa3 9JIeMeHTa a Onpe,lLeJIHeTCH KaK
MHO}KeCTBO Bcex 9.neMeHTOB y E Y, ,lL.nH KO-
TOpbIX HCTHHHO, 'ITO aRy. HanpHMep, npH
COOTBeTCTBHH «npHMaH X KacaeTCH OKPY)\{HO-
CTH y» oopa30M ,lLaHHOll npHMoll a HBJIHeTCH
MHO}KeCTBO Bcex OKpY)\{HOCTell, KaCalOIlI.HXCH
3TOH npHMoll (pHC. 27). PRC. 27
61

B03bMeM Tenepb K3KOA-HII6Y}I,b 9JleMeJIT
83 MXO.>KecTBa Y H3 pHcYHKe 26, HanpHMep
t, H Bce CTpeJIKH, KOTopwe OKaHIIHBalOTCH IJ
9TOM 3JIeMeHTe. MHO.>KecTBO HallaJI 9THX CTpe-
JIOK H8SbIB81OT nOAHblM npo06pa30 8AeMeHma
t npH COOTBeTCTBHH R. 3TOT nOJIlibIA npoo6-
pa3 0003HallaJQT R-I (t). OH COCTOHT H33.lle-
MeHTOB a, c, d MHO)\{ecTBa X, R-I (t)=
= {a; c; d}. B 06Il1.eM BH,lLe nOJIHbIH npoo6paa
P. 28 9JIeMeHTa bEY onpe,lLeJIHIOT KaK MHO)\{ecTBO
TaKHX 9JIeMeHTOB x EX, 'ITO HCTHHHO xRb.
HanpHMep, npH COOTBeTCTBHH «npHMaH X KacaeTCH OKpY)\{HOCTH y»
I10JlHbrit np006pa3 OKPY)KHOCTH y COCTOHT H3 Bcex npHMbIX, KacaIO-
HXC" 3mB OKpYJKHOCTH (pHC., 28).
MO>KeT CJIYIIHTbClI, liTO H3 }l,aHHOH TO'IKH a E X He BbIXO}l,HT HH O}l,Ha
ape.1lK8. Tor}l,a 06pa3 JIeMeHTa a "YCT, R (a) = 0. COBoKynHocTI> A
Bcex 3JIeMeHTOB H3 X, HMelOUJ.HX HenYCTbIe 06pa3'bf (T. e. TaKlIX, liTO
)13 H)lX BbIXO}l,HT XOTb O}l,Ha CTpe.n:Ka), Ha3h1BalOT .MHO;m;eCtn60M onpeiJeAe-
HUfl COOf1'l1JefnCm8UJl R. MHmKecTBO B BCex 3JIeMeHTOB H3 Y, HMeIOI:IJ,HX
nenYCTOH nOJIHbIA npo06pa3 (T. e. T3KHX, 'lTO B HHX KOHIIaeTCH XOTb O}l,-
Ha CTpe.1IKa), Ha3bIBaIOT MHo;m;eemeoM 3Hatte1iUU coomeemcmeUfl R. .D.JIfl
COOTBeTCTBHH, rpa.p KOToporo H306pa)KeH H3 pHCYHKe 26, MHOJKecTBO
onpe}l,eJIeHHH COCTOHT H3 9JIeMeHTOB a, b, c, d, 3 MHO>KeCTBO 3HalleHH" -
H3 3JIeMeHTOB x, y, z, t.
YnpaacBeB..
l._.J].iHW MOJKecTBa ,,= {2; 4; 6} / B= {5; 7; 9; ]]}. MeJK,D,Y HHMH YCTaHOBJIeHO
oooTBeItlJ"BHe «thlCJlO a OoJII.lIle '1Hc.n b», aEA, b EB. DOCTpoiiTe rpaq, H rpaq,HK
9Ti:>ro ((I()TBeTCTBHH.
2. Ha pHCYHKe 29 H306paJKeH rpaq, COOTBeTCTBHH MCJK)(y MHOJK!:cTBaMH X H Y.
a) KSII:OBW 06JIacTI. OrnpaBJIeHHH H o6JIaCTI. npH6blTHH 3TOro cooTBeTcTBHH1
6) J(al!0Bw MHOJKeCTBO onpeAeJIeHHH H MHOJKecTIIO 3Ha'leHHH 3TorocOOTBCTCTBH1I?
a) HalWITe rpaqtHK TOro COOTBeTCTBHH.
3. rfYCTI. X={-I; -2; -3; 1; 2; 3; O} H Y - MHOJKCCTBO Bex HaTypaJII.HblX
'lHceJI. KaJK.AoMy 'lHCJlY xE X CTaBHTCH.B COOTBeTCTBHe ero KBa.npaT. BblnH111HTe Bee
napl>I  n h8Jl.JleJKBI1-'He rpaq,HKY wroro COOTBeTCTBHH. DocrpoiiTe ero rpaq,. KaKoBO
MHO' 3H8'1epHH !!TOre COOTBeTmHH?
4. YCTI. X={«MSMa»; «nana»; «PIIM8»; «SlMI»}, Y {a; M; p; n; $I}. CoCTaBI.Te
;l.eKapTOBO npOH3BeACHHe X X Y. OrMeTI.Te B HeM napw, CBH3aHHblt COOTBCTCTBHeM
«B CJlOBO x BXOAHT 6YKBa y». 3aAaliTl ire JKe cooi-BeT-
X Y crBHe rpa4JoM.
5. ,[\JIH MHOJKeCTB X H Y H3 npHMepa 4 nocTpot-
Te rpaq,HK Ii rpaq, COOTBeTCTBHH «B CJlOBO x HC BXOAHT
6YKBa y».
Pac. . Ilt
"
2. HeKoTopwe TRnbi COOTBeTCTBHH. One-
pau,HH H COOTBeTCTBHJlMH. ECJiH rpa.pHK F
COOTBeTCTBHH R MelK}I,Y MHOJKecTBaMH X H Y
COBn3.l{aeT CO BceM ,lJ.eKapToBbIM npOH3Be}l,eHH-
eM X X Y, TO 9'1'0 COOTBeTCTBHe Ha3blBaIOT ntJA-
H.blM. ECJIH JKe rpaq,nK COOTBeTcnHH R "YeT

(r= 0), TO R H83h1B81OT ttyCltlhlM COORt1lmtCftl8UeM. HanpH, COO'I'-
BeTCTBHe «X nWfaeTCR TeM >Ke, 'IeM y» nYCTO, ec.nH X - MHO>K 3a-
U,eB, a Y - MHOJKeCTBO THrpoD..
nOJIHOe H nycroe COOTBeTCTBHH cYIlI.ecnylOT Memy JllOOblMH).BY-
MH MHO)\{OCTBaMH X H Y. ECJIH 3a,lLaHbI HenYCTble MHO>KecTsa X H Y,
TO Me)\{y HHMH MOlKHO YCTaHOBHTb U,eJIbIH pH HHbIX COOTBeTCTBHH,
BblOHpaH no-pa3HOMY nO,lLMHO}KOCTBa r B ,lLeKapTOBOM npOH3Be,lLeHHH
X X Y. Ha,lL nO,lLMHOlKeCTBaMH ,lLeKapTOBa npOH3Be,lLeHHH MO)\{HO BnOJI-
HHTb pa3JIHlfHbIe onepaIlHH: 06'be,lLHHHTb HX, nepeceKaTb H T. ,lL. Ka)\{)I,OH
H3 9THX onepaIlHH OTBeLJaeT onepaIlHH Ha COOTBeTCTBHHMH.
HanpHMep, eCJm MelK,lLY X H Y 3aaHbI COOTBeTCTBHH xPy H xQy,
TO HX nepeCe'ieHUeM R=P f) Q H<l3bIBaIOT COOTBeTCTBHe xRy, rpacpHK
KOTOpOro HB.nHeTCH nepecetleHHeM rpa<j:JHKOB 3THX COOTBeTCTBHH. HHbI-
MH CJIOBaMH, xRy B TOM H TOJIbKO B TOM c.nY'Iae, KOr,lLa xPy H xQy.
ECJIH COOTBeTCTBHe xPy 3a,n:3HO npe,lLHKaTOM P (x, y), COOTBeTcrsHe
xQy - npe,lLHKaTOM Q (x,y), TO xRy 3a,lLaeTCH npe,lLHKaTOM R (x, 11)=
=P (x, y) /\ Q (x, y).
06'beOUHeHUeM S=P U Q COOTBeTCTBHH xPy H xQy Ha3bIBalOT COOT-
BeT(:TBHe xSy, rpa<j:JHK KOTOpOro HBJlReTCH 06'b'e,lLHHeHHeM rpa$HKOB CO-
OTBeTCTBHH xPy H xQy. HHbIMH CJJ!)BaMH, xSy B TOM H TOJIbKO BTOM CJIY-
llae, KOr,lLa xPy HJIH xQy. EcJ1H COOTBeTCTBHe xPy 3a,lL<lHO npe,lLHKaTOM
P (x, y), a xQy- npe,lLHK3TOM Q(x, y), TO xSy 3aaeTCH npe,lLHK3TOM
S (x, y)=P (x, y) V Q (x, y). HanpHMep, COOTBeTCTBHe x4=y ,lL.nH LJHCeJI
HB.nHeTCH OO'be,lLHHeHHeM COOTBeTCTBHH x<y H x>y - ec.nH x4=y, TO
H.nH x<y, H.nH x>y. OopaTHO, eCJIH x<y HJIH x>y, TO x4=y. A COOT-
BeTCTBHe x<y .nH QHCeJI - nepeCe'IeHHe COOTBeTCTBHH xy H x =l=Y-
ec.nH xy H x4=y, TO x<y, a ec.nH x<y, TO xy H x4=y.
MO)\{eT C.nY'IHTbCH, LJTO rpaq,HKH cooTBeTcTBHH xPy H xQy - ,AonoJl-
HlfTeJIbHbIe MHOlKeCTBa B X X Y (T. e. OHH He nepeceKalOTCH, a HX OO'be-
}f.HHeHHeM HB.nHeTCH Bce Xx V). TaKHe COOTBeTCTBHH Ha3bIBalOT npOo
mU80nOAOJ/CHbtMU. HanpHMep, COOTBeTCTBHe «l.JHC.nO x 6o.nbllie LJHCJIa yt
npOTHBono.nOlKHO COOTBeTCTBHIO «l.JHCJlO x He npeBOCXO,lLHT 'lHCJIa {p.
Ec.nH COOTBeTCTBHe xPy 3a,lLaHO npe,lLHKaTOM P (x, y), TO npOTHBOnOJIOlK-
Hoe eMY COO TBeTCTBH e 3a,n:3eTCR OTpHlI,aHHeM 9Toro npe,QHKaTa, T.. e.
npe,lLHKaTOM P (x, y).
CooTBeTCTBHH xPy H xQy H83b1BalOT Hec08MeCmUMIJlMU, ecJIH He CYBYeT HR
O)(UOi! napbI 9JIeMeUTOB (x; y), ,II,JIH KOropoH O)(HOBpeMeHHO BbinOJIUHJlHCb 6b1 YCJlOBHJI
xPy H xQy. HanpHMep, )(JIH npHMblX HeCOBMecTHMbI COOTBeTCTBHH xlly H x.lY - )(1Ie
npHMblC fie MorYT 6bITb O)(UOBpeMeuuo H napaJIJIeJII.UblMH, H nepneH)(IIKYJIHpRblMH.
ECJIH rpaq,HK COOTBeTCTBHH xPy HB.nHeTCH nO,lLMHO)\{eCTBOM rpa<j:JHKa
COOTBeTCTBHH xQy, rOBOpHT, tlTO xQy - CAeOcmeue xPy. B 3TOM CJIY-
tlae ,lLJIH .n1060R napbI (x; y), TaKOR, LITO xPy, HMeeM H xQy. HanpliMep,
COOTBeTCTBHe «Tpeyro.nbHHKH X Ii Y nO,lL06HbI» - C.ne,lLCTBHe COOTBeTCT-
BHH «Tpeyro.nbHHKH X H Y KOHrpY9HTHbI» - JII06aH napa KOHrpY9HTHblX
Tpeyro.nbHHKOB nO,lLOoHa.
Ka)\{,lLOMY oHHapHOMY COOTBeTCTBHIO Me)\{y MHO)\{ecTB3MH X H Y
MBeqaeT 6HHapHoe COOTBeTCTBHe Me>KAY Y H X. HanpHl\dep, MOJK-
It

HO CKa3aTb «npHMaH X KacaeTCH OKPY)\{HOCTH y» - 9TO oy,n:eT COOT-
BeTCTBHeM MelK,n:y MHOlKeCTBOM X npHMbIX H MHOlKecTBOM Y OKPYlKHOC-
TeH. Ho eCJIH npHMaH KaCaeTCH OKPYlKHOCTH, TO OKPYJKHOCTb, B CBOIO
Oqepe,n:b, KaCaeTCH npHMOH, H MbI nOJIYLJaeM COOTBeTCTBHe «OKPY)\{-
HOCTb y KacaeTCH npHMOH X», ,n:JIH KOToporo OOJIaCTblO OmpaBJIe-
HHH CJIYlKHT Y, a OOJIaCTblO npHOblTHH X. HHbIMH CJIOBaMH, npH
nepexo,n:e K HOBOMY COOTBeTCTBHIO o6JIacTH OTnpaBJIeHHH H npHObITHH
nOMeHHJIHCb pOJIHMH - OOJIaCTb OTnpaBJIeHHH CTaJIa OOJIaCTblO npl-!-
ObITHH H, Hao60poT, OOJIaCTb npHOblTHH CTaJIa OOJIaCTblO OTnpaBJIeHHH.
B napax (X; y), BXO,n:HIlI.HX B rpacpHK COOTBeTCTBHH, KOMnOHeHTbI TO)\{e
MeHHIOTCH MecTaMH - BMecTO (X; y) Ha,n:o nHcaTb (y; X).
BoooIll.e, nYCTb 3a,n:aHO COOTBeTCTBHe xRy Me)l{,n:y MHO)\{eCTBaMH X
H Y. 06pamHbiM eMY Ha3bIBalOT COOTBeTCTBHe ySx Me)l{,n:y MHO)l{eCTBa-
MH Y H X, TaKoe, 'ITO ySx B TOM H TOJIbKO B TOM CJIY'Iae, Kor,n:a xRy.
OObILJHO BMeCTO ySx nHllIYT yR-lx. HanpHMep, ,n:.nH COOTBeTCTBHH «lIHC-
JIO X ,n:eJIHTCH Ha LJHCJIO y» oopaTHbIM oy,n:eT COOTBeTCTBHe «lIHC.nO y
HBJIHeTCH ,n:eJIHTeJIeM LJHCJIa X» - X ,lLeJIHTCH Ha y B TOM H TO.nbKO B TOM
CJIYLJae, Kor,n:a y - ,n:eJIHTeJIb X.
lITOObI nOJIY'IHTb rpacpHK COOTBeTCTBHH R-1, oopaTHoro COOTBeT-
CTBHIO R, Ha,lLO B3HTb rpacpHK COOTBeTCTBHH R H nOMeHHTb MeCTaMH KOM-
nOHeHTbI B KalK,n:OH nape. A LJTOObI nOJIYLJHTb rpacp COOTBeTCTBHH R-1,
Ha,AO H3MeHHTb Ha oopaTHoe HanpaBJIeHHe Bcex CTpeJIOK B rpacpe COOT-
BeTCTBHH R.
YopamUeUHJI
t. ,l],JIH COOTBeTCTBHH, YKa3aUUblX B npHMcpax 1, 2 H 4 n.l, nocTpoHTC 06paTHblc,
npoTHBOnOJIOJKUblC H 06paruble npoTHBOnOJlOJKUblM. HaH)(HTC rpa<jJHKH H rpa<jJbI
9THX COOTBCTCTBHH.
2. ,l],JIH HHJKeCJIc,n;YlOlLI,HX COOTBCTCTBHH afJoPMYJIHpyiiTe 06paTublc, npoTHBOno-
JlOJKHble H 06paTHblc npoTHBOnOJlOJKUblM: a) TO'IKa a JIeJKHT ua npHMoH b; 6) '1HCJIO a
HBJlHCTCH KOpHCM ypaBHcHHH b; B) npHMaH a ncpCCCKaCT OKPYJKUOCTb b; r) npHMaH a
nepeCCKaCT npHMYIO b; ,n;) .9JICMCHT a npHHB)(JICJKHT MUOJKCCTBY M.
3. CoOTBCTCTBHH xRy H xSy MCJK)(Y MHOJKCCTBaMH X= {I; 3; 5; 7} H Y= {2; 4; 6; 8}
aa)(aUbi CBOHMH rpa<jJHKaMH:
R: {(1; 2); (1; 4); (1; 6); (I; 8); (3; 4); (3; 6); (3; 8); (5; 6); (5; 8); (7; 8)},
S:{(I; 2); (3; 2); (3; 4); (5; 4); (5; 6); (7; 6); (7; 8)}.
a) nocTpoiiTC rpa<jJHKH 06'bC)(HUCHHH H nepCCC'IeHHH .9THX COOTBCTCTBHiL
6) nOCTpOHTe rpa<jJbI 06'be)(HHCUHH H ncpece'leHHH .9THX COPTBCTCTBHH.
4. 5IBJIHeTCH JlH COOTBCTCTBHe r=F!f CJIC)(CTBHCM COOTBCTCTBHH X=FY?
5. 5IBJIHCTCH JlH COOTBCTCTBHe x3=Fy3 CJIC)(CTBHCM COOTBCTCTBHH X=FY?
3. OTHOllIeHHSI Ha MHO)KeCTBe. BHHapHbIe COOTBeTCTBHH Me)l{,n:y
X H X Ha3bIBalOT 6UHapHblMU omHomeHUflMU Ha MHOJ/CeCf1l8e X. Ha-
npHMep, ecJIH X - MHO)l{ecTBO JIIO,n:eH, TO COOTBeTCTBHH «lIe-10BeK X -
,n:Pyr 'IeJIOBeKa y», «X lKHBeT B O,lLHOM ,n:OMe C y», «lIeJIOBeK X - OTeIl LJe-
JIOBeKa y» HB.nHIOTCH OTHOllIeHHHMH MeJK,n:Y JIIO,n:bMH. OTHOllIeHHe Ha
MHO)\{eCTBe X 3a,n:aHO, eCJIH YKa3aHO MHO)\{ecTBO r, HBJIHIOIlI.eeCH nO,lL-
MHO)l{eCTBOM ,n:eKapTOB3 npOH3Be,n:eHHH X X X. OTHOllIeHHH, KaK H COOT-
BerCTBHH, npHHHTO 0603HaLJaTb OYKBaMH R, S, T, Q H ,n:p. H nHCaTb
xRy, xSy H T. ,n:. HTaK, OHHapHoe OTHOllIeHHe R Ha MHolKecTBe X -
72

9T0 napa MHO>KetTB (X; r), r,lLe rcXX x. MH()}!{CTBO X Hat,yBalOT
06AacmblO 3aOaHufl omHOUleHUfl R, a MHO)\{ecTl30 r -- cpa:{J..,IWM om-
HOUleHUJl R.
PaCCMOTpHM, HanpHMep, Ha MHOlKeCTBe X :- {I; 2; 3; 4} OTHollIeHHe
«x>y». rpacpHK 9Toro OTHOllIeHHH - MHO)\{ecTBO r= {(2; 1); (3; 1);
(3; 2); (4; 1); (4; 2); (4; 3)}, COCTOHIlI.ee H3 Bcex Tex nap (x; y), x e: X,
yE X, ,lLJIH KOTOpblX x>y.
'lJacTo npH 3a,lLaHHH Toro HJIH HHoro OTHOllIeHHH npH nOMOIll.H npe-
,lLHKaTa ,lLonycKalOT COKpaIll.eHHH. HanpHMep, 00 OTHOllIeHHH «x<!p
Ha MHO)\{eCTBe ,lLeHCTBHTeJIbHbIX LJHCeJI rOBopHT, LJTO 3a,lLaHO OTHollIeHHe
«MeHbllIe», a OTHOllIeHHe «<I>Hrypa F KOHrpY9HTHa cpHrype cD» Ha MHO-
lKecTBe reoMeTpHLJeCKHX cpHryp MO)\{HO 3a,lLaTb C.nOBaMH «OblTb KOHrpy-
9HTHbIMH» H.nH npocTo rOBopHT 00 OTHOllIeHHH KOHrpY9HTHOCTH Ha MHO-
)\{eCTBe reoMeTpHLJeCKHX cpHryp.
B MaTeMaTHKe BCTpeLJalOTCH OTHOllIeHHH, OT.nHLJHbIe OT oHHapHbIx..
I1x 3a,lLalOT npH nOMOIll.H TpexMecTHbIx, LJeTbIpexMecTHbIx If T. ,lL. npe-
,lI.HKaTOB. HanpHMep, npe,lLHKaT «llHc.no Z HB.nHeTCH CYMMOI'f LJHCeJI x H y»,
3a,lLaHHbIH Ha MHo)\{ecTBe N, BbIpa)l{aeT TpexMecTHoe (TepHapHoe) OT-
HOllIeHHe B MHO)\{eCTBe HaTypa.nbHblx LJHCeJI. '
Ha .n1060M MHO)\{ecTBe X paccMaTpHBalOT OTHOllIeHHH TO)\{,lLecTBa H
pa3JIHLJHH. OmHoUleHu£ mOJ/Coecmea x==.y 3a,lLaeTCH MHO)\{ecTBOM r nap
BH,lLa (x; x), x E X.I1HbIMH CJIOBaMH, x = y B TOM H TO.nbKo B TOM c.nYLJae,
Kor,lLa x COBna,lLaeT C y. llacTo BMecTO x = y OTHOllIeHHe TO)\{,lLeCTBa 0603-
HaLJalOT x=y H Ha3bIBalOT omHOUleHueM pa8eHCmea. OTHOllIeHHe, npOTH-
Bono.nOlKHoe OTHOllIeHHIO TOlK,lLeCTBa, Ha3bIBalOT omHomeHueM pa3AU'iUJl
H 0603HaLJalOT x:=I=y H.nH x=l=y. rpacpHK OTHOllIeHHH pa3JIHLJHH COCTOHT
H3 TaKHX nap (x; y), LJTO X H Y pa3.nHLJHbI.
KaK H Ha.n JlI06b1MH COOTBCTCTBHHMH, Ha.n OTHOlIlCHHHMH B MHOJKeCTBC X MOJKHO
BblnOJlHHTb OnCpal.lHH nepCCe'leHHH, 06'bC)(HHeHHH H T. )(. B pe3YJlbTaTe CHOBa 6y1J,YT -
nOJlY'IaTbCH OTHOlIlCHHH B X. OrHolIlcHHe ySx, 06paTHoe OTH0111CHHIO xRy B MHOJKe-
CTBC X, TaKJKC HBJlHeTCH OTUOlIlCHHeM B X. HanpHMep, OTUOlIleHHC y>x 06paTHO
OTHOlIleHHIO x<y B MHOJKeCTBe panHOUaJIbHblX '1HCeJI. OrHolIleHHC «y - Ha'laJII.HHK .n
06paTHO OTHOlIleHHIO «x - nO)('1HHCHHblH y».
KaK H .lI.JIH JlI06b1X COOTBeTCTBHH, OTHOlIlCHHe xRy Ha3b1BaeTCH cAeiJCtn8UeM OT-
HOlIlCHHH xSy, ecJlH rpa<jJHK OTHOlIleHHH S - nO)(MHOJKeCTBO rpa<jJHKa OTH0111CHHH R.
HanpHMCp, B MHOJKeCTBC HaTypaJII.UblX '1HCeJl OTHorueHHe nm - CJle1J,CTBHC OTHO-
lIleHHH «n )(eJIHTCH Ha m» - eCJlH n )(eJIHTCH Ha m, TO HaBepHHKa n'Pm.
YupamueuuJI
I. )].JlH KaJK)l.OrO H3 CJle)(YIOlI1.HX OTHOlIleHHA YKaJKHTe MHOJKCCTBO, napbl 9JJe-
MCHTOB KOTOporo OHO MOJKCT CBH3b1BaTb:
a) «x paBHO 1/»;
6) «x HBJIHCTCH O)(HOKJlaCCHHKOM y»;
B) «x 3HaKOM C y»; r) «x BbilIle, '1eM y».
2. B MHOJKecTBe A= {3; 6; 9; 12} 3a)(aHO OTHOlIleHHe R: «l!HCJlO x )l.eJIHTCH HI
'1RCJlO 1/», npH'IeM xEA H yEA. BblnHlIlHTe Bce napbl '1HCeJI H3 MHOJKeCTBa A, Haxo-
)l.HlI1.HXCH B OTHOlIleHHH R. 0603Ha'lbTe MUOJKeCTBO Bcex TaKHX nap 6YKBOH r. BbI-
HCHHTe, KaKHe H3 CJle)(YIOlI1.Hx BblCKa3b1BaHHH HCTHHHbI:
a) (9; 3)Er; 6) (3; 9)Er; B) (12; 12)Er; r) (9; 9)r.
3. )].aHo MHOJKecTBO B={I; 3; 5; 7; 9}. 3.1eMeHTbI 3TOro MHOJKCCTBa CBH3aHbI OT-
H0111eHHeM S: «'-IHCJlO x Ha 2 60JlblIlC '1HCJla y». 3anHlIlHTe MHOJKeCTBO nap. npHHBA-
73

.!Ie1KlllnXK rPa41HKY OTHOWeHHSI. BwSlCHHTe, KaKHe H3
CJleAYIOlI1.HX BblCKB3W1aHHH HCTHHHbI:
8) 381; 6) 385; B) 7S3; r) 987.
4. BblSlCHHTe, KaKHe H3 CJleAYIOUI,Hx MHO)KecT8 Sl8-
JlSlIOTCR rpaq,HKaMH OTHOUIeHHH 8 MHO)KecTBe X= {2; 4;
6; 8; I  :
8) (6; 8); (6; 10); (6; 6); (2; ,.)};
X 6) (4; 4); (4; 2); (6; 2); (8; 2); (8; 4)};
B) (8; 10); (10; 9); (6; 2)};
r) (2; 2); (4; 4); (3; 3); (6; 6)}.
5. B CJlC)(Y1OlI1.HX 3a.na'laX I) BbI,lJ.eJIHTe Ha3BaHHSI
OTH0111eHHH H 06'bcKTOB, CBH3aHHblX 3THM OTH0111CHHeM.
a) B Mara3HHe 3a 3 )(HH npOAaJIH 1 m caxapy; ..
nepBbli )J.eHb npOAaJlH 300 K2, 80 BTOpoR ,!(eHb - 8
 plI3a 6oJII.lIle, '1eM B nepBblH. CKOJIbKO KKJlorpaMMOB caxap8 npoAaJIH B rpeTHB
.I1eHI>?
6) AJIb60M CTOHT 8 Korr.; KHHra B 5 pa3 )(opO)Ke aJlbOOMa. CKOJIbKO CTOHT KHHra?
B) OT!.!,Y X JICT; CblH B 4 pa3a MOJIO)KC OTlI,a. CKOJII.KO JIeT CblHY?
6. npHBC,n;HTC npHMcpbl 3a.na'l H3 Y'Ie6HHKOB Ha'laJIl>HblX KJIaCCOB, B KOTOpblX
paccMaTpHBalOTC1I OTH0111CHHH «JIerqe». «TH)KeJIee». «OOJII.lIle B ... PB3».
7. nocTpoiiTe Ha KOOP)(HHa,THOH nJIOCKOCTH rpaq,HK OTH0111CHHH R: «x>y», ec.n.a
OHO 3B)(aHO Ha MUO)KCCTBe:
a) X= {x!xEZ, -2-<:x<:2}; 6) X=Z; B) X=R.
8. Ha MHO)KCCTBC R )(CHCTBHTeJIbHblX '1HCeJI 3BAaHbI OTH0111CHHSI Q: «x+y= h,
S: «x+y>1» H T: «x+y<h. IlocTpoHTe HX rpaq,HKH H3 KOOPAHHaTHOH nJIOCKOCTH.
9. Ha pHcYHKe 30 H306paJKeH rpacpHK OTHOlIleHH1I T B MHOJKecme Z. nepe'lHCJlHTe
lICe napbl, npHHB)(JJ:e)KalI1.Hc rpaq,HKY OTHOlIleHHH T.
10. rpa4>HK OTHOlIlCHHH 8 Ha MHO)KCCTBC Z BCCX lJ,eJIbiX '1HCeJI HMeeT BH,n;: f=
.... {(x; y) I -1 -<:x-<:5 H 0-<:y-<:4}.
nGCTpOHTe ero Ha KooPAHHaTHoH IIJIOCKOCTH.
II. IlocTpoiiTe rpaIjJHK OTH0111eHHSI 8 (CM. 38,lta'lY 10), ecJJH OHO 3BAaHO Ha MHO-
:lKecrBe R )(CHCTBHTeJIbHblX '1HCeJI.
12. Ha MHO)KCCTBC JIIO)(CH 3B)(aHbI OTH0111eHHSI
R: «x H y )KHBYT B O)(HOM H TOM JKe ropo)(e»,
Q: «x H y Y'IaTCH B O)(HOH H TOH )Ke 111KOJIe» H
S: ex H y 3aHHMalOTCR B oAHOM H TOM )Ke KPY)KKC B AaHHOH wKOJIe».
bblHCHHTC, KaKOC H3 OTH0111eHHH HBJIHCTCSI CJlC)(CTBHeM APyroro.
13. Ha MHO)KCCTBC Z lI,CJIbiX 'IHCeJI OTHOlIleHHe R HMeCT rpaq,HK
f l = {(3; 0); (-2; 0); (-I; 0); (0; 0); (I; 0); (2; 0); (3; 0);
(-3; I); (-2; I); (-I; I); (0; I); (1; I); (2; I); (3: I)},
8 OTH0111CHHC Q - rpa4>HK
f 2 ={(-1;0); (0;0); (1;0); (-I; I); (0; I); (1; I); (-1;2); (0;2); (1;2)}.
it) I1CTHHHO JIH BblCKB3b1B8HHe f 2C:: f l?
6) HaHAHTe rpaljlHKH 06'be;1;HHeHH!I H nepecC'leHHR AaHHblX oTH0111eHHH H Hm-
pa3HTc HX Ha Koop,n;HHaTHoH nJIOCKOCTH.
14. Ha MHO)KCCTBC HaTypaJIbHblX '1HCeJI 3a)(aHbI OTH0111eHHSI
T: «y>x-2» H 8: «y=x».
Ii) nOCTpoHTe Ha KooPAHHaTHOH nJIOCKOCTH rPa4mKH 3THX oTH0111eHHH.
6) HaH)(HTe ncpCCC'ICHHC H 06'bC)(HHCI:lIIe oTHOweHHA T H S.
15. HaH)(HTC ncpece'lCHHC H 06'bCAHHeHHe oTH0111eHHii R H S Hli NHO)KecTBe AeH-
CTBHTeJII.HblX '1HCCJI, CCJlH
a) R: «y>x2», .,8: «y>x+3»;
6. R: «!/'<4"':""'x2», 8: cy>x+2».
16. 3alIlTpHxoBaHHble 06JI8CTH. H306p8>KeHHwe HI pHCYHKe 31 (a, 6), npeACT3B-
JUHOr COOTBeTCTBeHUO rpaIjJHKH OTHomeHHIi R R Q R8 MHO)KecTBe .I1eiiCTBRTeJlbHblX
'lHCeJl.
9
.
2
.
_ -1
.
2
.
PHC. 30
.
1.1 3aNl'lH B3HThI H3 Y'Ie6HHKa IllleJIKO A. c., naHroBa M. A.. Mopo M. 11., Ilw111-
ORO A. M. MaTeMaTRKa, 3 KJIacc. M., 1974. .
1.

R y y Q
'It

II

0 X
0 X
a) 0)
PIIC. 31
a) OTMCTI.TC 111TPliXOBKOi'l rpaq,HK OTH0111CHHB. npoTHBOnOJlO)l{HOrO OTH0111eHHIO R.,
6) 3a)(aiiTe OTH0111eHHe R H eMY npOTHBOllOJlO)l{HOe npH nOMOUl,H HCpaBeHCTB3,
C ABYMII nepeMeHHblMH.
B) H306pB3HTe Ha KOOPAHHaTHOH llJIOCKOCTH nepece'lCHHe H 06'be)(HHeHHe A3HHblX
OTH0111eHHii R H Q.
17. Ha MHO)l{CCTBC R Bcex )(eiiCTBHTeJII.HblX '1HCeJI 3a.naHbI OTH0111CHHII:
a) x2+y2=9; 6) x2+y2.c;;;:9 H x;>O; B) ,y;>Q. y.c;;;:x H x+y<l.
nOCTpoiiTe Ha KOOP)(HHaTHOH nJlOCKOCTH rpaq,HKH 3THX OTH0111CHHii.
ncpece'lCHHeM CKOJlbKHX OTH0111CUHii HBJlIle'l'ClI OTH0111CUHe B)?
18. Me>K)I,Y 3JleMeHTaMH MHO}l{ecrBa X={-2; -I; 0; ]; 2} 3a)(aHO OTU0111CHHe S:
c1lHCJlO x )(eJIHTCH Ha '1HCJIO y».
a) 1ITO 3Ha'lHT yS-IX?
6) 3anH111HTe rpaq,HKH OTHOl1IeHHii S H S-l H nOCTpoiiTC HX Ha KOOp)(HHaTHOii
nJlOCKOCTH (Ha O)l,HOM H TOM }l{e '1epTe}l{e).
19. X - MHO)l{ecTBO pa3J1H'IHblX PYCCKHX CJIOB B HCKOTOpOM CJIOBape, Y - MHO-
}l{ecrBO '1aCTeH pe'lH B PYCCKOM H3b1KC. McJKJI.Y 3THNH MUO)l{CCTBaMH 3a)(auo COOT-
BCTCTBHe: «OiOBO x HBJlHCTCH '1aCTblO pe'lH y» (x EX, yEY). OjJopMYJlllpyiire CooT-
BCTCTBIIe, npoTHBOnOJlO)l{HOC )(aUUOMY.
2{). X - MHO)l{ecTBO reoMeTpH'IecKIiX 4>Hryp, H306pa)l{CHUbiX Ha PHCYUKC 32,
Y = R - MUO}l{CCTBO )(CiiCTBHTeJIbHblX '1HCeJI. McJK,D;Y MHO)l{CCTBaMH X H Y 33)(alIO
COOTBCTCTBHe s: «<l>Hrypa x HMeeT nJlOlIl.B.!\b. paBHYIO y».
a) 3a.naiiTe COOTBCTCTBHC, 06paTHOC )(aHHoNY.
6) KaKOBbi MHO)l{ecrBO onpe)(eJICUHH H MUO)l{ecTBO 3U3'1CHHii COOTBCTCTBHH S-J1
21. Ha PHCYHKC 33 M306pa)l{eH rpaq,HK OTH0111eHH!I R H MHO)l{ecrBC Z u.e.'1b1X '1HCeJI.
a) nepC'IHCJIHTe ace napbl, npHua)(JJe>KalIl.He rpaq,HKY OTH0111eHHH R.
6) 3anHlIlHTe MHO)l{ecrBO onpe)(eJICHHIiI H MHOJKecTBO 3Ha'lCHHii OTHOlIlCUHIiI R.
22. nOCTpoiiTe rpaq,HKH oTH0111eHHii, 06paTHbix )l.aHHblM (pHC. 34).
I
I
'1
I I I
I I I
I I
mp 1 13
 I
 
 t'/ 
I I I I '"  ,.-t'j
IF.
r
I ri I
PHC. 32
.

4. fpact» 6HHapHoro OTHOllIeHHSI. llTOObI
Har .nH,lLHO npe}I.CTaBHTb OTHOllIeHHe R B MHO-
)KeCTBe X, H306pa3HM TOqKaMH 9.neMeHTbI
9Toro MHO)KecTBa, a 3aTeM npOBe,lLeM cTpe.nKH
OT x K Y ,lL.nH Bcex nap TOqeK (x; y) TaKHx,
qTO xRy. TIo.nyqeHHbIH qepTe)\{ Ha3bIBalOT
cpacjJOM omHOUl£HUfl R, a TOqKH, H306pa)\{alO-
IlI.He 9.'1eMeHTbI MHO)\{eCTBa X, 8epmuHaMu
c pacjJa.
TIOCTPOHM, HanpHMep, rpacp OTHOllIeHHH
R : «x>y» , 3a,lLaHHOrO Ha MHO)\{eCTBe X =
= {2; 4; 6; 8; 12}. .LI..nH 9Toro pHcyeM ,lLHarpaM-
MY MHO)\{eCTBa X, H306pa3HB 9.neMeHTbI 9Toro
MHO)\{ecTBa TOqKaMH. 3aTeM npOBO,lLHM CTpe.n-
KH OT X K Y ,lL.nH Bcex nap (x; y) TaKHX, qTO
x>y. nO.nyqaeM rpacp OTHOllIeHHH «x>y», 3a-
,lLaHHOrO Ha MHo)\{eCTBe X = {2; 4; 6; 8; 12}
(pHC. 35).
PaccMoTpHM B TOM )\{e MHO)KeCTBe X = {2;
4; 6; 8; 12} OTHOrneHHe Q: «lJ:HC.nO x HB.nHeT-
CH ,lLeJIHTeJIeM qHc.na y» H nocTpoHM ero rpact».
11300pa3HB 9.neMeHTbI MHOlKeCTBa X TOqKa-
MH, npOBO,lLHM cTpe.nKH OT x K y ,lL.nH Bcex
nap (x; Y) TaKHX, qTO x HB.nHeTCH ,lLe.nHTeJIeM y (pHC. 36). HanpH-
Mep, npOBO,lLHM CTpeJIKY OT qHc.na 2 K qHCJIY 4, nocKo.nbKY qHCJIO 2 -
,lLeJIHTeJIb qHCJIa 4. Ho, KaK H3BecTHo, .n1060e qHC.nO HB.nHeTCH ,lLe-
JIHTeJIeM caMoro ce6H. TI09TOMY,lL.nH Ka)\{,lLOH TOqKH x, H306pa)\{aIOIlI.ei'I
9JIeMeHT MHO)\{ecTBa X, pHcyeM CTPeJIKY, Haqa.no H KOHeIl KOTOpOH COB-
na,AalOT C x. CTpeJIKY Ha rpacpe, y KOTOpOH Haqa.no H KOHeIl COBna,lLaIOT,
Ha3bIBalOT nemAeu. Cne,lLOBaTeJIbHO, rpacp OTHOllIeHHH Q B KalK,lLOH Bep-
llIHHe HMeeT neTJIH.
nYCTb X - MHO)\{eCTBO npHMblX, H306pa)KeHHbIX Ha pHcYHKe 37.
PaccMoTpHM B 9TOM MHO)\{eCTBe OTHorneHHe napa.n.neJIbHOCTH 1).
I) npHMI.Ie a H b, npHHa)(ne/KalI1.He O,ll.HOH nJ10CKOCTH, Ha3I.IBalOTCH napaJInenI.-
IIb/MH, ecnH OHH He HMelOT 06lI1.HX TO'leK HJIH COBna)(aIOT. OrH0111eHHc napaJIneJII.IlOCTH
0603Ha'laIOT: a II b. .
171
o
PHC. 33
PHC.
PHC. 36
76
x
x
0)
PHC. 34

a 9:
;/' f
d
.
b . 0
c e e
PHC. 37 PHC. 38
JIerKo BH,lLeTb, qTO allb, blla, allf, fila, bllf, !lIb, clie, ellc, alia, bllb, flit,
dlld, elle, cllc, T. e. rpa<j:JHK OTHOllIeHHH napaJIJIeJIbHOCTH B MHO)\{eCTBe X
npe,lLCTaBJIHeT CODoH TaKoe MHO)\{ecTBO:
f={(a; b;); (b; a); (a; f); (f; a); (b; f); (f; b); (c; e); (e; c); (a; a); (b; b);
(f; f); (d; d); (e; e); (c; c)}.
llToobI nocTpoHTb rpa<j:J OTHOllIeHHH napa.n.ne.nbHocTH, H300pa3HM
Ka)\{,lLYIO npHMYIO TOqKOH Ha nJIOCKOCTH H coeHHHM CTpe.nKoH Ka)\{,lLYIO
napy napa.n.neJIbHbIX Me)\{,lLY CODoH npHMbIx (pHC. 38). TaK KaK npHMaH a
napaJIJIeJIbHa npHMoH b, TO Be,lLeM CTpeJIKY OT a K b. Ho ecJIH allb, TO
H blla; CJIe,lLOBaTeJIbHO, npOBO,lLHM CTpeJIKy OT b K a. .lJ.aJIee OKOJIO
Ka)\{,lLOH BepllIHHbI pHcyeM neTJIIO, nOToMY qTO Ka)\{,lLaH npHMaH napa.n-
JleJIbHa caMoH ce6e.
YupamHeHBJI
1. nOCTpOHTe rpaeJ>bI OTU0111CHHH, paCCMOTpCUHbIX B 3a)(a'laX 2 H 3 npelJ,bIlI.YlI1.cro
nyuKTa.
2. B MUOJKecTBe npHMblx, H306paJKcHHblx Ha pHcyuKe 37, pacCMOTpHTe OT/to-
111eUHe nCpneUIJ,HKYJlHpHocrH H nocTpoHTe cro rpa<jJ.
3. nocrpOHTe rpaeJ> OTH0111CHHH Q: «4HCJlO x lJ,eJIUTCH 6C3 OCTaTKa Ha '1HCJlO /p,
r)(e xEX, yEX H X={-6; -4; -2; 0; 2; 4}.
a) 3anH111HTe MHOJKecTBO nap (x; y) TaKHX, 'ITO xQy. 0603ua'lbTe 3TO MUOJKecTIIO
6yKBOH f.
6) Q6'bHCUHTe, nO'leMY HCTHUUbI CJlClJ,ylOlI1.He BblCKa3b1BaUHH:
(0; 2)Ef; (4; -4)Ef; (-4; 4)Ef; (0; O)Ef.
4. Ha pHcyuKe 39 CnJl0111UblMH crpeJIKaMH 3a.nau rpaeJ> OTH0111CHHH «6b1Tb ceCT-
peH:' B MUOJKecTBe )(eTeH ua111ero )(BOpa. ,l],CTH 0603ua'leHbI TO'IKaMH A, S, B, r, 11., E,
)J(. OnpelJ,eJIHTe, KTO H3 UHX HBJlHCTCH IJ,CBO'IKOH, KTO MaJlb'lHKOM. 0 KOM UeJIb311 no
9TOMY rpa<jJy HH'IerO CKa3aTb?
5. nocrpOHTe no PHCYUKY 39 rpaeJ> OTU0111CHHSI «6b1TI. 6paToM», eCJIH H3BecTIIO,
'ITO II. H )J(, a TaKJKC E H )J( ue 6paTI.H.
6. Pe111HTe CJle)(ylOlI1.He 3alJ,a'lH H3 Ha-
'IaJII.UOrO Kypca MaTeMaTHKH ), HcnoJlI.3YH
rpaeJ>bI oTu0111eHHH, paccMaTpHBaeMbix B
3THX 3a.na'lax.
  -
rA \£
B
)/(
.
1) Mop 0 M. Ii., B aUT 0 B a M. A.
MaTeMaTHKa. Y'IOOUHK JIH BTOpOro KJIaCca.
M., 1973.
PHC. 39
<17

a) CocraBI.Te Bce B03MO>KHble npHMepbl Ha AeJIeHHe (6ea OCT<fl'Ka) C qHCJlaMH:
]2, 26, 4, 96, ]3, 78, 3.
6) 3anHWUTe )(Ba paBCHCTBa H ABa HepaBeHCTBa, HCnOJIb3YH Bblpa>KeUHH 23+ 12,
40-16, 12+23, 40-5. CKOJlbKO Bcex B03MO>KHblX paBeucTB MO>KeT 6b1TI. nOJlY'IeHO
H3 )(aUUblX Bblpa>KCHHii?
i 2. OTHomEHIIH 3KBBBAJIEHTHOCTH
t. Pa36HeliHe MH01KeCTBa Ha nonapHO HenepeceKalOHeCH no,A-
MH01KeCTBa. MHOlKeCTBO X Bcex cTY,lLeHTOB ne,lLHHCTHTYTa MO)\{HO pa3-
6HTb Ha nO,lLMHO)\{eCTBa, COCTOHIlI.He H3 CTY,lLeHTOB, 06YLJaIOIlI.HXCH Ha
O).l.HOM H TOM )\{e Kypce. Ec.nH OOYLJeHHe ,lL.nHTCH LJeTbIpe rO,lLa, TO no.ny-
tlaeM LJeTbIpe nO,lLMHO)\{eCTBa: CTY,lLeHTOB nepBoro Kypca, CTY,lLeHTOB
IJfOporo Kypca, CTY)J.eHTOB TpeTberO Kypea H cry,neHTOB '1eTBepTOro
-Kypca. HHKaKHe ,lLBa H3 3THX MHO)f{ecTB He. HMeK7I' OOm.HX 3JIeMeHTOB
(CTY,lLeHT He MO)\{eT cpa3Y YLJHTbCH H H8 BTOpoM, H H8 TTbehvKypce),
a 06De,lLHHeHHeM 3THX MHO)f{e0TBSlBJIHeTCJI MHO}KeCTOO X Bcex CTY,AeH-
TOB. rOBOpHT, LJTO X pa30HTO Ha LJeTbIpe nonapHO HenepeceKaIOIIJ.HXCH
no)J.MHO)f{eCTBa Xl> X 2 , X a , X.. To :>t<e MHO>KecTBO X MO>KHO pa30HTb
Hi HenepeceKalOHeCH OO).l.MHO>KeCTBa If ).I.pyrHMH cnoc06aMH, HanpH-
MeP, Ha IOHOllIeH H ).I.eBYllIeK, no B03pacry, Ha KOMCOMOJIbn:eB H He KOM-
COMOJIbIleB H T. ).I..
BoooIll.e, MO>KHO rOBOpHTb 0 pa36ufflUU iJaHHoeO .MHCY.JICeCm6a Ha
nonapHO HenepeceKalOIlI.HeCH nO)J.MHOlKecTBa HJIH KJIacc.bI TOr,lLa, KOr,lL8
O,lLHOBpeMeHHO Bbmo.nHHIOTCH c.ne,lLYIOIlI.He YC.nOBHH:
1. Bce noiJMHoJ/Cecmea, o6pa3Y1014Ue pa36ue1iue, HenYCf1lbt.
2. JI106ble o8a maKUX noiJMHOJ/Cecmea He 11RpeceKalOmcfl.
3. 06"6eOUHeHue 8cex noiJMHoJ/Cecme ecmb oaHHoe MHOJ/Cecmeo u.
TaK, MHO>KecTBO HaTypa.nhHbIX LJHCeJI MO)KHO pa36HTb Ha TPH no,lL-
MHO>KeCTBa - MHO>KeCTBO npocTbIX LJHce.n, MHO)\{ecTBO COCTaBHbIX LJH-
CM H MHOlKecTBO, COCTOHIIIee H3 e,lLHHHIJ,bI. ::no >Ke MHOlKeCTBO MOlKHO
pa36HTb H Ha ).I.Ba K.nacca - KJIaCC LJeTHbIX H KJIaCCHeLJeTHbIX HaTypa.nb-
HhIX LJHce.n.
Pa36HeHHe MH02KecTBa Ha nonapHo-HenepeceKaiOIIIHecH nO,AMHO>Ke-
CTBa .ne)\{HT B OCHOBe BceB03MO)\{HbIX K.naccHcpHKaIlHH. nOHHTHe «KJIaCr»
H 'ero CHHOHHMbI «THn», «ceMeHCTBo», «pot(», «BH,A», «coPT» llIHpOKO npH-
MeHHIOTCH BO BCex 06JIacTSlx LJeJIOseqecKOH ,lLeHTeJIbHOCTH. TaK, B OHO-
,JIOrHH Bce )j{HBhIe opraHH3Mb1 pacnpe,lLeJIHIOT Ha THnbl, B CeJIbCKOM xC)-
3HHcTBe COpTKPYIOT no pa3Mepy H.nH Becy tPPYKTbI, CJIOBa B c.noBapHX
pa36HBalOT Ha nO,lLMHO)\{ecTBa, pacno.naraH HX B a.ncpaBHTHoM nOpH,lLKe,
H T. ,lL.
Pa30HeHHe MHO>KeCTBa Ha nonapHo-HenepeceKaIOll\HecH nO,lJ.MHO-
>KecTBa LJCTO npOH3BO,lLHT no HeKOTOpOMY CBOHcrBY, KOTOpoe MOJKeT
npHHHMaTb pa3JIHtlHbIe 3HatleHHH. HanpHMep, MO>KHO npoH3BO).l.HTb pa3-
(JMeHHe Ha K.naCCbI no I1BeTY, 06be,lLHHHH B O,lLHH KJIaCC npe,lLMeTbI o,nHoro
I) ¥CJlOBHe 1 HHOf,lta onYCK31OT. HanpHMep, npH paooHeHHH IIIHO}KeCTBa yqa-
XCH Ha KJlaCCbI flO nepBoH 6YKBe HX HH U3CC. COO'I'IIeTCTBYlOllViii 6YKBe
,Q .
18

H TOro )l(e u,BeTa. npH 3TOM KpaCHbte iIpe,1tMeThl nona,.!lyr B O,ll,HH MaCC,
3eJIeHbte - B ,lLpyroii. 'IepHbIe - B TpeTUH HT. ,lI,. MOHO CKa33Tb, 'fro
pa36HeHHe npOH3BO,lLHTCH Ha OCHOBe OTHOllIeHHH «x HMeeT TOT lKe U&er.
'lTO H !p. Toqoo TaK )Ke pa36ueHHe CTYAeHTOB no KypCaM npoH3BOJ{HTCJI
Ha OCHOBe OTHOIUeHHH «X Y'IHTC.II H3 TOM }Ire Kypce, 1fI'O H !p.
Ho He BCHKoe OTHOllIeHHe R MelK,lLY 3JJeMeHTaMH MHoecTBa lJ,aer
B03MO>KHOCTb pa36RTb 3TO MH02KeCTBO H3 KJIacCbI. Hen:b3H. H3npHMep,
pS36HTb H3 nonapSO-HenepeceKaIOIll.HeCH DO,ll,MHOJKecTBa MHroKeCTBO
CTY,lLeHTOB HeKOToporo HHCTHTyr8 npH DOM01D.H OTHOIIIeHHH «CTY)J.eHT x
3H3KOM co CTYJJ.eHTOM !p. JI.eAcrBHTeJJbJfO. ecJIH x 3HaKOM C y, TO X Ii Y
OK3)KYTCH B O,ll,HOM I1OJl;MH01KeCTBe. ECJlH 11 3H8KOM C Z. TO Z ,l(OJJeH 88-
XOJI.HTbCH B O}(HOM 11O)f.MHOBe C Y H, CJIe)J.OB3TeJlblfO. C X. Ho MOJKeT
CJlyqHTbCH, qm x Be 3HaKOM C Z. Tor)J.3 OKIDKeTCH, 'ITO B 01J,HOM no,umo-
>KecTBe ecTh JlIO)J.H, KOTOpbIe Apyr C .ltPyroM He 3H8KOMhl. a TOro He
.n.OJIHO 6btTb npH pa36HeHHH MHolKecTBa no YKa3aHHOMY OTHorneHIlIO.
C .n.pyroii CTOpOHbI. TaKHe OTHomeHHH. KaK «6bITb 'O,l(HOKypcUHXOM»
B MHOlKeCTBe CTY.lI.eHTOB HeKOTOpOro HHCTHTyra, «pO)J.HTbCH B O)J.HON If
TOM lKe ro)J.Y» B MH01KeCTBe JHO,!I.eH, «HMeTh O).t,HH H TOT JKe OCT8TOK npH
)l.eJleHHHH3 )J.aHHoe 'DICJlO» B MHWKeCTBe H3Typ'aJIbHbIX IIHceJI, ,n:alOT BOO-
MOlKHOCTb pa36HTb MHOlKecTBO, B KOTOpOM OHB p8CCM3TPHB3IOTC:II. Ha
KJIaCCbI. B "YHKTe 3 MhI paCCNOTpHM. KaKHMH }I.oJDKHW tl.rrb CBoHCTBa
OTHOmeHII5I, rrro6Y: c ero DOMOID.bIO MO)KH() 6bL110 paa6HTb MHO)KecTBO
Ha MaCCbI.
YnpaZHeHBJI
I. MeJK,D;Y 3J1eMeUTaMH MUOJKecTBa X= {ne'lb; Ca)(OBblH; CHerOBHK; noca./UI11>;
CUCJKHblH; uaIIeKJIa; CHeJKOK} 3BA<lUO OTH0111eUBe «CJIOB.B X H Y IlMeIOT O).tHU M TOT JKe
KOPCUb». nOK3JKHTe, 'ITo npH nOMOlI1.H )(aUHOrO OTU0111eHHH MHOJKecTBO X MOJKuO
pB36HTb Ha nonapHO-HenepeceK3JOecll nO)(MHOJKecTBa.
2. MoJKHO JIll pas6MTb H3 nonapHO-HenepecexBlOlI1,HeCH nOAMHOJKecma MHO__
CTIIO X (CM. 3a.na'lY 1) npH noMOlI1,U OTUOUJeHHH Q: «C.ooBa x II Y npHH8).tJ1eJKaT Ok
BOii H Toii JKe '1aCTH pe'lH»?
3. npHBeAHTe npHMepbl pa36HeHHH ua KJlaCCbl CJIe)(YIOlI1.HX MUOJKeCTB:
a) MftOIKecTB8 )"IalI1,IIXCJI HeKOTOpOH IDKOJIbI; 6} MHOJKecTB8 CJIOB B CJIO&ape;
B) MHOJKecTBa )(CpCBbeB B ca.ny.
4. Ha JIJIOCKOCTH npoBe,It,eHs HpSMWi 1. Ha KaKHe IIQnapHO-HenepeceKaIODUlecJl
no).tMHOJKeCTBa pas6H.J1OCb MHO:ai:ecTIIO Bcex TO'IeK IIJIOCKOCTH?
5. J{auo MBOJKecTIIO A= {alaEN, a-<:18}. BbI).teJIHTe H3 MHOJKeCTBa A noJl,MllO-
JKecTBa, KaJK,D;oe H3 K<m:>pblX COC1'OHT H3 'IHCM, .lUHOIf\HX O)l;HH H roT JKe OCTaTOK Dpil
AMeUHH Ha 4. npoH30HAer JIB npH 31'OM pa36HeHHe MUOJKeCTB.B Ha KJlaCChl?
6. npHBeAHTe npHMepw uaCCH4»tKan;HH, BWnOJlHHeMblX B pB3JJH'WbIX 06JIacTHx
'IeJIOBe'leCKOH )(TeJIbHOCTH.
7. npHBeAHTe npHMepbl KJlaccmpHKaIl,HH, BWnOJlUHeMbIX B U<I'IaJlbHOM KyJltt
M<lTeMaTHKH.
2. OcHoaHwe CBOiicTBa OTHomeHHH. nYCTb Ha MHO)I{ecrse X 3a,n.aHO
HeKoTopoe OTHOllIeHHe R.
1. OTHollIeHHe R H83bIBaeTCSI pet/J.AeKCU81iblM, ecJJH ,lLJIH JII060ro X
H3 MHOlKeCTBa X HCTHHHO xRx. .lJ.PYrHMH CJJOBaMH, OTHOllIeHHe R Ha
MHO)KeCTBe X peq>JIeKCHBHo, ecJIH KaJlC,lJ,blH 3JIeMeHT X EX HaXO)l;HT-
Cft B OTOOWeHHH R C CaNnN co6oH.
ft

TaK, pe<I>JIeKCHBHO OTHOUIeHHe KOHrpY3HTHOCTH Ha MHO>KeCTBe re-
OMeTPHqecKHX <j:JHryp, nOCKOJIbKY Ka)\{,lLaH <j:JHrypa KOHrpY3HTHa CaMOn
ce6e.
2. OrHollIeHHe R Ha3bIBaeTCH aHmupecjJAeKCU8Hb/.M, eCJIH HH O,lLHH
3.lIeMeHT x H3 MHOlKeCTBa X He HaXo,lLHTCH B OTHOUIeHHH R C caMHM
CoOOH.
HanpHMep, OTHOllIeHHe «npHMaH X nepneH,lLHKYJIHpHa npHMoH !p
H8 MHOlKeCTBe npHMbIX nJIOCKOCTH aHTHpe<I>JIeKcHBHo, TaK KaK HH O,lLHa
npHMaH He nepneH,lLHKYJIHpHa caMOH ce6e.
CYIlI.eCTBYIOT OTHOllIeHHH, He HBJIHIOIlI.HeCH HH pe<I>JIeKCHBHbIMH, HH
aHTHpecp.neKCHBHbIMH. npHMepOM TaKOrO OTHOllIeHHH MO)\{eT CJIYlKHTb
OTHOllIeHHe «ToqKa x CHMMeTpHqHa TOqKe y OTHOCHTeJIbHO npHMOH l»,
3a,AaHHOe Ha MHolKecTBe TOqeK nJIOCKOCTH. .lJ.eHcTBHTeJIbHO, Bce TOqKH
npHMOH l CHMMeTpHLJHbI caMHM ceDe, a TOqKH, He JIelKaIIlHe Ha npHMOH l,
ceDe He CHMMeTpHqHbI.
3. OTHollIeHHe R Ha3bIBaeTCH cUMMempU'iHbtM, eCJIH ,lLJIH JIIOObIX
3JIeMeHTOB x H y H3 MHO>KeCTBa X H3 xRy CJIe,lLYeT yRx.
TaK, CHMMeTpHqHO OTHOllIeHHe napaJIJIeJIbHOCTH Ha MHOlKeCTBe npH-
MbIX nJIOCKOCTH: eCJIH npHMaH x napaJIJIeJIbHa npHMoH y, TO H npHMaH
y napa.n.ne.nbHa npHMOH x.
4. OTHollIeHHe R Ha3bIBaeTCH acuMMempU'iHbtM, eCJIH HH ,lL.nH KaKHX
3JIeMeHTOB x H y H3 MHO)\{eCTBa X He MO)\{eT C.nyqHTbCH, LJTO O,lLHOBpeMeH-
HO H xRy, H yRx.
npHMepOM aCHMMeTpHLJHOrO OTHOllIeHHH HBJIHeTCH OTHOllIeHHe
«x<y» , 3a,n:aHHoe Ha MHO>KeCTBe ,n:eHCTBHTeJIbHbIX LJHCeJI, TaK KaK HH
npo KaKYIO napy qHCeJI x H y HeJIb3H CKa3aTb, qTO o,n:HOBpeMeHHO H
x<y, H y<x.
5. OTHollIeHHe R aHmucuMMempU'iHO, ecJIH xRy H yRx o,lLHoBpe-
MeHHO BbInOJIHHIOTCH B TOM H TOJIbKO B TOM CJIyqae, KOr,lLa x=y.
AHTHCHMMeTpHqHOe OTHOllIeHHe - OO'be,lLHHeHHe aCHMMeTpHqHOrO oT-
HOllIeHHH C OTHOllIeHHeM TO)\{,lLeCTBa.
6. OTHollIeHHe R Ha3bIBaeTCH mpa1i3UmU81iblM, eCJIH ,lLJIH JIIOObIX
3.lIeMeHTOB x, y H Z H3 MHolKecTBa X H3 Toro, LJTO xRy H yRz, cJIe,lLYeT,
'IT0 xRz.
HanpHMep, TpaH3HTHBHO OTHOllIeHHe «OTpe30K X ,lL.nHHHee oTpe3Ka
y» B MHO>KeCTBe oTpe3KoB: eCJIH oTpe30K x ,lL.nHHHee oTpe3Ka y H oTpe30K y
)I.JIHHHee OTpe3Ka z, TO OTpe30K x ,lLJIHHHee OTpe3Ka z.
CBOHCTBa pCIjJJIeKcHBHOCTH, CHMMeTpH'IHOCTH H TpaH3HTHBHOCTH HarJIH)(HO HJI-
JII0crpHPYlOTCH npH H306paJKeHHH oTHolIleuHH rpa<jJaMH. ECJIH OTHOlIleHHe R B MHO-
JKecTBe X pcqmeKcHBuo, TO rpaljJ STOro OTHOlIleHHH B KaJKIJ,OH BeplIlHHe HMeeT neTJIIO.
ECJlH OTH0111eHHe R CHMMCTpH'IHO, TO rpaljJ BMeCTe C KaJKlJ,OH CTpeJIKOH, HIJ,YlI1.eH H3
TO'IKH X B TO'IKY y, IJ,OJIJKCH cOlJ,epJKaTb CTPeJIKY, COeIJ,HHHIOlI1.YIO Te JKe TO'IKH, HO
HJl.YUI,YIO B 06paTHoM HanpaBJIeHHH. fpaljJ TpaH3HTHBHoro OTHOlIleHHH BMeCTe co CTpeJI-
KaMH, HIJ,YlI1.HMH OT X K Y H OT Y K Z. )(OJIJKeH COlJ,epJKaTb H crPeJIKY, H)(YUI,YIO OT X K Z.
YnpamHeHHJI
J. OnpelJ,eJIHTe CBOHCTBa CJlelJ,YIOUI,HX OTHOlIleHHH:
a) «npHMaH x nepeCeKaeT npHMYIO Y» (B MHOJKeCTBe npHMblx);
6) «<1>Hrypa F KOHrpY3HTHa ljJHrype G» (B MHOJKecTBe reoMcTpH'IecKHx ljJHryp);
8Q

B) «4HCJlO X (5oJlblIle 'IRCJla Y Ra
2» (B MHOIKCCTBe HaTypaJIbHblX 'IRCeJI);
r) «4HCJlO x )(eJIHTCH Ha 'IRCJlO Y
6e3 OCTaTKa:t (B MHOIKCCTBe HarypaJIb-
HbiX 'IRCeJI).
2. Ha MHOIKCCTBe JIIOIJ,CH 3a.naHO
OTHoweHHe R: «x ceCTpa y». 8b1HC'
HHTe, KaKHe H3 CJlCIJ,YIOlI1.HX BblCKa3b1.
BaHHH HCTHHHbI: a) OTH0111CHHe R CHM-
MeTpHtIHO; 6) OTHOWCHHe R aCHMMeT-
pH'IHO; B) OTH0111eHHe R pt$JICKCHB-
HO; r) OTHOlIlCHHe R aHTHpt$JICKCHB-
HO; D,) OTHOlIlCHHe R TpaH3HTHBHO. PHC. 40
3. .LJ:aHHoe B nYHKTe 2 onpclJ,eJIc-
HHe pt$JIcKcHBHoro OTHoweHHH MOIKHO
3anHcaTI. npH nOMOlI1.H CHMOOJIOB B TaKOM BH)(C: (VxEX)xRx.3anH111HTe OCTaJIbHble
onpCD,eJICHHH B aHaJIOrH'IHOH <jJopMe.
4. nOCTpOHTe rpa<l> OTHoweHHH «x>y» B MHOIKCCTBe X= {3; 4; 5; 6}. 4TO MOIKHO
CKa3aTI. 0 CBoHcTBax 3a.naHHoro OTHOWCHHH no rpa<l>Y? nOCTpOHTe rpa<l>HK 3TOro
OTHOWCHHH Ha KooplJ,HHaTHoH nJIOCKOCTH.
5. Ha PHCYHKC 40 H306paIKcHo MHOIKCCTBO oTpe3KoB. nOCTpOHTe rpa<l> OTHowe-
HUH KOHrpY3HTHOCTH B 3TOM MHOIKCCTBe, 0603Ha'lHB KaJK,Zl,blH OTpC30K TO'IKOH. Ka-
KHMH COOHCTBaMH 06JIalJ,aCT 3TO oTH0111eHHc?
6. npHBCIJ,HTC npHMcpbl OTH0111CHHH H3 Ha'laJIbHOrO Kypca MaTCMaTHKH H onpe-
lJ,eJIHTC ux CBoHcTBa.
7. Ha MHOIKCCTBC X= {3; 4; 5; 6} 3alJ,aHO OTHOlIlCHHC «;(;3y». OnpC)(eJIHTe COOK-
CTBa 3TOro OTH0111CHHH H nOCTpOHTe ere rpa<l>HK Ha KOOp)(UHaTHOH nJIOCKOCTH. CpaB-
HHTe rpa<l>HKH OTH0111CHHH «x>y» (ynp. 4) H «x';;i:y».
a IL
ct _f
e
ib ,
--
c
3. OTHollIeHHSI 9KBHBaJIeHTHOCTH. ECJIIf OTHOllieHIfe R B MHOlKeCTBe
X oo.na,lLaeT cBoikTBaMII pe<j:JJIeKCIIBHoCTIf, cIIMMeTpIIllHoCTII II TpaH-
3IITIIBHOCTII, TO ero Ha3bIBalOT OTHOllieHIIeM aK8U8aAeHmHocmu. npHMe--
paMII OTHOllieHIIH 9KBIIBaJIeHTHOCTII HBJIHIOTCSI OTHOllieHIIH, paCCMOTpeH-
HbIe HaMII paHbllIe: «6bITb O,lLHOKYPCHIIKOM» (Ha MHOlKeCTBe CTY,lLeHTOB
HeKOToporo IIHCTIITYTa), «IIMeTb O)l,IIH II TOT lKe KopeHb» (Ha MHO)\{eCTBe
CJIOB) II )l,p. C KalK)I,bIM TaKIIM OTHOllieHIIeM CBH3aHO pa36IIeHIIe MHOlKeCT-
Ba Ha HenepeceKalOlIIIIecH nO,lLMHO)\{eCTBa. 11 9TO He CJIYlIallHO. I1MeeT
MeCTO CJIe,lLYlOlIIaH TeopeMa.
D./lfl moco t.tm06bt omHoUl£Hue R n0380AflAO' pa36Umb MHOJ/CeCm80 X
Ha K.1aCCbl, Heo6xooUMO U oocnwmOt.tHO, t.tm06bt R 6blAO OmHOUleHueM at(,-
8U8aAeHmHocmu.
11. 0 K a 3 aTe JI b C T B 0 H e 0 6 x 0 )I, HMO C T II Y C JI 0 B H SI.
TIYCTb 3a,lLaHO pa36IIeHIIe MHolKecTBa X Ha KJIaCCbI. TIoKalKeM, tp'O
9TO pa36HeHIIe n03Bo.nHeT onpe,lLeJIIITb B MHOlKeCTBe X HeKOTOpoe OTHO-
llIeHHe 3KBIIBaJIeHTHOCTII.
3a)l,a)l,HM B MHOlKeCTBe X OTHOllieHIIe R: «.3JIeMeHT x HaXO,lLIiTCH B TOM
)Ke KJIaCce pa36HeHIIH, liTO II 9JIeMeHT y». 3T0 OTHOllieHIIe pe<j:JJIeKcHBHo.
nOCKOJIbKY BCHKHH 9JIeMeHT x HaXO,lLIiTCH B O,lLHOM KJIacce C caMHM co-
Doll. 11.aJIee, ,lLJIH JII06bIX 9JIeMeHTOB x II y II3 MHolKecTBa X II3 TOro, lITO x
JIelKliT B TOM lKe KJIaCCe, liTO II y, CJIe,lLYeT, 'lTO Y JIelKliT B TOM lKe KJIaCce.
liTO II X. TaKIIM 00pa30M, OTHOllieHIIe R CIIMMeTpIIllHO. KpoMe TOro, OT-
HOllieHIIe R TpaH3HTIIBHO, TaK KaK )l,JIH JII06bIX Tpex 9JIeMeHTOB II3 MHO-
)KecTBa X cnpaBe,lLJIIIBO CJIe,lLYlOlIIee YTBeplK,lLeHIIe: ecJIH x JIe)KHT B
81

TOM )\{e KJIaCCe, tJTO H y, a y - B TOM )\{e KJl8CCe, tJTO H l, TO X HaXO,lLHTCJI
B TOM lKe KJIaCCe, tJTO H Z.
I1TaK, OTHOlIIeHHe R: CHaXO,lLHTbCH B O.D;HOM H TOM )Ke KJIacce» peep..
JleKCHBHO, CHMMeTpH'IHO H TpaH3HTHBHO. Cn:e.D;OBaTeJIbHD R - OTHO-
llIeHHe KBHBaJIeHTHOCTH Ha MHolKecTBe X.
JI. ° K 8 3 aTe JI beT B 0 .D; 0 C TaT 0 tJ HOC THY C JI 0 B H 9.
.l{oX3JKeM Tenepb, 1JTO Ka>K,lLoe OTHOllIeHHe 9KBHB8JIeHTHOCTH (T. e.
JIJ060e pe4JJleKCHBHoe, CHMMeTpHtJHoe H TpaH3HTHBHoe OTHOllIeHHe)
B MH02Kecrne X onpe,lLMHeT pa30HeHHe 9TOro MHOlKecTBa Ha nonapHO-
HenepeceKalOIlI.HecH nO,lLMHolKecTBa. .lJ.JIH 3Toro nOCTaBHM B COOTBeTCTBHe
KalK.D;OMY 9JIeMeHTY a MHolKecTBa X era 06pa3 npH OTHOllIeHHH R.
T. e. MHOlKecTBO R (a) TaKHX 3JIeMeHTOB x, tfrO aRx (HanpHMep, ecJIH
xRy - OTHOllIeHHe «npMa1l x nap8JIJleJIbHa npHMOH y», TO KJlaCC
R (a) COCTOHT H3 Bcex npJlMbIX, napaJIJIeJIbHblX npHMOH a). TaK KaK
OTHOllIeHHe R pe$JIeKCIiBHO, TO .D;JIH JIlo60ro a EX HMeeM aRa, a 3TO
3Ha'IHT, tJTO KalK.D;bIll 3JIeMeHT a npHHa,1l.Jle>KHT COOTBeTCTB)'IOIlI.eMY eMY
KJlaccy R (a), a E R (a).
nOKalKeM, 'lTO ecJIH b E R (a), TO KJIaCCbI R (a) H R (b) COBnaalOT.
B caMOM ,lLMe, B 3TOM CJIYtJae HMeeM aRb. ECJIH c E R (b), TO bRc, a
Tor,lLa H3 aRb H bRc B CHJIY TpaH3HTHBHOCTH R HMeeM aRc. Ho Tor,n:a
c E R(a). 3Ha'lHT, KaIlW>lH 3JIeMeHT NHOlKecTBa R (b) npHHa..'Ie)\{HT
R (a), T. e. R (b) c R (a). 06paTHo, nyCTb c E R (a), T. e. nYCTb aRc.
TaK KaK aRb, TO B CH.ny CHMMeTpHtJHOCTH OTHOllIeHHH R HMeeM bRa,
a TOr,n:a H3 bRa H aRc CJIe,n:yeT, tJTO bRc, H nOTOMY c E R (b). 3HatJHT,
JIIo60ii 3JIeMeHT H3 R(a) npHHa)l,JIe>K:HT R(b), T. e. R(a)cR(b). Tal(
KaK R(b)cR(a) 8 R(a)cR(b), TO R(a) = R(b).
113 ).{OKa3aHHoro )'TBep>K,lLeHHSI CJle,!\ye-r, '1TO ecJlH MHO)\{ecTBa
R (a) H R (b) HMelOT XOTb O,n:KH 06rn.HH 3.neMeHT c, TO OHH COBna,lLaIOT.
B caMOM ,n:eJIe, B 3TOM cJIytJae 06a MHO>KecTBa R (a) H R (b) COBna,lLaIOT c
R (c), a nOTOMY paBHbI ,n:pyr ,lLpyry. TaKHM 06pa30M, .m06bIe ,lLBa MHO-
)KecTBa R (a) H R (b) .'1860 COBna)l.alOT )l.pyr c ,n:pyroM (ec.nH bE R (a»,
JlH60 He nepeceKalOTcH ,n:pyr C pyroM (ecJlH bR (a». A TaK KaK K8)K-
,lLblll 3JIeMeHT a npHHa,n:JIelKHT CBoeMY KJIaccy R(a), TO MbI OOJIYtJHJ1H P83-
6HeHHe X Ha nonapHo-tIenepeceKalOl.IlHecH KJlaccbI.
'PacCMOTpHM JreCKO.nbKO npHMepoB OTHorneHHH 3KBHBaJIeHT-
HOCTH.
1. OTHoIIIeHHe «Bblpa)KeHHS1 x H y HMeIOT o,n:HHaKOBbie tJHC.lIOBbIe
3HaLJeHHSI» Ha MHOlKecTBe tlHCJIOBbiX Bblp8>KeHHH SlBJIHeTCH OTHomeHHeM
3KBHBaJIeHTHOCTIf, DOCKOJlbKY 080
a) pe<PJIeKCHBHo: 3HatJeHHe BbipalKeHHH x COBna,n:ae-r co 3Ha'IeHHeM
Bblpa:>KeHHSI x;
6) CHMMeTpH'lHO: ec.nH 3HatJeHHe BblpalKeHHSI x COBna,lLaeT co 3Ha'le-
HHeM BbIpa>KeHHSI y, TO H 3HatJeHHe Bblpa)\{eHHH y COBna;I.aeT co 3Hatle-
HHeM Bblpa>KeHHSI x;
B) Tp8H3HTIIBHO: ecJIH 3HaqeHHe Bblpa)KeHHSI x COBna;I.8eT CO 3H3-
'1J.eHHeM BbIpaEeHHSI y, a 3HaLJeHHe Bblp3MeHHSI y coena)l.ae-r CO 3Ha..-e-
1I1teM BbipalKeHHH Z, TO 3HalJeHHe BbrpaJKeHHH XCOBnaJl,3er co 3HatJeHHeM
r3l"pa1!reHHH Z.
82

MHO)\{ecTBO BCeX 'IIfCJIOBblX 8b1pa)KeHHH pa36HBaeTCJI 3THM 01'H0UIe-
HHeM Ha KJIaCCbI, B Ka)!{,AOM II3 KOTOpbIX HaXO)l,HTCH BbIpa)\{eHHH, 3Ha-
qeHlU1 KOropbIX rrOnapHO COBrra)l,aIOT. TaK, Bblpa)KeHIIH 5+3, 2 3 ,
2+2+2+2 HaXO)l,HTCH B O)l,HOM KJIaCCe (IIX 3HaqeHHR paBHbl BOCbMII),
a Bblpa)\{eHHSI 7-3,21, 16: 4 - B )l,pyroM (lix 3HaIJeHHH paBHbI lJeTbI-
peM).
2. B MHO)\{eCTBe rrpHMbIx Ha rrJIOCKOCTII arHonreHHe napaJIJIeJIbHOCTH
SlBJIHeTCH OTHOllieHIIeM 3KBHBaJIeHTHOCTII. npHMble x H y, JIaKaIWle B
O)l,HOH nnOCKOCTH, napaJIJIeJlbHbI, ecJ1H OHH JIH60 He nepeceKaIOTCH JIIIOO
COBna)l,alOT. n03TOMY OTHorneH lie napaJIJIeJIbHOCTH
a) pe<j:JJIeKCIIBHO: xlix )l,JISI JIIOOOll npHMOll x;
0) CIIMMeTpHLJHO: ecJIII xlly, TO Y!lx;
B) TpaH3IITIIBHO: eCJIH xlly II yllz, TO xllz.
OTHoweHHeM rrapaJIJIeJIbHOCTH MHO)\{ecTBO BCex npSlMbIX nJIOCKOCTH
pa36HBaeTCH Ha 'KJIaCCbI, COCTOHIIJ,lIe H3 rrapaJIJIeJIbHbrX )l,pyr )l,pyry
rrpH:YlbIX. TaKIIe K,'1aCCbI Ha3bIBalOT nY'ilWMU napa,'1JIeJIbHbIX npHMblx.
3. B MHolKecTBe reOMeTpIIqecKIIX cpHryp OTHOllIeHH5IMH 3KBHBaJIeHT
HOCTH HBJIHIOTCH, HanpHMep, TaKlie OTHOllIeHHSI:
a) «<I>lIrypa x KOHrpY3HTHa cpIIrype y». , '
6) «<I>Hrypa x nO)l,OoHa cpHrype y».
OTHOWeHIIe «<I>lIrypa x KOHrpY9HTHa cpIIrype y» pa30HBaeT MHOlKe-
CTBO Bcex reOMeTpH'IeCKIIX cpIIryp Ha KJIaCCbI 3KBIIBaJIeHTHOCTH, COCTOH-
IIJ,lIe II3 norrapHo KOHrpY3HTHbIX cpIIryp.
B UaqaJIbHOM Kypce MaTeMaTIIKII TaK)\{e BCTpe'IaIOTCH OTHOllIeHHH
3KBIIBaJIeHTHOCTH, HanpHMep OTHoweHHe (Bblpa)\{eHIIH x II y HMeIOT
O)l,HHaKOBbIe qHCJIOBble 3HaQeHHSI» B MHO)\{eCTBe qHCJIOBbIX BblpalKeHHH.
Bblpa)\{eHIIH, rrpHHa)l,JIe)KaIIJ,lIe O)l,HOMY II TOMY )\{e KJIaccy 3KBHBaJIeIIT-
HOCTH, Ha3bIBalOT pa81ibIMU. nOHHTHe paBeHcTBa )l,po6ell HBJI5IeTC5I OT-
HOllIeHHeM 3KBHBaJIeHTHOCTH B MHO)\{eCTBe ,lLPOOell. ¥i3Y'IaH reoMeTpII'Ie-
CKHll MaTepIIaJI, M,'1a)l,WIIe llIKo.'1bHIIKH CTaJIKIIBalOTCH C TaKHMH OTHO-
IIHMH 3KBIIBaJIeHTHOCTH, KaK OTHoweHIIe KOHrpY3HTHOCTH cpuryp
(B Ha'IaJIbHOM Kypce MaTeM8THKII rOBopHT 0 paBeHCTBe cpHryp), OTHD-
meHHe paBHOBeJlHKOCTII (Jmryp.
YupaZHeUR»
t. Ha MHOJKeCTBe X= {xlxEN, x<12} 3a)(auo OTH0111CHHe R: «x H Y H;eIOT O.!\I\H
H TOT )KC OCTaTOK npH )(eJICHHH Ha :» (xEX, YEX).
a) KaKHe 113 'lHcen MHOJKecTBa X 06palI(3IOT npe,l{HKaTbl xR7, 8Ry 8 HC'I'HHHbIe
BblCKa3blBaHHH? '
, 6) nOKaJKHTe, '1TO R - OTHOmCHHe !tKBHB<IJ1eHTHOCTH.
B) 3anUlIlHTe Bce KJlaCCbI !tKBHBaJlCUTHOCTH, Ha KOTOpble pa36HBaeTcH MHOJKecTIIO
X ,llaHHblM OTHoweHueM.
r) nOCTpoiiTe rpa<P OTHOlIlCHHH R.
IJ,) l1306pa3HTe rpa<PHK OTHoweHHJI R Ha KOOplJ,HHaTHOH IJJIOCHOCTB.
2. A - MHOJKecTBO npHMliX Ha IlJ1OCKOCTH. BblJlCHJlTe, KaKHe H3 CJle,llYIOW.HX
OTHOweHUH B MHOJKeCTBe A JlBJUllO'J'CH OTH0111eHHJlMH 9KBHBaJleHTHOCTH.
a) «npHMaH a nCpnCH,llHKYJIHpH8 npHMoH b».
6) «np!lMaH a napaJIJleJIbH8 np1lMoir "".
B) «IlpHMaH a nepeCCKaer npJlM)'1O "".
Ii. LlOKaJKHTe, '1TO KOlffpy!tHTHOCTb TpeyroJlI.HHKOB JlBJlHeTCJI OTH0111eHHeM !H{-
,83

BHBaJICHTHOCTH. 061>IICHHTe, DO'IeM)' BYI1OJ1HlleTCIi Ka)KJ(oe us CBOAcTB OTHOlIleHHII
9KBHBalieUTHOCTH.
4. PacCMOTpHTe B MHOJKecTBe JIIO)(CH CJlC)(YIOIUHe OTH0111eHHH, YKaJKHTe Cpe)(H
HHX OTH0111eHHH 3KBHBaJIeHTHOCTH.
a) «xnOXOJK Ha !p.
0) «x H y JKHBYT B O,ll,HoM AOMe».
B) «x H Y - )(PYSI.H».
r) «x JKHBeT 3TaJKOM BY111e, '1eM y».
5. B MHOJKecTBe X= {3; 4; 5; 8} sa.naHO OTH0111eHHe R, rpa<jJHK KOTOporo HMeeT BH)(:
r= {(3; 3); (3; 4); (4; 3); (4; 4); (5; 5); (5; 8); (8; 5); (8; 8)}.
a) .lJ.oKIDKHTe, '1TO R - OTHOlIlCHHe 9KBHBaJIeHTHOCTH.
6) nOCTpOHre rpacpHK OTH0111eHHH R Ha KooPAHHaTHoH nJIOCKOCTH. YKaJKHTe
ero oc06ellHocTH.
6. fpacpHK OTHOlIlCHHH S B MHOJKeCTBe X= {I; 2; 3} COCTORT H3 nap (1; 2), (1; I),
(2; 2), (2; I), (3; 1), (3; 3). 51BJIHeTCH JIH S OTH0111eHHeM 3KBUBaJIeHTHOCTH Ha MHOJKC-
CTBC X?
7. MHOJKCCTBO B= {I; 2; 3; 4; 5; 6; 7} pa36HJIH Ha KJlaCCbI 3KBHBaJICHTHOCTH
CJlCIJ,YIOIUUM 06pa30M: B={I; 3; 4}U{2; 7}U{5; 6}. 3anUlIlUTe rpa<jJHK COOTBeTCT-
BYIOIUcro 3TOMY pa36ueHuIO OTHOlIlCHHH 3KBHBaJICHTHOCTU U nOCTpOHTe cro Ha KOOp-
JJ,HHaTlIoH nJlOCKOCTU.
8. nYCTI. R U Q - OTH0111CHHH 3KBHBaJICHTHOCTH. .lJ.oKaJKHTe, '1TO ncpCCC'ICHUe
3TUX OTUOlIlCIIHH TaKJKC HBJIHCTCH OTH0111eHHeM 3KBHBaJICHTHOCTH.
9. npHBelJ,UTe npHMcpbl oTH0111eHHH:
a) peqJJ1CKCHBHoro U CHMMCTpU'IUOro, HO HC TpaH3HTHBHoro;
6) PE$JlCKCUBUoro H TpaH3HTuBHoro, HO He CHMMCTpH'IHOro;
B) CUMMCTpU'IUOro H TpaH3UTUBHoro. HO HC pE$J1CKCHBHoro.
9 3. OTHomEHlIH IIOPH):(KA
1. OTHOllIeHHH CTpororo nopH.lI.Ka. CJlOBO «ITOpHOK» 'IaCTO YITOTpe6-
JlHIOT H B 06bIeHHOll pe'IH, H B MaTeMaTHKe. MbI rOBopHM 0 ITOpH)I,Ke
CJlOB B ITpe.nOlKeHHH, 0 ITOpH)I,Ke 3aITHCH pellIeHHH ypaBHeHHH, 0 ITO-
pHKe BbIITo.nHeHHH ellCTBHll B ITpHMepe H T. )1,. TIpH 3TOM B CJlOBO «ITa-
pHOK» BK.nabIBaeTCH TaKOll CMbIC.n: OHO 03Ha'IaeT, KaKOll 3.neMeHT Toro
H.nn HHoro MHolKecTBa 3a KaKHM c.ne)l,yeT (H.nIt KaKOll 3.neMeHT KaKOMY
npellIeCTBYeT). HaITpHMep, rOBopH 0 ITOpH,lLKe BbIITo.nHeHH5I )l,ellCTBHH
B ITpHMepe (17-12).6+ 18 : 9, MbI 3HaeM, [{TO )l,ellCTBHe YMHOlKeHHH Ha
6 c.neyeT 3a eHcTBHeM BbI'IHTaHHH, a CJIOlKeHHe c.neyeT BbIIlO.nHHTb
IToc.ne TOro, KaK OYYT BbI ITO.n HeHbI YMHO)\{eHHe H ,lLe.neHHe.
TaKHM 00pa30M, HHTYHTHBHoe ITOHHTHe ITOpH,lLKa MelKY 3.neMeHTaMH
HeKoToporo MHolKecTBa CB5I3aHO C 3a)l,aHHeM Ha 3TOM MHOlKeCTBe OT-
HoweHHH «x c.neYeT 3a y» (H.nH «x ITpe)l,llIeCTByeT y»), ITpH'IeM 3TO OT-
HOllIeHHe O.nlKHO ObITb TpaH3HTHBHbIM: eCJIH x c.ne,lLYeT 3a Y H Y CJIe)l,Y-
eT 3a Z, TO x c.ne)l,YeT 3a Z. KpOMe TOro, OTHOllIeHHe «c.ne)l,yeT 3a» )l,OJllKHO
o6JIa.n.aTb CBOllCTBOM aCHMMeTpH'IHOCTH: H3 Toro, 'ITO x CJIe)l,yeT 3a y,
BbITeKaeT, 'ITO Y He c.neyeT 3a x.
Ho CBOllCTBOM TpaH3HTHBHOCTH H aCHMMeTpH'IHOCTH OOJla)l,alOT MHO-
rHe )l,pyrHe OTHOllIeHHH, HaITpHMep, OTHOllIeHHe «60.nbwe» Ha MHOlKeCTBe
HaTypa.nbHbIx 'IHCeJI H.nH OTHoweHHe «BbIllIe» Ha MHO)\{eCTBe .n1O,lLell, cpaB-
HHBaeMbIX ITO POCTY. 00 OTHOllIeHHHX «c.ne)l,yeT 3a», «60.n , bllIe» «BbIWe)
rOBopHT, LJTO OHH HB.nHIOTCH OTHoweHHHMH cTpororo I10pH)I,Ka.
BoooIll.e, OTHoweHHe R Ha MHolKecTBe X Ha3bIBaeTCH omHoUl£HueM
cmpococo nOpflOKa, ec.nH OHO TpaH3HTHBHO H aCHMMeTpH'IHO.
84

y
- 5
-f
-J
-2
-1
a 1 2 3 I.t x
I I
PHC. 42
BbIHCHHM ocoOeHHocTH rpa<j:Ja OTHOllIeHHH cTpororo nopH,lLKa. .lJ.JIH
3Toro nocTpoHM rpacp OTHoweHHH «x<y» Ha MHOlKeCTBe X = {3; 1; 5; 2; 4}
(pHC. 41). MbI BH,lLHM, 'ITO rpacp )l,aHHOrO OTHoweHHH He HMeeT rreTeJIb
H JIIOOYIO napy 'IHCeJI (x; y), TaKYIO, 'ITO x<y, COe,lLHHHeT TOJIbKO O,lLHa
CTpe.nKa, H)I,YIlI.aH OT x K y. ECJIH H3 x H)I,eT CTpe.nKa B y, a H3 y--
B Z, TO H3 X H,lLeT CTpe.nKa B Z.
nOCTpOeHHbIH rpacp n03BOJIHeT paCrrOJIOlKHTb 3JIeMeHTbI MHO)\{eCTBa
X B TaKOM nopH,lLKe: 'IHC.nO 1 (OHO MeHbwe Bcex OCTa.nbHbIX), 3aTeM 'IHCJIO
2 H T. )1,. TIO.nY'IHM MIW)\{eCTBO X =(1; 2; 3; 4; 5), npo KOTopoe rOBO-
pHT, 'ITO OHO ynopH,lLO'IeHO OTHOllIeHHeM «x<y».
rpacpHK OTHoweHHH «x<y» Ha MHO)\{eCTBe X = {3; 1; 5; 2; 4} ecTb
MHO)\{eCTBO G= {(I; 2), (I; 3), (1; 4), (I; 5), (2; 3), (2; 4), (2; 5),
(3; 4), (3; 5), (4; 5)} (CM. pHC. 42).
YupamHeHHII
I. Cpe.ll,H CJICIJ,YIOIl.lUX OTHOWCHHH, 3alJ,aHHblX Ha MHO/KCCTBC OTpe3KOB, YKa/KHTe
OTHOWCHH5I CTpororo nOpHIJ,Ka: a) «OrpC30K X )(JJHHHCC OTpC3Ka Y»; 6) «OTpC30K X
KOpO'leOTpe3Ka Y B 2 pa3a»; B) «Orpe30K X IJ,JIHHHee OTpC3Ka Y Ha 5 CM»; r) «OrpC30K X
KOHrpY::!HTeH OTPC3KY Y»; .11,) «OTpC30K X KOpO'le OTpC3Ka Y».
2. Ha MHO/KCCTBe JIlOlJ,eii 3alJ,aHI.l CJIC,lI,YIOIllHC OTHOlIlCHHH: a) «6blTI. cTapwc no
B03pacTY»; 6) «6blTI. cccTpoii»; B) «6blTI. BblWC no POCTY»; r) «/KHTI. B OIJ,HOM H TOM /Ke
/tOMe». BbIHCHHTe, KaKHC H3 HHX HBJIHIOTCH OTHOlIlCHHHMH crpororo nOpH,lI,Ka.
Ta6J1HIla 2
<1>aMHJIHH I POCT (D CM)
HHKo.aCB I 157
A.1eKcceB I 165
HBaIloB I 172
K03JIOB I 169
TypKHH I 172
3r;cpcB I 179
85

PHC. 43
y
I-j
1-'1
J
I.-
f-1
0 1 234- x
I I
a. TIoCTpoilTe rp8I}J OTHOlIleHHH «x Bbl111e y», S3,1J,IIH-
. MOro Ha MHO)l{eCTBC JlIO)(eii (CM. Ta6JIHLI,Y 2). KaK HII
rp8I}Je OTpa3HJIC5I TOT 4JaKT, '1TO B )(aHHOM MI10)l{eCTBC
HMeeTCH )(Ba '1eJIOBeKa O)(HHaK080ro pocTa?
4. a) Ha MHO)l{CCTBC X= {3; 5; 11; 9; 7} 3a)(aHO
OTHOIJ.IeIlHC «x>y». nocrpoiiTe rp8I}J )(aHHOrO OTH0111C-
HHH. HaHIJ,YTCH JIH lIa IICM TO'IKH, KOTOpble lie COeIJ,HHC-
Hbi CTpeJIKOH?
6) nOCTpOHTe rp8I}JHK G .:noro OTHOWCHIIH lIa KO-
OpJl.HHaTHOii nJIOCKOCTH. BepHo JIH, '1TO (7; 3) E Iff
5. B Ha'laJII.HOM KYpCC MBTCMaTHKH lIa MHO-
)t(eCTBC HaTypaJlI.HblX '1HCeJI paccMaTpHBalOT CJIeIJ,YIO-
mHe OTH0111CHHH: a) «60JIblIle»; 6) «60JlI.lIle Ha»;
B) «60JII.111e B»; r) «paBHO»; IJ,) «HenOCpC,lI,CTBCHHO CJIelJ,YeT
sa»; c) «CJ1ClJ,yeT 3a»; )I() «'1HCJIO X lJ,eJIHTCH Ha '1HCJ10
y 6e3 OCTaTKa»; 3) «'1HCJ10 X HaXO)(HTCH Me)l{)lY '1I1CJ1a-
MH Y HZ».
YKa)l(HTe Cpe)(H HHX OTH0111eHHH CTpororo no-
p5llJ,Ka.
2. OTHOllIeHHR HeCTpororo nOpRJI,Ka.
HapH)I,Y C OTHOllIeHHeM «x<y» (<<x>y») B
MaTeMaTHKe paCCMaTpHBalOT OTHOllIeHHH
«X!J» (<<xy»), npe,lLCTaBJIHIOlIIHe co60H
06De)l,HHeHHe OTHOllIeHHll «X<Y» H «x = y»
PHC. 44 (<<X>!J» H «x=y»). rOBOpHT, tITO «xy» HBJIHeT-
CH OTHOllIeHHeM HecTpororO nOpH)I,Ka.
BOOolIle, OTHOllIeHHe R Ha MHOlKeCTBe X Ha3blBaeTCH omHOUl£HueM
HeCmpOCOco nopflOKa, ecJIH OHO pe<PJIeKCHBHO, aHTHCHMMeTpHtIHO H TpaH-
3HTHBHO. TaKHe OTHOllIeHHH HBJIHIOTCH OO'be)l,HHeHHHMH OTHOllIeHHH
CTpOrOrO nOpH)I,Ka C OTHOllIeHHeM TOlK)I,ecTBa.
K 9TOMY THny npHHa,u,JIe)l{aT, HanpHMep, TaKHe OTHOllIeHlIH: «He
BblllIe) (Ha MHO)l{ecTBe JIIO)I,eH, cpaBHHBaeMbIX no POCTY); «He 6oJIbwe»
(Ha MHO)l{ecTBe )l,eHCTBHTeJIbHbIX tIHCeJI); «ObITb )l,eJIHTeJIeM» (Ha MHOlKe-
CTBe HaTypaJIbHblX tIHCeJI).
ECJIH Ha MHO)\{ecTBe X = {3; 1; 5; 2; 4} paccMoTpeTb OTHOlliNme He-
cTpororo nOpH)I,Ka «X!P, TO rpa<}:> 9Toro OTHOllIeHHH B Ka)l{)l,Oll BepllIHHe
6Y,lLeT HMeTb neTJIlI (pHC. 43). 4TOObi nOCTpOHTb rpacpHK OTHoweHlIH
«x!J» Ha Koop,lLHHaTHoit nJIOCKOCTH, Ha)l,O K rpa<}:>HKY OTHOllIeHHH «x<!J».
3a)l,aHHOrO Ha TOM )l{e MHolKecTBe X, )l,OoaBHTb TOtIKH (l; 1), (2; 2),
(3; 3), (4; 4), (5; 5), JIe)l{alIlHe Ha oHcceKTpHce npHMoro yrJIa (pHC. 44).
YnpaacueUHJI
I. CpCIJ,H CJICIJ,YlOmHX OTH0111CHHii YKa)l{HTC OTHOlIleHHH HCCTpororo nOpHIJ,Ka:
a) «'1HCJ10 X HC MeHI.111e '1HCJIa Y (Ha MHO)l(eCTBe R); 6) «'1eJIOBeK X - IJ,Pyr '1e.OBeKIi
Y» (Ha MHO)l(ecTBC JlIO,lI,cii); B) «'1eJIOBCK x 3HaKOM C '1eJIOBeKOM Y» (Ha MHO)l{eCTBe JlIO-
..cii); r) «'1HCJ10 x KpaTHo '1HCJ1Y Y» (Ha MHO)l{eCTBe N); IJ,) «IJ,OM x He BbilIle .l\OMa !p
(Ha MHO)l(ecTBC IJ,OMOB ropOlJ,a); e) «OKPY)I{HOCTI. X JIe)l(HT BHYTPH OKPY)I(HOCTH !p
(Ha MHO)l{CCTBe OKpY)I{HOcreii nJIOCKoCTH).
2. Ha MHO)l{CCTBC X= {2; 4; 6; 8; lO} 3alJ,aHbI OTH0111CHHH R: «x He OOJII.111e !p
H Q: «x - lJ,eJIHTeJII. y».
a) .lI.OKII)I(HTe, '1TO R H Q - OTH0111CHHH HecTpororo I1OpHI.KII.
«I) nocTpoAre rp8I}JbI iTUX Q.TH0111eHHH. B 'lei( HX 'OTJIH'IHe?
B6

. B) nOCTpOi\T8 rpa<PHKH OTHOlIleHHH R H Q H3 KOOP.!IMH3THOII IlJIOCKOCTH. Cpaa-
HHTe HK.
3. lI.aHo MHOJKeCTBO '1HCJlOBbiX BblpaJKeHHH A= {72-5; 19+8; 48: 6; 37.2},
Ha KOTOpOM 3BAaHO OTHOlIleHHe P: «3Ha'leHHe sblpIDKeHHII x MeHI.lIle HJlH paSHO 3Ha-
'1eHKIO BI.IpaJKeHHI1 !p.
a) OjJOPMYJlHpYHTe CBOHCTsa OTHOlIleHHII P.
6) 0603Ha'lHB KaJKIJ,Oe '1HCJlOBOe BI.lpaJKeHHe H3 MHOJKeCTBa TO'IKOi!:, nOCTpOHTe
rpacp ornOlIleHHI1 P.
4. Y!3Ct)PB3HTC Ha KOOplJ,HHaTHOH nJIOCKOCTH rpacpHK OTH0111eHHII «x;;'y» , ecJIH
ORO 3a,'.l.aHO H3 MHOJKOCTSC: a) lI,eJIbiX HeoTpHlI,aTeJII.HblX '1HCeJI; 6) .!Iei!:CTBHTeJIbHblX
qHCeJI; B) cpaBHHTe rpacpHK lJ,aHHOrO OTHoweHHH, 3alJ,aHHOrO Ha R, C rpacpHKOM OT-
HOlIlCHHII «x>y» (TaK)KC 3alJ,aHHOrO Ha R).
5. Bo BTOpOM KJlaCCC Y'IB1I1HCCII pcwalOT TaKYIO 3a.naQY: «CocraBb BCC B03MO)KHblC
npHMepbl Ha IJ,CJleHHe (6e3 OCT3TKa) C '1HCJlaMH 12, 26, 4, 96, 13, 78, 3 (.N'! 6, c. 154,
2 KJI.).
a) ECJlH Y'Iaw.HecH 6YIJ,YT pCijlaTb 3TY 33J{a'lY MeTOIJ,OM. «no.ll.60pa», TO KaK Bbi
npoBcpHTC npaBHJ1I.IlOCTI. ce pC111eHHH? CKOJlI.KO Bccro npHMepoB 1J,0Jl)KHO nOJlY-
'1HTbCII?
6) BblllcHHTe. 0 KaKOM OTH0111eHHH H.!ICT pe'lb B )(aHHOH 3alJ,aQe H Ha KaKOM MHO.
lKeCTSC OHO paCCMaTpHBaeTCH.
B) nOCTpOHTC rpacp OTHOlIlCHHH «x IJ,CJlHTCIi Ha y 6C3 OCTaTKa» Ha MHOJKCCTBe
{12; 26; 4; 96; 13; 78; 3}. CKOJlI.KO CTpe.10K OH co.nepJKHT? CoBna.naer JlH OTBCT H3
3TOT Bonpoc C OTBeTOM 3a.na'lH IJ,JlII BTOpoKJlaCCHHKoB?
3. YnopRJI.O'feHHble MHO)KeCTBa. MHO)\{ecTBO X, Ha KOTOpOM 3a,lLaHO
OTHOllieHIIe nopH,AKa R (cTpororo 1I,1lH HeCTpororo), Ha3b1BaeTCH 'taC-
mUtlHo ynopfloOtteHHblM M1iO:J1CecI1l6OM. l.JacTo rOBopHT TaK)I{e, tJTO B 9TOM
CJIY'Iae MHO)\{eCTBO X ynopH,lLotJeHo OTHOllieHIIeM R.
TaK, B n. 1 Mbi nOKa3aJIII, tJTO MHO)\{ecTBO X={3; 1; 5; 2; 4} MO)\{eT
ObITb ynopH,lLotJeHo OTHOllieHIIeM «X<!J». 3m )l{e MHO)\{ecTBO MO)\{HO
ynopH,lLO'IIlTb Ii npii nOMOIll.Ii OTHOllIeHliH «X!J».
PaccMoTpliM Ha MHO)l{ecTBe X={3; 1; 5; 2; 4} OTHoweHHe «l.JIICJIO X
KpaTHo 'IliCJIY y». 3T0 OTHOllieHIIe TpaH3IITIIBHO: ecJIIi X KpaTHo y H
'y KpaTHo Z, TO X KpaTHO Z. .lJ.aHHoe OTHOllieHIIe aHTIICIIMMeTplitJHO: X
KpaTHO y Ii Y KpaTHO X O,lLHOBpeMeHHO TOJIbKO B TOM CJIYtJae, KOr,lLa x=y.
'KpOMe Toro, OHO pecp.neKCIIBHO: .n1060e 'IliCJIO II3 MHO)\{eCTBa X KpaTHO
caMoMY ceDe. C.ne,lLoBaTeJIbHO, OTHoweHlie «ObITb KpaTHbIM» Ha MHO)\{ecT-
Be X = {3; 1; 5; 2; 4} HBJIHeTCH OTHOllIeHlieM HecTpororo nOpH,lLKa.
rpacp ero II300pa)\{eH Ha pIIcYHKe 45.
CpaBHHM STOT rpacp C rpa<j:JoM OTHOllieHIIH «X!J» (pIIC. 43).
5IB.nHHCb rpacpaMII OTHollieHIIll HecTpororo nOpH,lLKa, OHH TeM He
MeHee OT.nlitJaIOTCH )J.pyr OT ,lLpyra. Ha rpacpe
,0THOllieHIIH «xy» .nIOObIe ,lLBe BepllIHHbI coe-
IJ.IiHeHbI O)J.HOll crpeJIKOll, Ha rpa<j:Je)\{e OTHO-
weHliH «X KpaTHO y» IIMeJOTCH BepllIHHbI, He
coe,lLliHeHHbIC CTpeJIKOll - HanpIIMep, HeT
CTpe.'1KIi, Coe,lLIIHHlOlIIell BepllIlIHbI 3 Ii 5.
rOBOpHT, tJTO OTHOllieHIIe «x!J» ynopH,lLotJII-
BaeT MHO)KecTBO X AUHeuHo.
OTHoweHIIH cTpororo nopH)J.Ka TaK)I{e MO-
rYT ynopH,lLO'IliBaTb MHO)\{ecTBO no-pa3HoMY.
TaK, OTHOllIeHlie cTpororo nopH,lLKa «x BbIllIe
fP, paccMoTpeHHoe B 3a,lLatJe 3 (n. 1), ynopH)(O- Pac. 45
87'

IIIfBaeT .lI.aJ.!HOe MHomec'tl3O JIIO.ll.eH HeJlJlHeHHO, a oTHorneHHe «x<!p
MHOlKeCTBo X = {3; 1; 5; 2; 4} ynopH.lI.otlHBaeT JIHHeimo.
BooolIle, eCJIII OTHOllieHIIe nopH.lI.Ka R (crpororo IIJIII HecTpororo) Ha
MHOJKecTBe X oOJIa.ll.aeT CBOitCTBOM: .lI.JlH JlIOObIX 3JIeMeHTOB x H y II3
MHolKecTBa X JIII60 xRy, JlH60 yRx, TO R - omHoUl£Hue AUHeUHOCO
nOpflOKa, a MHOJKecTBO X - AUHeUHO ynopfliJO'ieHHOe MHOJ/CeCmBO. ECJIII
MHOlKeCTBO X KOHelIHO II COCTOIIT H3 n 3.neMeHTOB, TO JIIIHeitHoe
ynopH.lI.O\llIBaHIIe X CBO.ll.IITCH K HYMepaIlHH ero 9J1eMeHTOB IIHCJIa-
MII 1,2, . . ., n.
HanpIIMep, MHOlKeCTBO HaTypaJlbHbIx \lHCeJI OTHOllieHIIeM «MeHbllIe»
ynopH.lI.o\llIBaeTCH .nHHeitHo. OTHollieHIIe ,lLeJIHMOCTH ynopH.lI.O'IHBaeT
MHOJKecTBO HaTypaJIbHbIX \lIICeJI HeJIIIHellHo.
JlIIHeHHo ynopH.nO'IeHHbIe MHOlKeCTBa OOJIa.naIOT pH.nOM CBOHCTB.
PaccMoTpIIM HeKOTopbIe II3 HIIX.
TIYCTb a, b, C - 3JIeMeHTbI MHolKecTBa X, ynopH)I,O\leHHoro OTHO-
rneHHeM R. ECJIH H3BeCTHO, lITO aRb II bRc, TO rOBopHT, \lTO 3JIeMeHT
b AeJ/Cum Me:»eoy BAeMeHnwMU a II C.
HanpIIMep, ecJIH MHOlKecTBO HaTypaJIbHbIX \lHCeJI ynopH)I,O\leHO OT-
HOllIeHHeM «x<y» , TO H3 Toro, \lTO 3<5 H 1<3, c.ne.nyeT, \lTO \lHCJIO 3
JIelKliT MelK)I,y \llIc.naMH 5 II 1.
MHOlKecTBO' X, JIIIHeitHo ynopH)I,O\leHHOe OTHOllieHIIeM R, Ha3bIBa-
eTCH OUCKpemHbtM, eCJIII MelK)I,y JIIOObIMII )l,BYMH ero 3JIeMeHTaMII JIelKliT
JlHllib KOHe\lHOe MHOlKecTBO 3JIeMeHTOB.
JlHHeitHo ynopH)I,O\leHHoe MHOJKecTBO Ha3bIBaeTCH nAomHbtM, eCJIH
.AJIH JIIOObIX )l,BYX pa3JIII'IHbIX 3JIeMeHTOB 9Toro MHolKecTBa CYlIlecTBYeT
3JIeMeHT MHolKecTBa, JIelK3IlI.IIit MeJK.lI.Y HIIMII.
TaK, MHOlKeCTBO Q paIlIIOHaJlbHblX \lIICeJI, ynopH.lI.O\leHHoe OTHOllIe-
HIIeM «X<!p, nJlOTHO, nOCKOJIbKY )l,JlH JlIOObIX paIlIIOHaJlbHbIX \lHCeJI x
H Y (X=f;y) CYlIlecTBYeT \lIICJIO z= xt y TaKoe, \lTO Z JleJK:IIT MelK)I,y x II y.
Yupa2KBeBBJI
J. Ha MHO>KecTBC X={-2; -5; 1; 3; 7; -4} 331J,3HO OTH0111CHHC P: «x!p.
BblHcHHTe, K3KHC H3 CJlCIJ,YIOlI1.HX BblCKa3b1BaHHA HBJlHIOTCH HCTHHHI.IMH: a) P - OT-
H0111CHHC cTpororo nOpHIJ,K3; 6) P - OTH0111CHHe HccTpororo nOpHIJ,Ka; B) X - JlH-
HCHHO ynopH)(O'lCHHOC MHO>KecTBO; r) X - HeJlHHCHHO ynopHIJ,O'lCHHOC MHO>KCCTBO.
2. BblHCHHTe, K3KHC H3 CJlC)(YIOlI1.HX OTH0111CHHA HBJlHIOTCH OTH0111CHHHMH JlHHCA-
HOro nOpH)(Ka: 3) «6b1TI. IJ,JlHHHee» (Ha MHO>KeCTBe OTpC3KOB); 6) «6I.ITI. nOIJ,'1HHCHHblM)
(Ha MHO>KeCTBC BOCHHOCJIY>KaIl.lHX); B) «6b1TI. Bbl111C no POCTY» (Ha MHO>KCCTBC y'lall.lHXCJ'I
KJlaCca); r) «6b1TI. 60JlI.111C» (Ha MHO>KecTBC pau.HoH3J1I.HI.IX '1HCCJI); )() «HaXO)(HTI.CJ'I
JlCBee» (Ha '1HCJlOBOii npHMoii).
3. .lI.aHo: X - MHO>KCCTBO 6YKB JlaTHHCKoro an(j>3BHTa,
A= {a; b; c; d; e; f}. B= {c; f}. C= {a; b; d},
D= {d; k; 1; m; n}. E= {k; 1; m; n}.
PaCCMOTpHTC Me>KlJ,y HHMH OTH0111CHHC R «6b1TI. nOIJ,MHO>KCCTBOM».
a) BblnH111HTe BCC napbl MHO>KeCTB, HaXO)(Hll.lHXCH B OTH0111CHHH R.
6) .lI.OK3>KHTe, '1TO R - OTH0111CHHC HecTpororo nOp5llJ,Ka. JlHHCHHO JlH OHO?
4. Ha cOBoKynHocTH nOIJ,MHO>KCCTB MHO>KCCTB3 I 3alJ,3HO OTHOWCHHC «6b1TI. cOO-
CTBeHHblM nO)!.MHO)\\CCTBOM». OnpClJ,eJIHTe cBOACTBa ,noro OTH0111eHHH. 51BJlHeTCH JlH
88,

OHO OTHOlIlCHHeM nOpHJlKa? ECJ!H «.!la» , TO KaKoft MO nOp1tJ10K: tTporRA HJlR ReCTpO-'
rHH, JIHHCHHblH HJIH Her?
5. Y Ka}l{HTe cPe.zlH CJ!C)(YIOIUHX OTHOlIlCHHH Te, npH nOMoIUH KOTOpblX MO)I{II()
ynopHJJ,O'lHTI. MHO}l{ecTBO yqaIl\HXCH KJIaCca: a) «6b1TI. cTaplIlc no B03pacTY»; 6) «3a.
HHMaTI.CH R O)(HOM H TOM }l{C KPY}I{KC»; B) «6b1TI. HC BbilIlC no POCTY».
6. Ha30BHTe OTHOlIlCHHH, npH nOMOIUH KOTOpblX MO}l{HO KaKHM-JIH60 06pa30M
ynopH.!lOqHTI. MHO}l{ecTBO X={6; 7; 2; 4; 1; 3; 5}.
7. B HaqaJII.HOM KYpCC MaTeMaTHKH yqaIUHCCH pClIlalOT TaKHC 3a)(a'lH:
a) (N2857, 2 KJI.) PaCnOJIO}l{H no nopHJJ,KY:
]) BqCpa, n03aBqCpa, 3aBTpa, nOCJ1e3aBTpa, CerO)(HH;
2) BTOpHHK, qerBCpr, nOHCJJ,eJII.HHK, cpc){a, cy660Ta, n,HTHHLI,a.
6) (N2 10, 2 KJI.) PaCnOJIO}l{H CJIC)(YIOIUHC qHCJ1a:
1) B nopHJJ,Kc B03pacraHHH: 706, 670, 459, 661;
- 2) B nOpH)(KC y6b1BaHHH: 347, 503, 305, 827, 799.
BblHCHHTe, KaKHC MHO}l{ccrBa ynopH)(O'lHBaIOTcH B 9THX 3aJJ,aqax. Ha30BHTe
OTHOlIlCHHH, npH nOMOIUH KOTOpblX npOHCXO)(HT 9TO ynopH){OqCHHC.
fi 4. OTOI>PAiKEHlIH
1. OT06pa)KeHHR. PaccMoTpIIM eIll.e O,1l.IIH BalKHbIll qaCTHbIll CJIyqaH
OOIll.ero nOHHTIIH COOTBeTCTBIIH - oToopalKeHIIH MHOlKeCTB. TIpli COOT-
BeTCTBIIII R MelK,1I.Y MHOlKeCTBaMII X II Y oopa3 3JIeMeHTa a E X MOlKeT
OKa3aTbCH nycTbIM, a MOlKeT CO,1l.eplKaTb II HecKOJIbKO 3JIeMeHTOB. - Omo-
6pa:»eeHueM MHOlKeCTBa X B MHOlKeCTBO Y Ha3bIBalOT TaKoe COOTBeTCTBIIe
MelK,1I.Y 3TIIMII MHolKecTBaMII, qTO oopa3 JIlOooro 3JIeMeHTa a E X COCTOIIT
II3 O,1l.Horo II TOJIbKO O,1l.HOrO 9JIeMeHTa MHOlKeCTBa Y. I1HbIMII CJIOBaMH,
.lI.JIH JIlOooro a EX Hall,1l.eTCH O,1l.IIH II TOJIbKO O,1l.HH 3JIeMeHT bEY, TaKon,
liTO aRb.
TaKIIM oopa30M, rpacpIIK OToopalKeHIIH MHolKecTBa X B MHOlKecTBO Y
He MOlKeT CO,1l.ep)l{aTb .lI.BYX pa3JIIIqHbIX nap (a; YI) II (a; Y2) C O,lLHOll II Ton
)l{e nepBOll KOMnOHeHTon. TIpli 3TOM ,1I.JIH JII060ro a E X B HeM Hall,lLeTCfl
napa BII,1l.a (a; b), r,lLe bEY. ECJIIf H30opa3IITb oToopalKeHlIe MHO)l{ecTB3
X B MHOlKecTBO Y rpa<p<>M, TO II3 KalK,1I.On TOqKII MHOlKeCTBa X oY,lLeT
BblXO,lLIITb O,1l.Ha II TO.ITbKO O,1l.Ha CTpMKa (pIIC. 46).
OTOOpalKeHIfH MHO)l{ecTB 0603HaqalOT TaK:
f
f: X-.Y, X....Y.
3,11.ecb f - CIfMBOJI caMoro OToopalKeHIIH. ECJIII npii OTOOpa)l{eHIfII f
9JIeMeHTY x COOTBeTCTByeT 9JIeMeHT Y, TO nIfllIYT:
f : x-.y IIJIII x.!....Y IIJIII y=f(x).
TI P II Me p 1. nYCTb X - MHOlKecTBO X Y
CTY,lLeHTOB B aY,1I.HTopHIf, a Y - MHOlKecTBO
CTOJIOB B TOll lKe aY,lLIiTopIIII, npIfqeM KalK,1I.bIH
CTyn.eHT CIf,lLIiT <;\1 O,1l.HIIM II3 CTOJIOB. COOT BeT-
CTBIfe «CTY,1I.eHT x CII,lLIiT 3a CTOJIOM y» 3a,lLaeT
OToopalKeHIIe X B Y. Oopa30M CTY.lI.eHTa x
npH 3TOM OToopalKeHlI1I HBJIHeTCH CTOJI, 3a
KOTOpbIM OH CII,lLIiT.
TI p II M e p 2. TIYCTb X = Y = N - MHO-
lKecTBO HaTypaJIbHblX qIICM. CoOTBeTCTBIIe PHC. 46
81)'

cll.ecS1TI-fQHaS1 3anl-lCb IIHCJII,X COCTOKT H3 Y I1H$I' onpe,AeJIS1eT (HO'-
Opa)\{eHHe N B N. TIpH 9TOM OTOOpa)\{eHHH 'IHCJIY 39 COOTBeTCTBYeT
IJHCJIO 2, a 'IHCJIY 45 981 - IJHC.nO 5 (39 - .D.BY3HatlHoe IJHCJlO, a
45 981 - nHTH3HaIJHoe IJHCJlO).
n p H M e p 3. nYCTb X - MHO)\{ecTBO IJeTblpexyroJlbHRKOB, a
Y - MHO)\{ecTBO OKpYlKHOCTeH. CooTBeTCTBHe «qeTblpexyroJIbHHK X
BnHcaH B OKPYlKHOCTb !f» He 51BJ1HeTCH OTOopalKeHHeM X B Y. TaK KaK
ecTb tleTbIpexyrOJlbHHKH, KOTOphle HeJIh3H
X Y BnHCaTb B OKPYlKHOCTb. Ho ecJJH IJeTI"'pex-
yroJlbHHK Bnn:caH B KaKYIO-HHOY,lLb OKpy)\{-
HOCTb, TO HH B KaKYIO HHYIO OKPY)\{HOCTb
OH He MOJKeT oblTb BnHcaH (HHaIJe 9TH OK-
PY>KHOCTH HMeJIH Obi IJeTbIpe 06II1He TOIJKH -
BepllIHHbI IJeTbIpexyrOJIbHHKa).
0603HaIJHM IJepe3 A MHO)\{eCTBO IJeThIpeX-
yrOJIbHHKOB, KOTopbIe MorYT obITb BnHca-
aJ HbI B OKPY*HOCTb. YKa3aHHoe BbIllIe COOT-
BeTCTBHe on pe,lLeJIHeT OTOopa>KeHHe MHO-
X Y >KecTBa A B MHO)\{ecTBO OKpY)\{HocreH Y. f 0-
BOpsn, 'ITa nOJIY'IHJIOCb oToopalKeHHe U3 MHO-
:JICectn8a X 6 MHO:HCeCm60 Y (a He MHO:HCeCm6a
X BY).
n p H Me p 4. CooTBeTCTBHe «B OKPY)\{-
HOCTh x BnHcaH KBa,lLpaT y» He HBJIHeTCSI
OToopalKeHHeM MHO)\{ecTBa OKPY)\{HOCTeH B
MHOlKecTBO K8a.D.paTOB, TaK KaK B O,lLHy H TY
0) )\{e OKPY1KHOCTb x MO>KHO BnHcaTb 6ecKOHeIJHO
MHoro pa3JIHQHbIX KBa,lLpaTOB.
X Y Ka>K,nGMY nO.ll.MHO)\{ecTBY A B X COOTBeT-
cTByeT OO)l.MHO)\{eCTBO B Y, KOTopoe COCTOHT
H3 oopa30B scex 9.'IeMeHTOB x MHO)\{eCTBa A
(T. e. H3 KOHIIOB Bcex CTpeJIOK, Ha'IHHalO-
II1HXCH B 3.neMeHTax MHolKecTBa A). 3T0
MHOlKecTBO Ha3bIBalOT 06pa30M A npH oToopa-
}KeHHR f Ii 0603HaIJaIOT f(A). HanpHMep,
eCJIU Ha pHcYHKe 46 B3HTb A={a; b; c},
8) TO f(A)={y; z}. MllO)\{eCTBO f(X) Ha3bIBalOT
06pa30M X npH OTOOpa)\{eHHH f (OHO COCTOHT
X Y H3 KOHIIOB scex npOBe,lLeHHbIX crpeJIOK).
z)
Puc. 47
.
¥npamHeHHH
1. Ha PHCYHKC 47 H306plUKeHbI rpacpbl pB3J1H'I-
HbiX C(J()TBeTcrBHiI:. KaKHe 83 3THX COOTBeTCTBHH IIB-
nlllOTCli Or06pIUKCHHHMH?' Hail:)J,HTe )J,JIH !Ka)l{Jloro COOT-
BCTcrBHII cro 06naCTI. OrnpaBJIeHHII, 06nacTI. npH6b1.
THH, 06nacTI. onpC)J,eJICHHH H MHOJKCCT80 3HaQCHHH.
2. LlaHbI il:Ba MHmKecTBa: X= {napHJK; MocKBa;
8ap1113Ba; Ta111KeHT; KpaKoB; Pllfa} H Y= {CCCP;

4>P8HUHB, rrOJlI.Wa,8eJIHK06pRTWU1B}. rroerpoATe
rpaq, COOTBeTCTBHB NeJK)(Y HHNH «fopo.l1 x H8XO,IUITCB
B rOCY,lI,apCTBe y». 51BJ1BeTCH JlH !m> COOTBeTCTBHe 01'06-
pa11<eHHeN? HaHARTe 06pa3 I1O,lI,MH011<ecTBa fMotXB3;
BaplllaB3}. '
3. K8>KJlOi 'OO'IKe M OTpe3Ka AB noCTaBHN B
COOTBeTCTBHe ee npoeKll.HIO Ha npBMYIO 1 (pHCYHOK
48). »BJlBeTCB JlH !m> COOTBeTCTBHe ot'06plI}!{eHH-
eM? Haii,ll,HTe 06pa3 OTpe3Ka AB npH' 3TON ore6pa-
)KeHHH.
4. MHOJKeCTBO X COCTOHT H3 scex KBa,ll,paroB HS
nJIOCKOCTM, a MHOJKeCTBO Y - H3 Bcex OKpYJKHOCTeH
liS TOH JKe nJIOCKOCTH, nOCTaBHM B COOTBeTCTBHe KaJKl1-oMY KBa,ll,PItTY BnHcaHHYIO
B Hero OKPYJKHOCTI., 51BJlHCTCH JIM 3TO COOTBeTCTBHe oT06paJKeHHeM X B Y?
5. 51BJIHeTcH JIH COOTBCTCTBHe «ypaBHeHHe X HMeeT KopeHI. Y» oT06paJKeHHeN
8
l
A 1 11 , 8 ,
PHC. 48
f
x-+y MHOJKecTBa KBa.npaTHbiX ypaBHeHHH B MHOJKecTBO '1HCeJl?
6. 51BJIHeTCB JlH COOTBeTCTBHe «4eJI()BeK Y - CblH 'IeJIOBeK3 .\I» OT06p81KeHHeM
f
x-+y MHOJKecTB3 BceJ( JllO,lI,eH B Ce6H? 51BJ1"eTCB JlH TaKHM OT06pSJKeHHeM 006TBeT-
f
CTBHe «4eJIOBeK Y - cTapwHii CbllI '1eJIOBeKa x»? 51BJIHeTCSI JlH oT06psJKeHHeM x-+y
COOTBeTCTBHe «4eJIOBeK Y - OTell. '1eJlOBeKa x»?
2. nOJIHblH npoo6pa3. 06pa1Hoe oT06pa)l(eHHe. npH oT06pa)l{eHHH
X.!.. Y Ka)!{)1.bIH 9JIeMeHT MHO)l{eCTBa X nepexo)J,HT B OA"H " TOJIbKO
O)J,HH 3JIeMeHT "3 Y. Ho Ha 3JIeMeHTbI MHO)l{ecTBa Y OrpaHH1JeHHH He
HaKJIa)J,bIBaeTCH - B HeKOTopbIe H3 HHX He oT06pa)l{aeTCH HH O)l,HH
.9JIeMeHT H3 X, B ,APyme oT06pa)\{aeTCH pOBHO O,lLHH 9J1eMeHT, a B HeKOTO-
pble - HecKOJIbKO HJIH ,Aa)Ke 6ecKOHe'IHO MHoro 9JIeMeHTOB.
HanpHMep, ecJIH X - MHO)l{ecTBO KBa)l,paTOB, a Y - MHO)l{ecTBO
OKpY)\{HOCTeii H Ka)l{)J.OMY KBaJI,paTY CTaBHTCH B COOTBeTCTBHe BnHC8HHaH
B Hero OKPY)I{HOCTb, TO B O,lLHY H TY )l{e OKPY)\{HOCTb OTo6pap<aeTCH 6e-
CKOHe1JHO MHoro KBa,lLpaTOB, a HMeHHO, Bce KBa,lLpaTbI, KOTopbIe onHcaHbI
BOKpyr 9TOll OKPY)I{HOCTH (pHC. 49).
Ha30BeM nOAHtHM np006pa30M 3JIeMeHTa y E Y npH OTOopa)\{eHHH
f
X -+ Y MHO)\{ecTBO Bcex 3JIeMeHTOB H3 X, KOTopbIe npH 3TOM oT06pa)\{e-
HHH nepeXo)l,HT B y, T. e. TaKHX x EX, 'ITO f (x) = y. 3T0
MHO)\{ecTBO 0603H.:i'IaJOT f- l (y). MTaK, f- l (y)= {xix EX Af (x)=y).
06pa3 X npH OT06pameHHH f COCTOHT H3 Bcex 3JIeMeHTOB BY, HMe-
JOllI.HX HenYCTOA nOJIHbIH npoo6pa3.
x
y
J> HC. 49
PItC.50
It

x-
y
ECJIH nOBepHYTb B OOpaTHYIO CTOPOHY Bce
CTpe.nKH rpacpa OTOopalKeHHH X!.... Y, TO nOJIY-
lIHTCH rpacp COOTBeTCTBHH Me>K,ll.Y MHO)\{ecTBa-
MH Y H X (pHC. 50). Ho 9TO COOTBeTCTBHe,
BooolI1.e roBOpH, He OY,AeT OTOopa)\{eHHeM Y
B X. Bo-nepBbIX, MO)\{eT CJIYlIHTbCH, liTO f(X)
He COBna,lLaeT C Y. A TOr,Aa nOCJIe TOro KaK
MbI nOBepHeM CTpeJIKH, B Y oY,AYT TOLJKH, H3
KOTOpbIX He BbIXO,AHT HH O,AHOll CTpe.nKH.
OTOopa)\{eHHH X!.... Y, TaKHe, LJTO f (X) = Y,
Ha3bIBalOT OTOopa)\{eHHHMH X Ha 8ce MHOJ/Cecmeo Y. B COBpeMeHHOH
MaTeMaTHlIeCKOll JIHTepaType TaKHe OTOopa)l{eHHH Ha3bIBalOT ClOp'bet(,-
mU8HblMU (OT cppaHIIY3CKOro npe,lLJIora sur - «Ha»).
BTOpbIM oc.nO)\{HeHHeM npH H3MeHeHHH HanpaBJIeHHH CTpe.nOK HB-
JIHeTCH TO, liTO H3 HeKOTopbIX 9JIeMeHTOB MHO)\{eCTBa Y MorYT BbIXO,lLHTb
HeCKo.nbKO (a TO H oeCKOHellHO MHOro) CTpe.nOK. 3T0 npOH30ll,lLeT C TeMH
9,leMeHT3MH, nOJIHbIll npooopa3 KOTOpbIX CO,lLep)\{HT oOJIee O,lLHOrO 9JIe-
MeHTa. 4TOobI H3oe)\{aTb 9Toro OCJIOlKHeHHH, Ha,lLO nOTpeOOBaTb, 'ITOobI
no.nHbIH npo06pa3 KalK,lLOrO 9.neMeHTa y E Y CO,lLep)\{a.n He oo.nee O,lLHOrO
9.TleMeHTa (npH 9TOM OH MO)\{eT obITb H nycTbIM). TaKHe OTOopa)\{eHHH
B cOBpeMeHHoH MaTeMaTHlIeCKOll .nHTepaType Ha3bIBalOTCH UH'beKmU8HbIMU
(cpaBHHTe C.nOBO «HH'beKIIHH»). Ha pHcYHKe 51 H30opa)\{eH rpacp HH'beK-
THBHOro OTOopa)\{eHHH X BY.
f
ECJIH oT06pa)\{eHHe X-+ Y HH'beKTHBHO, TO Ka)\{,lLOMY 9JIeMeHTY Y
H3 f(X) COOTBeTCTBYT O,lLHH H TO.nbKO O,lLHH 9JIeMeHT x H3 X, OTOopa)\{a-
IOUJ.HllCH B Y (O,lLHH 9JIeMeHT HaH,lLeTCH nOTOMY, liTO Y E f (X), a ,ABYX ObITb
He MO)\{eT, TaK KaK f HH'beKTHBHO). 3HallHT, eCJIH H3MeHHTb HanpaB.neHHe
Bcex CTpe.nOK Ha oopaTHoe, nOJIYlIHTCH OTOopa)\{eHHe MHO)\{eCTBa
f (X) (HO He BCer,lLa MHO)\{eCTBa V!) B MHO)\{ecTBO X. 3T0 OTOopa)\{eHHe
Hil3bIBalOT 06pamHbiM K OTOopa)\{eHHIO f H 0003HallalOT f-l. OHO ClOp'beK-
THBHO -,lL0 H3MeHeHHH HanpaBJIeHHH CTpeJIOK H3 Ka)\{,AOrO x E X
BbIXO,lLHJIa CTpe.nKa, a nOCJIe Hero B Ka)!{,AOM x E X KOHlIaeTCH CTpe.nKa.
JIYlIllIe Bcero OOcTOHT ,lLe.no B CJIYlIae, KOr,lLa OTOopa)\{eHHe f oo.na,lLaeT
060HMH CBOHCTBaMH, T. e. H ClOp'beKTHBHO, H HH'beKTHBHO. TaKHe OTO-
opa)\{eHHH Ha3bIBalOT 83auMHo-oiJH031ta'iHbtMU. B COBpeMeHHOll MaTeMaTH-
qeCKOll .nHTepaType qaCTO npHMeHHIOT ,lL.nH HHX TepMHH «6ueKmU8Hble
omo6paJ/CeHUR.» HJIH, KOpOqe, «6tLeK4UU» (OT JIaTHHCKoro bis-
«!tBa)\{,lLbI») .
I
(.1TaK, oT06pa)\{eHHe X-+ Y B3aHMHO-0,lLH03Ha'IHO B TOM H TOJIbKO B
TO:'vl CJIyqae, eCJIH npH 9TOM OTOopa)\{eHHH no.nHbIll npooopa3 Ka)\{,lLoro
9J1eMeHTa Y E Y COCTOHT H3 O,lLHOrO 9JIeMeHTa x EX, T. e. eCJIH Ka)\{,lLbIH
9JIeMeHT Y E Y HBJIHeTCH oopa30M O!tHOro H TO.nbKO O,lLHOrO 9.neMeHTa
xEX.
TI p Ii Me p. TIYCTb X - MHO)\{eCTBO naJIbTO B rap,lLepo6e, a Y -
t.IHO)\{eCTBO KPlOqKOB B 9TOM rap,lLep06e. TIocTaBHM B COOTBeTCTBHe Ka)\{-
,lLOMY nanbTO KpIOLJOK, Ha KOTOpOM OHO BHCHT. ECJIH Ka)\{,AOe naJIbTO BH-
PHC. 51
92

CUT na' KplO1JKe (a He ne*UT, CK3*eM; - H3
no.ny), TO 9T0 COOTBeTCTBHe HB.nHeTCH OT06-
pa*eHHeM X B Y. 3T0 oT06pa)\{eHHe HH'beK-
THBHO, ecJIH HH Ha O,AHOM KplOIIKe He BHCHT
DoJIee O,AHoro naJIbTO (HO MorYT 6bITb H CBO-
Do,lLHbIe KpIOIIKH), H ClOp'beKTHBHO, ec.nH Bce
KplO'IKH 3aHHTbI (HO Ha HeKOTopbIX KplOIIKax
MOrYT BHceTb HecKO.nbKO na.nbTO). HaKOHeIl,
9T0 OT06pa)\{eHHe 6HeKTHBHO (B3aHMHO-0,AHO-
3HaIIHO), eCJIH Ha Ka)\{,lLOM KplOIIKe BHCHT O,lLHO
H TOJIbKO O,lLHO naJIbTO.
f
Ec.nH X-V - B3aHMHO-0,lLH03HaIIHoe OT06pa)\{eHHe X Ha Y (6u-
eKTHBHoe oT06pa)\{eHHe), TO oopaTHoe OTOOpa)\{eHHe f- l TO)\{e B3aHMHO-
0,lLH03HaIIHO (pHC. 52). n09TOMY B TaKOM CJIYIIae rOBopHT TaKme 0
83auMHo-OOH03Ha'iHOM coomeemcm8uu Me)\{,lLY MHO)\{eCTBaMH X H Y.
npH B3aHMHO-0,lLH03HaIIHOM COOTBeTCTBHH Ka)\{,lLbIH 9JIeMeHT x H3 X,
paBHo KaK H Ka)\{,lLbIll 9.neMeHT y H3 Y, nOna,lLaeT B O,lLHY H TO.nbKO B O,AHY
napy, npHHa,lLJIe)\{alI1.YIO rpacpHKY oT06pa)\{eHHH.
YupamOeOHJI
x
y
PHC. 52
1. KIDK)lOMY '1eJIOBCKY x CTaBHTCSI B COOTBCTCTBHe cro MaTI. y. SIBJlSleTCSI JlH
9TO COOTBeTCTBHC oT06paJKCHHeM MHOJKCCTBa JIIO)lCH X B Ce6S1? ClOp'bCKTHBHO JIH OHO?
I1H'bCKTHBHO JIH 9TO oT06paJKCHHe? I1MCCT JIH 3TO OT06pIDKCHHC 06paTHoc?
2. Ha pHcYHKe 53 CTpeJIKaMH nOKa3aHbI oT06paJKeHHSI MHOJKCCTBa TpcyrOJII.HH-
KOB B MHOJKCCTBO KpYJKKOB. KaKHe H3 3THX oT06paJKCHHH ClOp'bCKTHBHbl? KaKHe H3
HHX HH'bCKTHBHbI? ECTI. JIH CpClIH 3THX oT06paJKCHHH 6HCKTHBHOC?
3. Ka)f{)(OMY 3pHTeJIlO KHHOTeaTpa nOCTaBHM B COOTBCTCTBHC KpCCJIO, Ha KOTOpOM
OH CH)lHT. npH KaKOM YCJlOBHH 3TO COOTBCTCTBHC SlBJISleTCSI oT06paJKCHHCM MHOJKe-
CTBa X 3pHTeJICH B MHOJKeCTBO Y KpCCeJI? npH KaKOM YCJlOBHH 3TO oT06paJKeHHe
ClOp'beKTHBHo? Bcer)la JIH OHO HH'beKTHBHo?
4. MHOJKCCTBO X COCTOHT H3 KOeTeH )lOMHHO, a MHOJKCCTBO Y - H3 '1CTblpeX Hr-
poKOB. Ka)f{)(oH KOCTH ,qOMHHO nOCTaBJICH B COOTBeTCTBHC HrpoK, )lepJKalllHH 9TY
f
KOCTI.. SIBJlSleTCSI JIH 9TO COOTBeTCTBHC ClOp'bCKTHBHblM oT06paJKCHHCM X_V? I1Hb-
eKTHBHO JIH OHO? liTO SlBJlSleTCSI nOJIHblM npoo6pa30M )laHHOrO HrpoKa?
5. MCJK/{Y MHOJKCCTBOM HMeH X= {AH)lPCH; BOPHC; MHXaHJI; A.ncKceH; KOH-
CTaHTHH; BaCHJIHH; KHpHJIJI; BaJICHTHHa; CeMCH; CotpI.Sl; MapHSI; Q,1I.ra; THMYP;
Tp<xj>HM; IOpHH; SIKOB} H MHOJKCCTBOM Y 6YKB PyccKoro aJI<jJaBHTa YCTaHOBJIeHO
COOTBCTCTBHC, npH KOTOpOM KaJK)lOMY HMCHH conOCTaBJISlCTCSI ero ncpBaSi OYKB8.
PHC. 53
00

x y Z $lBJlSleTCR nil MO, COOTBeTCT8He OT06pa-
>KeHHeM MHO>KeCTBa X B MHOxreCTBO Y?
I1H'beKTHBHO JlH OHa? ClOp'beKTHBHO JlH
OHO? Haii,ltHTe nOJlHblii np006pa3 MHO-
>KCCTBa r JI 3CHbiX 6YKB.
3. 3KBH8a.11eIlTHWe MHO)l[ecT-
Ba. 06a MHO)\{ecTBa, H306paJKeH-
Hble Ha pHCYHKe 52, COCTOHT H3
lIeTblpeX 9JIeMeHTOB. Boo6m,e, KG-
PHC. 54 HellHbIe MHO/KeCTBa, Me/KJ\Y KOTo-
pbIMH CYlI1.ecTBYeT B3aMMHO-0,lLHO-
3HallHOe COOTBeTCTBHe, HMelOT nopOBHY 9JIeMeHTOB. Ha30BeM JIJ06t,re J\Ba
MHO}KecTBa, Me)\{,lLY KOTOpbIMH MO)\{HO YCTaHOBHTb B3aHMHO-0,lLH03HaLJ-
Hoe COOTBeTCTBHe, 9K8U8aAeHmHblMU. BYJ\eM nHCaTb B 9TOM CJIYtlae
X........ Y. HanpHMep, MHO)\{ecTBa HrpOKOB JIJ06bIX ,lLBYX CPYTOOJIbHbIX
KOMaH,lL 9KBHBaJIeHTHbI. B 9TOM MO)\{HO y6eJ\HTbCH, H He nepectlHTbIBaH
HX HrpOKOB, a BbICTpOHB 06e KOMaH,lLbI Ha nOJIe H 3aMeTIIB, 'lTO npOTHB
KalK,lLOrO HrpOKa O,lLHOll KOMaH,AbI CTOHT O,lLHH H TOJIbKO O,lLHH HrpOK BTG-
poft: KOMaH,lLbI.
nOKa)\{eM, liTO OTHOllIeHHe 3KBHBaJIeHTHOCTU MHO)\{ecTB 06JI3J\3er
CBOllCTBaMH pe<PJIeKCHBHOCTH, CHMMeTpHlIHOCTH H TpaH3HTHBHOCTH.
I1HbIMH c.nOBaMH, ,lLOKa)\{eM CJIe,AYIOmHe YTBep)\{J\eHHH:
a) ,lLJIH .nlOooro MHO)\{ecTBa X HMeeM X........ X;
6) eCJIH X........Y, TO Y........X
B) eCJIH X........Y H Y........Z, TO X........Z.
qToobI ,lLOKa3aTb nepBoe yYBep/K,lLeHHe, J\OCTaTOtlHO nocraBHTb K3/K-
,lLOMY 9JIeMeHTY x E X B COOTBeTCTBHe TOT /Ke caMbIH 9JIeMeHT: x-+x.
3THM Onpe,lLeJIHeTCH B3aHMHO-0AH03H311HOe COOTBeTCTBHe Me/K,lLY X H X,
a nOTOMY X........ X.
.lJ.oKa)\{eM Tenepb YTBep/KfteHHe 6). OrHollIeHHe X........ Y 03HallaeT, 'lTO
CYlI1.ecTByeT oHeKTHBHoe OTOOpa)\{eHHe f: X-+Y. ECJIH nOBepHYTb
CTpe.rIKH, TO nOJIYtlHTCH oHeKTHBHoe OToopameHHe Y -+X (Be,lLb B
Ka/K,lLOH TO'IKe y E Y KOHlIaeTCH pOBHO O,lLHa CTpeJIKa). 3HaLJHT, ecJIH
X........Y, TO Y........X.
HaKOHeu, ,lLOKameM YTBep)\{,lLeHHe B). TIYCTb X........ Y M Y........Z. 3T0
3HaIlHT, liTO CYIlI.eCTBYIOT 6HeKTHBHbIe oT06pa/KeHHH f : X-+Y H q> : Y-+
-Z. B03bMeM KaKOll-HJ.f0Y.IJ.b 3JIeMeHT x EX. EM}' COOTBeTcTByeT 9JIe-
MeHT y= f (x) H3 Y, a 9JIeMeHTY y COOTBeTCTBYeT 3JIeMeHT Z=q> (x) H3 Z.
TorJ\a OTOOpa)\{eHHe x-+z Ii 6YJ\eT OHeKTHBHbIM OTOOpa)\{eHHeM X Ha Z
(plte. 54). 3HaLJHT, X........Z.
nOCKO.nbKY OTHOllIeHHe X........ Y HB.nHeTCH 9KBHBaJIeHTHOCTblO, co-
BOKynHocTb BCex MHOlKeCTB pa36HBaeTcH Ha KJIaCCbI 9KBHBaJ1eHTHbIX
,lLpyr ,lLPyry MHO)\{eCTB. npH TaKOM pa30HeHHH, HaUpJlMep, MHO)\{ecTBO
yrJIOB KBa,n,paTa nona,n,eT B O,lLUH KJIaCC C MHOJKeCTBOM BpeMeH ro,lLa,
MHO)\{ecTBOM Hor Y JIHCbI, MHOJKeCTBOM KOJIec aBTOMOOHJIH H T. ,lL.-
Bee 9TH MHO)\{ecTBa 9KBHBaJIeHTHbI ,lLPyr ,lLPyry, TaK KaK CO,lLep/KaT no
lIeTbIpe 9.neMeHTa. I1HbIMH CJIOfIaMH, npH TaKOM pa30HeHHH ,lLBa KOHellHblX
MHO)\{ecTBa nOna,lLalOT B O,lLIIH KJIaCC B TOM H TOJIbKO B TOM CJIytlae,
KorJ\a OHH HelOT O,lLHHaKOBoe LJHCJIO eMeHTOB.
f

1 2 J ... 6
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2 '" 6 8 10
PHC. 55
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· · : _ 1 2 -i  :  ...
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7 5 J 1 2 * 6
PHC. 56
...
mo n03BOJIHeT onpe,AeJIHTb HaTypaJIbHOe '1HCJIO KaK 06wee CBOHCTBO,
KOTOpblM 06.nMaeT KJIaCC 3KBHBC!-JIeHTHblX ,npyr ,npyry KOlfeqHbIX MHO-
)KeeTB. no CYTH ,AeJIa, HMeHHO TaKHM nYTeM BBO,nHT HaTyp8JIbHbie LJHCJI3
B HaLJaJIbHOll llIKo.ne. 4T06bI 06MICHHTb, LJTO TaKoe LJHC.nO «,ABa», PHCY-
JOT ,ABYX II,bInJIHT, ,ABa aBTOM06HJIH, ,nBYX pe6HT H T. ,A. To 06Il1.ee, LITO
eCTb Y Bcex 3TH X MHO)KecTB, H ecTb HaTypaJIbHOe LJHCJIO «,ABa». TaK )Ke
MO)\{HO BBeCTH LJHCJIa «O.I\HH», «TpH» H T. .1\. 4HCJIO «HYJIb» COOTBeTCTBY-
eT npH 3TOM nycTOMY MHO)f{eCTBY (3TOT KJI3CC CO.l\ep)KHT JIHWb o)tHO
MHO)KecT80, TaK KaK He CYIlI.ecTBYeT pa3JIHLJHbIX nYCTblX MHO)\{eCTB).
OnHcaHHblH no.l\XO.l\ K nOHHTHIO HaTypaJIbHoro LJHCJIa Tpe6yeT npe.ll.-
BapHTeJIbHoro onpe,AeJIeHHH IIOHHTHH KOHe'lHoro MHO)KeCTBa, a 3TO
onpe,AeJIeHIIe TPy.nHO Jl,aTb, He 3HaH, LJTO TaKoe HaTypa.nbHoe LJIICJIO.
n09TOMY BMecTO Hero npHMeHHeTC5I HHOH no,AXo,n, KOTOpblH 6Y,lLeT onN-
caH B rJI3Be IX.
IloHnHe 9KBHBaJleHTHOCTH n03BO.1lHeT cpaBHIIBaTb Me)\{,lLY co6oii
He TOJIbKO KOHe'lHble, HO H 6ecKOHetlHble MHO)KecTBa. ECJIH ,ABa MHOJl<e-
CTBa X H Y 9KBHBaJIeHTHbI, TO rOBopHT, LJTO OHH HMelOT OOUHaK08ytO
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9TO TO )\(e C8MOe, qro '1HCJIO HX 3.neMeHTOB.
Moo)\{ecI'BO, HMelOlII.ee TY )Ke MOUJ,HOCTb, tlTO H MHO)KeCTBO N H8-
TYPaJIbHbIX LJHCe.n, Ha3blBalOT C'lemHbtM. BCHKoe CLJeTHOe MHO)\{ecT80
6ecKOHeLJOO, HO KaJK.I\OMY 3JIeMeHTY TaKoro MHO)KecTB8 MO)\{HO nOCTa-
BHTb B COOTBeTCTBHe HaTypaJIbHOe LJHCJIO TaK, tlTO see 3JIeMeHTbI OKa-
JKYTCH 3aHYMepoBaHbl.
HanpHMep, MHO)\{ecT80 'IeTHbIX LJHCeJI C'IeTHO. Ha pHcYHKe 55 no-
Ka3aHO, KaK YCTaHoBl:ITb B3aHMHO-0)J.H03Ha'iHOe COOTBeTCTBHe Me)I{.ny
3THM MHO)KecTBOM H MHO)KeCTBOM N seex HaTypaJIbHbIX 'IHCeJI. Boo6w.e,
At060e 6ecKOHeIlHoe nooMHo:J/Cecmeo MH03freCTTUJa N cqemrw. CtleTHO H MHO-
)KecTBO Z I1eJIbiX qUCeJI. Ha pHCYHKe 56 YKa3aHO, KaK ero 3aHYMepo-
BaTb.
YnpaacHeHBJI
L AOKaJKHTe C'lernOCTb CileAYJOlI1.RX MHO)l(ecJ'B: a) He-
qeTHbiX HarypaJIbHblX qHCe.n; 6) Ky60B HaTypaJIbHblX '1H-
ceJI; B) TO'leK n.llOCKOCTH, 06e KOOp.!\HHaTbi KOTOpblX-
HaTypaJIbHble '1HCJla.
2. AOKaJKHTe, '1TO MHO)l(eCTBO TO'IeK oTpe3xa AB
9KBHBaJIeHTHO MHO)I(ecTBY TOqeK m06oro APYroro <rrpe3Ka. 0
3. AOKIDKHTe, qTO MHO)l(ecTBO TO'IeK rpaHHlI.bI KBa,D.-
paTa 3KBHBaJIeHTHO MHO)l(eCTBY TO'IeK OKPYJKHOCTH.
4. AOKaJKHTe, '1TO MIIO)l(eCTBO TO'IeK nyqa OA ItKBH-
lIaJIeHTHO MHOiKeCTIIY ,'O'leK nY'Ia DB (pRC. 57).
A
8
PRC. 57


f.ll8Ba V
KOOPHHATLI, YPABHEBHH H HEPABEHCTBA
 1. KOOPJ(RHATbI HA IIPHMOR:
t. KOOPAHHaTbi Ha npSlMOH. B 9TOM nYHKTe MbI nOKa)\{eM, KaK 3a-
AaTb nO.nO)\{eHHe TOqeK npHMOll C nOMOIll.blO 'IHCeJI. B03bMeM npHMYIO I
H Ha Hell TOqKH 0 HE. Ha30BeM TOqKY 0 Ha'iaJlOM KoopouHam, a E -
eiJUHU'iHOU mO'iKou. TIpHMeM ,lL.nHHY oTpe3Ka OE 3a eouHu4Y U3MepeHUJI.
iJJlUH, a HanpaB.neHHe I OT TOqKH 0 K TOqKe E - 3a nOAOJ/CumeAbHoe Ha-
npa8AeHue. TIocTaBHM Ka)\{,lLOll TOqKe M npSlMOH I B COOTBeTCTBHe'ee
IWOpowwmy, T. e. TaKoe qHC.nO x, qTQ:
a) MO,lLYJIb qHc.na x paBeH paccToHHHIO OT 0)1.0 M : 1 x 1= I OM I;
6) ec.nH M*O, TO qHC.nO x nOJIO)\{HTeJIbHO, KOr)l.a TOqKa M JIe)\{HT
Ha JIY'Ie OE, H oTpHIIaTe.nbHo, KOr,lLa M JIe)\{HT Ha npoTHBono.nO)\{HOM
JIY'Ie.
YCJIOBHe 6) MO)\{HO CcpOpMy.nHpoBaTb nO-)l.PyroMY, CKa3aB, qTO KOOp-
.l1HHaTa x TO'IKH M no.nO)\{HTeJIbHa, ecJIH HanpaBJIeHHe OT TO'IKH 0
K TOqKe M no.nO)\{HTe.nbHO, H oTpHIIaTeJIbHa, eCJIH 9TO HanpaB.neHHe
OTpHIIaTeJIbHo. 113 ,lLaHHOrO BbIllIe onpe,lLeJIeHHH c.ne,lLyeT, LJTO KOOp,lLH-
HaTa TO'IKH 0 paBHa HY.n1O (paCCTOHHHe I 00 I paBHO HYJIIO), a KOOp,lLH-
HaTa TO'IKH E paBHa e)l.HHHIIe.
TIpHMYIO C 3a,lLaHHOll Ha Hell CHCTeMOll KOOp,lLHHaT Ha30BeM KOOpOU-
ftamHOU npflMOU. ECJIH TOqKa M HMeeT KOOp)l.HHaTY x, TO 6Y,lLeM nHcaTb:
M (x) (pHC. 58). Ka)\{)l.oii TOqKe M KOOp,lLHHaTHOll npHMoll COOTBeTCTByeT
onpe!l,e.neHHoe 'IHCJIO x - KOOp,lLHHaTa 9TOll TOqKH, a Ka)\{,lLOMY qHCJIY
x - O,lLHa TO'IKa M, HMelOw.aH 9TO qHC.nO CBoell KOOp,lLHHaToH. TaKHM
o6pa30M, coomeemcmeue x-+M MeJ/Coy MHOJ/CecmeOM oeucm8ullleAbHbix
'iUCeA U At/iOJ/Cecm80M mO'ieK npJlMOU 83aUM/iO-OOH03/ia'i/io.
TIocTpoeHHoe COOTBeTCTBHe Me)\{!I,Y TO'IKaMH npHMoii H qHc.naMH n03-
Bo.nHeT reOMeTpHqeCKH H306pa)\{aTb COOTHOllIeHHH Me)\{)l.Y qHCJIaMH H,
Hao60poT, CBO,lLHTb pemeHHe reOMeTpHqecKHx BonpocoB Ha npHMoll
K OnepaIHHM Ha)l. qHc.naMH. B nepBYIO Oqepe,lLb BbIHCHHM, KaK H306pa-
lI(alOTCH Ha npHMoll HeKoTopble qHc.nOBbIe MHO)\{eCTBa, T. e. MHO)\{ecTBa,
cocToHw.He 113 )l.ellCTBHTe.nbHbIX qHCeJI.
MHO)\{eCTBO qHCeJI x, Y,lLOB.neTBOpHIOW.HX HepaBeHCTBY a:::;;; x, Ha3b1-
BaeTCH 'i1lG.108bIM AY'iOAt H 0603HaqaeTCH [a; +00[, Ha KOOp,lLHHaTHOH
npHMOll 3TO MHO)\{eCTBO 11306pa)\{aeTCH .nyqOM AX, H,lLYIlI.IIM B no-
JIO)KHTe.lbHOM HanpaB.neHHH OT TO'IKH A (a), npHqeM caMR ToqKa A
96

npI1Ha,lLJleiIO!T .'IY'IY (puc. 59). 411c .
JIOBOIf .nY'I ]-00; a] COCTOHT 113 a
qHce.n, YlloB.neTBOpHIOIlI.HX Hepa-
BeHcTBY xa. OH 11306pa)\{aeTCH
.nyqOM, H,lLYIlI.HM OT TOqKH A B OTpH-
uaTe.nbHOM HanpaB.neHHH (pHC. 60).
MHO)\{eCTBO qHce.. Y 1l0B.neTBO-
pHlOlIIHX HepaBeHcTBY a<x, Ha3bI-
BaeTCH omKpbtmbtM liUCA08btM AYliOM
H 0603HaqaeTC5I Ja; + 00[. OHO
H306pa)\{aeTCH omKpbtmbtM AYliOM
Ha KoopllHHaTHoll npHMoll, T. e.
JIyqOM AM C oT6poweHHoll TOLJ-
KOll A. 4T06bl OT.nHqaTb Ha PH-
cYHKax .1Y'I OT OTKpblTOrO JIYLJa, a)
MbI 6YlleT 0603HaLJaTb Haqa.no AYlia
qepHOIf TOqKOH, a Haqa.no omKpbt- 0)
moco .nyqa - KPY)\{KOM (pHc.61).
LJucA08blM ompe3KOM Ha3bIBalOT
MHO)\{ecTBO qHce.n, KOTopble Y,lLOB-
.neTBopHIOT llBOHHOMY HepaBeH-
CTBY axb. Ero 0603HaqalOT (a; bJ. Ha KOOp,lLHHaTHOll npHMoi
LJHC.nOBOll oTpe30K H306pa)\{aeTCH ompe3KoM C KOHIIaMH A (a) H B (b)
(KOHIIbI npHHa,lL.ne)\{aT OTpe3KY, pHC. 62, a). HepaBeHCTBo a<x<f1
3a,lLaeT liUCA080U npoMeJ/CymoK, 0603HaqaeMbIH ) a; bl. OH H306pa-
)KaeTCH omKpbtf1lbtM ompe31WM AB, T. e. OTpe3KOM AB C oT6pollIeR-
HblMH KOHIIaMH A H B (pHC. 62, 6).
.
E
'.
M(x)
PHC. 5H
.
a
:.
.
A(a)
PHC. 59
a
I
A (a)
PRC. 60

a
o
A(a)
PRC. 61
...
A(a)
B(b}
:.
:J
A (a)
o
B(b)
..
PHC. 62
YupamHeHHH
I. OrMCTbTe Ha Koop.!\HHaTHOH npS!Moii CJle.l1YIOlI1.He TO'IKH!
1 1 1
A (-3. 5), B (4-f), C(6')' D ().
2. OTMCTbTe Ha KOOP.l1HHaTHOii npS!Moii TO'IKH A (-3) H B (5). qeMY paBHo p
CTOS!HHC MeJKJ{Y 9THMH TO'IKaMH?
3. 3a.!\aHbI 'IHCJIOBbiC MHO}KeCTBa M=]-4; -I[, K=[-2; 5]. l1306pa3HTe Ba
KOOp.l1HllaTHoii npS!Moii MHO}KCCTB3:
a) M n K;
6) M UK; B) M ' U K'; r) M ' nK. ():{onOJlHeHHSI 6epyrcSi B MHO}KCCTBe .l1eiiCTDIl.
TeJIbHbIX 'IHCeJI.)
4. 11306pa3HTe Ha KOOP.l1HH3THOii npS!Moii MHO}KCCTBa
1 1
Nn[-4; 36'] H Zn[-4; 7]'
5. B KaKYIO TO'IKY ncpeii.l1eT TO'IKa A (-I), eCJIH OHa CHa'laJI3 CJl,BHHeTCSI BDpaoo
Ha 7 e,l1HHHI.{, a nOTOM BJICBO Ha 10 e.n;HHHI.{?
6. 11306pa3HTe Ha KOOp.l1HHaTHoii npS!Moii CJIe.n;yrolI1.He MHO}KecTB3:
a) {xlxERtd<x<5}; 6) {xlxEZA-3<X<4};
B) {xlxERA-4<x<2}.
7. .lI.aHbI MHO}Kec'l'Ba A= {xJxER A-4<:x<24}. B= {xlxER J\-2<X<5}
C={x/xERA-5<x<1}.
11306pa3HTe Ha KOOp.l1HHaTHoii npS!Moii CJlc.!\YIOlI1.He MHO}KecTBa;
a) AnBnC; 6) AUBUC; B) (AnB)UC; r) (A / UB / )nC;.l1) (A"B)UC.
4 N. 702
m

I
a
I
e
I I , I"
Ora) f. .M(x)
I
O/(a)
!
a
j ..
M(x)
PIIC. 63
PHC. 64
2. npe06pa30BaHHe KOOpJl.HHaT Ha npHMoH JIHHHH. HaLJa.no KOOp,lLH-
HaT 0 H e,AHHHtlHYIO TOLJKY E MO)\{HO BbI6HpaTb Ha npHMOH npOIl3Bo.nb-
110 (.nHllib 6bI TOLJKH 0 H E He COBna,lLa.nH ,lLpyr C ,lLPyrOM). npH nepeXO,Ae
OT O,lLHOH CHCTeMbI KOOp,lLHHaT Ha npHMOH K ,lLpyroH CHCTeMe Koop,J,IfHaTbI
TOLJeK MeHHIOTCH. PaCCMOTpHM cJle,lLYIOIIIHe npeoopa30BaHHH CHCTeMbI
KOOp,lLHHaT Ha npHMOH.
a) nepeHOC HaqaJla KOOp.lI.HHaT. TIpH 9TOM npeoopa30BaHHI! Ha1JaJIO
KOOp,lLHHaT 0 nepeHOCHTCH B llpyrylO TOLJKy 0 ' , a 1l.TJHHa ellHHII1JHOrO
OTpe3Ka H HanpaB,leHIfe Ha npmlOH OCTalOTCH HeH3leHHbIMII (T. e. pac-
CMaTpHBaeTCH HOBaH CHCTeMa KoopJJ,lfHaT (0 ' ; E ' ), rlle 10E 1= 10' E ' I, a
HanpaB.neHHH OT 0 K E H OT 0 ' K E ' COBna,lLalOT (pHC. 63».
0003Ha1JHM IIepe3 a KoopllHHaTY TO'IKJI 0 ' B cTapoH CHCTeMe. TIYCTb
TOIIKa M HMe.na KOOp,lLHHaTY x. HaH.iI.eM ee HOBYIO KoopllHHaTY x'.
EC.TJH TOLJKa M JIe)\{IIT cnpaBa OT 0, a TOLJI<a 0 ' pacnOJIO)\{eHa Me)\{,lLY
011 M, TO 10M I=x, 100 ' I=a, 10 ' M I=x-a. 113 pHcYHKa 63 BH,lLHO, LJTO
x' = 10 ' M 1= 10M 1-100 ' I, H nOToMY x' =x-a.
MO)\{HO )l.OKa3aTb, '1TO paBeHCTBO
I
X =x-a
BbI!10.TJHHeTCH npll .n1060M pacnOJ10)\{eHIIH TO'IeK 0, 0 ' H M I).
npOBC.!\CM .!\OKa3aTc.bCTBO .!\JliJ CJlY'IaH, H306pa)KCHHOrO lIa PHCYHKC 64. B 3TOM
CJIy':ac TO'lKa 0' pacllo.1mKcHa cJlcna or 0, H nOToIY cc Koop.!\KHaTa oTpHuaTc.%Ha:
a=-IOO'I. 4Hc.1a)Ke x H x' nOJlO)KHTeJIbHbl: x=IOMj, x'=IO'MI. Ho B 3TOM c.1Y'lae
10' MI=IOMI+IOO'I, H n09TOMY CHOBa x'=x+ (-a)=x-a.
(1)
TI P H Me p 1. HaH)l.eM HOBYIO KOOp,lLHHaTY TO'lKH A (5), eC.111 Ha-
IIa.no KOOp,lLHHaT nepeHeceHo B TOLJKY 0 ' (-4). TIo cpopMYJIe (1) HaXO,lLHM:
x' =5-(-4)=9.
TI P H M e p 2. nOCJIe nepeHoca HaIIa.na KOOp,lLHHaT B TOtIKY 0 / (3)
KOop,lLHHaTa TOtlKH A CTaJIa paSHa -7. HafuI.eM Koop,AHHaTY 9TOii TOLJ-
KH B nepBOHaLJa.nbHoH CHCTeMe KOOp,lLHHaT.
B ,lLaHHOM c.nYIIae a=3 H x' =-7. 3HaLJHT, -7=x-3, H n03TOMY
x=-4.
n p H M e p 3. TIepBOHaLJa.nbHaH KOOp,lLIIHaTa TO'IKH A 6blMJ paB-
Ha 5, a nOCJIe nepeHoca HaLJaJIa Koop,lLBHaT OHa CTa.na paBHa -2. B Ka-
KylO TOLJKY nepeHec.nH HaLJaJIO KOOp,lLHHaT?
TaK KaK x=5, x ' =-2, TO -2=5-a. n09TOMY a=7.
6) H3MeHeHHe HanpaBJIeHHH OCH. OcTaBHM Tenepb TOtlKY 0 HenO)l.-
BH)\{HOH, a TOtlKy E 3aMeHHM T01JKOH E', CHMMeTpHLJHOH E OTHOCflTeJIb-
HO O. TIpH 3TOM npeo6pa30BaHliH ellHHHIIa H3MepeHHH .AJlHH OTpe3KOB
Re MeHHeTC", HO no.nO)\{HTeJIbHoe HanpaBJIeHHe CTaHOBHTCR OTpHIIaTeJIb-
HbIM, 1\ Hao60poT. n09TOMY Koop1!.HHaTbl Bcex TO'{eK MeHHIOT 3J!DR. Ha-
1) 3,QCCb H AaJlee x' - eC03Ha'ICIiHC HOI!OH, a x - CT';[ioii KOOp)(IIHaTL! flJ'iKH,
98

npHMep, HOBaH KOOp,lLHHaTa TO'I- I I ,
KH A (4) paBHa -4, a TOtlKH a E E'ra)
B (-5) paBHa 5. TOJIbKO KOOp,lLH-
HaTa TO'IKH 0 oCTaeTCH HeH3MeH- PRC. 65
HOH. ¥lTaK, B 9TOM CJIYtlae HOBaH
KOOp,lLHHaTa X' 3aBHCHT OT cTapOH CJIe,lLYIOIlI.HM oopa30M:
X'=-X.
,.
M(x)
B) H3MeHeHHe MaCIllTa6a. npH 9TOM npe06pa30BaHHH TO'IKa 0
OCTaeTCH HenO,lLBHmHOH, a TOtlKa E 3aMeHHeTCH TOtlKOH E ' , HaXO,lLHIlI.eH-
CH no TY me CTOPOHY OT 0, tlTO H TOtlKa E (pHC. 65). nYCTb KOOp,lLHHaTa
TOtlKH E ' paBHa a>O. B03bMeM TOtlKY M, CTapaH KOOp,lLHHaTa KOTOpOH
paBHa x, a BOBaH paBHa X'. npe,lLnO.nOlKHM }J)IH npOCTOTbl, tlTO 9Ta TOq-
Ka .ne}!(HT cnpaBa OT Ha'IaJIa KOOp,lLHHaT. Tor,lLa tlHC.nO x nOKa3bIBaeT,
BO CKO.nbKO pa3 oTpe30K OM ,lL.nHHHee oTpe3Ka OE, a - BO CKO.nbKO pa3
[OE'J .1JIHHHee [OE], X'- BO CKO.nbKO pa3 [OM) ,lL.nHHHee [OE'J. I1HblMfi
CJIOBaMH, eCJIH BblopaTb 3a e,lLHHHIIY H3MepeHHH ,lL.nHH oTpe30K OE, TO
10M I=x, 10E'I=a H x' = :g;,\ = : . I1TaK, x' =  . TaK lKe ,lLOKa3bI:
BaeTCH, 'ITO x' =, ecJIH TOtlKa M JIelKHT c.neBa OT O. 3HatlHT, eCAU
a
dAUHa eOUHUliHOcO Ompe3Ka yeeAuliUBaemCfl 8 a pa3, KOOpOUHambt 8cex mOo
'leK npflhWU YMeHbutalOn1Cfl 8 a pa3.
HanpHMep, ec.nH ,lL.nHHa e,AHHHtlHOrO OTpe3Ka Y,lLBOHTCH, TO HOBaSi
KOOp,lLHHaTa TO'IKH A (8) CTaHeT paBHa 4, a HOBaH KOOp,lLHHaTa TO'IKR
5
B (-5) craHeT paBHa -"2' ECJIH lKe ,lLJ1HHa e,lLHHH'IHOrO OTpe3Ka YMeHb-
1
llIHTCH B 3 pa3a, TO HOBaH KOOp,lLHHaTa TOtlKH C (7) CTaHeT paBHa 7 : 3=
=21.
YupamHeHHJI
1. nocnc nepeHOCB HB'IBJlB KOOPAHHBT KOOP)(HHBTB TO'lKH A (-7) CTaJIB pBBHa 2.
B KaKYIO TO'lKY nepeHeceHo na'laJIO KOOp.!\HHaT?
2. no KaKOH <jx>pMYJIe npeo6pB3YIOTCH Koop)(RHaTbl, ecnR HOBaH CRCTeMB KOOp-
)(HHaT no.1Y'leHa nocne nepeHOCB HB'IaJlB KOOp)(HHaT B TO'lKY A (5) H YBeJIH'IeHRs[
e)(HHHLI,bI 11-.1RHbl B 4 pa3B?
3. no KaKoH <jJopMyne npco6pB3YlOTCH KOOP)(HHBTbl, ecnH CHa'laJIB YBeJIH'IHTb
el1-HHHLI,Y M<JcwTa6a B 4 pa3B, a nOTOM nepeHecTH Ha'lanO KOOPI1-HHaT B TO'IKY, KOOp-
)(HHaTa KOTOpOH paBHa 5?
4. YI3BecnIo, 'ITO 5 rpa)(ycoll lIlKaJlbl lI.eJIbCHH paBHbl 9 rpa.nycaM lIlKaJIbl <Pa...
peHreHTa, npH'ICM Ha'lanO OTC'ICTa lIlKaJIbl lI.eJIbCHH HaXO.1\HTCS! B TO'IKC, HMelOlI1.eii' m}
lIlKanc <1>apeHrcHTa Koop.!\HHaTY 32. HanHlllHTc <jJopMynbl ncpCXo)(a OT lIlKaJIbi lI.eJIb.
CHS! K 111Ka.1e <1>apcHreHTa n 06paTHo.
3. HeKOTopble 3a,lLaLJH aHaJIHTHLJeCKOH reOMeTpHH Ha npHMOH JlHHHH.
nOKameM Tenepb, KaK C nOMOIll.blO Koop,lLHHaT pellIalOTCH reoMeTpH''Ie.
CKHe 3a,J,atlH.
3 a ,lL a 'I a 1. n ycmb oaHbl oee mOliKU A (a) U B (b) Ha IWopiJuHa.m.HOtl
npflMou. HauoeM paccnlOflHue MeJ/Coy amUMU mOliKaMu.
4*
99

P e III e H II e. PaCCMOTpIIM CHa'Ia.na 'IaCTHbIH CJIY'Iai'l, KOr,lLa O,lLHa
H3 9TIIX TO'IeK COBna,lLaeT C Ha'IaJIOM KOOp,lLIIHaT. 0603Ha'IHM KOOp,lLIIHa-
TY BTOpOH TO'IKII 'Iepe3 C. TIo Onpe,lLe.neHIIIO KOOp,lLHHaT Ha npHMOH
(C. 96) MO,lLYJIb 'IIICJIa c paBeH paccToHHHIO OT O,lLO A, T. e. lOA 1= lei.
HanpHMep, paccToHHIIe OT Ha'Ia.na KOOp,lLIIHaT ,lLO TO'IKII A(-4) paBHo 4.
Tenepb MbI y)\{e MO)\{eM peIllIITb 3a,lLa'IY B OOIll.eM cJIY'Iae. .lJ.JIH 9Toro
nepeHeceM Ha'Ia.no KOOp,AHHaT B TO'IKY A (a). Tor,lLa no cpopMy.ne (1)
n. 2 HOBaH KOOp,lLIIHaTa b ' TO'IKII B CTaHeT paBHa b ' =b-a. 3Ha'IIIT,
paCCTOHHHe OT A ,lLO B paBHO 1b'1, T. e. Ib-aj.
I1TaK, MbI ,lLOKa3MII, 'ITO paccmOflHue MeJ/COY mOliKaMU A (a) U
B (b) pa8HO Ib-a/:


IABI= Ib-al.


(1)


3Ta cpopMYJIa CnpaBe,lLJIIIBa npii JIIOObIX npe,lLnOJIO)\{eHIIHX 0 3HaKax
KOOp,lLIIHaT a II b.
TI p II M e p 1. HaH,lLeM paCCTOHHIIe Me)\{,lLY TO'IKaMII A (4) II B (-6).
no cpopMYJIe (1) HaXO,lLIIM:
IABI= 1-6-41= 1-10/=10.


TI p H Me p 2. PeIllIIM HepaBeHcTBO Ix-41
7.
reoMeTpH'IeCKHH CMbICJI 9Toro HepaBeHCTBa TaKOB: HaHTH KOOp,lLH-
HaThI TaKIIX TO'IeK B, 'ITO paccToHHIIe OT B ,lLO A (4) He oo.nbIlle 'IeM 7.
reOMeTpH'IeCKH HCHO, 'ITO Bce TaKHe TO'IKII .ne)\{aT Me)\{,lLY TO'IKaMH C KO-
op,AHHaTaMII 4-7=-3 II 4+7=11. .3Ha'IIIT, -3
x
1l.
3 a,lL a 'I a 2. F08opflm, limo mOliKa C ompe3/w AB oeAum eco 8 om-
HomeHUU m: n, eCAU lAC!: !CBI=m : n. HauoeM KoopouHamy x
mOliKU C ompe3Ka c KOH/-faMU A (Xl) U B (X 2 ), Komopafl oeAum eco 8 om-
Homef-tuu m: n.
.lJ.JIH npOCTOTbl 6Y,lLeM C'IIITaTb, 'ITO XI<X 2 (<j:JopMYJIa, KOTOPYIO MbI
BblBe,lLeM, BepHa II npii XI>X 2 ), TaK KaK TO'IKa C (X) JIe)\{HT Me)\{,lLY TO'I-
KaMII A (Xl) H B (X 2 ), TO XI<X<X 2 , TI09TOMY X-XI>O, II, 3Ha'IHT, IACI=
= IX-XII=x-XI' TO'IHO TaK )\{e X 2 -X>O, II n09TOMY ICB I =X 2 -X
I ACI m X-Xl =.!!!:..
n09TOMY II3 YCJIOBIIH I CB I = n nOJIY'IaeM, 'ITO X 2 -X n 113.
9Toro ypaBHeHIIH HaXO,lLHM n (X-XI)=m (x 2 -x), II nOTOMY X=
nXI + mX 2
m+n
I1TaK, KoopOUHama mOliKU C, oeAflU¥!U ompe30K AB 8 omHomeHUU
m : n, 8blpaJ/Caemcfl cjJOPMYAOU
X = nXI +mx 2 ( 2 )
m+n '



oe X\- /wopouHama mOliKU A, a X 2 - KOOpOUHatlW mOliKU B.
B 'IaCTHOCTH, Cepe,lLIIHa oTpe3Ka ,lLe.nHT ero B OTHOIlleHHII 1: 1.
3Ha'IIIT, KoopouHama cepeouHbl Ompe3Ka 8blpaJ/Caemcfl cjJOPMYAOU


X\+X 2
x cep =



(3)


]00




n p H M e p 3. HaH,lJ,eM Koop,lJ,IIHaTY cepeAHHbI oTpe3Ka AB, eCJIII
A = A (5), B = B (-9). no $opMYJIe (3) nOJIYtJaeM:
_ 5+(-9) --2
xcep - 2 - .


n p H M e p 4. HaHAeM KOOp,lJ,HHaTY T01JKH, AeJIHIII.eH OTpe30K AB II
OTHOIIIeHHH 2 : 3, eCJIH A =A (1) H B=B (-9).
no <fx>pMYJIe (2) nOJ1Y1JaeM:
_ 3.1+2.(-9) _ -15 _ 3
x- 2+3 -----s-- - .
3 a ,lJ, a 1J a 3. IJ,eH.mp f7lJlJ/Cecmu iJeyx M{1f;C ml U m 2 , nOMeU¥!H.H.bl

coonwemcnweH.H.O 8 mOttKu A u B, lleJ/Cum H.a ompe3Ke AB u iJellum e20'
omH.omeH.UU m z : mi. HaiiiJeM KOopiJUH.amy /{eH.mpa mflJ/Cecmu, eCllti
M{1f;Ca ml H.axoiJumcfl 8 mottKe A (Xl)' a M{1f;Ca m z - 8 mottKe B(x.).
no <fx>pMYJIe (2)


mlx1 +m2 x Z
X -
1!.T - ml +mz .


YupalKHeHHH
1. Haii:AHTe paCCTOSIHHe Me/KllY TO'IKaMH A (a) H B (b), eCJIH:
a) a=-6, b= 12; 6) a=-12, b=-6;
B) a=7, b=-4; r) a=-3, b=-19.
2. Haii:AHTe TO'IKH, OTCTOSl1II.He OT T01jKH A (3) Ha 7 eAHHHl{.
3. PeI1IHTe HepaBeHcTBa:
a) Ix-31<6; 6) Ix+5k9; B) 12x+121..;;=8;
r) 13x-151 <9.
4. Haii:AHTe KooMHHaTY TO'lKH, AeJIH1II.eii: oTpe30K C KOHl{aMH A (a) H B (b) B 0'1'-
HOI1IeHHH m: n, eCJIH:
a) a=8, b=22, m=3, n=4; 6) a=27, b=9, m=2, n=l.
5. TO'lKa C(4) AeJIHT oTpe30K AB B OTHOI1IeHHH 5: 2. Haii:AHTe KoopAHHaTY (I
TO'IKH A, eCJIH KoopAHHaTa b TO'IKH B paBHa 8.


i 2. KOOP,l),lIHATbI HA IIJIOCKOCTII


1. ,l\eKapToBa npBMoyrOJlbHaB CHCTeMa KoopJl.HHaT Ha nJIOCKOCTII.
4T06bI nOCTaBHTb B COOTBeTCTBHe Ka)!{AOH TotJKe npHMoH ee KOOp)J.H

HaTY, MbI 3aAaJIH Ha npHMoH cHCTeMY KooPAHHaT, T. e. napy TotJeK
(0; E). nOJIO)!{eHHe TOtJKH Ha nJIOCKOCTH 3aAaeTCH ABYMH tJHC

JIaMH - a6CUHCCOH: H oPAHHaTOH:. 4T06bI OnpeAeJIHTb 3TH tJHCJIa,
nOCTpOHM CHatJaJIa CHCTeMY KOOp,lJ,HHaT Ha nJIOCKOCTH. l{JIH 3TOro
BbI6epeM Ha nJ10CKOCTII ynopHA01JeHHYIO TPOHKY T01JeK (0; E 1 ; E.),
TaKYlO, 1JTO npHMbIe OE I H OE 2 nepneHAHKYJIHpHbI, a AJ1HHbI OT:'
pe3KOB OE 1 H OE z paBHbI eAHHHue: IOE 1 1=IOE z l=1 (pHC. 66). Tor)J.3
Ka)!{)J.aH H3 npHMbIX OE I H OE 2 CTaHeT KooPAHHaTHoH npSIMoH C Ha-
1JaJ10M OTC1JeTa O. nepBYIO H3 HHX Ha3hIBaIOT OCblO a6c/{ucc H 0603Ha-
1JalOT Ox, a BTOpylO - OCblO opiJuH.am H 0603HatJaIOT Oy. COBoKynHOCTb
06eHx oceH Ha3bIBaIOT iJeKapmoeoii npflMoY20llbH.oii cucmeMoii KoopiJUfUltTl
xOy. 4aIlJ,e Bcero OCb a6cUHCC pacnOJIaralOT Ha pHcYHKe ropH30HTaJIb-
HO II BbI6npalOT nOiIOIKUTe,1JbHOe HanpaBJ1eHHe Ha HeH CJIeBa HanpaBd.


101




y I
I
8 I
I
I!l.(em{3(!pmb E 2 - IIIemfJepmb +--- -
.
I I I I I I I I 1/ .
I I I I 1 I I I
I I I a I E, I I I X
1l1l.(emt3epmb IY l.(em8epmb -
I I I I I I 1/ I I
I I I Y I I
-M(-2j4) I i
- fj I I
:5 I
2
1
A .1
I
-"2 -1 a 1 '; I X
L
I I
PHC. 66
PI1C. 67
3 OCb Op.n:HH3T - BepTHK3JIbHO H BbI6HP31OT rrOJIOIKHTeJIbHOe HanpaBJIe-
HHe Ha Hell CHlI3Y BBepx. DJIOC1<OCTb, Ha KOTOpOll 3a.n:aHa npBMO)TOJJb-
HaB ,lJ,eKapTOBa CHCTeMa Koop.ZJ:IIHaT, Ha3hIBalOT Koopi)UH.amH011 n.1OCKO-
cmblO. OCH KOOp..'l.HIIaT .n:eJISIT KOOpJI.HHaTHYIO rrJJOCKOCTb Ha 4 llacTH,
Ha3bIBaeMbIe tlemeeplWlMu.
Terrepb MbI YIKe MOIKeM YKa3aTb 1JHCJIa, orrpe)J.eJISIIOIII.He rroJloJ!{eHH
T01JKH M Koop.n:HHaTHoll rrJIOCKOCTH. l{JISI 3Tora orrycTHM H3 TO'IKII M
nepneH,lJ,HKYJISIpbI Ha OCH a6CUHCC H op.n:HHaT. nYCTb A - TOIIKa rrepe-
celleHHB OCH a6cUllcc C rrepneH,lJ,IIKy.'JSIpOM, orrYIlJ,eHHbIM Ha 3TY OCb, a
B - T01JKa nepeCe1JeHHSI OCH op.n:HHaT C orrYIII.eHHbIM Ha Hee rrepneH.n:H-
KYJISIpOM. Ha30BeM a6c/{uccot1 T01JKH M Koop.n:HHaTY T01JKH A (Ha KOOp-
)l.HHaTHoll rrpSIMoH Ox), a opiJuH.amot1 3TOH T01JKH - KOOp.n:HHaTY T01JKH
B (Ha Koop.n:HHaTHoll npHMoll Oy). 4HCJIa X H Y Ha3hIBaIOT iJeKapnW8bIMU
npJlMoY20AbH.bUtU KoopiJuHamaMu T01JKH M. ECJIH TOliKa M HMeeT KOOp-
.lI.HHaTbI x H y, TO rrHWYT M (x; y).
3HaKH KOOp.n:HHaT BH.n:HbI H3 CJIeAYIOIII.eH Ta6JIHUbI:
1J eTBepTb I A 6cHCC3 IOPAHH3T3
I + +
II - +
III - -
IV + -
Op.n:HHaTbI Bcex T01JeK OCH a6cuHcc paBHhI HYJIIO, TaK )!{e KaK H a6cu,HCCbI
Bcex T01JeK OCH Op.n:HHaT. Y Ha1JaJIa KOop,lJ,HHaT 0 06e KOOp,lJ,HHaTb! paB-
RbI HyJIIO.
n p H Me p I. Ha1JepTHM TOliKY M (-2; 4). l{JISI 3TOro OTJIOIKHM Ha
OCH aOCI.I.HCC OTpesOK OA .n:JIHHbI 2 B JIeBYIO CTOPOHY, a Ha OCH Op,lJ,HHaT
OTpe30K OB .n:JIHHbI 4 BBepx. npoBe.n:eM tJepe3 rrOJIYlIeHHble T01JKH
nepneH.n:HKYJISIphI K OCSIM (pHC. 67). I1cKOMa5I TOQKa M - T01JKa
nepeCe1JeHHSI 3THX rrepneH.n:HKYJISIpOB. TOT )!{e pe3YJIbTaT MO)!{HO
no.1JYQHTb HHbIM 06pa30M. CHa1JaJIa CTpOHM T01JKY A, a rrOTm.l npo-
BO.n:HM lIepe3 Hee rrepneH.n:HKy.1JBp KOCH a6cu,HCC H OTKJIa,lJ,blBaeM
Ha HeM BBepx OTpe30K ,lI.JIHHbI 4.
102

n p If M e p 2. HaHAeM KOOp-
AIfHaTbI T01JKIf M (pIfC. 68).
113 T01JKH M OnYCTliM nepneH-
AHKYJISIpbI Ha OCH KOOpIIHaT.
TaK KaK T01JKa A HaXOHTCSI
cnpaBa OT T01JKH 0 Ha paccToSIHHH
5 OT :nOH T01JKH, a T01JKa B-
HH)!{e TO'IKIf 0 Ha pacCToSIHHH 3 OT
Hee, TO KoopAIIHaTbI T01JKH M paB-
HbI 5 H -3, M =M (5; -3).
OTMeTIIM, 1JTO eCJIH T01JKa M
HMeeT KOOpIIHaTbI a H b, M =
=M (a; b), TO ee npoeKUHSI A Ha
OCb a6cUHcc (OCHOBaHHe nepneHAHKY.1JSIpa, onYIII.eHHoro H3 T01JKH M
Ha OCb a6CUHCC) IIMeeT KoopAHHaTbI a H 0, A =A (a; 0), a ee npoeK-
UHSI B H3 OCb opHHaT - KOOpIIHaTbI 0 H b, B=B (0; b).
- Ij
1
1\
- -1 0 1 213 . ,x
-1 I
I
2.2
"
.!.}.l1 rM
I'
PIIC. 68
YnpamueUHH
I. 11306pa3HTc J/a KOOp.l\Hl!aTHOH nJlOCKOCTH TO'lKY M (x; y) H TO'IKH M 1 (lxl; y),
M 2 (x; Iyl). M3 (Ixl; Iy/). KaKHMH .l\OJDKHbI 6bITb KOOp.l\HHaTbI TO'lKII M, '1T06bI Bee
3TH TO'lKH COBna.l\aJlH? B KaKOM CJIY'Iae nOJlY'IaTCH '1eTbipe pa3J1H'IHbIe TO'IKH? BbI-
nHWHTe KOOp.l\HHaTbI npoeKUHH TO'leK M, MI. M 2- Ms Ha OCH Koop.l\HHaT.
2. OocTpoihe TpeyrOJlbHHK ABC e BepWHl!aMH A (-4; 0). B(O; 6), C(-I; -I).
3. OOcTpoHTe OKPY>KHOCTH C ueHTpoM A (3; -4), O.l\Ha 113 KOTOpbIX KacaeTcn OCR
a6CUHCC. a BTopaH - OCH Op.l\Hl!aT.
4. ITocTpoiiTe OKPY>KHOCTb C ueHTpOM A (-3; 4), npOXO.l\HWYIO 'Iepe3 Ha'laJlO
KOOpI!HaT.
5. OocTpoihe TpeyrOJlbHHK ABC C BepwHHaMH A (2; 7), B (6; 5), C (8; 1). HaH.l\HTe
Koop.'\HHaTbl TOIJeK M, N, P, .l\eJlH1II.RX nOnOJlaM COOTBeTCTBeHHO oTpe3KH AB, BC, CA,
II nocTpoiiTe rpeyroJlbHHK MNP.
6. OpoBe!J;HTe npHrYIO '1epe3 TO'lKH A (I; 2). B (-4; -3) H OKPY>KI!OCTb C u.el!TpOM
C(l; I) II paHycoM 5. no '1epre>KY onpe.l\eJlHTe Koop.'\I!HaTbJ TO'!eK nepCCe'lCHHJI
npHMoH II OKPY>KIIOCTH.
2. npC06pa30BaHIfSI KOOpJl.IfHaT Ha nJIOCKOCTIf. Ha OHOH H TOH IKe
nJIOCKOCTII MOIKHO no-pa3HoMY BhI6HpaTb CHCTeMbI npm-royrOJIbHbIX
AeKapTOBb!X KoopAHHaT.
a) nepeiloc Ha'HlJIa KOOpJl.HHaT. B03bMeM CHC€MY KoopHHaT
xOy If BbIoepeM Ha KoopHHaTHoH nJIOCKOCTH T01JKY 0' (a; b). lJepe3 :JTY
T01JKY npOBeeM np5JMbIe, napaJIJIeJIbHble OCSIM KoopHHaT, H BhI6epeM
Ha HIfX HanpaBJIeHIISI, cOBnaAalOIII.He C HanpaBJIeHIfSIMH oce" a6cUIfcc
If opJ.I!HaT ClICTeMbI xOy (pHC. 69). ECJIH BbI6paTb TOT IKe eAIfHH1JHbIH
OTpe30K, 1JTO H JISI CIICTeMbI xOy, TO nOJIY1JIIM BTOpylO CIICTeMY KOOpH-
HaT x'O'y'. rOBOpSIT, 1JTO OHa nOJIY1JeHC.J H3 CHCTeMhI xOy nepeH.OCOM.
H.atf.aAa KoopOUHam B T01JKY 0'.
B03bMeM Ha KoopAIfHaTHOH nJIOCKOCTH KaKYIO-HIf6YAb T01JKY M.
nYCTb ee KOOP.D.HHaTbI B HCXOAHOH CHCTeMe KoopAIfHaT paBHbI x If y,
a B HOBoil ClICTeMe Koop.'J:HHaT OHH paBHbI x' H y'. BbIBeAeM <fx>PMYJIbI,
BbIpaiKaK>llUIe x' H y' 1Jepe3 x H y. LI.JISI 3Toro onycTHM H3 TotJeK 0' H M
nepneHJ.lIKYJISIpbl Ha OCH a6cUHcc If OP.D.HHaT. Ha OCIf a6cUIfcc nOJIY1JaeM
103

IY Y' I I I
I-M f1(X;yJ - I-
0' a'b'
'I}' X'
0 01 M X
I
T01JKH 0 1 H M1' HMelOIII.He a6CUHC-
ChI a H x. Do opMYJIe nepeHoca
Ha1JaJIa KoopHHaT Ha KOOpIIHaT-
HOH: npm,IOM (opMYJIa (1) n. 2
S 1) nOJIY1JaeM: x' =x-a. T01JHO
TaK)!{e OKa3bIBaeTC5I, 1JTO y' =
=y-b. y!TaK, CPOPMYAbl, Bblpa-
JlcalOlI.1ue x' H y' If.epe3 x U y, UMe-
10m BUi):
{ <=x-a,
y =y-b,
ei)e a U b - Koopi)UH.ambl HOBOCO
H.alf.a/la Koopi)UH.am.
6) H3MeHeHHe HanpaBJIeHHSI oceH. DycTb Ha1JaJlO KOOp,J,lJH3T oc-
TaeTCH HenOBH)J{HbIM, a HanpaBJIeHHH oceH H3MeH5IIOTC5I Ha npoTlJBo-
nOJIO)!{HbIe. 5ICHO, 1JTO npH 3TOM 06e KoopHHaTbI MeH5IIOT CBOlf 3HaK,
T. e.
(1)
PHC. 69
{ x;=-x,
y =-y.
B) H3MeHeHHe MaCmTa6a. HaKOHeU, nOCMOTpIlM, KaK npe06p33YIOTCH
KoopHHaTbI, eCJIH, OCTaBJI5IH Hell3MeHHbIMH nOJIO)!{eHHe Ha1JaJIa KOOpH-
HaT Ii HanpaBJIeHlI5I OceH:, H3MeHHTb B k pa3 ;J.,lHHY eIIHli1JHOrO OTpe3Ka.
TaK )!{e KaK II B CJIY1Jae CHCTeMhI KOOpIIHaT Ha np5IMOH:, BbIBOHM, 1JTO
B 3TOM CJIY1Jae
(2)
{ ,x
y X, _ ; ,
- k'
(3)
YnpaiKHeHHH
1. HaqaJIo KOOp,l\HHaT nepeHeceHo B TO'lKY 0' (4; -3). KalwBbl HOBbIe Koop.'\HHaTbI
TOqeK A (5; 2), B(-3; -1), C(2; -6)?
2. OOCJIe nepeHoca Ha1laJIa KOOp,l\HHaT HOBbIe KOOp,l\HHaTbl TO'lKH A (-4; 6)
paBHbI 3 H -5. Haii,l\HTe HOBbie Koop,l\HHaTbI T01IeK B (3; 1), C(-I; 8), D (-12; -3).
3. Haii,l\HTe KOOp,l\HHaTbI TO'lKH A (x; y), eCJIH nOCJIe nepeHoca Ha1laJIa Koop.QHHaT
H T01IKY 0' (-5; -I) ee HOBbie Koop.QHHaTbl paBHbI 2 II 4.
4. HanHwHTe <PoPMYJIbI npeo6pa30BaHHH KOOp.QHH8T, eCJ1H Ha1la,10 Koop.J.HHaT
OCTaJlOCb HenO,l\BH/KHblM, a HanpaBJIeHHe OCH a6CII.HCC H3MeHHJIOCb Ha 06p<JTHoe.
5. KaKoBbI HOBbie KOOp,l\HHaTbI TO'IeK A (9; -3), B(-6; -12), C(-3; 15), eCJJH
HaqaJIO KOOP.IlHHaT H HanpaBJIeHHH oceii He H3MeHHJIHCb, a .QJIHHa e,l\HHH'IHOrO OTpe3Ka
YBeJIHqHJIaCb B 3 pa3a? KaKHMH CTaHYT 9TH Koop:\HHaTbl, eCJIH ,l\JIHHa e,l\HHHlJHOro
OTpe3Ka YMeHbWHJIaCb B 5 pa3?
3. HeKOTopble 3a)1.a'!H aHaJIHTH'!eCKOH reOMeTpHH Ha nJIOCKOCTH.
nOKa)!{eM, KaK C nOMOIII.blO KoopHHaT MO)!{HO pewaTb reOMeTpHlIeCKHe
SaAallIi Ha nJIOCKOCTH.
3 a  a 1J a 1. BblIf.UCAU.f paccmOfLHue MeJICi)y mOIf.K(UfU A (Xl; Yl)
u B (Xi; Yi) Jia Koopi)UH.amH.OU flAOCKocmu (pHC. 70).
104

OnYCTHM nepneH.ZI.HKYJIHpbI H3
T01JeK A H B Ha OCH KoopAHHaT.
Orpe30K AB HBJIHeTCSI rHnoTeHY-
30H npHMoyrOJIbHOrO TpeyrOJIbHH-
Ka ACB H no TeopeMe nHaropa
HMeeM: IABI2= IACI2+ IBCI2. HaM
HaAo HaH:TH AJIHHbI OTpe3KOB A C
H BC. Ho OTpe3KH AC H A1Bi
HMeIOT oAHHaKoBYIO AJIHHY, a AJIH-
Ha OTpe3Ka A 1 B 1 paBHa IX2-X11
(T01JKa Ai HMeeT Ty)!{e a6 C UHCCY PRC. 70
Xl> tJTO H T01JKa A, a T01JKa Bl -
Ty)!{e a6cUHccy X2, qTO H B).
n03TOMY IACI2= IAIBlI2= IX2-XlI2= (X2-X1)2. T01JHO TaK)!{e YCTaHaB-
JIHBaeM, 1JTO IBCI2= (Y2-Y1)2. 3Ha1JHT, IABI2= IACI2+ IBCI2=
= (x 2-X1)2+ (y 2-Yl)2.
HTaK,
y I I I
f- B(xz;yz)
",..
....
4- .......... 4C
A(X 1 ;Y1)
10 A IR. X
IA-?I=V (X2-X1)2+ (Y2-Yl)2. (I}
n p H M e p 1. HaH:AeM paccToHHHe Me)!{AY T01JKaMH A (3; 6) H
B (6; 2). no opMYJI e (1) nOJIY1JaeM:
IABI=V (6-3)2+ (2-6)2=V 3 2 +4 2 =V 25 5.
n p H M e p 2. HaH:,lJ,eM Ha OCH 0pAHHaT T01JKY, HaxoAHIII.YIOCH H8
paccToHHHH 10 OT TOtJKH A (6; 3).
A6cUHcca JI1060H: TOtJKH Ha OCH 0pAHHaT paBHa HYJIIO. n03TOMY
HCKOMYIO T01JKY MO)!{H O 3anHcaTb B BHA e B (0; Y), a ee paccToHHHe OT
T01JKH A (6; 3) paBHo V (0-6)2+ (y-3) 2 . no YCJIOBHIO 3aAatJH nOJIyqaeM
ypaBHeHHe
V36+ (y_3)2= 10.
H3 3Toro ypaBHeHHH HaxoAHM, 1JTO 36+ (y-3) 2 = 100. n03TOMY (y-3)2=
=64 H, 3Ha1JHT, y-3=:l::8, OTKYAa Yi= 11, Y2=-5. 3Ha1JHT, Ha OCH op-
.ZI.HHaT eCTb ABe TOtJKH, HaXOAHIII.HecH Ha paccToHHHH 10 OT T01JKH A:
T01JKH B 1 (0; 11) H B 2 (0; -5).
3a A a 1J a 2. HaiioeM KOOpOUH.ambl cepeOUH.bl C ompe3Ka c KOH./{aMU
A (Xl; Y1) H B (X2; Y2) (pHC. 71).
l{JIH 3Toro onYCTHM H3 T01JeK
A, B H C nepneHAHKYJIHpbI Ha
OCb a6cuHCC. TOtJKa C 1 HBJIHeTCSI
cepeAHHOH: OTpe3Ka A1Bi.3Ha1JHT,
ee a6CUHcca paBHa Xl t X2 . TaK)!{e
onpe.ll.eJIHeM, tJTO opAHHaTa T01JKH C 2
paBHa Yl t Y2 . n03TOMY TOtJKa C
HMeeT KoopAHHaTbI
. Xl+X2 Yl+Y2 (2) '
Keep = --r. Ycep = --r .
qj I I I I
2 8(xz;yz)
",..
",.. "lC
I
!A(x 1 ;!/t) I
i
i
0 A. z )(
I
PHC. 71
]05

n p H 1\1 C P 3. Hai1:.n:eM Koop.n:IfHaTbI cepe.n:HHbI OTpe3K3 C KOHuaMH
A (-6; 5) II B (3;-7).
no opMYJIaM (2) nOJIY1.IaeM:
x cep = -+3 = 1,5, Ycep = 5;7 =-1.
YupamHeHBJl
1. HaH;UITe l\JIHHY OTpe3Ka MN C KOHLI,aMf!:
a) M (-3; 2), N (5; -2); 6) M (2; 7), N (6; 4).
2. D.aH TpeyrOJIbHHK ABC C BepWHHaMH A (0; 5), B (-5; 3), C (4; -5). HaHl\HTe
JVlHHY OTpe3Ka, COeiJ.HHSlIOw:ero cepel\HHbI CTOpOH AB HAC.
3. Hai\iJ.HTe 0PiJ.HHaTY TO'IKH M, 3HaSl, '1TO ee a6cLI,f!cca paBHa 7, a pacCTOHHHe
OT M iJ.O TO'lKH N(-I; 5) paBHO 10.
4. D.aHa TpaneLl,HH ABCD: A (I; 3), B(-2; 8), C(O; 7), D(5; I). Haiil\HTe JVlHHY
ee cpe.J.Hei\ JIf!HIIH.
5. D.aHbI TOlJKf! A (- I; 2) H B (3; 5). HaiiiJ.HTe nJI01Il.3Jl.b H nepHMeTp KB3Jl.p3Ta
ABCD.
6. D.oKa)KI1Te, '1TO TpeyrOJIbHHI{ C BepWHHaMH A (I; I), B (2; 3), C (5; -I) npRMo-
yrOJIbHbIii.
7. 113BeCTHbI KOOpiJ.HHaTbI Tpex BepwHH KBal\paTa ABCD: A (2; 6), B(5; 6),
C(5; 3). HaiiiJ.f!Te KOOpiJ.HHaTbI Ll,eHTpa KBaiJ.paTa, ero '1eTBcpTOii BepWf!HbI H IJJIOlU3Jl.b
KB3iJ.paTa.
 3. QIICJ10BbIE II BYKBEHHbIE BbiPAiKEHlUI
1. 4HCJlOBble Bblpa>KeHHSI. PewHM 3a.n:a1.lY: «PaccToSIime Me)!{.n:y ro-
po.n:aMII A II B paBHo 240 KM. H3 ropo.n:a A BhIeXaJI BeJIOClIne.n:HCT co
CKOpOCTblO 20 KM/Ii, a 1.Iepes 3 Ii HaBCTpe1.lY eMY BbIeXaJI H3 ropoAa B
aBTO!\106H.lI1CT co CKOpOCTblO 70 lOt/ii. 4epes CKOJIbKO 1JaCOB nOCJIe BbI-
e3Aa aBTOM06HJI5I OHII BCTpeTSITCSI?»
CHa1JaJIa HaH:.n:eM, KaKOH: nYTb npo.n:eJIaJI BeJIOCHneAIICT 3a 3 Ii.
l{JISI 3Toro Ha.n:o Yl\IHOIKHTb 20 Ha 3. He BbInOJIHSIH 3Toro .n:eH:CTBIIH, JIHWb
0603Ha1JH1\I ero: 20.3. nOcJIe 3Toro HaH:.n:eM, Ha KaKOM paccToHHHH OT
ropo.n:a B BeJIOCHne.n:IICT I1aXO.n:HJICSJ. 1Jepe3 3 Ii nOCJIe BhIe3.n:a: 240-20.3.
l{a.'Iee onpe.n:eJl5IeM CKOpOCTb C6JIH)!{eHHSI Be.lOCllne.n:IICTa H aBTOM06HJIH:
20+ 70. 11 HaKOHeu, Haxo.n:HM, 1.Iepes CKOJIbKO 1JaCOB nOCJIe BbIe3.n:a aB-
Tm106l1JISI npOlI30H:.n:eT BCTpe1Ja: (240-20.3) : (20+70).
B pe3Y JIbTaTe peWeHH5I 3aAa1.lH MhI nOJIY1JHJIH 1.IHCJIOBoe BbIpa)!{eHHe
(240-20.3) : (20+70). B 3TOM BbIpa)!{eHHH 0603 Ha1.leHbI AeH:CTBH5I,
KOTopbIe Ha.n:o c.n:eJIaTb Ha.n: .n:aHHbIMH B YCJIOBHH 3aAa1.lH 1.IHCJIaMH. 1.IT06bI
nOJIY1JJ1Tb OTBeT, .n:aHa npocpaMMa BbI1.IHCJIeHHSI 3Toro OTBeTa. BbInOJIHSIH
3TY nporpaMMY, nOJIY1.IaeM 3Ha1.leHHe 1.IHCJIOBOro BbIpa)!{eHHH:
(240-20.3) : (20+70) = (240-60) : 90= 180: 90=2.
3Ha1.llJT. BCTpe1Ja npOH30H:.n:eT 1Jepes 2 Ii nOCJIe BbIe3.n:a aBTOM06HJIH.
B 06III.eM BH.n:e nOHHTHe 1.IHCJIOBOrO BbIp3)!{eHHSI onpe.n:eJISIIOT TaK:
a) K GJlCiJoe liUCJlO fl6JlJlemcfl liUCJl06blM BblpaJICeH.UeM.
G) ECJlu A U B - 'IUCJl06bte BblpaJICeHufl. mo (A)+ (B), (A)-(B),
(A) . (8) H (A) : (B) mOJICe fl6JlfllOmcfl liUCJlOBblMU BblpQ,:JICeH.UflMU.
106

BhIIIOJIH5iH )'](333HHbIe OnepaUHlI, HaXOAH;\1 3Ha1JeHUe 1JHC.l0BOrO BU-
pa)!{eHHH. EC,lH T01JHO CJIeAOBaTb 3TOMY OnpeAeJIemnO, TO npmJJJ10Cb 6b1
fIHCaTb CJIHWKOM MHoro CK060K, HanpHMep (2)+(3) HJIH (7) .(9). l{JIH
COKpaIII.eHHH 3anHCH YCJIOBH.'IHCb He 3aKJII01JaTb B CK06KII OTl(eJIbHble
'IHCJIa. KpOMe TOro, YCJIOBHJ1HCb He nHcaTb CK06KH, eCJ1l1 HeCKOJIbKO
Bb:pa)!{eHHH CKJIal(bIBaJOTCH HJIH Bbl1JHTaIOTCH, npU1JeM 3TH onepaUJUI
BbInO.1HHIOTCH no nOpH.l1.KY C.leBa HanpaBO. TO!.fHO TaK )!{e He nHIIIYT
CK060K H Tor,J,a, KorAa nepeMHOlKaJOTCH H:aJ ,lJ,elIHTCH HeCKO.1bKO qU-
ceJI, IIpH1JeM 3TH onepalJ,lm BbIITO.'IHHJOTCH 110 nopHJI.KY CJIcsa HanpaBO.
HalIpHMep, nHWYT TalC
25-4+61-14-42 HJIH 60: 3.5.15: 25.
HaKoHcu, YC.l0BHJIHCb CHa1JaJIa BhIIIOJIH5JTb AeHcTBHH BTopoH CTY-
neHH (YMHO)!{eHHe H ,II.eJIeIUle), a nOTOM IIepBOH cTYlIeHH (c.l0)!{eHHe I{
Bbl1JHT3HHe). n03TOMY BbIpaIKeHue (12.4: 3)+(5.8: 2.7) 3anllCbIBaIOT
TaK: 12.4 : 3+5.8 : 2.7.
B COOTBt'TCTBHH C 3THM IIOpHAOK ,II.efICTBHfI IIPH BbI1JHC.It'HHrI 3Ha1Je-
HHH tJHCJIOBOrO Bblpa)!{eHHH TaKOB:
1) EC.llU 'iUCAOBoe Bblpa:JfCel-lUe He Cooep:JfCum CKo60K, mo Hado pa3oe-'
AUfllb eeo Ha 'lacmu, omoeAeliHble opye om opyea 3HaKa.MU CM:JfCeHUfl U 6bl-
'lUmaHUfl, U Bbl'iUCAUmb 3IW'le/-lUe Ka:JfCoou. mmwu. ttllcmu, BbtnOAHflfl YM-
HOJICe/-lUfl U OeAe/-lllfl no nopfloKY CAeBa HanpaBO; nOCAe amoco, 3llM.e/-lU8
Ka:JfCoylO 'lacmb ee 3/-latteHueM, Hau.mu 3Ha'le/-lUe Bbtpa:JfCeHUfl, BblnOAHJIJJ
onepatfuu C,1OJICeHUfl U 6bL'lUmaHUR no nopfloKY CAeBa HanpaBo.
2) ECAU 'lUC,WBOe Bblpa:JfCeHUe cooep:JfCum cKo6KU, mo Haoo 63Rmb 'ILlCmu
8blpaJ/CeHll51, 3aT{.J110'leHHble AteJICoy ,7eeOu. U npaeorl cK06Ka/l.fU U He coiJep-
:JfCaU{ue UHblX CK060K, Hat'imu llX 3Hal(£HUe no npaBUAY 1) U 3a.MeHumb no-
AYI(£HHblM 3N.Q'leHUeJ.t., onycmlle cK06Ku. ECAU IlOCAe amoeo 11O,1!J'lUmCfl Bbl-
pa:JfCeHUe 6e3 cKo6oK, Bbl'lUCAUnlb eeo no npaBU,1Y 1). B npomUBHoAt CAY-
"tile CHoea npll,HeHumb npaeUAO 2).
BbItJHCJIHM, HanpHMep, 3Ha!.feHHe BbIpa)!{eHHH
«36 : 2-14) .(42.2-14)+20) : 2.
CHatJaJIa HaXO,illlM, tJTO
36:2-14= 18-14=4, 42.2-14=84-14=70.
3aMeHHB 36 : 2-14 H 42.2-14 HX 3HatJeHHHMH, nOJIyqaeM:
(4.70+20) : 2=(280+20) : 2=300: 2= 150.
11TaK, 3Ha1JeHHe 3a,II.aHHOrO BbIpa)!{eHHH paBHo 150.
OTMeTHM, 1JTO He BCHKoe 1JHUIOBOe BbIpaIKeHue HMeeT 3Ha!.feHHe. Ha-
IIpHMep, BbIpa)j\eHHH 8 : (4-4) H (6-6) : (3-3) 1JHCJIOBOrO 3!1a1JeHHH
He HMelOT, TaK KaK ,lJ,eJIeHHe Ha Hy.1b HeB03;\!0)j\H0.
YupamUeUHJI
1. BbIno.'JHIiTe Bce YKa3aHHbIe t\eHCTBHB H Har:.IlHTe 3HaqeHHB Bb!paJl\eHHJi:
a) O,039:( 2 '(2,31 :0,077»);
)07

6 5,2+ 17,25-(3,36:0,3)
) (2,7:0,18)+(0,65:0,13) :0,05;
(2,1-1,965): (0.12.0,45) 1 :0,25 .
B)
0,0325:0,13 O,16.b,25'
/ 1 3 1 \ "
r) -.!.... ( 2,5+3 3 .4 5 + 2'3 ) . 1 0,5- 0,2
17 2 5-3-.!... . 4 G-2 .\  -0, [25
, 3 ' 3
O'2}
2. CY1Il.eCTBYIOT JlH lIHCJlOBble 3HalleHHH c,1e.l\YI01II.HX Bbipa/KeHHH, eCJlH B Ka'le-
CTBe .l\OnYCTHMblX pe3YJlhTaTOB Bcex npOMe/KYTOllHblX .l\eJ!:cTBHJ!: paccMaTpHBaTb JlHlllb
UMble HeoTpHuaTe.lIbHble '1HCJla:
a) «(4-7)+3.5)). (8-6); 6) «(5+7): 24).16-5;
B) (3.7-6.8)+15-1O?
I1MelOT JlH 3TH Bblpa/KeHHH 3HalleHHH, eCJIH ClIHTaTb .l\onYCTHMblMH Bce ueJlbre '1HCJla?
2. 4HCJlOBble HepaBeHcTBa. O.n:HHM 1I3 Ba)!{HeHWHX npllMepoB OT-
HoweHIISI nopH.n:Ka HBJIHeTCH OTHoweHlIe {(MeHbwe» B MHO)!{eCTBe .n:eHcT-
BIITeJIbHbIX 1JHCeJI, 0603Ha1JaeMOe 3HaKOM <. MO)!{HO .n:OKa3aTb, 1.ITO 3TO
OTHoweHlIe HBJIHeTCH OTHoweHlIeM CTpororo JIIIHeHHOro nopH.n:Ka, T. e.
1JTO OHO aCIIMMeTpH1JHO II TpaH3l1TlIBHO, npll1JeM AJISI JIl06bIX .n:BYX pa3-
JIH1JHhIX .n:eHcTBHTeJIbHbIX 1JlIceJI X II Y BbInOJIHHeTCH O.n:HO H TOJIbKO O.n:HO
H3 OTHoweHIIH x<y IIJIII y<x. LI.aJIee MO)!{HO .n:OKa3aTb, 1.ITO x<y B TOM
H TOJIbKO B TOM CJIY1.Iae, Kor.n:a y-x>O. npH 3TOM H3 a>O II b>O CJIe-
.n:YIOT HepaBeHCTBa a+b>O II ab>O.
H3 YKa3aHHbIX CBOHCTB 1JIICJIOBhIX HepaBeHCTB MO)!{HO BbIBeCTII BCe
OCTaJIbHbIe HX CBOHCTBa.
1) OmHoUl£Hue x<y coxpaH.flemCfl npu npU6aBAeH.UU K 06e;uI "laCmflM
OaH.020 U mo20 OICe tf.UCAa (3TO CBOHCTBO Ha3hIBaeTCH PIi0f1'lOHH.OCmb10
OTHoweHIIH nopH.n:Ka OTHOCIITeJIbHO Co10)!{eHIIH). HHbIMII CJIOBaMlI,
eCAU x<y, mo aAfl AI06020 tf.UCAa a BbtnO.1HRemCfl HepaeeH.Cm60 x+a<y+a.
B caMOM .n:eJIe, H3 x<y cJIe.n:yeT, 1JTO y-x>O. Ho (y+a)-(x+a)=
=y-x>O, H nOTOMY
x+a<y+a.
nOCKOJIbKY x-a=x+(-a), y-a=v+(-a), TO 1I3 x<y CJIe.n:yeT,
1JTO x-a<y-a.
2) ECAU x<y U a<b, mo x+a<y+b.
B caMOM .n:eJIe, B 3TOM cJIY1.Iae y-x>O H b-a>O, a nOToMY II
(y+b)-(x+a) = (y-x) + (b-a»O.
3) OmliOUl£H.Ue x<y coxpaH.flemCfl npu YMH.OOlCeH.Uu 06eux tf.acmeu H.e-
paeeH.CmBa Ha OaH.O U mo CJlCe nOAOOlCumellbHoe tf.UCIlO, m. e. U3 x<y U
a>O clleayem HepaeeHcmBo ax<ay.
B caMOM .n:eJIe, 1I3 x<y BbITeKaeT, 1.ITO y-x>O. TaK KaK npoIl3Be.n:e-
HHe ABYX nOJIO)!{HTeJIbHbIX 1JlIceJI nOJIO)!{IITeJIbHO, TO H a (y-x»O.
TaK KaK a (y-x)=ay-ax, TO OTCIO.n:a BbITeKaeT, 1.ITO ax<ay.
4) ECAU X, y, a, b - nOIlOOICUmeAbHbI.e tf.UClla, mo U3 HepaeeHcmB
x<y U a<b cAeayem HepaBeHcmBo ax<by.
B caMOM .n:eJle, H3 x<y II nOJIO)!{IITeJIbHOCTH a CJIe.n:yeT, 1JTO ax<ay,
a H3 a<b H nOJIO)!{HTeJIbHOCTII y cJIe.n:yeT, 1.ITO ay<by. Ho TorAa B CIIJIY
108

TpaH3HTHBHOCTH OTHOWeHH5I HepaBeHCTBa H3 ax<ay H ay<by CJIeAYeT,
tJTO ax<by.
HepaeeH£meo y>x 8K8UeaAeHmHO H.epaeeHcmey x<y - 06a H.epa-
6eH.CfI18a OaHOepeMeH.H.O ucmUHHbt UAU AOJ/CHbt. 3HaKU HepaeeHCfI18a < U
> e3aUMHO-06pamHbt.
5) n pu U3MeHeHUU 3HaKoe uceA HepaeeHcmeo 3aMeHRemCR Hepa-
6eH.CfI180M 06pamH020 CMblCAa: eCAU x<y, mo -x>-y.
B caMOM AeJIe, x<y 03Ha1JaeT, 1JTO y-x>O. Ho y-x=(-x)-(-y),
a nOToMY (-x)-(-y»O, T. e. -y<-x.
6) npu YMHoJ/CeHuu 06eux acmeu HepaeeHcmea Ha ompuUpmeAbH.Oe
UCAO 3HaK HepaeeHcmea 3aMeHflemCR 3HaKO,1t 06pamHoco CJtbtCAa: eCAU
x<y U a ompul{ameAbHo, mo ax>ay.
B caMOM AeJIe, YMHO)!{eHHe Ha OTpHuaTeJIbHOe 1JHCJIO a l\1O)!{HO 3a-
MeHHTb YMHO)!{eHHeM Ha nOJIO)!{HTeJIbHOe 1JHCJIO Ja I, npH KOTOpOM 3HaK
HepaBeHcTBa cOXpaHHeTCH, H YMHO)!{eHHeM Ha -1, npH KOTOpOM 3TOT
3HaK MeH5IeTCSI Ha 3HaK 06paTHoro CMbICJIa.
1 1
7) ECAU O<x<y UAU x<y<O, TO -<-.
y x
l{JIH AOKa3aTeJIbCTBa AOCTaT01JHO 3aMeTHTb, 1JTO J.._ = y-x . TaK
x y xy
KaK 1JHCJIa x H y no yCJIOBHIO HMelOT O.ll.HHaKOBbIe 3HaKH, TO xy - nOJIO-
1 1
)!{HTeJIbHOe 1JHCJIO, a n03ToMY 3HaKH--- H y-x o.ll.HHaKoBbJ. TaK KaK
x y
1 1 I I
Y - x nOJIO)!{HTeJIbHO, TO H -- - nOJIO)!{HTeJIbHO, T. e. - <-.
x y y x
Hap5IAY C OTHoweHHHMH x<y H x>y paCCMaTpHBalOTC5I OTHoweHHH
xy H xy. HepaBeHCTBO xy 5IBJISIeTC5I AH3'bIOHKUHeH HepaBeHCTB
x<y H x=y H nOTOMY HCTHHHO, eCJIH HCTHHHO XOT5I 6bI O,lJ,HO H3 HHX:
xy= (x<y) V (x=y).
HanpHMep, HCTHHHO, 1JTO 41O, nOCKOJIbKY HCTHHHO, 1JTO 4<10. TOtJHO
TaK )!{e HCTHHHO, 1JTO 44, TaK KaK HCTHHHO, 1JTO 4=4. A HepaBeHCTBO
43 JIO)!{HO, nOCKOJIbKY JIO)!{HbI 06a BbICKa3bIBaHH5I 4<3 H 4=3.
l{BoHHoe HepaBeHcTBo x<y<z SIBJIHeTC5I KOH'bIOHKUHeH Hepa-
BeHCTB x<y H y<z - OHO HCTHHHO JIHWb npH YCJIOBI1H, 1JTO HCTHHHbI
06a 3THX HepaBeHcTBa. HanpHMep, HCTHHHO .lI.BOHli0e HepaBeHcTBO
4<8<10, TaK KaK HCTHHHbI 06a HepaBeHcTBa 4<8 H 8<16. A .lI.BOHHoe
HepaBeHcTBo 4<10<8 JIO)!{HO, TaK KaK, XOTH HepaBeHcTBo 4<10 HC-
THHHO, HepaBeHcTBo 10<8 JIO)!{HO.
YnpaIKUeUU1I
I. KaKHe H3 CJIe.l\YI01II.HX HepaaeHCTB HCTHHHhI:
a) 59; 6) -4>5; B) 2>0;
r) 00;.l\) V"7>VS; e) V"2> V3?
2. KaKHe H3 cJIe.l\YI01II.HX .L!BOHHhIX HepaBeHcTB HCTHHHbI:
a) -6-6<0; 6) 8<311; B) -404;
r) 7 0-7?
109

3. PaBcHCTBo If HepaBeHCTBO "IHCJIOnbiX BblpaJKeHlfH. DYCTb ,lJ,an>l
ABa 1JIIC,10BbIX BblpaIKemI5I A H B. Mbl MOIKeM COCTaBIITb H3 3THX BhIpa-
>KeHHH paaeHcmso A =B H H.epaBeHcmBa A>B, A<B H T. A. 31'H pa-
BeHCTBa II HepaBeHcTBa HBJIHIOTCH BblCKa3bIBaHMHMH, KOTopbIe MorYT
6hITb KaK HCTIIHHbIMH, TaK II Jl0IKHbIl\IH. PaBeHCTBO A =B C1JHTaeTCSI HC-
THHHbIM B TOM If TOJIbKO B TOM CJIY1Jae, Kor.n:a 06a BbIpa)!{eHHSI A H B
HMelOT 1JIICJIOBbIe 3Ha1JeHH5I, npH1Je!\1 3TH 3Ha1JeHHSI o.n:HHaKOBbJ. HanpH-
Mep, HCTHHHO paBeHcTBo 2+ 7=3.3, TaK I(aK H JIeBaSI H npaBaSI 1JaCTH
3Toro paBeHcTBa HMelOT 3Ha1JeHHe 9. A paBeHcTBo 7+5=4.5 JIO)!{HO,
TaK KaK 3Ha1JeHHe JIeBOH 1JaCTH paBHo 12, a npaBaSI 1JaCTb HMeeT 3Ha1Je-
Rife 20. J10IKHO H paBeHcTBo 6: (2-2) =5, TaK KaK BblpalKeHHe
6 : (2-2) He HMeeT 1JIICJIOBOrO 3Ha1JeHHSI.
OTMeTHM, 1JTO ecJIH C1JHTaTb H3BeCTHblM JIHIIIb MHO)!{eCTBO HaTypaJIb-
HbIX 1JIICeJI, TO paBeHcTBo 4-8+ 10=2.3 JIOIKHO, TaK KaK B MHoIKecTBe
N He onpe.n:eJIeHO 3H81JeHHe BbIp8IKeHHSI 4-8. O.n:HaKo nOCJIe pacIIIHpe-
HlfSI MHOlKeCTBa HaTypaJIbHbIX 1JHCeJI H BBe.n:eHHSI oTplluaTeJIbHbIX 1JH-
C 3TO paBeHcTBo CTaHOBHTCSI HCTHHHblM, rrOCKOJIbKY 06e ero 1JaCTH
HMelOT 3Ha1JeHHe 6. .
OTHoIIIeHHe paBeHcTBa 1JHCJIOBbIX BblpaIKeHHH 06JIa.n:aeT CBOHCTBaMH
pJIeKCHBHoCTH, CHMMeTpH1JHOCTH H TpaH3HTHBHOCTH, T. e. SIB,5IeTCH
OTHOIIIeHHeM 3KBHBaJIeHTHOCTH. n03TOMY MHOIKeCTBO Bcex qHCJIOBbIX
BbIp3)!{eHIIH pa36HBaeTcSI Ha KJI2CCbI 3KBHBaJIeHTHOCTH, COCTOSIIII.He JI3
BbIp, aIKeHIIH IIMelOIII.HX O.n:HO H TO IKe 3HaqeHIIe. HarrpllMep, B O.n:HH If
TOT me KJIaCC 3KBJmaJIeHTHOCTH nona.n:alOT BbIpa)!{eHIIH 5+ I, 9-3,
2.3, 12 : 2 H T. .n:.- Bce OHH HMelOT 3Ha1JeHHe 6.
H3 .n:aHHoro BbIllie orrpe.n:eJIeHHSI BbITeKaeT, 1JTO eC.TJH IICTHHHbJ
paBeHcTBa A =B H C=D, r.n:e A, B, C, D - 1JHCJIOBbIe BbIpaIKeHHSI, TO,
npH YCJIOBHH BbmOJIHHMOCTH COOTBeTCTBYIOIII.HX orrepaUHH, HCTHHHbI H
paBeHCTBa
(A)+(C)=(B)+(D);
(A) .(C)=(B) '(D);
(A)-(C)=(B)-(D);
(A) : (C) = (B) : (D).
HepaBeHCTBO A<B, r.n:e A If B - 1JHCJIOBbIe BbIpa)!{eHHSI, MbI 6y.n:eM
C1JHTaTb HCTIIHHbIM, eCJIH A H B HMelOT 1JHCJIOBbIe 3Ha1JeHHSI, npH1JeM
QHCJIOBOe 3Ha1JeHHe BbIpaIKeHHSI A MeHbwe 1JHCJIOBOrO 3Ha1JeHH5I Bblpa-
>KeHH5I B. HarrpHMep, BCTHHHO HepaBeHcTBo (18-3) : 5<3+4, TaK
KBK (18-3) : 5 BMeeT 3Ha1JeHHe 3, 3+4 HMeeT 3Ha1JellHe 7, a 3<7.
nOCKO.lbKY 3anHCH BH.n:a A=B, C<D H T. .n:., r..'l.e A, B, C, D-
QHCJIOBbIe BbIpaIKeHHSI, SIBJISIIOTCSI BbICKa3bIBaHHSIMH, MbI MOIKeM BbirrOJI-
HSITb Ha.n: HHMH JIOrH1JeCKHe orrepaUHH KOH'bIOHKUHH, .n:H3'bIOHKUHH, HMrr-
JIHKaUHH II T. .n:. HanpHl\lep, HepaBeHcTBo AB HBJISIeTCSI .n:H3'bIOHKUHeH
HepaBeHcTBa A<B H paBeHcTBa A =B:
AB= (A<B)V(A=B).
QHO BCTHHHO, eCJIH HCTHHHO XOTSI 6bI O.n:HO H3 BbIcKa3bIBaHHH A<B,
A=B. HanpllMep, HCTHHHO, tJTO (2.4+ 15). 235+ 19, TaK KaK 3HaQe-
HHe BblpmKeHHSI (2.4+ 15).2 paBHo 46, 3Ha1JeHHe BblpaIKeHHH 35+ 19
oaBHO 54, a HepaBeHcTBo 46<54 HCTHHHO..
[10

.LLBoHHoe HepaBeHCTBO A<B<C 51BJHleTCSI KOH'bIOHKUHei:j Hepa-
BeHCTB A<B H B<C. OHO HCTHHHO B TOM H TOJIbKO B TOM C.1Y!-Iae, KorAa
liCTHHHbI 06a 3TH HepaBeHCTBa. HarrpHMep, HCTHHHO, tITO 16+4<
< 125: 5<3.10. B caMOM .D,e.le, 3Ha1JeHHe 16+4 paBHo 20, 3!IatIeHH€
125 : 5 paBHO 25, a 3HaIJeHUe 3.10 paBHO 30. TaK KaK 20<25 H
25<30, TO ABOllHoe HepaBeHCTBO HCTliHHO.
YupamHeHDH
I. OpOBcpbTe IICTUHHOCTb paBeHcTB:
1917 I V - 3 -
a) 852 = 2"4; 6) - 64 = -4; B) V-64= -4;
r) 17-91= 19-71; ll) 131=3; e) (4+ Y7) (4- Y7)=3 2 ;
/K) 13-1-511=13-51; 3) 33+43+53=63;
n) 8833=88 2 +33 2 ; K) 4626.9396=6939.6264.
2. npoBepbTe nCTHHIlOCTb HepaBeHcTB:
a) 675+872< (63+73+53)+ (83+73+23);
6) 1973> (1+9+7+2) (12+9 2 +7 2 +22)-197.2- (197-2);
B) 1971>19.72+ 197.2+ (197-2)+ (1+9+7+2)
4. BbIpa)KeHHSI c nepeMeHHbIMH. HeKoTopbIe YCJIOBIf51 3a;r3lJlI :-.mrYT
6hITb 0603HaIJeHbI He 1JHCJIaMIf, a 6YKBaMH. HarrpHMep, eCJI!I B 3aAaqe
li3 rr. 1 paCCT05lHHe Me)!{AY ropo,lJ,aMH paBHO a KAt, TO OTBeT npm!eT BH,A:
(a-3.20) : (20+70). (1)
ECJIH :>Ke paccToSIHHe paBHo a KM, a CKOpOCTIf Be.lOCHrre,lJ,HCTa 11 3BTOMO-
6HJISI COOTBeTCTBeHHO paBHbI b H C, TO OTBeT rrpIfMeT BH,A:
(a-3b) : (b+c). (2)
MbI rrOJlyqHJIH 8blpa3lCeH.UJl c nepeMeliliblMu. B BbIp8)!{eHIfe (1) BXOAHT
OAHa rrepeMeHHa51 a, a B BbIp8)!{eHHe (2)- TPH nepeMeHHbIe a, b u C..npH-
AaBaSI 3THM 6YKBaM pa3JIHIJHbIe 3HaIJeHHSI, rrOJIYIJHM pa3JIHIJHbIe 3aAa1JH.
4T06b1 rrOJIY1JHTb OTBeT Ka:>K,lJ,oH H3 3THX 3a,lJ,aIJ, Ha.Zl;O rrO,lJ,CTaBHTb COOT-
BeTCTBYIOIII.He 3HaqeHHSI 6YKB B BbIpa:>KeHHSI (1) IfJIH (2). Hanpm!ep, eCJIII
paCCT05lHHe Me:>K,lJ,Y ropoAaMH 240 KM, CKOpOCTb BeJIOCHne.D,HCTa paB-
Ha 15 KM/It, aBToM06HJIH 50 KM/It, TO Ha,lJ,O 3aMeHHTb B BbIpax:eHHH (2)
6YKBya Ha 240, 6YKBY b Ha 15, a 6YKBY c Ha 50. nOJIYIJaeTCSI IJHCJIOBO€
BhIpa:>KeHHe (240-3.15): (15+50), 3HaIJeHHe KOToporo paI3lO 3. B
STOM CJIYIJae BCTpeIJa rrpOH30H,lJ,eT IJepe3 3 It nOC.le BbIe3,lJ,a aBTOM06HJIH.
06lilee nOHSITHe BbIp8)!{eHHSI C rrepeMeHHhIMIf orrpe,lJ,eJIHeTCH TOqHO
TaK :>Ke, KaK H nOHSITHe IJUCJIOBOrO BbIpa:>KeHHSI (CM. c. 106), TOJIbKO,
KpOMe qHCeJI, BbIpa:>KeHH5I C rrepeMeHHbIMIf cO,lJ,ep:>KaT H 6YKI3bJ. MbJ
C1JHTaeM, 1JTO IJHTaTeJIIO H3BecTHbI rrpaBHJIa 3arrHCH TaI{HX BbIpa-
IKeHHH.
ECJIH AaHO BbIp8)!{eHHe, cO,lJ,ep:>KaIII.ee, HarrpIfMep, rrepeMeHHbIe x H y,
TO Ka)K,lJ,OMY KOpTe:>KY (a; b), COCTOSIW.eMY H3 IJHCeJI, COOTBeTCTB)'eT qHC'
JIOBoe BbIpa:>KelciHe. OHO rro.'IYIJaeTCH H3 6YKBeHHoro BbIpamE'H!ISI, ecJIli
3aMeHHTb 6YKB)' x IJHCJIOM a, a 6YKBY Y 1JHCJIOM b. ECJIH rrOJIYIJeHHOe qHC-
JIOBOe BbIpa:>KeHHe HMeeT 3HaqeHHe, TO 3TO 3HaIJeHHe Ha3bIBalOT 3liatteH.U'
111

eM 6blpaJlCeHUJl IIpli x=a li y=b. BblpaIKeHI1SI C II€peMeHHbIMI1 0603Ha-
1JalOT TaK: A (x), B (x; y) 11 T. ,lJ,. ECJIli B BbIpa:>KeHl1li B (x; y) 3aMeHI1Tb
x 1JliCJIOM 15, a y 1JliCJIOM 4, TO IIOJIY1JeHHOe 1JHCJIOBOe BbIpaIKeHlie 060-
3Ha1Ja1OT B (15; 4).
Bblpa:>KeHliSI C nepeMeHHbIMli He SIBJIHIOTCSI IIpe.ll.l1KaTaMli, TaK KaK
IIpli IIO,lJ,CTaHOBKe BMecTO 6YKB 1JliCJIOBbIX 3Ha1JeHliH IIOJIY1JaeTCSI He BbI-
CKa3bIBaHlie, a 1JliCJIOBOe BblpalKeHl1e. 3Ha1JeHl1eM 3Toro Bblpa:>KeHItH
SIBJISIeTCH He «I1CTliHa» liJIli «JIO:>Kb», a HeKOTopoe 1JliCJIO.
Ka:>K,lJ,OMY BbIpa:>KeHlilO, cO,lJ,ep:>KaIII.eMy O,lJ,HY 6YKBY x, COOTBeTCT-
BYeT MHoIKecTBo, COCTOHIII.ee H3 1JliCeJI, KOTopbIe MO:>KHO IIO,lJ,CTaBJISITb B
3TO BbIpaIKeHl1e, T. e. 1Jl1ceJI, IIPli KOTOpblX 3TO BbIpaIKeHlie liMeeT on-
pe.ll.eJIeHHOe 3Ha1JeHlie. 31'0 MHOIKeCTBO 1Jl1ceJI Ha3bIBaIOT o6.1aCmblO on-
peOeAeHUJl oaHHozo 6blpaJlCeHUJl. HaIIpliMep, 06JIaCTb onpe,lJ,CJIeHliSI BbI-
pa:>KeHliSI 4 : (x-3) COCTOI1T 113 Bcex 1JliCeJI, KpOMe 1JliCJIa 3. 06JIaCTb IKe
;-
OIIpe,lJ,e.1eHI1H BbIpaIKeHI1H J; x-5 COCTOI1T U3 Bcex 1Jlice.'1 x, IIpl1 KOTOpbIX
X - 5>0, T. e. li3 1JliCeJI, npliHa,lJ,JIeIKaIII.I1X 1JliCJIOBOMY JIY'IY [5; 00[.
B HeKOTopbIX CJIy1JaSIX 06JIaCTb X 3Ha1JeHHfl x 3apaHee OrpaHli1JeHa He--
KOTOpbIMli yCJIOBliSIMli. Hanpl1Mep, MOIKC'f 6bITb, 1JTO X - HaTypaJIbHoe
1JliCJIO (HanpliMep, 'IliCJIO rO.10B CKOTa). Tor,lJ,a H B cooTBeTcTBYJOlltee
BbIpa:>KeHlie C IIepeMeHIlOM MOIKHO IIO,lJ,CTaBJIHTb JIliWb 3Ha'IeHliSI, npl1-
Ha,lJ,JIeIKaIII.l1e COOTBC'fCTBYIOIII.eMy MHoIKecTBY (HanpliMep, MHO:>KeCTBY
HaTypanb-HbIX 'IliCeJI). EC,111 BbIpa:>KeHHe CO,lJ,epIKI1T HecKOJIbKO 6YKB,
HaIIpl1Mep 6YKBbI x If y, TO no,lJ, 06JIaCTbIO onpe,lJ,eJIeHliH 3Toro BblpalKe-
HliSI IIOHIfMalOT MHO:>KeCTBO nap 1JliCeJI (a; b) TaKI1X, 'ITO IIpli 3aMeHe x
Ha a li y Ha b IIOJIY1JaeTCH 1JI1CJIOBOe BbIpaIKeHlie, liMelOIII.ee 3Ha1JeHl1e.
HaIIpIfMep, 06JIaCTb onpe,lJ,eJIeHliSI BbIpaIKeHliSI 7: (x-y) COCTOliT li3
Bcex IIap 'IHCeJI (a; b) TaKIfX, 1JTO a=Fb.
B 6YKBeHHbIx BblpalKeHliSIX MO:>KHO 3aMeHSITb IIepeMeHHble He TOJIbKO
1JHCJIaMli, HO li ,lJ,pyrliMli 6YKBeHHbIMH BbIpa:>KeHliSIMli. HanpliMep, ecJIH
B BbIpa>KeHliH 3x+2y 3aMeHliTb x Ha 5a-2b, a y Ha 6a+4b, IIOJIY1JliTCH
6YKBeHHoe BbIpaIKeHlie
3(5a-2b)+2(6a+4b).
Ero 3HaqeHIIe npu 3a,lJ,aHHbIX 3Ha1JeHliSIX a It b MO:>KHO COC1JliTaTb, CHa'Ia-
JIa HaM,lJ,H 3Ha'ieHIfH x 11 y, a nOTOM nO,lJ,CTaBIfB 3TI1 3Ha'IeHliH B I1CXO,lJ,Hoe
BbIpaIKeHIfe. Hanpl1Mep, eCJIli a= 12, b= 10, TO CHa1JaJIa HaXO,lJ,liM, 1JTO
x=5.12-2.1O=40, y=6.12+4.10=112, a IIOTOM, 1JTO 3x+2y=
=3.40+2.112=344.
.ILBa BbIpaIKeHI1H C nepeMeHHoH A (x) li B (x) Ha3blBalOTCH mOJICoecm-
6eH.H.O pa6HblMU, ecJIli OHH IIpliHliMalOT O.ll.HHaKOBbIe 3Ha1JeHliSI IIpli JIIO-
6bIX ,lJ,OIIYCTI1MbIX 3Ha1JeHliSIX BXO,lJ,HIII.liX B HliX 6YKB. HaIIpliMep, TOIK-
x XZ
,lJ,ecTBeHHO paBHbI BbIpa:>KeHliSI (x+3)Z li x 2 +6x+9. BbIpaIKeHI1SI 4' li 4x
He SIBJISIIOTCH TO:>K,lJ,eCTBeHHO paBHbIMH - ecJII1 x = 0, TO IIepBoe H3 HI1X
npliHliMaeT 3Ha1JeHlie 0, a BTopoe He liMeeT 1JliCJIOBOrO 3Ha1JeHliH. Ho B
06JIaCTli OTJIli1JHbIX OT HYJISI qHCeJI 3Tli ,lJ,Ba BbIpa:>KeHHSI TO:>K,lJ,ecTBeHHO
paBHbI. YTBeplK,lJ,eHl1e 0 TOIK,lJ,eCTBeHHOM paBeHCTBe ,lJ,BYX BbIpaIKeHliH C
nepeMeHHOH HBJIHeTCSI BbICKa3bIBaHlieM. HaIIpHMep, YTBeplK,lJ,eHlie,
112

1JTO BbIpa)!{eHHe (x+3)Z TO:>K.lI.eCTBeHHO paBHO BbIp8)!{eHHIO xZ+6x+9,
MO:>KHO 3anI1CaTb TaK:
(Vx)«x+3)Z=x Z +6x+9).
06b['IHO ,lJ,JIH KpaTKOCTI1 OnyCKaJOT KBaHTOp Vx H nHWYT KOpO'Ie:
(x+3)z=xz+6x+9. O,lJ,HaKO TaKaH 3anHCb He COBceM TOtJHa - 3TO
paBeHCTBO MO:>KHO paCCMaTpUBaTb H KaK ypaBHeHHe (CM. 9 4).
YupalKHeHBJI
1. npOBepbTe HCTHHHOCTb paBeHCTB:
3a 2 -b Z (4b-a) b+a
a) Ja-b (a+4b) - aZ+b Z npH a=3, b=2;
6) ( + 6q 2 ) . ( PZ+4qz +l ) __ npH p=_I,
p-2q 4qZ_pZ p+2q' pZ_4qZ - 2p
q=-2.
2. D.oKa/KHTe CJIe.l()'IOIl1He TO/K.l(eCTBa:
a) (a 2 +1J2) (x2+yZ)= (ax-by)z+ (bx+ay)Z;
6) x + y + Z - 0
(x-y) (x-z) (y-x)(y-z) (z-x) (z-y) .
3. npll KaKHX 3Ha'leHH5IX x HBJl5IIOTC5I TO/K.n,eCTBaMH CJIe.n,YI01II.He paBeHcTBa:
1 I
a ) (X+2)(x-3) =x + 2' 6 ) x=1-X-tT _ I ?
x-3 ' I I x
x-l + x+l
 4. YPABHEHIUI II HEPABEHCTBA
1. YpaBHeHHH C OJl.HOH nepeMeHHoH. PaccMOTpHM 3a.ll.atJY: «B KJIeTKe
CH.lI.HT a3aHhI H KpOJIHKH. Y HHX 19 rOJIOB H 62 HorH. CKOJIbKO ct>a3aHOB
Ii CKOJIbKO KpOJIHKOB B KJIeTKe?» Ee MO:>KHO peWHTb apHMeTHtJecKH.
Ho 60JIee npocToH cnoc06 peWeHI1SI COCTOI1T B COCTaBJIeHIiH ypaBHeHHH.
0603Ha'II1M tJHCJIO a3aHoB 6YKBOH x. Tor.ll.a ya3aHoB 2 X Hor. 4HCJIO
KpOJIHKOB paBHo 19-x H y HHX 4(19-x) HOr. no YCJIOBIiJO 3a.ll.atJH ,lJ,OJI)!{-
HO BbInOJIHSITbCH paBeHCTBO 2x+4(l9-x)=62, T. e. 76-2x=62. Pe-
waH 3TO ypaBHeHHe, HaX0.ll.HM: 2x=76-62= 14, H nOToMY x=7. 3Ha-
tJHT, B KJIeTKe 6bIJIO 7 a3aHoB H 12 KpOJIHKOB.
ECJIH 6bI B YCJIOBI1H 3a.ll.a1JH 6bIJIO .lI.aHO, 'ITO y KpOJIHKOB H a3aHoB
61 Hora, TO nOJIY1JI1JJH 6bI ypaBHeHHe 2x+4(19-x)=61 , H3 KOToporo
I
HaWJIH 6bI x=7 2 . 3TO npOTIiBOpe1JHT CMbICJIY 3a,lJ,a'IH, TaK KaK X - Ha-
TypaJIbHOe 'IHCJIO. K npoTIiBopetJHIO MbI npHWJIH 6bI H pewaH 3MatJY,
B KOTOpOH 06III.ee 1JHCJIO Hor paBHSIJIOCb 80. KopHeM ypaBHeHHH 2x+
+4(19-x)=80 HBJISIeTCSI 'I HCJIO x=-2, a 1JHCJIO a3aHOB He MO:>KeT 6bITh
OTpHuaTeJIbHbIM. Bo06III.e, X Jl.OJI)!{HO 6bITb HaTypaJIbHbIM tJHCJIOM, He
npeBOCXO,lJ,SIIII.HM 18 (eCJIH C'I, HTaTb tJTO B KJIeTKe eCTb XOTb O,lJ,HH KpO-
JIHK), T. e. X ,lJ,OJI)!{HO npI1Ha,lJ,JIe:>KaTb MHO)KeCTBY X={I; 2; 3; 4; 5; 6; 7;
8; 9; 10; II; 12; 13; 14; 15; 16; 17; 18}.
113

npli peWeHlIIf ypaBHeHliH MhI ,lJ,eJIaeM HeKOTOpbIe npeo6pa30B3HliSI.
HanpliMep, pewaH ypaBHeHlie 76-2x=62, MbI npli6aBliJIli K 06EEl\I1JaC-
THM ypaBHeHliH 2x li BhI1JJIli li3 06elix qaCTeH ypaBHeHliH 62. B pe-
3YJIbTaTe nOJIY1JIfJIOCb ypaBHeHIJe 2x= 14. 4T06bI pewliTb ero, npli-
W.'IOCh pa3,lJ,eJlliTh 06e 1JaCTli ypaBHeHliH Ha 2. nOcJIe Ka)K)J,Oro li3 3TIfX
npe06pa30BaHliH nOJIY1JaJIOCb HOBoe ypaBHeHlie, HO Bce nOJIY1JeHHbie Ha-
Mli ypaBHeHliH - li 76-2x=62, li 2x= 14, li x=7 (a 3TO TOIKe yp2BHe-
Hlie!) - liMeJIli O,lJ,HO li TO IKe peweHlie, a liMeHHO 1JHCJIO 7.
BbISICHliM, Ha KaKOM OCHOBaHlili MbI ,lJ,eJIaJIli 3Tli npe06r;a30B3HliSI
ypaBHeHliH li n01JeMY 6bIJIli YBepeHbI, 1JTO npli 3TliX npe06pa30B3HliHX
He li3MeHHIOTCH KOpHH peW3eMoro ypaBHeHliH. 11HOr,lJ,a 3TO 06'bSCHHIOT
CJIe,lJ,YlOlII.liM 06pa30M. npe,lJ,nOJIaraIOT, 1JTO x - O.n.HO li3 peweHliH ,lJ,aH-
HOro ypaBHeHIfH. Tor,lJ,a npli 3TOM 3Ha1JeHlili x ypaBHeHl!e npenpalIl.a-
eTCH B liCTIfHHoe 1JHCJIOBOe paBeHcTBo. A TaK KaK 1JliCJIOBOe p3EOJCTBO
He H3PYWliTCH, eCJIli K 06eliM ero 1JaCTHM npli6aBliTb O,lJ,HO li TO llie 1JliC-
JIO liJIH OT 06elIX 1JaCTeH OTHHTh O,lJ,HO li TO IKe 1JIJCJIO, TO MO)!{HO .n.eJIaTb
)'K33aHHbIe BbIwe npe06pa30BaHliSI li B KOHue HaHTli, qeMY IKe paBHO
1JHCJIO x. npli 3TOM nO,lJ,xo,lJ,e x paCCMaTpliBaeTCSI KaK qliCJIO. O,lJ,HaKO
CYIueCTBYIOT ypaBHeHIfH, He liMelOlII.lie peIIIeHliH, Hanpl!Mep 2x=2x+6.
3,lJ,eCb, ,lJ,eJIaSI TaKlie IKe npe06pa30BaHliSI, MbI nplixO,lJ,liM K JlO)!{HOMY
paBeHcTBY 0=6. 3TO nOKa3bIBaeT, qTO peIIIeHlie ypaBHeHliH HeJlh3H Ha-
qJ!HaTb co CJIOB «nycTb x - KopeHb ypaBHeHliH».
KpoMe Toro, YKa3aHHhIH no,lJ,xo.n. K peIIIeHlilO ypaBHeHliH MOLKET npw
BeCTIf K nOHBJIeHlilO JIliWHliX KopHeH, T. e. KopHeH, KOTOpbIM Y.lJOBJIeT-
BOpHIOT ypaBHeHliH, nOJIY1JaeMbIe B XO,lJ,e npe06p330BaHliH, HO He Y,lJ,OB-
JIeTBOpSIeT licxo.n.Hoe ypaBHeHlie. HanpliMep, pelliaSI ypaBHeHlie V x+9 =
=-5, MbI B03BO,lJ,liM 06e qaCTli B KBa,lJ,paT (eCJIli ,lJ,Ba qliCJIa p3BHbI, TO
paBHbI li liX KBa,lJ,paTbI). B pe3YJIbTaTe nOJIY1JaeM ypaBHeHlie x+9=25,
H3 KOToporo HaXO,'I.liM, qTO x=16. Ho 16 y,lJ,OBJIeTBOpHeT JIliWb ypaBHe-
BlilO x+9=25 li He y.n.OBJIeTBOpSIeT ypaBHeHHIO V x+9 =-5. ECJIH
nO,lJ,CTaBIfTb B 3TO ypaBHeHlie 16 BMeCTO x, nOJIY1JaeM JIO:>KHOe paBeHcTBo
1/25=-5 (npli peweHlili lippaUliOHaJIbHbIX ypaBHeHliH Bce KOpHl! no-
HliMalOT B CMbICJIe apliMeTli1JeCKOrO 3Ha1JeHli5l, T. e. CqliTalOT HeoTpli-
IJ.aTeJIbHbIMli qliCJIaMli). TaKIfM 06pa30M, npli peIIIeHlili ypaBHeHliH no
YKa3aHHoMY cnoc06y HaAo nOCJIe OThICKaHHH KopHeii npoBepHTh Ka:>K-
AbIH li3 nOJIY1JeHHhIX KopHeR, nO,lJ,CTaBJIHSI l!X B. ypaBHeHlie, 3 3TO He
Bcer,lJ,a JIerKO CAeJIaTb.
lla)J.liM n03TOMY 60JIee T01JHOe onpeAeJIeHlie ypaBHeHliSI li ero KOp-
HeM. nYCTb 3a)J.aHhI ,lJ,Ba BbIp8)!{eHIISJ fl(X) li f2(X), CO,lJ,ep:>KalIl.lie nepeMeH-
HYIO x, KOTopaSI np06eraeT HeKOTopoe MHO:>KecTBO X. Ha30BeM ypae-
H.eH.ueM O,lJ,HOMecTHbIH npe,lJ,liKaT fl(X)=f2(X), x EX. PeWliTb ypaBHeHIfe-
SHatJHT HaHTli 3HatJeHHSI nepeMeHHoH x, npH nO,lJ,CTaHOBKe KOTOphIX B
ypaBHeHlie nOJlyqaeTCSI liCTliHHoe paBeHCTBO, T. e. HaHTli MHO)!{ecTBO
HCTHHHOCTli T .n.aHHoro npe,lJ,liKaTa. B Aa.1JbHeHWeM MbI 6y.n.eM Ha3hIBaTb
IlHO:>KeCTBO HCTliHHOCTli npe.ll.liKaTa Mx)=f2(X}, x EX, MH.O:JICeCmBOM pe-
UJJ!H.UU ypaBHeHliSI, a qliCJIa, BXOAHIII.Ife B 3TO MHO:>KeCTBO,- KOpHJlMU
ypaeH.eH.UJl.
114

ILnpHMep, ypaBHeHHe (x-I)(x-3)=0 HMeeT 11,Ba lWpHH, 1 H 3.
3Ha1JHT, MHO:>KeCTBO peweHHH 3Toro ypaBHeHHSI HMeeT BH,lJ, T={I; 3}.
CYU1.eCTBYIOT ypaBHeHHSI, HMelOlII.He 6eCKOHeqHOe MHO)!{eCTBO peWeHItIf.
HarrpIfMep, JIlo60e HeOTpHuaTeJIbHOe qHCJIO y,lJ,OBJIeTBOpSIeT ypaBHeHIIIO
x= Ixl. B 3TOM cJIyqae MHO)!{ecTBO peweHIfH COCTOHT H3 Bcex HeoTpHua-
TeJIbHbIX 1JHCeJI.
MO)!{eT CJIy1JHTbCH, qTO fdx) IfJIH f2(X) He HMeeT 3HaqeHHSI npH He-
KOTOpOM a H3 MHQ)!(eCTBa X. Tor,lJ,a paBeHcTBo f1(X)=fz(x) CqHTaeTCH
JIO:>KHbIM, H nOToMya He MO:>KeT 6hITb O,lJ,HHM H3 KopHeH ypaBHeHHH fl(X) =
=fz(x). HanpHMep, HH 4, HIt 6 He MorYT 6bITb KOpHSIMH ypaBHeHHH x+
I I I
+- 4 = 7+ - 6 - npH x = 4 He HMeeT 3HaqeHHH ,lJ,p06b _ 4 ' a
x- x- x-
I
"PH x =6- ,lJ,p06b x _ 6 ' OTCIO,lJ,a SICHO, qTO, npe)!{)J,e 1JeM peIIIaTb ypaBHe-
HHe fl(X)=f z(x), nOJIe3HO HaHTH MHO:>KeCTBO A, Ha KOTOpOM H fl(X) If f 2 (X)
HMelOT onpe,lJ,eJIemIbIe 3HaqeHHSI. 31'0 MHO:>KeCTBO Ha3bIBalOT o6AaCJnbIO
iJonYCJnu.HblX 3liat1RHUU nepeMeHHOH x HJIH 06AacmblO onpeOeAeHWl ypafl-
H.eliWl. LI.JISI yp aBHeHHSI x +  4 = 7 +  6 TaKOH 06JIaCTbIO HBJ1SIeTCH
x- x-
Bce MHO>KeCTBO ,lJ,eHCTBIITeJIbHbIX 1JHCeJI, H3 KOToporo HCKJIlOqeHbI TOqKII
4 H 6. 31'0 MHO:>KeCTBO l\!O:>KHO 3a,lJ,aTb cJIe,lJ,YlOlII.HM 06pa30M:
A=]-oo; 4[ U ) 4; 6[ U )6; +00[.
PaJ,H npocToTbI TeOpeTHqeCKHX BbIBO,lJ,OB MbI 6Y,lJ,eM C1JHTaTb B 113JIb-
HeHWel\!, 1JTO H f1(X) H fz(x) onpe.ll.eJIeHbI Ha BceM MHO)!{eCTBe X.
B c.1yqae, KOr,lJ,a 06JIaCTb onpe,lJ,eJIeHHSI X npe,lJ,HKaTa fl(X)=f 2(X)
KOHeqHa (HanpHJI.fep, B 3a,lJ,a1Je 0 Kpo.1IfKax H $a3aHax), KOpHH ypaBHe-
HHSI MO:>KHO HaHTH nepe60poM, nO,lJ,CTaBJISISI no 01Jepe.ll.H B ypaBHeHIfe
1JHCJIa H3 MHO:>KeCTBa X. Ho KOr,lJ,a MHO:>KeCTBO X 6eCKOHeqHO, TaKoH
nYTb CTaHOBHTCSI HeB03MO:>KHbIM H ypaBHeHHSI Ha,lJ,O pewaTb HHbIM ny-
TeM. npIf 3TOM HCnOJIb3YlOT nOHSITHe paBHOCHJIbHOCTH ypaBHeHHH.
LI.Ba ypaBHeHHSI fl (x)=f z(x) H F l(X)= F z(x) Ha3hIBalOTCSI pa81i0CUAb-
H.blAtll, eCJIH HX MHOIKeCTBa peweHHlf paBHbI, T. e. eCJIIt Ka)!{)J,oe peweHlIe
nepBoro ypaBHeHHH SIBJIHeTCSI peweHHeM BToporo ypaBHeHHH H, 06paT-
HO, BCHKoe peweHHe BToporo ypaBHeHHH y,lJ,OBJIeTBOpSIeT H nepBoMY
ypaBHeHYIIo. npH 3TOM MbI C1JHTaeM, qTO 06a ypaBHeHHSI HMelOT O.ll.HY II
TY )!{e 06JIaCTb onpe,lJ,eJIeHHSI X. 11HbIMH CJIOBaMH, ypGJJI-leH.UJl pa8IiOCUAb-
libl, eCJlU 3K8ueaAeHmHbl npeOUKambl fl(X)=f2(X) U F 1 (x)=F z (x).
ECJIH l\!HO:>KeCTBO peweHHH yp aBHeHHSI ft(x) = f 2 (x) SIBJIHeTCSI no.ll.-
MHQ)KeCTBOM MHO:>KeCTBa peweHHH ypaBHeHHSI F l(x)=F 2 (x), TO F l(X)=
=F z(X) Ha3bIBalOT cAeocJneueJ'd ypaBHeHHH fl (x)=f z(x). 11HbIMH CJIOBaMH,
F 1 (x)=F z(x) - CJIe,lJ,CTBIfe ypaBHeHHSI ft(X)=f2(X), ecJIH Ka:>K,lJ,bIH
KopeHb ypaBHeHHSI f1(X)=f2(X) Y.ll.OBJIeTBOpSIeT ypaBHeHHIO F l(X)=
=Fz(x).
HanpHMep, ypaBHeHHe (x+ 1)2= 16 SIBJISIeTCSI CJIe,lJ,CTBHeM ypaBHe-
HHSI x+ 1 =4. B caMOM ,lJ,eJIe, ypaBHeHHe x+ 1 =4 HMeeT TOJIbKO O,lJ,HH
KopeHb x=3. nO,lJ,CTaBJISISI 3TOT KopeHb B ypaBHeHHe (x+ 1)2= H5,
nOJIyqaeM HCTHHHoe paBeHCTBO (3+ 1 )2= 16, nOKa3bIBalOII.I:ee, tJTO 3 y,lJ,OB-
JIeTBOpHer H ypaBHeHHIO (x+l)2=16.
115

J1.8a ypa8HeHUJl paeHOCUAbHbl 8 mOM U f1WAbKO 8 mOM cAyttae, Kozoa
KQJfCooe U3 HUX Jl8AflemCfl CAeOCtn8UeM opYZOZO.
11Hor,lJ,a ypaBHeHlie paBHOCliJIbHO ,lJ,li3'bIOHKUIfli ,lJ,BYX liJIIf 60JIbWerO
tJliCJIa ypaBHeHliiL B03bMeM, HanpliMep, ypaBHeHlie (x-I) (x-3)=0
li ,lJ,If3'bIOHKUlilO ,lJ,BYX ypaBHeHliH (2x-2=0) V (7x-21)=0. .MHO:>KecT-
BOM peweHliH ypaBHeHliH (x-I) (x-3)=0 HBJIHeTCH {I; 3}. TaK KaK
npoli3Be,lJ,eHlie ,lJ,BYX 1JliCeJI paBHo HYJIIO JIliWb B cJIyqae, Kor,lJ,a XOTH 6hI
O,lJ,HH li3 MHO:>KliTeJIeH paBeH HYJIIO, TO ,lJ,li3'bIOHKUIfH ypaBHeHliH (2x-2=
=0) V (7x-2I =0) CTaHOBliTCH IfCTliHHbIM BbICKa3bIBaHlieM npIf Bcex 3Ha-
1JeHliHX x, ,lJ,JIH KOTOpbIX CTaHeT liCTliHHO XOTH 6bl O,lJ,HO li3 BbICKa3b1Ba-
HliH 2x-2=O liJIli 7x-21 =0. nepBoe li3 HliX HCTliHHO, eC.'l1l .1.'=1, a
BTopoe - eCJm x=3. 3Ha1JliT, MHO:>KeCTBOM liCTIfHHOCTli ,lJ,li3'bJOHKUlili
HBJIHeTCH TO:>Ke caMoe MHOIKeCTBO {I; 3}. 3'1'0 li nOKa3bIBaeT, qTO ypaB-
HeHlie (x-I) (x-3)=0 paBHOCliJIbHO (2x-2=O) V (7x-2I =0).
npolIl.e Bcero HaHTli peweHlie ypaBHeHliH x = a - MHOIKeCTBO ero
peweHliH COCTOIfT li3 e,lJ,liHCTBeHHoro 1JliCJIa a, T={a}. n03TOMY npli pe-
llIeHlili ypaBHeHliH liX CTapalOTcH 3aMeHHTb paBHOCliJIbHbIMli liM ypaB-
HeHliHMli, liMelOlII.liMli 60JIee npoCToH Bli,lJ" nOKa He npli,lJ,YT K ypaBHe-
HlilO Bli,lJ,a x=a liJIli ,lJ,li3'bIOHKUlili x=al V x=a z V. . . V x=a n TaKliX
ypaBHeHliH. Tor,lJ,a MHOIKeCTBOM peweHliH 3a,lJ,aHHoro ypaBHeHliH HBJIH-
eTCH T={a l ; a z ; . . .; an}. 11Hor,lJ,a npliXO,lJ,liTCH nepeXO,lJ,liTb OT ,lJ,aHHOrO
ypaBHeHliH He K paBHOCliJIbHOMY eMY ypaBHeHIfIO, a K ero Clle,lJ,CTBlilO.
npli 3TOM MHmKeCTBO peweHliH pacwlipHeTCH, li nOTOMY Ha;r.o B KOHue
npOI!epliTb Bce HaH,lJ,eHHbIe KOpHli, nO,lJ,CTaBliB liX B HCXO,lJ,HOe ypaBHeHlie.
YupalKHeHDH
I. HaH)J.HTe MHO)KecTBO perneHHH ypaBHeHHH 2xz-7x+4=O, eC.1H: a) xER;
6) xEQ; B) xEZ; r) xEN.
2. YKa)KHTe MHO/KeCTBO, Ha KOTOpOM )J.aHHbIe )J.Ba ypaBHeHH5I paBHOCHJlbHbI:
a) (x+4)(x-l)=5(x-l) H x+4=5;
x 2 2x+l 2
6) 4x z +3 = 4x z +3 H x =2x+ 1.
3. PaBHOCHJlbHbI JlH CJle)J.YIOll1He ypaBHeHH5I Ha MHO/KecTBe )I.eHcTBHTeJJbHblX
QHCeJl:
B) (x-2) (x+3)=O H x-2=O;
6) IOx-2=4 II 5x-l=2?
4. 51BJl5IeTc5I JIll QHCJlO 5 KopHeM ypaBHeHlI5I:
6-x 1
a)- 5 =6+- 5 ;
x- x-
6) 6-x=6 (x-5)+ I? PaBHOCHJlbHbI JIll 9TII ypaBHeHH5I?
5. llaHbI ypaBHeHH5I:
a) i"=azx; 6) ax2-4=O;
B) ax-a2=4-2x; r) a+x=a z x-1.
OpH KaKHX 3HaQeHH5IX napaMeTpa a 9TH ypaBlleHH5I HMeIOT: I) e)J.HHCTBeHHoe perneHHe;
2) 6ecKOHe'lHOe MHO)KecTBO pernemli\; 3) He HMelOT pelUeflllH?
6. HaH)J.HTe 06J1aCTb onpe)J.eJJeHH5I H MHO/KecTBO perneHIIH ypaBlleHHH:
B) 2 2 1-  + 2x 3 -1 1 : 6) 2(x 2 + 1) 3:-13 1.
x - x + x + 1 x + 2x - I x. - 1
7. HaiI)J.HTe MHO/KeCTBa HCTHHHOCTH O)J.HOMeCTHblX npe)J.HKaToB:
a) i"+4=O, xER; 6) x=x, xEZ;
B) Ixl=lx+2I, xER;
116

3x-2 = 15x-3 _ x-4 . E R'
r) x-3 x2-9 x+3' x .
II ) 3 7 4- :Wx 2
2X-I + 2X+I - I_4x 2 =O, xER.
2. TeopeMbi 0 paBHOCHJlbHOCTH ypaBHeHHH. B 3TOM nYHKTe MbI .11.0-
Ka:>KeM TeopeMbI, nOKa3bIBalOlII.Ife, npIf KaKIfX npe06pa30BaHHHx .n.aH-
Hoe ypaBHeHIfe nepeXO,lJ,IfT B paBHOCHJIbHOe eM)' ypaBHeHHe.
T e 0 p e Mal. ECAU K 06eUM t[QcmflM ypa8H.eH.Ufl
fl(X)=f2(X), xE X,
(I)
npu6a8umb 8blpaJICeliUe F(x), UMelOU{l!e3H.atteHUe npu 8cex x U3 X, mo no-
A.yttUmCfl Ii080e y pa81ieH.Ue
fl(X)+F(x)=f2(X)+F(x), xE X,
(2)
Jl8AfllOU{l!eCfl CAeOCtn8UeM OaliH.020.
Ll 0 K a 3 aTe JI b C T B O. B caMOM .n.eJIe, nYCTb a - KopeHb ypaB-
HeHIfH (I), T. e. nYCTb fl(a)=f2(a). npIf6aBIfB K 06eIfM 'IaCTHM 3TOro
paBeHCTBa O.n.HO H TO :>Ke 'IIfCJIO F(a), nOJIY'lIfM HCTHHHoe paBeHCTBO
ft(a)+F(a)=f2(a)+F(a). OHO nOKa3hIBaeT, 'ITO a HBJIHeTCH If KopHeM
ypaBHeHIfH (2). 3Ha'lHT, Ka:>K.n.bIH KopeHb ypaBHeHHH (1) HBJIHeTCH KOp-
HeM If ypaBHeHIfH (2), T. e. ypaBHeHHe (2) - CJIe.n.CTBIfe ypaBHeHIfH (1).
HanpIfMep, ypaBHeHlIe 76-62=2x HBJIHeTCH CJJe.n.CTBIfeM ypaBHe-
HHH 76-2x=62, OHO nOJIY'IaeTCH H3 Hero npIf6aBJIeHHeM K 06eIfM '1ac-
THM O.n.HOro If TOro :>Ke BbipaIKeHHH 2x-62.
YpaBHeHHe ft(X)=f2(X), B CBOIO O'lepe.n.b, nOJIY1JaeTCH H3 ypaBHeHIfSl
f1(X)+F(x)=f2(X)+F(x) npIf6aBJIeHHeM K 06eIfM 1JaCTHM o.n.Horo H TOro
)!{e BbIpaIKeHIfH - F (x). n03TOMY He TOJIbKO ypaBHeHIfe (2) - CJleJl.-
CTBHe ypaBHeHIfH (1), HO H (I) - CJIe.n.CTBHe ypaBHeHHH (2), H, 3Ha'lHT,
3TH ypaBHeHHH paBHOCHJIbHbI.
TaKIfM 06pa30M, cnpaBe.n.JIHBa CJIe.n.YlOlII.aH TeopeMa:
T e 0 p e Mal'. ECAU 8blpaJICeH.Ue F(x) onpeoeAeH.O OAfl 8cex 3liatte-
H.uii x EX, mo y paeH.eH.Ufl
f 1 (x) = f 2(X), x EX,
H
f1(X)+F(x)=f2(X)+F(x), x E X,
paeflOCUAbflbl.
113 3TOH TeopeMbI BblTeKaeT, B 1JaCTHOCTH, TaKoe CJIe,lJ,CTBIfe:
JII060e ypaelieliue paeHOCUAbH.O ypaelieliulO 8uoa <D (x)=O. B caMOM
.n.eJIe, 'IT06bI npIfBecTIf ypaBHeHIfe fl (x)=f 2 (x) K BH.n.y <D (x)=O. .n.OCTa-
T01JHO npu6aBHTb K 06eHM 'IaCTHM 3TOro ypaBHeHHH BbIpa:>KeHHe -f 2 (x)
H nOJIOIKIfTb <D (x)=fdx)-f2 (x).
TO'lHO TaK :>Ke .n.OKa3bIBaeTCH CJIe.n.YIOllJ,aH TeopeMa:
T e 0 p e M a 2. ECAU 06e ttaemu ypa8H.eH.Ufl
fl (X)==f2(X), xE X,
(I)
117

y.MfiOJICumb fia 6btpaJICefiUe F (x), UJ,IOOU{ee 3lia'leliUe npu BeeX x EX,
mo nOAY'lUtnCfl I-/OBoe ypaBlieliue
fdx).F(x)=f2(X).F(x), xEX, (3)
flBA5IloU{eeCfl CACiJCtn6ucM ypaslieliufl (1).
YpaBHeHHe (1) 6Y,lJ,eT CJIe)l.CTBHeM (3) JIHWb npH YCJIOBHH, 1JTO H
1
F(x) HMeeT 3Ha1JeHHe npH Bcex X EX. E,lJ,HHCTBeHHbIM npenSITCTBHeM ,lJ,JIH
3Toro MO)!{eT 6bITb 06palIl.eHHe F (X) B HYJIb npH HeKOTopOM 3H2'H?HHH x.
n03TOMY BepHO CJIe,lJ,YlOlII.ee YTBep:>K,lJ,eHHe:
T e 0 p e M a 3. ECAU Bblpa:J1CeliUe F (x) UAleem 3lia'iefi ue iJUl Bcex
x E X U He o6paU{aemCfl 6 HYAb HU npu OaHOM 3Ha'lCHUU x U3 X, no ypa6-
H.eH.Ufl fdx)=f2(X) H fl(X).F(x)=f2(X).F(x), xEX, paC!t/ICWlbHbl.
B 'lllCmliOCmU, eCAU a*O, mo pasliOCUAbHbl ypa6Hel-lUfl fdx) = f 2 (x) H
afdx)=af 2 (x).
11HbllllII C.1JOBi1:lm, npu !JAlfLOJlCCHUU AI060zo ypa6lieliUfl Ha O'iL1U'lHOe
om HYAfl 'lUCAO nOJ1Y V .aemCfl ypaBHeliue, pa6HocuAbiioe aaHIiOldY. H3npH-
Mep, vpaBHeHHe 2x= 14 paBHOCHJIbHO ypaBHeIIHIO x=7.
npH peweHHH ypaBHeHHH npHMeHSIeTcH T3K)!{e MeTO)l. pa3.TJO)!{eHHSI
Ha MHO)!{HTeJIH. npe,lJ,n, OJIO)!{HM 1JTO BbIpaIKeHH5I fl (x), f Z (x), ...
..., fn (x) HMelOT 3Ha1JeHHH npH Bcex x EX. Tor,lJ,a 'lUCAO a EX JuoJICem
6blmb KoplieM ypaBlieHUfl
fdx)'f 2 (x)..... fn (x)=O (4)
6 mOM U mOAbKO B mOM CAY'lae, KoeiJa xomfl6bl OOHO U3 BblpaJiCel-lut1 fl (x),
f 2 (x), ..., fn (x) 06paw,aemcfl6 HYJlb npu x=a. 3T0 3Ha1JHT, liTO !J!](i6HeHue
(4) pasHOCUAbliO aU37JIOHK/{UU ypa6lieliut1 f:(x)=Ovf2(X)=0 V...
... V fn (x)=O.
HanpHMep, ypaBHeHHe x (x-4) (x+6) (x-8)=0 paBHOCHJIbHO ,lJ,H31>-
IOHKUHH ypaBHeHHll x=OVx-4=OVx+6=OVx-8=O, H nOToMY
MHO)!{eCTBOM ero peweHIIH SIBJI5IeTC5I {O; 4;-6;8}.
YupalKHeHHH
1. )J.oKa,KIITe, '1TO ypaBHeHHe (4x-5) (x 2 -4)=7 (x 2 -4) !lBJI!leTC!I C.'Ic.n:CTBHeM
ypaBHeHII!I 4x-5=7.
2. )J.oKaIKIITc, '1TO c.1Ie.n:YJOlUHe napbl ypaBHeHHi\ paBHOCII.1IbHbI:
a) 2x-l+xL9=3x-6+x2+9 II 2x-l=3x-6;
6) (2x-l) (x2+9)= (3x-6) (x2+9) H 2x-l=3x-6.
3. )J.oKaIKIITe, '1TQ ypaBHeHlle f2 (X)=g2 (x) !lE.1I!lCTC5I CJle.n:CTEHeM yp3BHeHH!I
f (x)=g(x). Bcer.n:a J1H paBIlOClI.1IbHbI 9TH ypaEHemISI?
3. HepaBeHCTBa C O)l.HO" nepeMeHHOH. HepaBeHCTBa, CO,lJ,EpJE3llI.He
nepeMeHHYIO x, HanpHMep fdx)<f2(X), xEX; fdx» f2(X), xEX H
T.,lJ,., TaK:>Ke HBJISIIOTCSI O,lJ,HOMeCTHblMH npe,lJ,HKaTaMH. PeIIIHTb T3IWe He-
paBeHCTBO 03Ha1JaeT HaHTH MHO:>KeCTBO T 1JHCeJI, npH nO,lJ,CTaHOBKe Ko-
TOpbIX BMeCTO X nOJIY1JaIOTC5I HCTHHHhIe HepaBeHCTBa. 3T0 MHO)!{eCTBO
1JHCeJI Ha3bIBalOT MliOJICeCtn60J11 peweliuii liepaseliCmBa.
MO:>KeT CJIy1JHTbC5I, 1JTO K3)K)J,Oe peWeHJJe O)l.HOrO HtT3EeHCTBa
YJl.OBJIeTBOpHeT ,lJ,pyrOMY HepaBeHCTBY. Tor,lJ,a BTOpoe HepaBeHCTBO Ha.
118

""f'::.
3bIBalOT CAeacmBueM nepBOrO. B03bMeM, HanpIiMep, HepaBeHCTBa x>4
Ii x>2. 5ICHO, 'Iro eCJIIi KaKOe-HIi6Y,lJ,b 1JIiCJIO 60JIbWe 1JeTbIpex, TO OHO
60JIbIIIe ,lJ,BYX. n03TOMY HepaBeHCTBO x>2- CJIe,lJ,CTBIie HepaBeHCTBa
x>4. MHOIKecTBO Q peWeHHH CJIe,lJ,CTBIfH ,lJ,aHHOrO HepaBeHCTBa BKJIIOIJaeT
B Ce6H MHOIKeCTBO T peWeHIfH 3a,lJ,aHHOrO HepaBeHCTBa, Q=:J T.
ECJIH ,lJ,Ba HepaBeHCTBa HMelOT O,lJ,HO Ii TO :>Ke MHO:>KeCTBO pellieHIiH,
IiX Ha3hIBaIOT pasHOCUAbNbiAtU. B 3TOM CJIY1Jae 06a HepaBeHCTBa HBJIHIOT-
CH CJIe.lI.CTBIiHMH ,lJ,pyr ,lJ,pyra. HanpIiMep, CKa3aTb, 1JTO KaKoe-HIi6Y,lJ,b
qIiCJIO a 60JJbIIIe 5, paBHOCIiJIbHO YTBep:>K,lJ,eHIiIO, 1JTO qIiCJIO a+ 1 6oJIb-
IIIe 6. n03TOl\lY HepaBeHcTBa x>5 Ii x+ 1>6 paBHOCH<1bHbI.
nOCKO.'IbKY HepaBeHcTBa, CO,lJ,epIKalIl.Iie x, HBJIHIOTCH npe,lJ,HKaTaMIi,
MOIKHO rOBopHTb 06 HX KOWblOHKUIiIi Ii ,lJ,H3"bIOHKUIiH. HanpIiMep, 1JIiCJIO
a Y,lJ,OBJIeTBOpHeT KOH"bIOHKUIiIl HepaBeHcTB
(3x-8>1)j\ (2x+5<15),
ec,1H OHO Y,lJ,oBJIeTBOp HeT Ii HepaBeHcTBY 3x-8> 1, Ii HepaBeHcTBY 2x+
+5<15. TaKHM 1JHCJIOM HBJIHeTCH, HanpIiMep, 4. B wKoJIe06b1IJHO rOBO-
PHT He 0 KOH"bIOHKUHH, a 0 CIiCTeMe HepaBeHCTB H 3anIiCbIBalOT ee TaK:
{ 3X-8> 1 ,
2x+5<15.
JJ.Ii3"bIOHKUIiH ,lJ,BYX nJIIi 60JIbWerO 1JIiCJIa HepaBeHcTB IiCTIiHHa npIi
HeKoTopoM 3Ha1JeHIiIi a, ecJIIi npIi 3TOM 3Ha1JeHIiIi IiCTIiHHO XOTb O,lJ,HO
83 3TIiX HepaBeHCTB. HanpIiMep, 1JIiCJIO - 2 npIiHa,lJ,JIe:>KIiT MHO:>KecTBY
peweHIfH ,lJ,H3"bIOHKUIiIi HepaBeHCTB
(2x>8) V (3x<-3). (5)
B caMOM ,lJ,eJIe, eCJIIi nO,lJ,CTaBIiTb 3TO 1JIiCJIO B nepBoe Ii3 HepaBeHcTB, no-
JIY1JIiTCH JIOIKHOe HepaBeHcTBo 2.(-2»8. Ho npIi nO,lJ,CTaHOBKe Toro
)!{e 1JIiCJIa BO BTopoe HepaBeHCTBO nOJIY1JaeM IiCTIiHHoe HepaBeHcTBo
3. (-2)<-3. 3HaIJHT, IJIfCJJO -2 npIIHa,lJ,,1eMIfT MHoIKecTBY peweHHH
,lJ,Ii3"bIOHKUHH (5). A IJHCJIO 0 He npHHa,lJ,JIe:>KHT 3TOMY MHO:>KeCTBY, TaK
KaK npH nO,lJ,CTaHOBKe :onoro IJIiCJIa B 06a HepaBeHCTBa, BXO,lJ,HlII.Iie B (5),
nOJIYIJaeM JIOIKHbIe HepaBeHCTBa 2.0>8 Ii 3.0<-3.
KaK npaBIIJIO, MHQ)KeCTBa pellieHIiH HepaBeHcTB 6eCKOHeIJHbI; ,lJ,JISI
Har.TJH,lJ,HOCTIi 3TIi MHOIKeCTBa 306paIKaIOT H a KOOp,lJ,IiHaTHOH ocv:.
JJ.JIH 3TOro MHOJKeCTBO peweHHH npe,lJ,CTaB.TJHIOT B BIi,lJ,e 06"be,lJ,IiHeHIiH He-
CKOJIbKHX nonapHo-HenepeCeKalOlII.HxCH T01JeK, OTpe3KOB, npoMe-
:>KYTKOB HJIM JIY1JeH. JJ.JIH 3Toro IiC{JOJIb3YIOT CJIe,lJ,YlOlII.Iie TeopeMbi 0 He-
paBeHcTBax C O,lJ,HOH nepeMeHHoH.
T e 0 p e Mal. ECAU ebtpa;){CeHue F (x) uAteem 3H.atteHue npu AI060M
xEX, mo HepaeeHcmea fdx)<f2(X) H fdx)+F(x)<f2(X)+F(x) pae-
H.OCUAbHbt.
JJ. 0 K a 3 aTe JI b C T B O. B caMOM ,lJ,eJIe, eCJIIi a npIiHa,lJ,JIe)!{HT
MHO:>KeCTBY peweHHH HepaBeHCTBa fdx)<f 2 (x), TO BblCKa3b1BaHIie
fl(a)<f 2 (a) HCTliHHO. npIi6aBJIHSI K 06eIiM 1JaCTHM 3Toro HCTliHHOrO He-
paBeHCTBa Or1!IO If TO IKe IJHC.TJO F (a), nOJJYIJaeM IiCTHHHoe HepaBeHCTBO
fd a ) + F (a)<f 2 (a)+ F (a). OHO nOKa3bIBaeT, 1JTO a HBJISIeTCSI peweHIieM
119

H HepaBeHCTBa fl (x)+F (x)<f 2 (x)+F (X). T01JHO TaK IKe ,lJ,OKa3hIBaeTCSI,
1JTO JII060e pellIeHHe HepaBeHCTBa fdx)+F (x)<f 2 (x)+F (X) Y,lJ,oBJIeT-
BopSIeT 11 HepaBeHcTBY fdx)<f 2 (X). 3Ha1JI1T, 3TH HepaBeHCTBa paBHO-
CI1JIbHbI.
AHaJIOrl11JHO MO)!{HO ,lJ,OKa3aTh paBHOCI1JIhHOCTh HepaBeHCTB
fdx»f 2 (X) 11 fdx)+F (x»f 2 (x)+F (X), a TaK)J{e fdx)f 2 (X) H
fdx)+F(x)f2(X)+F(x) H T.,lJ,.
T e 0 p e M a 2. ECAU IlUCAO a nOAo;){CumeAbI-lO, a> 0, mo I-lepaeel-lcmea
fdx)<f2(X) 11 afdx)<af2(x)paeI-lOCUAbHbl. ECAU ;)(Cea<O, mopaeHocuAb-
I-lbl Hepaeel-lcf1113a f dx)<f 2 (X) U af dx»af 2 (X) -- npu YMHo;){CeI-lUU I-la
ompu/{ameAbl-lOe IlUCAO I-laao U3MeHumb 31-laK I-lepaeel-lcmea.
TeopeMa 2 SIBJISIeTCH 1JaCTHhIM CJIY1JaeM CJIe,lJ,YIOIll.eH TeOpeMhI.
T e 0 p e M a 2'. ECAU eblpa;){Cel-llie F (x) onpeaeAeHo npu ecex 3l-lalle-
I-lWlX xE X U nOAo;){CumeAbl-lO I-la X, mo Hepaeel-lCmea fdx)<f2(X) H
fl (x). F (x)<f 2 (x). F (x) pael-lOCUAbl-lbl.
B CJ1Y1Jae eCJII1 F (x) HeOTpliuaTeJIhHO, paBHOCI1JIhHbI HepaBeHCTBa
fdx)f2(X) 11 fdx).F(x)f2(X).F(x).
OTMeTliM elIl.e CJIe,lJ,YlOlII.YIO TeopeMY.
I I
T eo p eM a 3. Hepaeel-lcmea O<fl (X)<f2 (x) li 0< f2(X) < fl(X)
pael-lOCUAbl-lbl apyz apyzy.
n p 11 M e p 1. PellIl1M HepaBeHcTBo 5x--5>2x+ 16. no TeopeMe 1
OHO paBHOCI1J1hHO HepaBeHcTBY 5x-2x> 16+5, T. e. 3x>21. no Teope-
Me 2 3TO HepaBeHcTBo paBHOCliJIbHO HepaBeHcTBY x> 7. 3Ha1JI1T, !\fHO-
IKeCTBOM pellIeHI1H 3a,lJ,aHHOrO HepaBeHcTBa SIBJISIeTCH 1JI1CJIOBOH JIyq
J 7; +00[.
n p 11 M e p 2. PellIl1M KOH'bIOHKUI1IO HepaBeHcTB
(3x-5< 4)!\ (2x-3>-I).
llJISI 3Toro CHa1JaJIa pellIHM nepBoe li3 3THX HepaBeHcTB:
3x--5<43x<9x<3.
T01JHO TaK, pellIaSI BTopoe HepaBeHcTBo, HaXo,lJ,I1M:
2x--3>--1 2x>2x> 1.
4I1cJIa, Y,lJ,OBJIeTBOpHlOlII.Ife KOH'bIOHKUIfIf 3TI1X HepaBeHcTB, ,lJ,OJI)!{HbI
y,lJ,oBJIeTBOpSITh 06011M HepaBeHcTBaM, T. e. MHO)!{eCTBO pellIeHI1H KOH'bIOHK-
11,1111 SIBJIHeTCH rrepeCe1JeHIfel\1 HaH,lJ,eHHhIX MHOIKeCTB pellIeHI1H. Ho rrepe-
Ce1JeHl1eM 1JI1CJIOBhIX JIyqeH x < 3 11 x> 1 SIB.1HeTCSI 1JIICJ10BOH rrpOMe)!{y-
TOK 1 <x<3. 3ToT rrpOMe)!{YToK li SIBJISIeTCH I1CKOMbIM MHOIKeCTBOM pe-
meHliH ,lJ,aHHOH KOH'bIOHKUI1I1.
npl1 pellIeHlil1 HepaBeHcTB BI1,lJ,a (x-a l ) (x-a 2 ).. . (x-a,.»O, r,lJ,e
a1<a 2 <...<a,., rrOJIh3YlOTCH CJIe,lJ,YlOlII.I1M paccYIK,lJ,eHl1eM: MHOIKli-
TeJIb X-ah OTpliuaTeJIeH rrpli x<a/i. 11 rrOJIOIKliTeJIeH npl1 x>ah.
HHhIMI1 CJIOBaMli, OH MeHSIeT 3HaK JIl1llIh rrpl1 x=ah' np0l13Be,lJ,eHHe
(x-a l )(x-a 2 )...(x-a,.) MOIKeT li3MeHliTb 3HaK, J1l1llIb KOr,lJ,a MeHHeT 3HaK
O,lJ,HH H3 MHOIKHTeJIeH, T. e. B T01JKaX ah a 2 , ..., a,..3TH T01JKI1 pa36HBalOT
qHCJIOBYIO OCb Ha npOMe:>KYTKH )-00; al[,Ja1; a 2 [, ...,] a,.-1; a,. [,)an;
120

 I
-6
II 11-0
1 35 x
PHC. 72
+00[, Ha KaIKAoM H3 "miX npOMe:>KYTKOB BbIpa:>KeHHe (X-al) (x-a 2 )...
...(x-a n ) HMeeT nOCTOHHHbIH 3HaK. n03TOMY AOCTaT01JHO Y3HaTb 3HaK
BbIpaIKeHHSI B OAHOH H3 TOlJeK npOMeIKYTKa, lJT06bI Y3HaTb ero Ha BceM
npOMeIKYTKe.
Onpe,lJ,eJISISI 3HaK BbIpaIKeHIfSI Ha KaIK,iJ.OM If3 npOMeIK)'TKOB, MbI OT6H-
paeM Te H3 HHX, Ha KOTOpbIX 3TO BbIpaIKeIlIfe nOJ10)KIJTe.lbHO. I1x 06'be-
AHHeHHe If SIBJISIeTCSI MHO)!{eCTBOM pellIeHHH HepaBeHCTBa
(X-al)(x-a2)... (x-an»O.
n p H M e p 3. PellIHM HepaBeHcTBo (x+6)(x-l)(x-3)(x-5»O.
TOtJKH -6, 1, 3, 5 pa36HBaIOT lJHCJIOBYIO npSIMYIO Ha npOMe)!{YTKH 1-00;
-6[,]-6; Il,I 1; 3[,] 3; 5[,15;+00[. BbI6epeM Ha JIYlJe 15;+00[ lJIfCJIO
10. nOACTaBJIHSI 3TO tJHCJIO B BbIpaIKeHIfe (x+6}(x-l )(x-3) (x-5),
nOJIYtJHM npOIf3BeAeHHe lJeTbIpex nOJImKHTeJlbHbIX MHoIKHTeJleH (10+6) X
X (10-1). (10-3). (10-5), KOTopoe nOJlOIKIITeJIbHO. TaK IKe Onpe,lJ,e.lIH-
eM, tJTO Ha npoMeIKYTKe )3; 5[ BbIpaIKeHHe OTpIfuaTeJIbHO, Ha npo-
MeIKYTKe ] 1; 3 [ nOJIOIKHTeJIbHO, Ha npoMeIKYTKe 1-6; 1 [ OTpIfuaTeJIbHO,
Ha JIYlJe ]-00; -6[ nOJIO)!{HTeJIbHO. 3Ha1JHT, pellIeHIfeM HepaBeHcTBa HB-
JIHeTCSI 06'be,lJ,HHeHHe JIYtJeH ]-00; -6[,15;+00[ II npoMeIKYTKa )1;3[:
T=]-oo; -6[ U] 1;3[ U ]5;+00[.
MOIKHO 6bIJIO orpaHHtJIfTbCSI onpeAeJIeHIfeM 3HaKa BbIpa)!{eHHSI Ha
JIYlJe ]-00;-6[. TaK KaK npH nepexo,lJ,e lJepe3 KaIK,lJ,YIO If3 TotJeK -6, 1,3,
5 OAHH If3 MHO)!{IfTeJIeH x+6, x-I, x-3, x-5 MeHSIeT 3HaK, TO H Bce
npOH3Be,lJ,eHIfe MeHHeT npIf 3TOM 3HaK. OTCIOAa cpa3Y BHAHO, 1JTO Ha
)-6; 1 [ BbIpaIKeHHe oTpHuaTeJIbHO, Ha 11 ;3[ nOJIOIKHTeJIbHO H T. A.
4T06bI CAeJIaTb 3TO paccYIKAeHHe 60JIee JlarJISI,lJ,HbIM, tJepTSIT KPIfBYIO,
KOTopaSI HAeT BbIllIe OCH a6cUHCC TaM, r,lJ,e Hallie BbIpaIKeHHe nOJIOIKH-
TeJIbHO, H HIf)!{e TaM, rAe OHO OTpHuaTeJIbHO (pHC. 72). 3Ty JIHHHIO Ha-
3 bIB a lOT KPUBOr1 3HaKOB.
YupamHeHHH
I. f1306pa3HTc Ha GHCJlOBOH OCll MHO}KCCTBa TOGCK. Koop.l\HHaTbI KOTOpblX Y.l\OB.
JleTBOp5!IOT C.1C.l\YlOllUIM YCJIOBH5IM:
!
a) x= {xlx-42}; 6) X= {xl-5<x<O};
B) x= {xl-3,2<x<3}; r) X= {xI3.6<x<8};
.11.) x= {xl l,7<x<4,5}.
2. 3a.l\aHre HcpaBeHcTBaMH n0.ll.MHO}KCCTBa '1HCJIOBOH OCH, H306pa}KeHllblC Ha
pHCyHKc 73.
3. l1CTHHHbI HJlH JlO}KHbI CJIe.ll.YlO1II.Hc HMn.1HKaUHH:
a a
a) bO -) (aO) /\ (bO); 6) (aO) /\ (b>O)  b O?
121

4. PaBHOCHJlbHbI Jill CJIe.n.YKl1Il.IIe
HepaBeHcTBa Ha MHOmeCTBe .n.eiiCTBII-
TeJlbHblX '1I1CM:
a) x 2 +3x-2>2 H r+3x-4 >0;
6) -3x+4<0 H 3x-4>0;
x-3
B) x2+1 >0 H x-3>O; r) 3x 2 <6x
H x<2?
5. Pew liTe CJIe.n.YIOIIlHe HepaBeHCT'
Ba (xER):
a) (x-2) (x+3) (x-4) >0;
6) (x 2 -4) (r-9) >0;
B) (x2+4) (x 2 -16»O.
4. YpaBHeHHH C ,lJ,BYMH nepe-
MeHHblMH. ECJIH B KJIeTKe Haxo-
AHTCSI X a3aHoB H y KpOJlHKOB, TO
06w.ee tJHCJIO HOr paBHO 2x+4y.
n03TOMY eCJIII B 3a;J:atJe CKa3aHO, tJTO 06w.ee tJHCJIO HOr paBHO 62, TO
MbI MOIKeM 3anllCaTb ypaBHeHlIe 2x+4y=62. O,lJ,H03HatJHO onpeeJIIITb
H3 3TOro ypaBHeHHSI 3HatJeHHSI X H Y He.1Jb3SI. Lla)!{e eCJIH OrpaHHtJHTbCSI
HaTypaJIbHbIMII 3HatJeHIISIMH X H y, 3,lJ,eCb MOrYT 6hITb TaKHe CJIYtJaH:
x=I, y=15; x=3, y=14; x=5, y=13 H T. ,lJ,.
YpaBHeHHe, COAep)!{aw.ee ABe nepeMeHHbIe X H y, SIBJlSIeTCSI ByxMeCT-
HblM npe,lJ,IIKaToM. napa tJIIceJI (a; b) Ha3hIBaeTCSI peUlerLUe};! 3Toro ypaB-
HeHHSI, eCJlH npH 3aMeHe X Ha a H y Ha b nO.1JytJaeM IICTHHHOe paBeHcTBo.
HanpIlMep, napa tJHCeJI (3; 4) SIBJISIeTCSI OHHM H3 pellIeHHH ypaBHeHHSI
X 2 +y2=25, TaK KaK 3 2 +4 2 =25. Ho 3T0 ypaBHeHHe IIMeeT H ApyrHe pe-
rneHHSI, HanpHMep (5; 0), (-3; -4) H T. A.
KaIKAoMY ypaBHeHHIo C ,lJ,BYMSI nepeMeHHhIMH cooTBeTcTByeT MHO)j{e-
CTBO ero pellIeHI!H, T. e. MHOIKecTBO, COcToSIw.ee 113 Bcex nap tJlIceJI
(a; b), npll nO,lJ,CTaHOBI{e KOTOphIX B ypaBHeHHe nOJIytJaeTcSI HCTIIHHoe
paBeHcTBo. npll 3TOM, KOHetJHO, eCJIII 3apaHee YKa3aHhI MHO)!{ecTBa X H
Y, KOTophIe MorYT npHIIIIMaTb HeH3BeCTHbJe X II y, TO HaAO 6paTb JIHllIb
TaKHe naphI (a; b), AJISI KOTOpblX a E X II bEY.
napy tJllce.'1 (a; b) MOIKHO H306pa3HTb Ha nJIOCKOCTI! T04KOII M,
HMelOw.eH KOOp,lJ,HHaThI a H b, M =M(a; b). PaCCMaTpl!BaSI 1!306pmKeHIISI
Bcex TOlleK MHO)j,eCTBa peweHHH ypaBHeHHSI C ABYMH HeI!3BeCTHhIl\lH, IIO-
JIYIJIIM HeKoTopoe nOAMHOIKeCTBO nJIOCKOCTlJ. Ero Ha3bIBalOT cpacpuKOAt
ypaBH.eHufl. HanpllMep, TotJKa M(3; 4) npHHaAJIe)!{HT rpaHKY ypaBHe.
HHH X 2 +y2=25 - eC.111 3aMeHHTb B 3TOM ypaBHeHHH X Ha 3, a y Ha 4,
nOJIytJHTCSI HCTIIHHoe paBeHcTBo 3 2 +4 2 =25. TotJKa me N (4;6) He npH-
HaAJIe)!{IIT rpaHKY ypaBHeHHH, TaK KaK paBeHcTBo 4 2 +6 2 =25 JIO)!{HO.
06bItJHO ypaBHeHlIe C BYMSI nepeMeHHbIMII HMeeT 6ecKoHetJHoe MHO-
:>KecTBO pellIeHHH, a nOToMY ero rpaqmK cOAepIKHT 6ecKoHe1JHO MHoro
TOtJeK. nOCTpOHTb HX, OTMetJaSI OHY 3a pyroH, HeB03MOIKHO - Aa:>Ke
3a MIIJI.'IHOH JIeT MbI OTMeTHM JIlIllIb KOHetJHoe tJHCJIO TotJeK. n03TOMY
npH6eralOT K reOMeTpHtJecKoMY cnoc06y onHcaHHH.
n p H M e p 1. MHOIKecTBO pellIeHHH ypaBHeHHSI y-x=O COCTOIlT
H3 Bcex nap tJIIce.1J (a; b), B KOTOpbIX nepBaSI KoopAHHaTa paBHa BTOpOH,
T. e. H3 nap BHa (a; a), a E R. ECJIH OTMeTHTb Ha nJIOCKOCTH HeCKOJIbKO
.
-5
i
a
.
3,5

12
.
-If.!.
2
o
5
o
5l-
PIIC. 73
122
:.
--
..
..

TOtJeK BH,lJ,a M (a; a), HanpHMep
M (1; 1), M (2; 2) H T. A., TO YBH-
)J.HM, 1JTO Bce 3TH T01JKH JIe:>KaT Ha
npSIMOH, npOXOAHlII.eH 1Jepes Ha-
qao KoopAHHaT H HaKOHeHHOH
KOCH a6cUHCC no,lJ, yroM 45°
(pHC. 74). B03HHKaeT npeAnoo-
IKeHHe, 1JTO Boo6w.e Bce T01JKH rpa-
$HKa AaHHoro ypaBHeHHSI JIeIKaT
. Ha 3TOH npSIMoiL IloKaIKeM, 1JTO
3TO AeikTBHTe.'IbHo TaK.
ECH a6cUHcca T01JKH M paBHa
ee opAHHaTe, TO TpeyrOJIbHHK
OPM, rAe 0 - Ha1JaJIO KoopAHHaT, a P - npoeKIJ.HSI T01JKII M Ha OCb
a6CUHCC, paBHo6eApeHHbIH (pHC. 74). n03TOMY BeH1JIIHa yra MOP
paBHa 45°. 06paTHo, ecml npSIMaSI OM HaJ{OHeHa KOCH a6CUHCC no,lt
yrJIOM 45°, TO Tpeyro.'1bHHK OPM paBHo6eApeHHbIH, H nOTOMY a6cUllcca
TOtJKH M paBHa ee opAHHaTe.
B 3TOM npIlMepe rpaHKOM ypaBHeHHSI OKa3aaCb npSIM3SI HHHSI.
HHIKe MbI nOKaIKeM, 1JTO rpaHKOM 106oro ypaBHeHHSI BH.1.a y=kx+b
HBSIeTCSI npSIMaH JIHHHSI.
n p H M e p 2. rpaHKo M ypaBHeHHH
V(X-2)2+ (y-3)2 = 4
y l/!
1/11
/''M(QjQ) -
,/ I I
 I
'/M(2;2)
,/ (1j1)
" o I I I P X
1/ I I I I
./ I I I I
PHC. 74
(1)
HBJISIeTCH OKPY:>KHOCTb C ueHTpOM A (2; 3) H paAHYCOM 4. B caMOM AeJIe,
pa CCTOSIHHe OT T01J KH M (x;y) AO T01JKH A (2; 3) BbIpa:>KaeTCSI opMYJIoii
V(x-2)2+(y-3)2. n03TOMY paBeHCTBO (1) 03Ha1JaeT, 1JTO 3TO paCCTOH-
HHe AJIH Bcex T01JeK rpaHKa paBHO 4; a 3T0 II 3Ha1JHT, 1JTO T01JKa .1eIKHT
Ha YKa3aHHoH BbIllIe OKPYIKHOCTH. 06paTHo, eCH T01JKa M (x; y) JIeIKHT
Ha OKPYIKHOCTH C ueHTpoM A (2; 3) H pa,lJ,HycoM 4, TO paCCT05H!lIe OT A ,lto
M paBHo 4, H nOToMY BbInOJIHSIeTCSI paBeHcTBo (1).
IlBa ypaBHeHHSI C ABYMSI nepeMeHHbIMH, HMelOw.Ife O,lJ,I1HaKOBbIe rpa-
$HKH, Ha3bIBalOTCSI pa8H.OeUJlbH.btMIl. HanpHMep, paBHocHblIbl ypaBHe-
HHSI x+2y=5 H 3x+6y= 15 - JII06aSI napa 1JHCeJI, YAoB.HeTBopHlOlII.aH
OAHOMY H3 3THX ypaBHeHHH, YAOBJIeTBOpSIeT H BTOPOMY.
JIerKo AOKa3aTb CJIeAYlOlII.He TeopeMbI:
T e 0 p e Mal. EeAU 8blpaJICeH.ue f (x, y) onpedeAeH.O d.1Jl 8cex 3H.atte-
H.UU x U y, mo ypaeH.eH.UR F (x, y)=<D (x, y) U F (x, y)+f (x, y)=cD (x,y)+
+f (x, y) paBH.OeUAbH.bl.
T e 0 p eM a 2. EeAu BbtpaJICeH.ue f (x, y) onpede//£H.o dAR Beex 3H.atte-
H.uii x U y U H.U npu KaKUX 3H.atteH.UJlX x U y H.e o6paUipemeJl 8 H.YAb, mo
ypa8H.eH.Wl F(x, y)=<D(x, y) U F(x, y).f(x, y)=cD(x, y).f(x,y) paeH.G-
eU.7bH.bl.
YupalKHeHHH
1. ,Z:loKa/KHTe paBHOCHJJbHOCTb CJIe.n.YIOw.Hx nap ypasHeHHH C llBYMH HeU3BeCT-
HblMH:
a) x2+1f+6x=8y-4 H X2+1f+ 6x-8y+4= 0;
123

6) x+y=5x-2y+l H (x2+y2+1) (x+y)= (x 2 +y2+1) (5x-2y+l);
B) X 3 _y3+6x+8=9x 2 +5x+9 H x 3 _y3+X=9x 2 +1.
2. HapHcyihe rpa<j>HKH ypaBHeHHH:
a) Ixl=y; 6) Ixl=lyl; B) (X-4)2+ (y+5)2= 16; r) y=x 2 ; A) lyl=x8.
5. YpaBHeHHe OKPY)KH OCTH. B npllMe pe 2 n. 4 6hIJIO nOKa3aHO, tJTO
rpaHKoM ypaBHeHHH V (x-2)2+(y-3)2=4 HBJIHeTCH OKpY)!{HOCTb C
ueHTpoM A (2; 3) H paAHycoM 4. TotJHo TaK IKe AOKa3bIBaeTCH 06lI1.ee )'T-
Bep)KAeHHe:
r pacpuKO.M ypaBHeH.UR
V (x-a)2+ (y-b)2=R (1)
MARemCR OKPYJ/CH.OCtnb C /{eH.mpo.M A (a; b) U paiJuycO.M R.
B CaMOl\! ,lJ,eJIe, eCJm T04Ka M (x; y) JI e)!{HT Ha 3TO H OKPYIKHOCTH, TO
IAMI=R. Ho IAMI=V(x-a)2+(y-b)2, n03TOMY V(x-a)2+(y-b)2=
=R. 3Ha4HT, Bce T04KH OKPYIKHOCTH npllHaAJIeIKaT rpaHKY ypaBHe-
HHH (1). 06paTHo. nYC Tb T04Ka M (x; y) npHHa.lI.JIeIKHT rpaHKY ypaBHe.
HIIH (I). TaK KaK V(x-a)2+(y-b)2 - pacCToHHHe OT M (x; y) AO
A (a; b), TO paCCTOHHHe IMA I paBHO R, H nOTOMY T04Ka M (x; y) JIe)KHT
Ha OKp YIKHOCTH.
04eBH,lJ,HO, 4TO ypaBHeHlle (1) paBHOCHJIbHO ypaBHeHHIO
(x-a)2+ (y-b)2=R2. (2)
B 3TOM BIi,lJ,e 06b!4HO H 3anHCbIBalOT ypaBHeHHe OKPYIKHOCTH.
n p H M e p 1. HanHllIeM ypaBHeHHe OKPYIKHOCTH C ueHTpOM
A (7;-6) H paAHycoM 8.
no <jJopMYJIe (2) nOJIytJaeM ypaBHeHHe B BHAe:
(x-7)2+ (y+6)2=64.
(3)
3T0 ypaBHeHHe MOIKHO 3anHcaTb HHa4e, paCKphIB CK06KH H npHBeAH
nOA06HhIe 4.1leHbI:
x 2 +y2-14x+12y+21=0. (3')
06paTHbIH nepexo.1J: OT ypaBHeHHH BHAa (3') K ypaBHeHHIO BHAa (3)
,lJ,eJIaeTCH nYTeM BbI.1J:eJIeHIIH nOJIHbIX KBaApaToB.
n p H M e p 2. 110Ka)!{eM, 4TO ypaBHeHlIe
X2+y2-6x+8y--=-75 =0 (4)
HBJIHeTCH ypaBHeHHeM OKPYIKHOCTH, H HaHAeM ee ueHTp H paAHYc.
4T06bl BbIpmI\eHHe x2-6x CTaJIO nOJIHblM KBaApaTOM, HaAO npIf6a-
BHTb K HeMY 3 2 =9, a 4T06bl nOJIHbIM KBa.1J:paTOM CTaJIO BbIpaIKeHHe y2+
+8y, HaAo ITpH6aBHTb K HeMY 4 2 = 16. n03TOMY nepenHllIeM ypaBHeHHe
(4) CJIeAYIOw.IIM 06pa30M:
(x2-6x+9)+(y2+8y+ 16)-75=9+ 16.
OTclO;J:a (X-3)2+(y+4)2= 100.
Tenepb YIKe BH,lJ,HO, tJTO 3TO - ypaBHeHlIe OKPYIKHOCTH C ueHTpoM
A (3;-4) H pa,lJ,HycOM 10.
124

MO/KeT CJIYlJHTbCH, '1TO nOCJIe BbI.n.eJIeHHH nOJIHblX KBa.n.paTOB nOJlY'IHM ypaB-
HeHHe BH.n.a (x-a)Z+ (y-b)z=O. CYMMa KBa.n.paTOB MO/KeT paBHHTbCH Hymo JIHl1Jb
B CJIY'Iae, Kor.n.a 06a CJIaraeMbiX paBHbI Hymo, T. e. Kor.n.a x-a=O II y-b=O.I1HblMH
CJIOBaMH, rpa<j>HK ypaBHeHHH (x-a)Z+ (y-b)Z=O COCTOHT H3 o.n.Hoii TO'IKH M (a; b).
ECJIH /Ke nOCJIe BbI.n.eJIeHHH nOJIHblX KBa.n.paTOB nOJlY'IHTCH ypaBHeHHe BH.n.a (x-a)Z+
+ (y-b)Z=c, r.n.e c - OTpHuaTeJIbHOe '1I1CJIO, TO rpa<jJHK ypaBHeHHH nYCT - HH npH
KaKIiX 3Ha'ieHIIHX x H y CYMMa .n.Byx KBa.n.paTOB He MO/KeT 6b1Tb OTpHuaTeJlbHoii.
YupalKHeHH.II
1. HanHl1JIIT' ypaBHeHHe OKPY/KHOCTH C UeHTpOM A (a; b) H pai\lIycoM R, eCJIH:
a) a=-4, b=O, R= 10;
6) a=O, b=O, R=6.
2. Haii.n.IITe UeHTp H pa.n.HYc OKPY/KHOCTH, 3a.n.aHHoii CBOHM ypaBHeHHeM:
a) XZ+IP+ IOx-6y= 110; 6) x Z +y2-4x-12y=24;
B) x2+y'2+8x+8y+32=O; r) xZ+yZ-6x-4y+7=O.
3. Haii.n.HTe Ha OKPY/KHOCTH xZ+yZ-3x-4y-4=O TO'IKH, a6cullcca KOTOpblX
paBHa 4.
4. Haii.n.HTe Ha OKPY/KHOCTH xZ+yz+5x-7y-4=O TO'lKH, a6cUHcca KOTOpblX
p3BHa HX op.n.HHaTe.
6. rpa$HKH HepaBeHCTB. OTHollIeHHSI MeIK,lJ,Y 1JHC.'IaMH x H Y BbIpa-
IKaIOTCSI He TOJIbKO npH nOMow.H ypaBHeHHH, HO H npH nOMow.H Hepa-
BeHCTB. IlycTb 3a,lJ,aHO HepaBeHCTBO, CO,lJ,epIKalIl.ee x H y. MHOIKecTBOM
ero pellIeHHH SIBJIHeTCSI MHOIKeCTBO Bcex nap 1JHCeJI (a; b), npH no,nCTa-
HOBKe KOTOpbIX BMecTO 6YKB x H y nO.'Iy1JaeTCSI HCTHHHoe HepaBeHcTBo.
HanpHMep, napa (4; 2) npHHa,lJ,.'Ie)!{IfT MHO)!{eCTBY pellIeHIfH HepaBeHCTBa
x2+yz<25, nOTOMY 1JTO npH 3aMeHe x Ha 4, a y Ha 2 nOJIY1JaeTCSI HCTHH-
Hoe HepaBeHCTBO 4 z +2 Z <25. ECJIH Ka)!{,lJ,OH nape 1JHCe.'I (x; y) H3 MHO-
IKecTBa pellIeHIfH HepaBeHcTBa nOCTaBHTb B COOTBeTCTBIfe T01JKy
M (x; y), nOJIY1JHM MHOIKeCTBO T01JeK Ha nJIOCKOCTH, 3aAaBaeMoe 3THM
HepaBeHcTBoM. Ero Ha3bIBalOT epacpuKoM aaH.H.oeo H.epaBeH.CmBa. rpaHK
HepaBeHcTBa 06bI1JHO SIB.'ISIeTCH 06JIaCTbIO Ha n.'IOCKOCTH.
4T06bI If306pa3HTb MHOIKeCTBO pellIeHIfH HepaBeHcTBa F (x, y»O,
nocTynaIOT, KaK npaBHJIO, c.'Ie,lJ,YIOw.HM 06pa30M. CHa1JaJIa 3aMeHHIOT
3HaK HepaBeHcTBa 3HaKOM paBeH-
CTBa H HaXO,lJ,SIT JIHHHIO, HMelO-
1UYIO ypaBHeHHe F (x, y). 3Ta .'IH-
HHSI ,lJ,eJIMT nJIOCKOCTb Ha HecKO.'Ib-
KO 1JaCTeH. TIoCJIe 3Toro ,lJ,OCTaT01J-
HO B3SITb B KaJKAOH 1JaCTH no 0,lJ,-
HOH T01JKe H npoBepHTb, BbIno.'I-
HSIeTCSI .'IH B 3TOH T01JKe HepaBeH-
CTBO F (x, y»O. ECJIH OHO BbI-
nOJIHSIeTCSI B 3TOH T01JKe, TO OHO
6YAeT BbInO.'IHSITbCSI H BO BceH
1JaCTH, rAe JIeIKHT 3Ta T01JKa.06'b-
e,lJ,HHSIH TaKHe 1JaCTH, nOJlY1JaeM
MHOIKeCTBO pellIeHHH: 3a,lJ,aHHoro
HepaBeHcTBa.
n p H Me p 1. TIOCTpOHM rpa-
.pHK HepaBeHcTBa y>x.
II
J
!I
!1\11  .
2 l
-1 a 1
 ;i
2
x
PHC. 75
125

MbI YIKe 3HaeM, 1JTO ypaBHe-
HHe y=X H306paIKaeT npSIMYIO
JlHHHlO. npOXOASIW.YIO 1Jepe3 HatJa-
JlO KOOPAHHaT H HaKJIOHeHHYIO K
OCH a6cUHCC nOA yrJIOM 45°. 3Ta
npSIMaSI AeJIHT BCIO nJlOCKOCTb Ha
.lJ.Be nOJlynJIOCKOCTH (pHC. 75).
B03bMeM B BepxHeH nOJlynJlocKoc-
TH TOtJKY M (0; 3). nOAcTaBJlSISI ee
Koop)J.HHaThI B Hallie HepaBeHCTBO,
nOJIY1JHM HCTHHHoe HepaBeHcTBo
3>0. 3Ha1JHT, 3Ta T01JKa, a TeM
caMhIM H BCSI BepxHSISI nOJlynJloc-
KOCTb, npHHaMeIKHT rpaHKY He-
paBeHcTBa. T01JHO TaK IKe npo-
BepSIeM, 1JTO T01JKH HHIKHeH no-
JI ynJlOCKOCTU He npHHaAJIeIKaT
3TOMY rpaHKY. HaKOHeu, T01JKH caMOH npSIMOH Y=X He npHHaAJIe-
IKaT rpa(pHKY, TaK KaK Ha 3TOH npSIMOH Y=X, a He y>x. 11TaK, rpa-
HKOM HepaBeHCTBa y>x SIBJlSIeTCSI MHO)!{ecTBO T01JeK nJIOCKOCTH, JIeIKa-
lII.HX BbIllIe npHMOH Y=X.
n p H M e p 2. nOCTpOHM rpaHK HepaBeHCTBa
(x-2)2+(y-l)z;?:9.
!it
A(2;1)
.
o
PHC. 76
x
(1)
CHa1JaJla npoBe)J.eM JlHHHlO, HMelOlII.YIO ypaBHeHHe (x-2)2+(y-l)z=9.
3T0 OKPYIKHOCTb C UeHTpOM A(2; 1) H paAHycoM 3 (pHC. 76). OHa AeJIHT
nJIOCKOCTb Ha ABe 06JIaCTH - O,lJ,Ha JIeIKHT BHYTPH OKPYIKHOCTH, a APY-
raH - BHe 3TOH OKPY)!{HOCTH. B03bMeM UeHTp OKPYIKHOCTH, T. e. T01J-
KY A (2; 1). ECJlH nOACTaBHTb ee Koop)J.HHaTbI B HepaBeHcTBo (1), nOJlY-
1JHM JIOIKHOe HepaBeHcTBo 02+0z;?:9. 3T0 nOKa3blBaeT, 1JTO BCSI BHYTpeH-
HHH 06J1aCTb He npHHaAJlelKlIT rpaHKY HepaBeHcTBa (1). Bo BHellIHeH
06J1aCTlI BbI6epeM T01JKY B (100; 0). Ee Koop,Z!.HHaTbI Y)J.OB.'IeTBopSIIOT He-
paBeHcTBY (1): 98 z + (-1)2;?:9. 3Ha1JHT, rpacpHKoM HepaBeHcTBa SIBJISI-
eTCSI MHOIKeCTBO T01JeK, pacnOJlO)!{eHHbIX BHe OKPYIKHOCTH C ueHTpoM
A (2; 1) H paAHycoM 3. feOMeTpH1JeCKH 3TO 03Ha1JaeT, 1JTO rpaHK Hepa-
BeHCTBa (I) COCTOHT H3 T01JeK, paccToSIHHe KOTOpbIX AO T01JKH A 6oJIb-
llIe H.TI!I paBHo 3
YupamHeHHH
DOcTpoihe rpa<PHKH HepaBeHCTB:
a) yx+3; 6) y<x-4; B) (x-6)Z+ (y-5)2",4; r) (x+ l)z+ (y-3)29;
.11) (x-l)2+ (y+4)2<4.
7. CHCTeMhI ypaBHeHHH H HepaBeHCTB. 3a)J.a1JY 0 KpOJIHKaX H a3a-
HaX (CM. n. 2) MOIKHO 6hIJIO 6bI pellIHTb HHa1Je. 0603Ha1JHM 1Jepe3 X 1JHCJIO
a3aHoB, a 1Jepe3 y - 1JHCJIO KpOJIHKOB. TorAa no YCJIOBHIO 3aAa1JH MO)K-
HO COCTaBHTb )J.Ba ypaBHeHHSI: x+y= 19 H 2x+4y=62. KaIKAoe H3 3THX
ypaBHeHHH SIBJISIeTCH ABYXMecTHbIM npeAHKaToM, H MHO)KecTBa HCTUH-
126

HOCTH 3THX ypaBHeHHH 6eCKOHe1JHbJ. Ho HaM HYIKHO HallTH TaKJle 3Ha-
tJeHHH x H y, KOTopbIe YAOBJIeTBOpHJIII 6bI 060HM ypaBHeHI!H:.f,' T. e.
KOII'bIOHKUHH npe,lJ,HKaTOB x+y= 19 H 2x+4y=62:
(x+y= 19);\ (2x+4y=62).
B llIKOJIe 3TY KOH'bIOHKUHIO 3anHCbIBalOT B BHAe:
{ x+y= 19,
2x+4y=62
H Ha3bIBalOT cHcTer..mlf ypaBHeHHH x+y= 19 H 2x+4y=62.
Bo06lI1.e, cucme,'rtou ypaBHeHuu f(x, y)=O u F (x, y)=O H33bIBaIOT
KOH'bIOHKUHIO 3THX ypaBHeHHH
(f(x, y)=O);\(F(x, y)=O),
HJIH, KaK ee 3anHCbIBaIOT B llIKOJIe,
{ f(x, y)=,
F(x, y)-O.
(1)
PellIIITb CHCTeMY ypaBHelIHH (1) - 3Ha1JHT HaHTH MHOIKeCTI30 nap
(a; b), npIf nO)!.CTaHOBKe KOTOpbIX B 3TH ypaBHeHIISI no.1Y4aIOTCH
HCTIIHHbIe paBeHCTBa f(a, b)=O H F(a, b)=O.
MbI 3HaeM (CM. n. 3  1 r.1J. Ill), 1JTO MI1O)!{ecTBO HCTHHHOCTII I\8!1'bIOHK-
U,HH ,lJ,BYX npeAHKaToB HB.1JHeTCSI nepeCe1JeHHeM MHO:>KeCTB HCTJlHHOCTII
caMHX npeAHKaToB. T01JHO TaK )!{e MHO)!{eCTBO pellIeHHH CHCTe;LI ypaB-
HeHHH (1) HBJIHeTCH nepeCe1JeHHeM MHO)!{eCTB HCTHHHOCTH ypaBlIeHHH
f(x, y)=O H ypaBHeHHH F (x. y)=O. feOMeTpH1JeCKIf 3TO :,EW",fieCTBO
MO)!{HO HaHTH CJIeAYlOlII.HM 06pa30M: Ha1JepTHM rpaqmKH ypasHeHHH
f(x, y)=O H F(x, y)=O H II a HAeM T01JKH HX nepeCe1JeHHH. KOop,1J1HaThI
3THX T01JeK H 6YAYT HCKOMhIMH 3Ha1JeHHHMH x II y.
n p H M e p 1. napbI (2; 5) H (-5;-2) npHHaAJIe)!{aT MHO)!{ecTBY
pellIeHHH CHCTeMhI ypaBHeHHH
{ y-X=3,
(x+ 1)2+(y-l)2=25.
B caMOM AeJIe, eCJIH nOACTaBHTb B 06a ypaBHeHHSI 3Ha1JeHHSI x=2 H y=5,
no.1JY1JIIM HCTHHHbIe paBeHCTBa 5-2=3 H (2+ 1)2+(5-1)2=25. T01JHO
TaK)!{e npH nOAcTaHoBKe 3Ha1JeHHH x=-5 H y=-2 nOJIY1Jae:v, IICTHH-
HhIe paBeHCTBa -2-(-5)=3 H (-5+1)2+
+(-2-1)2=25. MOIKHo ,lJ,OKa3aTb, 1JTOHHhIX yj
pellIeHHH CHCTeMa ypaBHeHHH (2) He HMeeT.
11306pa3HM reoMeTp H1JeCKH pellIeHIfe CHC-
TeMhI ypaBHeHHH (2). fpaHKoM YPaJmeHHH
(x+l)2+(y-l)z=25 HBJIHeTCSI OKpy)!{HOCTb
c ueHTpoM A (-1; 1) H pa.n.HYCOM 5. fpa-
q>HKOM ypaBHeHHH y-x=3 SIBJIHeTCH npH-
MaH JIHHHH (pHC. 77). MhI BHAHM, 1JTO 3TH
JIHHHH nepeceKalOTcH B T01JKaX B(2; 5) H
C (-5; -2).
(2)
)(
PHC. 77
127

x
a)
PHC. 78
6)
PHC. 79
HapSI.lI.Y C CIfCTeMaMIf ypaBHeHHrf paCCl\faTpHBalOT H CItCTeMbl Hepa-
BeHCTB. HanpHMep, CIfCTeMa
jy>3,
) xz+ y225
SIBJlSIeTCSI KOH1>IOHKlI.IteH 3THX HepaBeHcTB - ee MOIKHO 3amlcaTb B
BH,lJ,e:
(y>3) 1\ (xz+yZ25).
JlerKo BH.lI.eTb, 1JTO rpmjJHKOM 3TOH CHCTeMbI SIBJISIeTCSI nepecetJeHHe
Kpyra C ueHTpoM B Ha1JaJle KOOp.lI.HHaT H pa.lI.HYCOM 5 C 1JaCTblO nJIOCKO-
CTH, JIeIKaw.eH BbIllIe npHMoH, napa.ll.lle.llbHOH OCH a6cuHcc H OTCTOHlII.eH
OT Hee BBepx Ha 3 e.lI.HHHlI.bI (pHC. 78, a). npH 3TOM 1JaCTb rpaHHuhI no-
J1Y1JeHHOH 06JIaCTH He BXOJ,HT B rpaHK. A CHCTeMa HepaBeHCTB
jy>3,
\xz+yZ25
HMeeT nOXO)!{HH rpaHK (pHC. 78, 6) C TOH JIHllIb pa3HHueH, 1JTO Tenepb
rPa<PHKY npHHa.lI..Ile)KHT rpaHHlI.a 06JIaCTH. (4HTaTeJIb BH,lJ,HT, 1JTO 1Jac-
TH rpaHHlI.bI, nplfHa.lI.JIe)!{aw.He 06.1laCTH, H306pa)!{aIOTCSI Cn.llOllIHOH J1H-
ImeH, a He npHHa.lI..IleIKaw.He 06.1laCTH - llITpHXOBOH JIHHHeH.)
MO)!{HO paCCMaTpHBaTb H CHCTeMbI, COCTOSIlII.He H3 lIepaBeHCTBa H
ypaBHeHHH. HanpHMep, CHCTeMa
{YZ36
HBJIHeTCH KOH1>IOHKUHeH: ypaBHeHHSI y=x H HepaBeHCTBa x"-t- y"<36.
rpa$HK 3TOH cHcTeMbI - OTpe30K npSIMOR y=x, COe,lJ,HHSIlOlII.HH T01JKH
A H B ee nepece1JeHHH C OKPYIKHOCTblO.
YupalKHeHHn
1. PewHTe CHcTeMY ypaBHeHHll:
a) J 2x+3y= 11, 6) { x2+if=25, B) { x2+if-6x+4Y=7,
\.3x-4y=-9; 3x-4y=O; x2+y2+6x-4y=3;
2. 11306pa3HTe 06JIaCTb, 3811aHHYIO CHCTeMOll HepaBeHcTB:
a) { yx. 6) { yX2, 2. B) { x2+if+6x-4y=3,
y9, y18-x , y3.
r) { y=9-XA.
y=x2-4.
128

3. Haii)J.llTc MHomecTBo, 3811aHHoe YCJIOBll!IMll:
a) { y=x, 6) { yx.
X2+y2-8x-6y.E;:!!; x2+yZ-8x-6y= II.
i 5. JlHHEBHLIE YPABHEHHJI
f. .YpaaueHHe npHMoii JlHHHH  yrJloBbiM Ko94NlJH",HeHToM. PaCCM<n:-
paM npHMYJci, npOXO,lJ,HlII.YIO 1Jepes HatJaJIO KOOp.l\HHaT (pHC. 80). BblOe-
peM Ha 9TOH npRMOH TOtJKY A (Xl; Yl), OTJIHtJHYIO OT TOtJKH 0 H JIe)!{auJ.}'IO
B nepROH qeTBepTH. nOKaIKeM, tJTO tJaCTHoe Yl He 3aBHCHT OT BbI60pa TOq-
Xl I
KH A npHMoH. B caMOM AeJIe, B03bMeM APyrylO TOtJKY B (x; Y2) TOR )!{e
npRMOR. 113 pHCYHKa 80 BHAHO, qro TpeyrOJIbHHKH OAA' H OBB' nOAOO-
Hbi (no TeopeMe 00 yrJIe, nepecetJeHHOM napaJIJIeJIbHbIMH npHMblMH).
nOm>MY 11:l = \::: . Ho IAA'I=Yh 10A'I=Xh IBB'I=y,
lOB' I=x, OTCIOAa Jf..1 = !!..!.. STa nponopUHH H nOKa3bIBaeT, tJTO 3Ha-
Xl X
tJeHHe OTHOllIeHHH !L He 3aBHCHT OT BbI60pa T01JKH M (X, y) Ha npHMOH JIH-
X
HHH, .JL=k. TOT IKe pe3YJIbTaT nOJIYtJHTCH, ecJIH B3HTb TOtJKY B TpeTbeii
X
1JeTBepTH - B .9TOM CJIYtJae 06e KOOPAHHaTbI X H Y OTpHuaTeJIbHbI, HO HK
OTHOllIeHHe paBHO TOMY IKe tJHCJIY k. 4HCJIO k xapaKTepH3yeT HaKJIOH
npHMoH K nOJIO:>KftTeJIbHOMY HanpaBJIeHlI10 O<;H a6cUHcc, Hero Ha3bIBaIOT
ye/lQ(Jbl.M KoacpcpU/{U£H.mo.M 3TOH npHMOH. OHO paBHO TaHreHCY yrJIa
Me)!{AY OCblO a6cUHCC H npHMOH OA, k=tg a.
11TaK, OAfl A1060ii mOtf.KU M (x; y), Ae[}fCaJ4eii H.a npfl.Moii, sa UCKJlIOtf.e-
H.UeM mOOKU 0 (0;0), BbllWAH.flemCfl paeeH.cmBof=k. YMHO:>KHB 06e tJaCTl{
paBeHCTBa -f = k Ha x, nOJIYtJHM: y=kx. 3'1'0 COOTHOllIeHHe cnpaBeAJIH-
BO H AJlH TOtJKH 0(0; 0), TaK KaK O=k.O.
l{oKa:>KeM Tenepb, 1JTO, 06paTHO, eCAU OAfl KOOpOUH.am KaKoii-H.U-
6YOb mOtf.KU M (x; y) npu H.eKOmOpO.M k BblIWAH.flemCfl coomH.OUl£H.ue
y=kx, mo ama motf.KQ Ae[}fCUm H.a npfl.Moii C yeA08blM Ko!K/xPul.4ueH.-
mOM k.
B caMOM AeJIe, Ha 3TOH npHMOH ecTb TOtJKa M l C a6cUHCCOH x. 0PAH-
HaTa Y1 3TOH TOtJKH paBHa, KaK MbI AOKa3aJIH BbIllIe, kx. Ho TorAa Y=Yt
H TOtJKH Iv! H M l COBnaAaIOT, a 3TO H 03HatJaeT, 1JTO tbtJKa M JIeIKHT
Ha npHMOH.
11TaK, MbI AOKa3aJIH, tJTO npu
saoaH.H.o.M k ypaeH.eH.UIO y=kx Y008-
Aetn80pfllOm KOOpOUH.afllbl motteK,
pcu:nOAo:J/CeH.H.btX H.a H.eKomopoii
npfl.Moii AUH.UU, npoxooflU{eii ttepes
H.attaAo KOOpOUH.am, U H.e YOOBAem-
BopfllOm KOOpOUH.atnbt H.UKaKUX
opyeux mOtf.eK.
y
A'
8'
x
PllC. 80
5 Nt. .70'2
129

YpaBHeHHe y=kx 6bIJIO BblBe,lJ,eHO ,nJIH npHMbIX, JIejKaIII.HX B I
H III 1JeTBepTHx. l{JIH TaKHX npHMbIx yrJIOBOH K03HUHeHT k nOJIO:>KH-
TeJIeH. B03bMeM Tenepb npHMyro, npOXO.lI.Hw.ylO 1Jepes HatJaJIO KOOp-
,nHHaT H JIe:>KalIl.YIO BO II H IV 1JeTBepTHx (pIle. 81). 3HatJeHHe 1JaCT-
Horo . ; ,lJ,JIH Beex TOlfeK .9T0i{ npHMoH O,lJ,HO If TO Me. npH 3TOM a6cUHeca
TOqKH M 3ToH npHMOH nOJ1OJKHTeJIbHa, a ee 0p.lI.HHa'Fa O'fp Hll,aTeJI bHa.
3Ha'lUT. qac1JHoe  OTpHuaTeJIbHO. n03TOMY H ,lJ,JIH 3TOH npfl:MOH Ypa&-
HeHHe HMeeT BH,lJ,y=kx, HO 3Ha'leHHe .k oTpHuaTeJIbHo. MbI GYJl.eM rOIK)-
PHTb. tJTO npHMbIe nepBoro BH,lJ,a 06pa3YlOT OCTpbIH yroJI C nOJIO:>KHTeJIb-
IIHM HanpaBJIeHHeM OCH aOCt:r.HCC, a npHMbre BTOpOro BFf;:I:a - TynoA
yron (CM. pHC. 80, 81). MmKHo C<PoPMYJIHpoBaTb: yeAOBOU 1(osq,cpulflU!H.m
k MA(X}ICU/neA.eH. y npJf.Mbl:X; 06paaylOtlfUX ocmpbul ye.O/l c nOAOOICU-
me.llbH.bl.M H.anpa8A.eH.Ul!.M ocu a6c/{ucc, U ompu/¥lmeAeH. Y npJ/.MbIX,
06pasYlOtlfux mynou Y20A c amUM H.anpa8AeH.UeM. B CJIY1Jae, KOr.lI.a k=O,
yp:mHeHue npHMOA npHIIRMa.eT BH,.!( y=O.. Eh'OMY yCJlOBHID Y.11.OB.lIeTBO-
pHIOT TOtJKH, pacnOJIO:>KeHHbIe Ha OCH a6eUHcc, II TOJIbKO OHH. OTMeTHM,
QTO H o,lJ,HoA I1pHMOi, II'pOxoAHlII.eH Qepe3- HaqaJIO KOOf},lI.HHaT, a "MeR-
80 MH OCH op.nHHaT, HeJIb3H HanHcaTb ypaBHeHIISI BII.lI.a y=kx. lJ.,nH
BCeX TOtreK .9Toft npHMoft BbIIIOiIHHeTCH ypamreHRe x=O.
TIpoBeAeM Tenepb npHMYlO, napaJIJIeJIbHYIO npHMOH y=kx, pacnOJlQ-
jKeHHYIO BhIill Hee H OTCeKaIOW.YIO Ha OCM Op,l{IIHaT OTpe30K )l.JIHHbI b.
BoobMeM H8 8TOtl: npHMOH JII06ylO T01JKY M (x; y) II onYCTHM H3 Hee nep-
neH,lJ,HKYJIHp MM' Ha OCb a6CUHCC. 0603HaqHM tJepes M 1 TOtJKY nepece-
'IeHHH 3Toro nepneH,lJ,HKYJIHpa C npHMOH y=kx. l13 pHcYHKa 82 BH,nHO.
'ITO IMM' 1= IMM11+ !M1M'I. Ho Op,lJ,HHaTa M 1 M' TOtJKH M i palma
kx, a IM 1 Mf=b. 3HatJHT, y=IMM'I=IMMd+/M1M'I=kx+b.
.MbI .lI;OKa3aJIH, 'ITO KOOp,lJ,HHaTbI JII060If TOtJKH M (x; y), JIe:>KameH: Ha
npSlMoH, y,lJ,OBJIeTBOpHIOT paseHcTBY y=kx+b. n03TOMY ypaBHeHHe
9T0J\ npHMOH: HMeeT BH,lJ, y=kx+b. TaKOH:>Ke BH.lI. HMeeT ypaBHeHHe npH-
MOH, paenoiJOIKeHHoH HH:>Ke npSlMoH y=kx, TO.1bKO B 3TOM cJfY1Jae b
OTpHll,aTMbHO. B 060Rx cnYQaHX b paBHo 0p.lI.HHaTe TOtmH nepece-
qeHHH npSfMOA C OCblO Op,lJ,HHaT. D03TOMY b na3bI.BalOT HCStf.8MbHoU op-
OUH.Q11JQii npH.MoU. lJBCJIO k, K2tK H B paecMOTpeHHoM l!!NJ,Ue qaeTHOM
y
PHC. 81
130
x
x
PHC. 82

CJIYtlae, Ha3bIB81OT yeA08blM fW8f/JcpUl{ueH.mOM y t
np!uwu. OHO nOJIO}KHTeJIbHO, ecJIH npHMaH
06pa3yeT ocrpYH yrOJI C nOJIOIKHTeJIbHbIM
HanpaBJIeHHeM OCR a6c£I.IIcc, OTpH£I.aTeJIbHO,
ecJIJf 31'OT yron TynoH, H paBHO HYJIIO, ecJIH
nptlMa" - napaJIJIeJIbHa OCR a6cUHcc. n PR-
MbU!, napa.t/.Ae.llbH.bli! ocu opounam, He M02ym
Oblmb 3aOaHbl ypCl8fleHueM 8uoa y=kx+b.
J].JIH nOCTpoeIUlSI npHMOH, 38.l1.aHHOH ypaB-
HeHHeM y=kx+b (B ,lJ,aJJbHeHIIIeM MbI 6Y,lJ,eM X
rOBopHTb KopOtle: (<I1pHMOH y=kx+b»), Ha,lJ,O
CHatlaJIa nOCTpOHTb Ha OCH opJl.HHaT TOtJKY
B(O; b). nOCJIe 3Toro Ha,lJ,O HaiTH elIl.e O,lJ,HY
TOtiKY npHMOH. l{JIH 3Toro B03bMeM mo6oe
3HatleHHe Xl H Hall,lJ,eM COOTBeTCTBYlOlII.ee 3Ha-
tleHHe Yl = kXl + b. T 01JKa M dXl; Yl) AOJIHCHa PHC. 83
JIeIKaTb Ha HCKOMOH ITpHMOH. n03TOMY np-
MaH npoxoJl.HT tJepe3 T01JKH B(O; b) H M 1 (Xl; Yl), rJl.e Yl=kxl+b.
nOCTpOHM, HanpHMep, npHMYIO y=2x-4. OHa npOXQ,1OIT 'lepe3
TOQKY B (0; -4). nOJIaraH x= 1, HaXOJl.HM, tJTO y=-2. 3Ha t UfT, npHMu
npOXO,lJ,HT II tlepes TOtfKY Adl; -2) (pRC. 83).
YupamUeRRJI
1. nOCTpoHTe npHMYIO, HMelOlLlYIO yrJI080H Ko$PHlUfeHT k H Ha'laJIbHYJO 0P)J.H-
HaTY b, eCJIu:
a) k=l, b=O; 6) k=2, b=O; s) £=-1, b=O;
r) k= I, b=2; JJ.) k= 1, b=-3; e) k=2, b=-4;
/K) k=-l, b=4; 3) k=-l, b=-2; u) k=-2, b=4.
KaKHe 83 3THX npHMbfX 06pa3YIOT OCTpbfii yrOJI C nOJIO/KHTeJlbHblM HanpaBJIeHHeti
OCH roCllHCC?
2. npOXO)J.HT JlH npHMa!l y=k%+b '1epe3 TO'IKY M (%0: Ye), eCJIH: a) k=4, b= 1,
.111=-2, Yo=-7; 6) k= I, b=6, "11=4, Yo=9; B) k=-3, b=4, Xo={), Yo=4?
3. H.amlJm1Te yp8l3HeHHi1 np"-MbIX, H306p.IDKeHlfHK Ha PHCYHJre 84.
4. Ha npHMoH y=2x-4 HaH)J.uTe: a) TO'IKY C a6cII.HccofI 3, 6) TO'IKY C 0p)J.HHa.
TOR 8, B) TO'IKY, Dp)J.HHaTa KOTOpOH B)J.Boe 60JlbllIe a6CI.(HCCbI, r) TO'IKY, 0p!l.HHaTa KOTO-
poil Ha 3 6oJIbllIe, '1eM a6CllHcca.
2. Y CJlMIftl napa.iJleJIbHOCTH H nepneHAHKYJlllpHOCTM npJlMWX.
EcJIH npHMhIe I. H Is napaJIJIe.1IbHbI .lI.pyr .lI.pyry, TO OHH napaJIJIeJIbHbI
O.lI.HOH H Toti lKe npSIMOH, npoxo.lI.HlII.efi 1Jepes HatlaJIO KOOP.lI. HHaT
x
x
a)
D)
PRC. 84
5.
y
131

XoOUMO U aOCmamO'lHO, 'lm06bl
paeHbl.
HanpHMep, npSIMbIe y=3x-4 H y=3x+ 1000 napaJIJIeJIbHbI, a npSI-
MbIe y=2x -1 H y=-x+2 He SIBJIHIOTCSI napaJIJIeJIbHhIMH.
BbIBe.lI.eM Tenepb YCJIOBHe nepneH.lI.HKYJISIpHOCTH .lI.BYX npSIMbIX.
npOBe.lI.eM tlepe3 HatlaJIO Koop.lI.HHaT .lI.Be nepneH.lI.HKYJISIpHhIe npHMbIe,
y=k 1 x H y=kzx (pHC. 86). Tor.lI.a O.lI.Ha H3 HHX (HanpHMep, y=k 1 x) 00-
pa3YeT OCTphIH yroJI C nOJIO)!{HTeJIbHbIM HanpaBJIeHHeM OCH a6C£I.HCC,
a BTopaH - TynoH yrOJI. n03TOMY KOHUHeHThI k 1 H k 2 HMelOT pa3-
HhIe 3HaKH. B03bMeM Ha nepBOH npSIMOH TOtiKY M 1 (Xl; Y1), aHa BTOPOH-
TOtiKY M 2 (X 2 ; Y2)' 113 pHcYHKa 86 BH.lI.HO, tlTO yrJIhI M 1 0N 1 H OM 2 N.
HMelOT paBHYIO BeJIHtlHHY, a nOToMY npHMoyrOJIbHhIe TpeyrOJIbHHKH
M10N i H M 2 0N 2 nO.lI.06HbI. 3T0 03HatiaeT, tlTO / Yl 1 = I X2 / . Ho Yl =kI,
Xl Y z Xl
a =k2' H n03TOMY Ik11 = IkI ' T. e. Ik 1 1./k 2 1=1. nOCKOJIbKY k i Hk.
HMelOT pa3JIHtlHhIe 3HaKH, TO k 1 , k.- OTpH£I.aTeJIbHOe tlHCJlO. n03TOMY
H3 paBeHCTBa Ik l l'lk 2 1= 1 CJIe.lI.yeT, tJTO krkz=-l. CnpaBe.lI.JIHBO H 06-
paTHoe YTBep1K.lI.eHHe: eC/lU kl'k z =-I, mo npflMbte y=k 1 x U y=k.x
ne pneH.aUKY/lRpH.bl.
npOBe.lI.eM Tenepb JII06bIe .lI.Be npSIMhIe y=k 1 x+b i H y=kzx+b..
TaK KaK nepBaH H3 HHX napaJIJIeJIbHa npHMoH y=klx, a BTopaSI - npH-
MOH y=k 2 x, TO 9TH npHMhIe nepneH.lI.HKYJIHpHbI B TOM H TOJIbKO B TOM
CJIYtlae, Kor.lI.a nepneH.lI.HKYJIHpHhl
npHMbIe y=k 1 x H y=k 2 x,T. e. Kor.lI.a
ki,k.=-l. 11TaK, npRMbte y=
= k1x+b t u y=k2X+b 2 nepneHiJu-
KY/lRpH.bl 0 mOM U mO/lbKO 0 mOM
C/lY'«le, KoeOa BblnO/lH.JtemcR yC/l06ue
k i .k 2 =-1.
HanpHMep, npHMbIe y = 2x+4
1
X H y=- "2 x+ 7 nepneH.lI.HKYJIHp-
1
HbI, TaK KaK 2'(--"2)=-1.
PHC. 85
PHC. 86
132
(pHC. 85). 003TOMY HX yrJIOBble
KOHUHeHTbI paBHbI yrJIOBoMY
K03H£I.HeHTY 3TOH npSIMoH, T. e.
paBHhI .lI.pyr .lI.pyry. 06paTHo, ecJIH
yrJIOBbIe K03H£I.HeHTbI .lI.BYX npH-
MbIX paBHbI, . TO 06e 3TH npHMhIe
napaJIJIeJIbHbI O.lI.HOH H TOH )J<.npH-
MOH, npOXO.lI.HlII.eH tlepe3 HatlaJIO
KOOP.lI.HHaT, H nOTOMY napaJIJIeJIb-
Hbl.
11TaK, aim moeo, 'lm06bl' aoe
npflMble (He napa/l/le/lbH.ble OCU OP-
aUH.am) 6bl/lU napaJl/le/lbHbl, He06-
ux YM08ble KOJcpcpu/{ueHmbl 6bl/lU

YnpalKBeHBJI
1. CpeAH YKa3aHHblx HH/Ke ypaBHeHHii HaHAHTe ypaBlleHHSI napamIeJlbHblX H
nepneHAHKynSlpHblx npSlMblx;
I I
a) y=3x-4; 6) y=- gX+7; B) y= gX+6;
r) y=3x; A) y=-3x+2; e) y=-3x+ll;
I 1 I
>K) y=2"x+ 4"; 3) y= gx-l; H) y=-2x+8.
2. HaH)l.HTe yrno80iI Ko3Cj$HllHeHT npJlMoii, napanneJlbHoii npHMoii:
a) y=4x+2; 6) y=-x+l;
1
B) y=x+4; r) y=-4x+7.
3. Hail)l.HTe yrno80H KQHllHeHT npHMoil, nepneH)l.HKynJlpHoil npJlMoii:
a) y=4x+2; 6) y=-x+ I;
1
B) y=x+4; r) y=-4x+7.
3. YpUHeHHe nYlfKa npRMblx, npoxOJl.SlIl\HX lfepe3 3a)1.aHHytO TOItK)';
ypaBHeHHe npRMOH, npoxOJl.RlII.eH lfepe3 Jl.Be TOlfKH. B03bMeM Ha nJIOC-
KOCTH T01JKY A (xo; Yo) H npOBe,lJ,eM 1Jepes Hee npHMYlO, He napaJIJIeJIb-
HYIO OCH 0P.lJ.HHaT. nYCTb y=kx+b - ypaBHeHHe 3TOH npHMOH. TaK
KaK TOtiKa A JIeIKHT Ha npOBe.lJ.eHHOH npHMoH, ee KOOp.lJ.HHaTbI Y,lJ,OBJIeTBO-
pHIOT ypaBHeHHIO npHMoH: Yo=kxo+b. n03TOMY b=Yo-kxo. ECJIH no.!l-
CTaBHTb B ypaBHeHHe npHMoH HaH.lJ.eHHOe 3HatleHHe b, TO nOJlYtlHTCR
ypaBHeHHe y=kx+Yo-kxo HJIH, HHatle,
y-yo=k (x-xo).
(I)
3HatlHT, ypaeHeHue /l1060ii npRMoii, npoxoiJRw,eii ttepe3 mO'lKY A (xo; Yo),
3a UCKJlIO'leHUi!M npRMoii, napaJl/le/lbHOii ocu opiJuHam, MOJICHO sanuca111b
B 8uiJe (1). npH H3MeHeHHH 3HatleHHH k 6Y.lJ.YT nOJIY1JaTbCH pa3JIHtlHbJe
npHMbIe, npoXo.lJ.HlII.He tlepes TOtiKY A. MHOIKecTBO npHMbIx, npoxo.lJ.SI-
lII.HX 1Jepe3 3TY T01JKY (pIlC. 87), Ha3bJBalOT ny'lKOM. npR.Mbtx C l{eHmpoM A.
YpaBHeHHe (1), eCJIH C1JHTaTb B HeM k nepeMeHHOH, 3a.lJ.aeT JII06ylO npSl-
MYIO 3Toro nY1JKa, 3a HCKJIlOtleHHeM npHMOH x=x o , napaJIJIeJIbHOH OCR
0P.lJ.HHaT. n03TOMY CPOPMYJIY (1) Ha3bJBalOT ypaeHeHUeM nY'lKa npRMblx C
/{eHmpoM A (xo; Yo).
n p H M e p 1. HanHIIIeM ypaBHeHHe npHMoH, npoxo.lJ.Sll!l.eit 1Jepe3
TOtiKY A (4; -7) napaJIJIeJIbHO npHMoH y=2x-3. TaK KaK HC-KOMaH
npHMaH npoXo.lJ.HT tJepes TOtiKY
A (4;-7), ee ypaBHeHHe .lJ.OJIIKHO Y
HMeTb BH.lJ. y-(-7)=k (x-4), T. e.
y+ 7 = k (x-4). Ho TaK KaK OHa
napaJIJIeJIbHa npHMOH y=2x-3, a y
napaJIJIeJIbHbIX npHMbIX yrJIOBbIe
Ko3$PHulleHTbI paBHbI, TO k=2, H
nOTOMY HCKOMoe ypaBHeHHe HMeeT BH.lJ.
y+7=2(x-4), oTKY,lJ,a y+7=2x-8,
IlJIH y=2x-15. 0 X
n p H M e p 2. HanHIIIeM ypaB- PIIC. 87
133

HeHHe npRMOH, npOxo,lJ,HlII.eii tJepes TOtJKY A (-1; 5J nepneH.lI.HI<YJIHpHO
npHMoH Y=  x+6.,
TaK KaK HCKOMaH npHMaH npoXo.lI.HT 1Jepes T01JKY A (-1; 5), ee
ypaBHeHHe HMeeT BH.lI. y-5=k (x+ 1). 113 YCJIOBHH nepneH,lJ,HKYJIHpHO-
1
CTH kl.kz=-1 BbIBO.lI.HM, tJTo a .k=-I, H nOToMY k=-3. 3Ha1JllT,;I)C-
KOMoe ypaBHeHHe HMeeT BH.lI. y-5=-3 (x+ 1) HJIH y-5=-3x-3,
T. e. y=-3x+2.
npHMaH tJalIl.e Bcero 3a.lI.aeTCH .lI.BYMH T01JKaMH, pacnOJIO)!{eHHblMH Ha
»TOR npHMOH:. HanHIIIeM ypaBHeHHe npHMOH:, npOXO,lJ,HIII,eH tJepes TOtJKH
A (Xl: Yl) H B (Xz; Yz). TaK KaK 3Ta npHMaH npOXO.lI.HT tJepe3 T01JKY
A (Xl; Yl)' ee ypaBHeHHe HMeeT BH.lI.
Y-Yl=k (X-Xl)'
HaM HMo HaHTH 3Ha1JeHHe yrJIOBoro K03H£I.HeHTa k. lI.JIH 3Toro BOC-
I1OJIb3yeMcH TeM, tJTO npHMaH npoXOHT H tJepes T01JKY B (Xz; yz). ECJIH
nO.ll,CTaBHTb B nOJIYtJeHHoe ypaBHeHHe KOOp.lI.HHaTbI Xz H Yz TOtJKH B,
nOJIYtJHTCH HCTHHHoe paBeHCTBo YZ-Yl=k (XI-Xl). 113 Hero HaXO,lJ,HM,
1JTO ecJIH xz:;6Xlt TO
k = Y2-!li . (2)
XZ-Xl
3HatJHT, ypaBHeHHe npHMoH, npoXO.lI.HlII.eH tJepe3 TOtJKH A (Xl; Yl)
H B (xz; Y2), HMeeT npH Xl:;6xz BH.lI.
, Y-Yi = YZ-Yl (X-Xl)'
Xz - Xl
ECJIH, KpOMe Toro, Yl:;6yz, TO 3TO ypaBHeHHe MO)!{HO sanHcaTb B BH.lI.e
npOnOp£I. HH :
Y-Y1 X-Xl 3)
YZ-Yl = X;J-Xl . (
B CJIY1Jae, )wr,lJ,a X1=X Z ' oOe T01JKH HMelOT O.lI.HHaKOBYIO a6C£I.Hccy H no-
TOMY JIe)!{aT Ha npHMoH X=Xlt napaJIJIeJIbHOH OCH Op,lJ,HHaT. T01JHO TaK
)!{e ecJIH Y1=YZ' TO 06e T01JKH JIe:>KaT Ha npHMOH Y=Ylt napaJIJIeJIbHOH
OCH aOCUHCC.
n p H Me p 3. Haii.zI.eM yrJIOBOH KOH£I.HeHT npHMoH, npOXO,lJ,HlII.eH
tJepes T01JKH A (4; -8) H B (7; -5). no pMYJIe (2) nOJIYtJaeM, 1JTO
k= -5-(-8) -5+8 == 1
7-4 7-4 3 .
n p H M e p 4. HanHIIIeM ypaBHeHHe npHMoH, npOXOHIII,eH 1Jepe3
TOtJKH A (5; 2) H B (-1; 4). lI.JIH 3TOro B ypaBHeHHH (3) 3aMeHHM Xl H Yl
1JHCJIaMH 5 H 2, a Xz H Y2 tJHCJIaMH -1 H 4:
y-2 _ x-5
4-2 - (-1)-5 .
3'1'0 ypaBHeHHe MOIKHO npeo6pasoBaTb CJIe.lI.YlOlII.HM 06pa30M:
y-2 = x-5 HJIH Y -2 = - ( x-5 )
2 -6 3'
134

T. e.
1 11
y=- gX+ g'
n p H M e p 5. HamfllIeM ypaBHeHHe npHMoH, npoxo.lI.HlII.eH tJepea
T01JKH A (6; 1) H B (6; 4). TaK KaK a6cU:HccbI 3TH X TOtJeK O.lI.HHaKOBbI, TO
ypaBHeHHe npHMOH HMeeT BH.lI. x=6. ECJIH 6b1 MbI 3anHCaJIH ypaBHeHHe
.u u 6 y-I x-6
3TOH npHMoH B BH,lJ,e nponopU: HH , TO nOJIY1JHJIH bI 4-1 = 6- 6'
y-l x-6 T
HJIH 3 = O' aKaSI 3anHCb HenpaBHJIbHa, TaK KaK .lI.eJIeHHe Ha
HYJIb HeB03MO:>KHO.
¥opaIKHeHBR
1. HanH!lJHTe ypaBHeHHe np5!Moil:, npoXO)\5!1II.eH Qepe3 TO'lKY A (6; -3): a) ITa.
paJIJIeJIbHO np5!MOH y=5x-l; 6) nepneH)\HKYJI5!pHO np5!MOH y=2x+4; B) nap3JlJlMb.
HO OCH a6CllHCC; r) napaJIJIMbHO OCH 0p)\HHaT.
2. HanH!lJHTe ypaBHeHHe ITp5!MOil:, npoxo)\5!1II.eil: '1epe3 TO'IKH:
a) A (-3; 2), B (-1; 6); 6) A (5; 7), B (0; 3);
B) A (4; I), B (4; 8), r) A (1; -3), B (8; -3).
3. lI.aH TpeyrOJIbHIIK ABC C Bep!lJHHaMH A (-1; 9), B (2; 5), C (-2; 1). HanHlllHTe
ypaBHeHHH BblCOT 1) 310<0 TpeyroJlbHHKa.
4. lI.aH TpeyroJlbHHK ABC C Bep!lJHHaMH A (1; 3), B (5; 7), C (3; I). HanHlllHTe
ypaBHeHH5! Me)\HaH 9Toro ,peyrOJlbHHKa H ypaBHeHHe nepneH)\HKYJI5!pa, onY1Il.eHHoro
H3 Bep!lJHHbI C Ha Me)\HaHY AM.
4. YroJI Me)K.lJ.Y Jl.0YMR npRMblMH. BblBe.lI.eM PMYJIY .lI.JIH yrJIa 'P
MelK,lJ,y npSIMbIMH y=k 1 x+b 1 H y=k 2 x+b z . l{JISI 3Toro Y1JTeM, tJTO
yrJIOBOH KOHU:HeHT k SIBJIHeTCH He 1JeM HHbIM, KaK TaHreHCOM yrJIa
CG, 06pa30BaHHoro npSIMOH y=kx+b C nOJIO:>KHTeJIbHbIM HanpaBJIeHHeM
OCll a6CUllCC, k=tgCG. n09TOMY k1=tgCGh k Z =tgCG 2 , r,lJ,e CG 1 - yrOJI Ha-
KJIoHa KOCH a6CUllCC nepBOH npSIMOH, a CG z - BTOpOH npSIMOH.
113 TpeyrOJIbHHKa ABC (pHC. 88) BH.lI.HO, tJTO CG z =CG 1 +cp (BHeIIIHHH
yroJI TpeyrOJIbHHKa paBeH cYMMe .lI.BYX BHYTpeHHHX He CMe)!{HbIX C HHM
yrJIOB). 3Ha1JHT, cp=CGz-CGi, H nOTOMY
t t g rJ. - t g rJ. 1
tg<p= g(CG-CG)= z
z 1 1 +tg rJ.z.tg rJ.1
11TaK, maHG£H.C Y2Aa MeJ/CiJy npR-
M.blM.U y=klX+b 1 H y=k 2 x+b 2 Bbl-
paJICaemcJl q;OpM.YAOU
t kz-k1
g cp = 1 + k 1 . k 2 .
k 2 -ki
1 + k 1 . ks .
y
x
n p H M e p. HaH.lI.eM yrOJI
MelK,lJ,y npSIMblMH JIHHHSIMH y=
=3x-l H y=-2x+4. B 3TOM
cJIY1Jae k 1 =3, k 2 =-2 H
o
1) l1MCIOTC5! B BH)\Y np5!Mble, Co)\ep'
/Ka1Il.He BblCOTbI 3Toro TpeyroJlbHHKa.
PHC. 88
135

t -2-3-5
g qJ = 1 - H - 2) . 3 = - 5 = 1.
Ho eCJIH TaHreHC yrJIa qJ paBeH e,lJ,HHH£I.e, TO yrOJI paBeH 45°. 3Ha1JHT,
npSIMbIe 06pa3YIOT yroJI B 45°.
YnpalKHeHHR
1. HaH!l.HTe TaHreHCbl yrJIOB Me/K!l.Y nplIMbIMH:
a) y=5x-1 H y=x+7; 6) y=-4x+3 H y=3x-1.
2. HaH!l.HTe yrJIbI TpeyrOJIbHHKa ABC C BepIllHHa1H A (-3; 1). B (-1; 7), C (2; I).
3. HaH!l.HTe yrJIb! Me)!(!l.y Me!l.HaHaMH AM H BN rpeyro,lbHHKa ABC, I!MelOIIJ:erO
BepIllHHbI A (I; 8), B (-3; 2), C (5; 4).
5. 06w,ee ypaBHeHHe npRMoH. HeAocTaTKoM ypaBHeHHSI npSIMoH
C yrJIOBbIM K03H£I.HeHTOM SIBJISIeTCSI TO, 1JTO OHO He OXBaTbIBaeT Bcex
npSIMbIX Ha nJIOCKOCTH - npSIMbIe, napaJIJIeJIbHbIe OCH opAHHaT, He MO-
rYT 6bJTb 3aAaHbI TaKHM ypaBHeHHeM. OHH HMelOT ypaBHeHHe BH,lJ,a
x=a. CeH1JaC MbI HanHIIIeM ypaBHeHHe, 1JaCTHhIMII CJIY1JaSIMH KOTOpO-
ro RBJISIIOTCSI ypaBHeHHH y=kx+b H x=a. OHO HMeeT BIIA:
Ax+By+C=O. (1)
B 3TO ypaBHeHHe x H y BXOASIT B nepBoH CTeneHH. Ero Ha3bIBaIOT 06-
UiUM /lUHeUHbIM ypa8HeHUeM. nOJIOIKHB B 3TOM ypaBHeHHH B= 1, A =
=-k, C=-b, nOJIY1JHM -kx+y-b=O, T. e. y=kx+b. ECJIH IKe nOJIO-
)!{HTb A=I, B=O, C=-a, nOJIY1JHM x-a=O, T. e. x=a. 3HatlHT, H
ypaBHeHHe npSIMoH C yrJIOBbIM K03H£I.HeHTOM, H ypaBHelme npSIMoH,
napaJIJIeJIbHOH OCI1 Op,lJ,I1HaT - 1JaCTHbIe cJIY1Jal1 06lI1.ero JII1HeHHOro
ypaBHeHHSI.
ECJIH A =0 H B=O, TO 06lI1.ee JIllHeHHOe ypaBHeHHe npHHHMaeT BIIA
C=O. B CJIY1Jae. KorAa H K03I1UHeHT C pi1BeH HYJIIO, no.nY1JaeM pa-
BeHCTBO 0=0, KOTOPOMY YAOBJIeTBOpSIIOT Koop)J,HHaTbI JII060H T01JKI1 Ha
nJIOCKOCTH (KaKHe 6bI x H Y MbI HII BbI6Hpa,lIf, paBeHcTBo 0=0 OCTa-
eTCSI HCTHHHbIM). 3Ha1JHT, npll A=O, B=O, C=O rpaqmKoM 06lI1.ero JIH-
HeHHoro ypaBHeHHSI SIBJISIeTCSI BCSI nJIOCKOCTb. ECJIlI IKe A =0, B=O,
HO C*O, TO nOJIY1JaeM TaKoe paBeHCTBO, KaK, HanpHMep, 1 =0. npl1
JII06bIX 3HatJeHHSIX x 11 y 3TO paBeHcTBo JIOIKHO, T. e. rpaqmK ypaBHeHHH
1=0 nYCT.
ECJIII oT6poCHTb CJIY1JaH A =B=O, Kor,lJ,a rpaqmKoM ypaBHeHI1H Ax+
+ By+C=O SIBJISIeTCSI BCSI nJIOCKOCTb HJIH nYCToe MHOIKeCTBO, MO)!{HO YT-
BepIKAaTb, 1JTO 3THM q:iaqmKoM SIB.TISIeTCSI npHMaSI JIIIHIISI. B caMOM ,lJ,eJIe,
B03MOIKHbI ABa CJIY1JaSI: B=O HJIII B*O. ECJ1H B=O, TO ypaBHeHHe HMe-
eT BH,lJ, Ax+C=O. npll 3TOM, nOCKOJIbKY MbI oT6poCHJIH CJIY1JaH, KOr,lJ,a
A H B oAHoBpeMeHHo 06palIl.aIOTCSI B HYJIb, A*O. n03TOMY ypaBHeHHe
C '='_ u
MOIKHO 3anllcaTb B BHAe x=- A" .:710 - ypaBHeHlIe npSIMOH, napaJI-
JIeJIbHOH OCII 0 p,lJ,HHaT. B cJIyqae IKe, KorAa B*O, ypaBHeHHe Ax+ By+
A C A
+C=O paBHOCIIJIbHO ypaBHeHHIO y=-Jjx--jj' nOJIOIKHB -Jj=k H
136

- = b, nOJIYIJHM: y=kx+b, T. e. ypaBHeHHe npHMOH C yrJIOBbIM KO-
9$$H£I. HeHT OM.
HTaK, 2pat/JUKOM 06U¥20 AUHeUH020 ypaeHeHUJI. Ax+ By+C=O MOo
2ym 6bl1nb:
a) 8Cfl nAOCKocmb (ecJIH A=B=C=O); .
6) nycnwe MHO:J/Cecm80 (eCJIH A=B=O; HO C*O);
B) npflMafl AUHUfl, napaAAeAbHafl OCU OpaUHatn (eCJIH B=O, A*O);
r) npflMafl AUHUfl, He napaAAeAbHafl OCU OpaUHam (eCJIH B*O).
B nOCJIe,lJ,HeM CJIytJae yrJIOBOH K03qxpH£I.HeHT npHMOH BbIpaIKaeTCH
cpOPMYJIOH k=- .
OTMeTHM elIl.e, tJTO npH A =0 np HMaH napaJIJIeJIbHa OCII a6C£I.HCC
(k=O). ECJIH IKe C=O, TO npHMaH npOXO,lJ,HT IJepe3 HatJaJIO KOOp,lJ,HHaT.
B caMOM ,lJ,eJIe, B 3TOM CJIYIJae ypaBHeHHe npHMOH HMeeT BH,lJ, Ax+ By=O
H KOOp,lJ,HHaTbI TOtJKH 0 (0; 0) y,lJ,OBJIeTBOpHIOT nOMY ypaBHeHHIO.
ECJIH 1..*0, TO ypaBHeHHH Ax+By+C=O H A.(Ax+By+C}=O
paBHOCHJIbHbI. 3TH ypaBHeHHH 3a,lJ,alOT O,lJ,HY H TY IKe npHMYIO JIHHHIO.
HanpllMep, ypaBHeHHH 2x-y+4=0 H 6x-3y+ 12=0 3aJ(alOT OJ(HY H
TY IKe npHMYIO JIIIHIIIO (BTopoe ypaBHeHHe nOJIYtJaeTcH 1I3 nepBoro YM-
HOIKeHlIeM 06eHx tJacTeH ypaBHeHHH Ha 3).
3anHllIeM YCJIOBHH napaJIJIeJIbHOCTH H nepneHJ(IIKYJIHpHOCTH ,lJ,JIH
npHMbIx, 3a,lJ,aHHbIX 06lI1.HMH JIHHeHHbIMH ypaBHeHIIHMH: AIX+BlY+
+Cl=O H A 2 x+B 2 y+C 2 =0. ECJIH B1*O, B 2 *O, TO yrJIOBOH K03qxpH-
u u Al u Av+. U
UHeHT nepBOH npHMOH paBeH - - B . a yrJIOBOH K03'P't'H£I.HeHT BTOpOH
1 ·
npHMOH paBeH - 2 ' 3Tlf. npHMble napaJIJIeJIbHbI npH YCJIOBHH, tJTO k i =::.
2
=k 2 , T. e. - : =- : . 3T0 YCJIOBHe MOIKHO nepenHcaTb cJIe,lJ,YlOlII.HM
06pa30M:
AIB2=A 2Bi'
(2)
B TaKOM BHJ(e OHO rO,lJ,HTCH H ,lJ,JIH npHMbIX, napaJIJIeJIbHbIX OCH op,lJ,HHaT.
TOIJHO TaK )!{e YCJIOBHe nepneH,lJ,HKYJIHpHOCTH k l .k 2 =-1 nepenH-
meTeH J(JIH npHMbIX Alx+Bly+Cl=O H A 2 x+B 2 y+C 2 =0 B BH)(e
( - ; ) . ( - : ) . -1. 31'0 YCJIOBHe MOIKHO 3anl1CaTb CJIeJ(YlOlII.HM 06-
pa30M: AlA 2=-B 1 B 2 HJIH :>Ke
AIA2+BIB2=0. (3)
B 3TOM BH,lJ,e OHO cnpaBeJ(JIHBO H J(JIH CJIYtJaH, KorJ( OJ(Ha H3 npHMblX
napaJIJIeJIbHa OCH Op,lJ,HHaT.
HanHmeM ypaBHeHHe npHMOH, npOXO,lJ,HlII.eH tJepe3 TOIJKY M (xo; Yo).
3ro ypaBHeHHe J(OJI)!{HO HMeTb BH,lJ, Ax+ By+C=O. lh06b1 3Ta npHMaJI
npOXOJ(HJIa tJepes TOtJKY M (Xo; Yo), J(OJIIKHO BbInOJIHHTbCH paBeHCTBO
Axo+Byo+C=O. OTclOJ(a HaXOJ(HM C=-Axo-Byo, H nOTOMY ypaBHe..
HHe npHMOH J(OJIIKHO HMeTb BH,lJ, Ax+By+ (-Axo-Byo) =0, 'II. e.,
A (x-xo)+B (Y-Yo) =0. (4)
137

Tenepb YIKe He npHXOlI.HTCH HCKJII01JaTb npHMYlO, napaJIJIeJIbHYIOOCH
°P.lI. HHaT .
n p H M e p. HanHIIIeM ypaBHeHHe npHMOH, npOXO.lI.Hm:eH 1Jepes
T01JKY M (4; -2) "' napaJIJIeJJbHoH npHMoH 3x-5y+6=O.
TaK KaK HCKOMaH npHMaH npOXO,lJ,HT 1Jepe3 T01JKY M (4; -2), ee
ypaBHeHHe .lI.OJI:>KHO HMeTb BH.lI. A (x-4)+B(y+2)=O. 113 YCJIQaHH
napaJIJIeJIbHOCTH 3TOH npHMoii K npHMOH 3x-5y+6=O nOJIY1JaeM, 1JTO
Ko$PHUHeHTbl A H B ,l{OJI1KHbl Y,lJ,OBJIeTBOpHTb ypaBHeHHIO -5A=
=3B. Mbl nOJIY1JHJIH OJ(HO ypaBHeHHe .lI.JIH HaXO)!{)J.eHHH ,lJ,BYX HeH3-
BecTHbIX A H B. Ho ypaBHeHHH Ax+By+C=O H A.(Ax+By+C)=O
3a.lI.aIOT o.lI.HY H TY )!{e npHMYIO, a nOTOMY HaM HYIKHb! He caMH K03H-
UHeHTb! A H B, a JIHIIIb HX OTHOIIIeHHe. I1Hb!MH CJlOBaMH, ,lJ,OCTaTOqHO
HaRTH XOTb O.ll.HY napy qHCeJI (A; B), He paBHblX O.lI.HOBpeMeHHO HYJIIO H
Y.lI.OBJIeTBOPHlOlII.HX ypaBHeHHIO -5A =3B. B Ka1JecTBe TaKOH naPb!
MO)!{HO BbI6paTb napy (3; -5). n03TOMY ypaBHeHHe HCKOMOH npHMoH
TaKOBO:
3 (x-4)-5 (y+2)=O.
YopaarHeHDR
1. Haii,n,IITe yrJIOBbIe Ko$PIIUHeHTbI H Ha'laJIbHbIe op.1l;lIHaTbI cJle.1l;YlO1II.IIx npllMbIx:
a) 2x+y-3=0; 6) -4x+2y-5=0;
B) -7x-2y+l=0; r) 3x-5y+15=0.
2. nocrpoiiTe CJIe.1l;YlO1Il.IIe npHMbIe:
a) 4x-5y+20=0; 6) x-3y+9=0; B) 3x+y-6=0; r) 4y-7=0; .11;) 2x-5=0;
e) 3x+8y=0.
3. 1.Jepe3 TO'IKY A (-5; 1) npOBe.1l;llTe npllMYIO, napaJIJleJIbHYIO npllMoii:
6x-6y+2=0.
4. 1.Jepe3 TO'IKY A (4; 8) npOBe.1l;llTe npllMYJO, nepneH.1I;llKYJlIIPHYIO npllMoii:
2x+3y-5=0.
5. HanlllllllTe ypaBHeHllII BbICOT TpeyrooilbHllKa ABC C BeplllllHaMll A (-4; 1),
B (0; -3), C (2; 4).
6. HanlllllllTe ypaBHeHlle npllMoii:, npoxo.1l;1I1II.eii: 'Iepe3 Cepe.1l;llHY oTpe3Ka AB
H nepneH.1I;llKYJlllpHoii: 9TOMY OTpe3KY, eCJIll A=A (-4; 2), B=B(6; 4).
6. TOqKa nepeceqeHHH npHMblx. nYCTb 3a.lI.aHbI npHMbIe A1X+
+B 1 y+C 1 =O If A 2 x+B 2 y+C 2 =O. ECJIH 3TH npHMble nepeceKaIOTcH, TO
KOOp,lJ,HHaTbI TOqKH HX nepeceqeHHH ,lJ,OJIJKHbI Y,lJ,OBJIeTBOpHTb CHCTeMe
ypaBHeHHH
{ A lx+B1y+Cl=O,
A IX+ B2.Y+C 2 =O.
(1 )
11HblMH CJIOBaMH, HaHTH TOIJKY nepece1JeHHH npHMblX 03HaIJaeT peIIIHTb
cHCTeMY ypaBHeHHH (1).
n p H M e p 1. HaH,lJ,eM T01JKY nepeceqeHHH npHMbIx 3x-4y+5=O
H 5x+2y-9=O. l{JIH 3Toro peIIIHM cHCTeMY ypaBHeHHH
{ 3x-4y+5=O, (2)
5x+2y-9=O.
138

YMHOIKaH BTOpOe ypaBHeHHe Ha 2, nOJIY1JaeM paBHOClfJIbHYIO CHCTeMY
ypaBHeHHH
{ 3x-4y+5=O,
IOx+4y-18 --: 0.
C3AhIBaH 3TH ypaBHeHHH, nOJIYtJaeM 13x-13=0, OTKY,lJ,a x= 1.
TIO.nCTaBJIHH 3T0 3HatJeHHe B nepBOe ypaBHeHHe CHCTeMbl (2), HaXOJ(HM,
tJTO 3-4y+5=0, H n03ToMY y=2. 3HatJHT, TOtJKOH nepecetJeHHH npH-
MblJ{, 3x-4y+5=0 H 5x+2y-9=0 HBJIHeTCH TotJKa M(I; 2).
npHMbIe A 1 x+B 1 y+C 1 =0 H A 2 x+B2.Y+C 2 =0 MorYT He nepece-
KaTbCH - OHH MorYT 6bITb napaJIJIeJIbHbIMH H pa3JIHtJHbIMH HJIH COB-
nMaTb. YCJIOBHe napaJIJIeJIbHOCTH npHMLIx HMeeT BH,lJ, A 1 B 2 -A 2Bi=
ABC
=0 (CM. n. 5), a yCJIOBHe HX COBnMeHHH - BH,lJ, A: = B: = C ' HJIH,
tJTO TO >Ke CaMoe, A1B1-AzB1=O, A1CZ-AzC1=0, B1C1-B1Ci=0.
HTaK,ecJIH A 1 B 1 -A 2 B 1 =0,HO A 1 C 1 -A 1 C 1 =F0 HJIH BICI-BICl=FO,
TO CHCTeMa ypaBHeHHH (I) He HMeeT pemeHHH. ECJIH )!{e AIBI-AIBi=
= AICI-CiAI=B1CI-C1BI=0, TO npHMble A1x+BtY+C1=0 H
AtX+B.y+C1=O COBna.l131OT. B 3TOM cJIyqae KoopJ(HHaTbl JII060H TOtJKH
npHMOH AgX+BtY+Cz=O YJ(oBJIe'l1JOpHIOT o6oHM ypaBHeHHHM CHCTeMbl,
H nOTOMY CHCTeMa ypaBHeHHH: HMeeT 6ecKoHetJHoe MHO>KecTBO perneHHii_
n p H M e p 2. HaHJ(eM TO'IKY nepeceqeHHH npHMbIX 2x-7y+3=0
H 4x-14y+11=0.
TaK KaK 2-(-14)-4-(-7)=0, HO 2-11-3-4=FO, TO 3TH npHMble
napaJIJIeJIbHbI, HO He COBna,zr.3IOT. TOtJKH nepecetJeHHH 3THX npHMbIX He
CYlIl.ecTBYeT.
n pH M e p 3. HaHJ(eM TOtJKY nepecetJeHHH npHMbIx 2x-7y+3=O
H 4x-14y+6=0. B 9TOM cJIYtJae2 .(-14)-4. (-7)=2 .6-3 -4=(-7) 
X 6-(-14) .3=0, H nOTOMY npHMble COBna,zr.aIOT. 3HatJHT, JII06aH TotJKa
npHMOH: 2x-7y+3=0 npHHaAJIeIKHT HCKOMOMY nepecetJeHHIO.
YupalKBeUBJI
1. HaH.D;H're TO'IKH nepece'leHHSI CJIe.D;YJ01II.Hx npSlMblx:
a) 6x-y-3=O H 3x-4y+9=O;
6) 3x-4y+I=O H 9x-12y+3=O;
B) 5x-By+7=O H IOx-I6y+l=O.
2. HaH.D;HTe sePWHHbI -rpeyrOJIbHHKa, CTOpoHbl KOTOporo HlllelOT ypaBHeHHSI;
2x-3Y+5=O (A B);
3x+4u+i=O (BC)i
x+2y+I=O (AC).

r 11 a B a VI
(1)YHKHJI, IIPMEJI, IIPOH3BOHASI. HHTErPAJI
f 1. 'IBCJIOBLIE <l>YHKD;IIB
1. 4»YHKQHH H Bblpa)KeHHii. OrBeT moooH apH$MeTlfIlecKoii 3MatJH
aaBHCHT OT .l!.aHHbIX. PaccMoTpHM, HanpHMep, 3a.zI.atJY: «3a 6y6JIHK H CTa-
aH MOJIOKa 3anJIaTHJIH 10 Kon. By6JIHK CTOI1T 6 Kon. CKOJIbKO CTOHT
CTaKaH MOJIOKa?». OTBeT TaKOB: «CTaKaH MOJIOKa CTOHT 4 Kon.». Ho
ec.rJH B TOll )!{e 3a.zI.atJe 3aMeHI1Tb 6 Kon. Ha 7 Kon., TO OTBeT CTaHeT:
«CTaKaH MOJIOKa CTOHT 3 Kon.». TaKHM 06pa30M, Ka)!{.l!.oMY 3HatJeHHIO
OHMOCTH 6y6.r1HKa OTBetJaeT CBoe 3HatJeHI1e CTOHMOCTH MOJIOKa. 0603Ha-
qHM CTOHMOCTb 6y6JII1Ka 6YKBOH x, a CTOHMOCTb CTaKaHa MOllOKa 6YK-
80H y. Tor,lJ,a .l!.OJI)!{HO BbmOJIHHTbCSI paBeHCTBO x+y= 10. 113 3Toro
paBeHCTBa no.1Y1JaeM BbIpaIKeHHe y tJepes x: y= 10-x. OHO n03BOJISIeT
,lVIH Ka)!{)J,oro 3Ha1JeHHSI x HaHTH COOTBeTCTBYlOlII.ee 3HatJeHHe y (HanpH-
1 1
Mep, eCJIH x=-2, TO y= 12, a eCJIH X=2' TO y=9 2 ). Ho CTOHMOCTb
(Sy6JIHKa, KaK H CTOHMOCTb CTaKaHa MOJIOKa, He MO)!{eT BbIp8)!{aTbCSI IIlI
OTpH£I,aTeJIbHbIM, HIf .l!.p06HbIM tJHCJIOM - 3TO HaTypaJIbHOe tJHCJIO. no-
;hOMY x MOIKeT npHHHMaTb JIHIIIb 3HatJeHHSI 1, 2, 3, 4, 5, 6, 7, 8, 9.
toOTBeTcTBHe Me)!{)J,y x 11 y 3a.l!.aeTCH Ta6JIH£I,eH:
x 234 5 6 789
y 9 8 7 654 3 2
aHa onpeAeJISIeT oT06paIKeHHe tJHCJIOBOrO MHOIKeCTBa X = {I; 2;
8; 4; 5; 6; 7; 8; 9} B MHO)!{ecTBO R .l!.eHcTBHTeJIbHbIX tJHCeJI, HJIH, KaK
rOBopHT, 1JHCJIOBYIO $YHK£I.HIO C 06JIaCTbIO onpe.l!.eJIeHHH X. Bo06lI1.e,
rOBopSIT, 1JTO Ha MHoIKecTBe Xc. R 3a,lJ,aHa 'lUC/loeaJl (jJYHK/{UJl, eCJIH
Ka)!{)J,oMY tJHCJIY x H3 3TorO MHOIKeCTBa nOCTaBJIeHO B COOTBeTCTBHe
HeKoTopoe 1JHCJIO y. <l>YHK£I.HIO 0603Ha1Ja1OT 6YKBOH f 11 nHIIIYT y=f(x),
x EX. MHO)!{eCTBO X Ha3bJBalOT 06/laCmblO onpeae/leHUJl 3TOll $YHK£I.HH.
ECJIH BbIpa)!{eHHe, CO,lJ,ep)!{alIl.ee 6YKBY X, npHHHMaeT onpe,lJ,eJIeHHoe
3HatJeHHe ,lJ,.'IH .1I1060ro x H3 MHOIKeCTBa X, TO OHO onpe.l!.eJIHeT $YHK£I.HIO
Ha X, CTaBSIlII.YIO B COOTBeTCTBHe Ka)!{)J,OMY x E X 3HatJeHHe 8bIpa)!{e-
HIISI 8 T01JKe X.
HanpHMep, eCJII1 3a.zI.aHO BbIpaIKeHHe x2+3, TO eMY COOTBeTCT-
ByeT <PYHK£I.HH 8H,n:a y=x 2 +3, xE X. npH STOM pa3JIHtJHbIM MHO-
140

)l{eCTBaM X COOTBeTCTBYIOT pa3JIH1JHbIe YHK-
[.!JIll: YHKUHH y=x 2 +3, X E R, OTJIH1JHa OT
CPYHKUHH y=x 2 +3, X E R +. nepBaSI H3 HHX
HMeeT npH x=-1 3Ha1JeHHe y=(-1)2+3=4,
a BTopaH B 3TOR T01JKe He on pe.ll.eJIeHa.
MbI 6y,neM B ,lJ,aJIbHeHIIIeM Ha3hIBaTb X
apeY.MeHmOM. YHKUHH y=f (X), X EX. 3Ha- 
1JeHHe YHKUHH, COOTBeTCTBYIOllI.ee 3Ha1JeHHIO
a aprYMeHTa X, 0603Ha1Ja1OT f (a). C Ka)K)J,OH:
$YHKlI.HeH y=f (X), X EX, CBSI3aHO 1JHCJIOBOe
MHO}KecTBO Y, COCTOSIlII.ee H3 3Ha1JeHHH :HOH
YHKlI.HH, T. e. H3 1JIiCeJI BH,lJ,a f (X), X EX. I1HbIMH CJIOBaMH, Y =
= {f (X) Ix EX}. MHO)l{eCTBO Y 0603Ha1Ja1OT TaKIKe f (X). Ero Ha3bIBa-
lOT AtHOJICeCm80M. 3HG'leHUU CPYHKl{UU.
LLJISJ Ka:>K,lJ,OrO Bblpa:>KeHHSI C nepeMeHHOH X CYlII.eCTBYeT HaH60JIbIIIee
lJHCJIOBOe MHO:>KeCTBO, COCTOSIllI.ee H3 Bcex ,lJ,eHCTBHTeJIbHbIX 1JHCeJI, ,lJ,JIH
KOTOpbIX 9TO BhIpa:>KeHHe HMeeT 3Ha1JeHHe. TaKoe MHO:>KecTBO Ha3bIBalOT
06AaCmblO CYUIf!cm808aHU5l 8blpa:JICeHU5l. HanpHMep, ,lJ,JISI BhIpa:>KeHHH
x 2 +3 06JIaCTbIO CYlII.eCTBOBaHHSI SIBJISIeTCSI BCSI 1JHCJIOBaSI OCb, a ,lJ,JISI Bbl-
p(l)!{eHHSI V X-9- JIHllIb JIY1J [9;+00) (KBa,lJ,paTHbIH KopeHb MO:>KHO U3-
BJIeKaTb JIHllIb H3 HeOTpHuaTeJIbHhIX 1JHCeJI).
MO:>KeT CJIY1JHTbCSI, 1JTO BhIpa:>KeHHe HMeeT CMbICJI npH 3Ha1JeHHSIX
aprYMeHTa, He npHHMJIe)!{alIl.HX 06JIaCTH onpe,lJ,eJIeHHSI $YHKUHH. Ha-
npHMep, nJIOlIl.a,lJ,b S KBa,lJ,paTa BbIpa)!{aeTCSI 1Jepe3 ,lJ,JIHHY X ero CTOpO-
HhI no $opMYJIe S=x 2 . BbIpa:>KeHHe x 2 HMeeT CMbICJI npH JII06hIX 3HatJe-
HHSIX x. Ho ,lJ,JIHHa CTOpOHhI KBa,lJ,paTa BCer,lJ,a nOJIO:>KHTeJIbHa, H nOTOMY
06JIaCTbIO onpe,lJ,eJIeHHSI YHKUHH SIBJISIeTCSI 1JHCJIOBOH JIY1J )0; +00[,
T. e. ee HMO 3anHCbIBaTb B BH,lJ,e S=x 2 ,O<x<+00.
BCTpe1JaIOTCSI CJIY1JaH, Kor,lJ,a O,lJ,Ha H Ta :>Ke YHKUHSI npH pa3HhIX
3Ha1JeHHSIX aprYMeHTa 3a,lJ,aeTCSI pa3JIH1JHbIMH BbIp(l)!{eHHSIMH. 0603Ha-
1JHM, HanpHMep, 1Jepe3 S (X) nJIOlIl.Mb HrypbI, OTCe1JeHHOH OT HrypbI F
npSIMOH, napaJIJIeJIbHOH npSIMOH 1 H OTCTOHlII.eH OT Hee Ha paCCTOSIHHH X
(pHC. 89). H3 tJepTe:>Ka BMHO, 1JTO npH x:::;;;a OTceKaeMaH Hrypa HMeeT
CPOPMY npHMoyrOJIbHHKa C nJIOlIl.MblO ax. ECJIH x>a, HO x<a+b, TO
cpHrypa - 06'be.ll.HHeHHe KBa,lJ,paTa CO CTOpOHOH a H npSIMoyrOJIbHHKa
C BblCOTOH x-a H OCHOBaHHeM b. n09TOMY nJIOlIl.Mb HrypbI paBHa
a 2 +b (x-a). ECJIH x>a+b, TO nJIOlIl.Mb HryphI o)J.Ha H Ta )!{e H
paBHa a 2 +b 2 . HTaK, $YHKII.HH S (X) BbIpa:>KaeTCSI CJIe,lJ,YlOlII.HM 06pa30M:
{ ax, ecJIH x:::;;;a,
S (x) = a 2 +b (x-a), ecJIH a<xa+b,
a 2 +b 2 , ecJIH x>a+b.
Mbl BH,lJ,HM, tJTO 06JIaCTb onpe.ll.eJIeHHSI 9TOH $YHKUHH COCTOHT H3 Tpex
ytJaCTKOB, H Ha K(l)!{,lJ,OM YtJacTKe $YHKUHH HMeeT CBoe aHaJIHTHllecKoe
BhIp(l)!{eHHe.
npHBe.lI.eHHble npHMepbI nOKa3bIBalOT, tJTO HeJIb3H OTO:>K,lJ,eCTBJIHTb
nOHSITHH $YHKUHH H ee aHaJIHTHtJecKoro BhIpa:>KeHHH. CYlII.eCTBYIOt
CPYHKIJ.UH, He HMelOllJ.He aHaJIHTHllecKoro BbIpa:>KeHHH (HanpHMep,
141
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PRC. 89

TeMnepaTypa B03Ayxa B AaHHOM MecTa
B MOMeHT BpeMeHH t).
FpacpUKOM cpynK/{UU y =- f (x),
x EX, Ha3bIBalOT MHO)!{eCTBO nap
(x; y) TaKHX, tJTO x E X, a y=f(x),
T. e. nap BMa (x; f(x», xE X.
K8)!{,lJ;OH TaKOH: nape COOTBeTCTBYCT
T01JKa M Ha KoopAHHaTHOH: nJIOCKo-
CTH, HMelOlII.aH Koop,lJ,HHaTbI x H f (x).
MHO)!{ecTBO TaKHX T01JeK TO)!{e Ha-
3bIBaIOT rpaHKoM YHKUHH y=f (x),
xEX.
06bI1JHO rpaHK YHKUHH H30-
6paaeTCH HeKOTopoH JIHHHeH: Ha KoopAHHaTHoH nJIOCKOCTH. O,lJ,HaKO
He BCHKaff JIHHHH MO)!{eT CJIY}I{HTb rpaqmKoM $YHKUHH. l{eJIO B
TOM, 1JTO np 3a,lJ,aHHOM 3Ha1JeHHH x EX Cym.ecTBYeT JIHmb OAHO
3ijatJeHHe $YHKUHH y, COOTBeTCTBYIOIIJ;ee 3TOMY 3Ha1JeHHIO x. n03TOMY
H! KIDK.ZJ.OH: IJPHMOH:, napaJIJIeJIbHOH: OCH opAHHaT, MO)!{eT JIe)!{aTb He
60JIee OAHOH: T01JKH rpa$HKa $YHKUHH. HanpHMep, OKPY)!{HOCTb He
HeTCH rpaHKOM KaKOH:-JIHOO $YHKUHH - CYlIl.ecTBYIOT npHMbIe,
napaJIJIeJIbHbIe OCH op,lJ,HHaT (pHC. 90), Ha KOTOpbIX JIe)!{aT ,lJ,Be T01JKH
OKPY)!{HOCTH.
n p H M e p 1. HaH:,lJ,eM MHO)!{ecTBO 3Ha1JeHHH: $YHKUHH y=x 2 ,
x eX=< {I; 2; 3; 4; 5; 6; 7; 8}. TaK KaK MHO)!{ecTBO X KOHe1JHO, TO MHO-
ecTBO Hfa1JeHHH: MO)!{HO HaRTH, nO,lJ;CTaBJIHH no 01Jepe,lJ,H 3Ha1JeHHH x.
I1oJIY1JaeMl
Y={I; 4; 9; 16; 25; 36; 49; 64}.
2. HaHAeM MHO)!{eCTBO 3Ha1JeHHH: YHKUHH y=x 2 ,
- 2:::;;;;x:::;;;;3. Ha pHcYHKe 91 H306pa-
}l{eH ee rpaHK. 113 Hero BH,lJ,HO, tJTO
Ha OTpe3Ke [-2; 01 YHKUHH y6bIBa-
eT OT 3Ha1JeHHH 4=(-2)1 AO 3Ha1Je-
HHH 0, a Ha OTpe3Ke [0; 3) B03paCTaeT
OT 3Ha1JeHHH 0 ,lJ,0 3Ha1JeHHH 9=3 1 .
OrcIOAa HCHO, tJT() MHO)!{eCTBOM 3Ha-
1JeHHH YHKUHH HBJIHeTCH 06'be,lJ,H-
Helme OTpe3KOB [0;41 H [0;91, T. e.
OTpe30K [0;91, Y=[O;9J.
n p H M e p 3. HaH,lJ,eM MHO)!{e-
u .... 4
CTBo 3Ha1JeHHH 't'yHKUHH y= l+x t ,
x E R. HaHOOJlbmee 3HaqeHHe na
YHKUHH npHHHMaeT, ecJIH x 2 HMeeT
HaHMeHbllIee 3Ha1JeHHe, T. e. npH
x=O (npH OCTaJIbHbIX 3Ha1JeHHHX x
1JHCJIO x 2 nOJIO)!{HTeJIbHO). 31'0 3Ha-
X 1JeHHe YHKUHH paBHO 4. HaHMeHb-
mere 3HalJeHHH YHKlI,HH He HMeeT,
y
PRC. 90
npHMep
!J
9
8
7
6
8(3;9)
U2
x

TaK KaK npH YBeJIHtJeHHH x
3HaMeHaTeJIb YBeJIHtJHBaeTCH, a
4
Ap06h l+x l YMeHblllaeTcH H
npH6JIl:\)!{aeTCH K HYJIIO, HH-
Korha He AOCTHraH 3Ha1JeHHSI O.
3'1'0 nOK33bIBaeT, tJTO MHO:>Ke-
CTBOM 3Ha1JeHHll: $YHKUHH SIB-
JIHeTCH npOMe)!{YToK ]0;4]
(pHC. 92).
!J 4
PHC. 92
YnpalKOeOBJI
1. O)J.H3 CTOpOH3 npHMoyroJIbHHKa HMeeT JJ,JIHHY 5 .41, 3 BTopaH - x M. "IeMy'
p3BHa nJIOw.a.n.b S (.41 2 ) 9TOro npHMoyroJIbHHKS? ¥KalKHTe 06JIaCTb 3a)J.aHHI'I H
MHO/KecTBO 3Ha'leHHH COOTBeTCTBHH Me/K)J.Y x R S (x), a TaK/Ke 06JIaCTb CY1Il.eCTBO-
BaHHH BbIPaIKeHiffi <j>YHKUHH.
2. Y CalliH 5 opeXOB, a y OJIH H3 X opexoB 6oJIbllie. CKOJIbKO opeXOB y CalliH H
OJIH BMecre? ¥Ka/KHTe MHO/KecTBa, COOTBeTCTBHe Me)K)l.y KOTOpbIMH ycT3H3BJIHBaeTCI'I
B 9TOH 33)J.a'le. OOKa/KHre, '1TO 9THM onpeAe.neHa <j>YHKUHH, 1'1 HaH,n.HTe ee 06JIaCTb
33)J.aHHH H MHO/KeCTBO 3Ha'leHHii. HaH,n.HTe 06JIaCTb CY1Il.eCTBOBaHHH 33)J.3IOw.ero ee
BbIpa/KeHHH.
3. HH/Ke YKa3aHbI COOTBeTCTBHH, 33)J.aHHbIe CBOHMH rpa<j>HKaMH. Onpe)J.eJIHTe,
KaKHe H3 3THX cooTBeTcTBHii HBJIHIOTCH <j>YHKUHHMH. .r:LJIH 3THX COOTBeTCTBHii Haii)J.HTe
06JIacTb 33)J.aHHH H MHO/KeCTBO 3Ha'leHHii:
a) R= ! (l; 2); (3; 4); (5; 6); (7; 8); (9; 1O)};
6) R= (1; 2); (1; 3); (1; 4); (1; 5)};
B) R= (1; 2); (2; 2); (3; 2); (4; 2); (5; 2)}; .
r) R= (I; 1); (2; 2); (3; 3); (4; 4); (5; 5)};
)J.) R= (1; 0); (-1; 0); (2; 2); (-2; 2); (-2; -2)}.
4. OocTpoiiTe rpa<j>HK <j>YHKUHH y=-4x, xEX={-2; -1; 0; 1; 2}. Haii)J.HTe
MHOMeCTBO 3Ha'leHHii 3TOii <j>YHKUHH.
5
5. OocrpoiiTe rpa<j>HK. H Haii)J.HTe MHO/KeCTBO 3Ha'leHHii .!I.JlH <j>YHKUHH y= x ,
xEX={-2; -1; 0; 1; 2}.
6. OocTpoiire rpa<j>HK <j>YHKUHH y=9-x2, xER, H YKIDKHTe MHO/KecTBO TeX
3Ha'leHHH x, )J.JlH KOTOpbIX y;;;;'O.
7. IlocTpoihe rpa<j>HKH CJIe)J.YI01II.HX OTHOllieHHiI: B MHO/KeCTBe )J.eiicTBHTeJIbHbIX
'1HceJI H YKa/KHTe cpe)J.H HHX rpa<j>HKH <j>YHKUHH:
a) R= {(x; y) Iy=x};
6) R= l (X; y) I y"",x};
B) R= (x; y) I y= Ixl };
r) R= (x; y) I Iyl=x};
lI) R= {(x; y) I X=y2}.
8. Oelliexo)J. 1'1 BeJIOCHne)J.HCT BbIeXaJIH HaBCTpe'lY )J.Pyr )J.Pyry 1'13 ceJI A H B.
Oelliexo)J. npoxollHT 3a 'lac 6 KM, a BeJIOCHne)J.HCT )J.eJIaeT Ha x KM 60JIbllie. 4eMY paBHo
pacCTOHHHe y MelK.nY CeJIaMH, eCJIH OHH BCTpeTHJIHCb 'lepe3 3 'I? 3anHlliHTe COOTBeT-
CTBHe MelK.nY x H y. Onpe)J.eJIHeT JIH aHa <j>YHKUHIO? KaKoBbI 06JIaCTb 33)J.aHHH R
MHO/KeCTBO 3Ha'leHHH 9TOii <j>YHKUHH, eCJIH BeJIOCHne)J.HCT npOe3/KaeT He 60JIee 15 KM
B 'lac? OOCTpoHTe rpa<j>HK 3TOro COOTBeTCTBHH.
9. Mallia HalliJIa 6 rpH60B, a KOJIH Ha x rpH60B 6oJIbllIe. 4eMY paBHo '1HCJIO g
rpH60B, KOTOpbIe OHH C06p8JIH BMecTe? 3anHlliHTe COOTBeTCTBlJe Me/K)J.Y x H y H Hail;-
)J.HTe ero 06JIaCTb 33)J.aHHH 1'1 MHO/KeCTBO 3Ha'leHHii, eCJIH KomI co6paJI He 60JIee 15
rpH60B. 11306pa3lJTe rpa<j>HK 3Toro COOTBeTCTBHB. 4eM OTJIH'IaeTCH 3TOT rpa<j>HK OT
rpa<j>HKa 33)J.a'lH 8?
J43

2. npJlMaSi nponopHOHaJIbHOCTbt JlHHeHHaH aaBHCHMOCTb H HK
rpa«l>HKH. 4aCTo paccMaTpHBaIOT Tp.ll BeJI.IItJ.IIHbI, O,lJ,Ha H3 KOTOpbIX
paBHa npoH3Be.lI.eHHIO ABYx APYruX. HanpHMep, nYTb npH paBHoMepHoM
,lJ,BH)!{eHHH paBeH npOH3Be.lI.eHHIO BpeMeHH Ha CKOpOCTb ABH)!{eHHH, CTO-
HMOCTb TOBapa paBHa npoH3Be,n.eHHIO ero eHbI Ha KOJIHqecTBO TOBapa,
nJIOlIl.a.zlb npHMoyrOJIbHHKa paBHa npoH3Be.lI.eHHIO AJIHH ero CTOpOH.
MaTeMaTHtJecKaH 3aBHCHMOCTb 3THX BeJIHqHH BbIp3)!{aeTCH paBeHCTBoM
y-zx. ECJlH O)(Ha H3 ,nepeMeHHbIx x HJIH z nOCTOHHHa, HanpHMep z=k,
TnyqaJI cJ>YHKIJ.HH BHAfl 1!=kx. BYAeM roi30pliTb., qTO B 3T()"CJlX-
qae BeJUi'qHHa 11 npJlMO nponop/{UOHaJlbHa BeJIHqHHe x. ECJIH ynp-HMo
nponopnHOHaJIbHO x, TO qaCTHoe .!L AJIH Bcex nap COOTBeTCTBYlOlII.HX
x
3HatJeHHH: x H y (KpOMe napbI (0; 0) ) npHHHMaeT OAHO H TO }l{e 3HaqeHHe
k. O 3Ha1JeHHe Ha3bIBalOT KOatfxPu/{uenmoM nponop/{uOHa/lbHOCmu.
.lI.JIH TOro tJT06bI HaihH K03IfxPHUHeHT nponopHOHaJIbHOCTH, AOCTa-
TotJiw 3HaTb OAHY napy (xo; Yo) COOTBeTCTBYIOllJ.HX APyr APyry 3Haqe-
HHH, OTJIH1JHYIO dr napbI (0; 0). l1MeHHo, H3 paBeHCTBa Yo=kxo HaXOAHM,
qTO k= Yo . HanpHMep, npH nOCTOHHHOH ueHe TOBapa ero CT<)HMOCTb
Xo
nponopUHOHaJIbHa KOJIHtJecTBY TOBapa, npHtJeM KOHUHeHTOM npo-
nopUHOHaJIbHOCTH HBJISIeTCH ueHa. 4T06bI HaHTH ueHY, AOCTaTOtJHO
3HaTb CTOHMOCTb HeKOToporo KOJIHtJeCTBa TOBapa. HanpHMep, eCJIH 5 KZ
TOBapa CTOSIT 15 PY6., TO ueHa TOBapa paBHa 15:5=3 (py6. 3a KZ).
MbI 3HaeM (CM. n. 1  5 rJI. V), tJTO rpaHKoM ypaBHeHHH y=kx SIB-
neTCH npSIMaH JIHHHH, npOXOASIlII.aH tJepe3 HatJaJIO KoopAHHaT H HMelO-
w;aH yrJIOBOll K03$lmU.IIeHT k. HTaK, KOacpcpu/{uenm nponopf.{uona/lbHO-
emu k coenaiJaem c yc/l08blM KOatfxPuf.{uenmoM cpacpuKa cpynKf.{uu y=kx.
BOJIee 06lI1.eH, tJeM npHMaH nponopUHOHaJIbHOCTb, HBJIHeTCSI JIHHeH-
HaH 3aB.IICHMOCTb MeM)J,y BeJIHtJHHaMH. PaccMOTpHM npHMep.
CTOHMOCTb noe3AKH Ha TaKCH Onpe.ll.eJIHeTCH no CJIeAYIOllJ.eMY npa-
BHny: OHa paBHa 20 Kon. 3a KIDK)l.bIH KHJIOMeTP'H elIl.e 20 Kon. 3a K3)!{-
.lI.yro noe3AKY. HHbIMH CJIOBaMH, CTOHMOCTb OAHOll noe3AKH Ha x KM
Bhlp3)!{aeTCSI $opMYJIOll y"",,20x+20. 3Ta $opMYJIa HBJIHeTCH tJaCTHbIM
CJIYtJaeM 60JIee 06lI1.eH 3aBHcHMoCTH BMa y=kx+b. TaKYIO YHKU.IIJO
Ha3bIBalOT /lunefinoii cpynKf.{uefi.
4T06bI HallTH 3HatJeHHe KOHUHeHTak, HMO 3HaTb ABe napbI COOT-
BeTCTBYlOlII..IIX APyr Apyry 3HatJeHHH, HanpHMep (Xl; Y1) H (XZ; yz).
nOCKOJIbKY roe 3TH napbI YAOBJIeTBOpSIIOT YCJIOBHIO y=kx+b, TO HC-
THHHbI 06a paBeHCTBa Y1=kx1+b H Y2=kx2+b. H3 HHX BbITeKaeT, tJTO
Y.-Yi=kx2-kxi=k (X 2 -X1), H n03TOMY
k= Yz- Yl.
Xz -xl
HanpHMep, eCJIH nOe3A BbIIIIeJI CO CTaHUHH, HaXoASIlII.eHCSI rAe-TO Me}/{-
AY MOCKBOll H JIeH.IIHrpMOM, H ecJIH tJepe3 2 It nOCJIe HatJaJIa ABH}/{eHHH
OH HaXOA.IIJICH Ha paccToSIHHH 270 K.M. OT MOCKBbI, a tJepe3 5 It nOCJIe
HatJaJIa AB.II}/{eH.IISI - Ha paCCTOSIH.IIH 510 KAt OT MOCKBbI, TO CKOpOCTb
510-270
,lI,BH:>KeHHSI BbIp3)!{aeTCSI TaK: v= 5-2 = 120 (KAt/it). Bo06llJ.e, CKo-
144

pOCTb npm'iIOJIHHeAHo H paBItOMepHO ,nBH>Kyru;erocH TeJIa, KOTOpoe sa
npOMe>KyTOK BpeMeHH [i 1 , i z ) nepellIJIO H3 TOtJKH Xl B TOtJKY X g , Bblpa-
>KaeTCH 4>oPMYJIOA:
XZ-;i
(1= .
1.- 1
YopalKHeHOR
I. 3aUHlliHTe Bbipa/KeHH6 JlHHeAHoA 4>YHHUHH, ecJ[H SlIAaHW II H SHa'leHHe fh.
9TOA tIIY,iIKI\HH B m'tiKe Xl: . '.
Ii) k==2, X1=-3, Y1=4; 6) k=-l, x1?'6, Yl=-2i
3 . 2
8) k = 2' x1=-7, Y1=-S; r) le=- 3' xl=4, Yl=2.
fiocrpoiiTe rpa<j>HKH 3THX <j>YHKHA.
2. HaiiAHTe Bbipa/KeHHe JlHHeiiHoii 4>YHKD;HH, SHaH ee 3Ha'leHHI'I Yl H Yz B TO'IKaJt
,,{ H Xz, eCJIH:
a) X1=3, xz=5, Y1=4, Ya=O; 6) xl=-6, xz=3, Yl=l, y."""19.
3. 4epea a 'I nOCJIe Bblxoa co CTBHD;HH A B Hanpa!'lJl\!,!HH K CTaHD;HH B nOe3A
XOAHJlCH OT B Ha paCCTOHHHH 51' a 'Iepea b 'I - Ha paCCTOI'IHHH 5z. Haii.u.HTe pac-
CToI'IHHe MeH<.ll.Y CTaHUHHMH H CKOpOCTb nOe3l(a.
3. 06paTHaSi nponOpHOHaJIbHOCTb H ee rpa«l>HK. MbI BH,lJ,eJIH, tffO
eCJIH TPH BeJIHtJHHbI CBH3aHbI COOTHOllIeHHeM y=ZX, TO npH $HKCHpOBaH-
HOM 3HatJeHHH Z BeJIHtJHHa y npOnOpUHOHaJIbHa X. 3a$HKCHPYeM Te-
nepb y, nOJIO>KHB y=k. Tor,lJ,a X H Z CBH38HbI COOTHOllIeHHeM k=zx, T. e.
k
z""" -. fOBOPHT, tJTO X HZ 06pamHO nponopf.{UOHa/lbHbl. HanpHMep, ecJIH
x
npOHAeHHbIH nYTb SIBJIHeTCH nOCTOHHHbIM, TO CKOpOCTb H BpeMH <;BH3a-
HbI COOTHOllIeHHeM vi=k H nOTOMY 06paTHO nponOpUHOHaJIbHbI. TOtJHO
TaK >Ke 06paTHo npOnOpUHOHaJIbHbI ueHa H KOJIHtJecTBO TOBapa, ecJIH
ero CTOHMOCTb nOCTOHHHa, ,lJ,JIHHa H BbICOTa npHMoyrOJlbHHKa, eCJIH ero
nJIoru;a,lJ,b nOCTOSIHHa, H T. ,lJ,.
KaK H B cJIytJae OpHMOH nponopUHOHaJIbHOCTH, ,nJIH OTbICKaHHH
K03qxJmUHeHTa 06paTHoH nponopUHOHaJIbHOCTH k ,lJ,OCTaTotJHO 3HaTb O,lJ,HY
napy COOTBeTCTBYlOlII.HX 3HatJeHHA: X H y. HanpHMep, ecJIH Ha 3@8HHyro
CYMMY ,lJ,eHer MO>KHO KynHTb 61<:8 TOBapa ueHoH 4 py6. 3a KHJIOrpaMM, TO
31'a CYMMa ,neHer paBHa 24 py6. H nOJ'OMY 3aBHCHMOCTb MeMAY ueHoA:
24
TOBapa X Hero KOJIHtJeCTBOM Z HMeeT BH,n xz=24 HJIH Z=-.
x
H3YtJHM CBOHCTB8 YHKII.HH y=.! H nocTpoHM ee rpa4>HK. BbIpa-
x
k
>KeHHe x onpe,lJ,eJIeHO npH Bcex 3HatJeHHHX x, KpoMe x=O. nYCTb k>O.
k
Tor,lJ,a npH nOJIO>KHTeJIbHbIX 3HatJeHHSIX X 3HatJeHHH ,lJ,p06H - YMeHb-
X
llIaIOTCSI npH B03pacTaHHH ee 3HaMeHaTeJISI x, H npH HeorpaHHtJeHHoM
YBeJIHtJeHHH X CTaHOBHTCH CKOJIb yro,lJ,HO 6JIH3KHMH K HYJIIO. HanpHMep,
2
ecJIH k=2, TO npH x>2 000000 HMeeM y< 2000000 =0,000001. ECJIH >Ke
x>2 000 000 000, TO y<O, 000 000 001. Kor,lJ,a 3HaMeHaTeJIb X npH6JIH-
k
)!{aeTCH K HYJIIO, TO 3HatJeHHH ,lJ,p06H - HeorpaHHtJeHHO BoapaCTalOT. Ha-
x
145

y
5/
4/
3/
/2
/
__/ 1
--
npHMep, ecJIH k=2, TO npH
2
0<x<O,002 HMeeM y> 0 002 =
= 1000, a npH O<x<O,OOo 002
HMeeM y>1 000000.
4T06bI nOJIYtJHTb tJaCTb rpa$H-
Ka, cooTBeTcTBYlOlUYIO OTpH:U.a-
TeJIbHhIM SHatJeHHSIM x, 3aMeTHM,
tJTO BMecTe C TOtJKOH M (xo; Yo) rpa-
x k
$HK $YHKUHH y= x ,lJ,OJI:tKeH co-
.n.ep,)j{aTb H TOtJKY N (-xo;-Yo).
3T0 CJIe.lI.yeT H3 Toro, tJTO ecJIH
k k
Yo= -, TO -Yo= -. HQ TOtJKH
Xo -x o
M (xo; Yo) H N (-xo;-Yo) CHM-
MeTpHtJHbI OTHOCHTe.llbHO HatJaJIa
Koop,lJ,HHaT. n03TOMY rpa$HK
k
YHKUHH y= - CHMMeTpHtJeH OTHOCHTeJIbHO HatJaJIa Koop,lJ,HHaT.
x
o k k
fpa$HK $YHKUHH y=- X nOJIYtJaeTCSI H3 rpaHKa YHKUHH y=x
H3MeHeHHeM 3HaKa y .lI.JIH K8)K)!,OH TOtJKH. 3TH rpaHKH CHMMeTpHtJHbi
,lJ,Pyr ,lJ,pyry OTHOCHTeJIbHO OCH a6cUHcc. Ha pHcYHKe 93 H306p8)!{eHbI
th th U 2 ( U U ) 2 ( U
rpa't'HKH 't'YHKUHH Y=i CnJIOllIHOH JIHHHeH H Y=- X llITpHXOBOH JIH-
HHeH).
k
ECJIH k>O, TO rpaq,HK $YHKUHH y= pacnOJIaraeTCSI B I Ii lIItJeT-
BepTHX, a eCJIH k<O, TO BO II H IV tJeTBepTSIX. Kor,lJ,a x CTpeMHTCH K
k
6ecKoHetJHocTH, KpHBaH Y =- HeorpaHHtJeHHonpH6JIH:>KaeTCSI K OCH
x
a6cUHcc, O,lJ,HaKO He ,lJ,OCTHraeT ee. fOBOPHT, tJTO OCb a6cUHCC - zo-
pU30Hma/lbHaJl acuMnmoma 3TOH KpHBOH. Kor,lJ,a x CTpeMHTCSI K HYJIIO,
k
KpHBaH Y= - HeorpaHHtJeHHO npH6JIM:tKaeTCH K OCH opAHHaT, HO He A0-
x
CTHraeT ee. fOBOPHT, tJTO OCb Op,lJ,HHaT - 8epmUKa/lbHa.R. acuMnmoma
3T0" KpHBOH. BOJIee TOtJHoe onpeAeJIeHHe aCHMllTOT 6YAeT AaHO B n. 5
 4.
Y -l.
-x
Pnc. 93
Y npaJICBeBBJI
OOCTpoHre rpa<j>HKH <j>YHKHH:
a) y=-t; 6) y=  ; B) Y= II ; r) Y=- II '
4. KOMIT03HHSI 4JyHKHii (CJlO)l(HaSi «I>YHKHSI). Maccy Ky6a BbItJHC-
JIHIOT ITO $opMYJIe m=qV, r,lJ,e V - 06'beM Ky6a, a q - nJIOTHOCTb Ma.
TepHaJIa, H3 KOToporo OH CAeJIaH. 06'beM Ky6a BbItJHCJISIIOT no $opMYJIe
V=x 3 , r,lJ,e x - ,lJ,JIHHa ero CTOpOHbI, nOToMY ero Maccy MO:>KHO BhJtJHC-
JIHTb no $opMYJIe m=qx 3 . BeJIHtJHHbI x H m cmi3aHbI .n.pyr C .n.pyroM TaK,
146

liTO no 3a,lJ,aHHOMY 3HatJeHHIO x CHaqaJIa HaXO,lJ,HM V, a nOTOM no 3Haqe-
HHIO V - 3HatJeHHe m. 3HaqHT, K8)!{,lJ,OMY 3HaqeHHIO x COOTBeTCTBYeT
onpeAeJIeHHoe 3HaqeHHe m, T. e. m SIBJISIeTC5I $YHKUHeH OT x. TaKYIO
4IYHKII.HIO Ha3bIBIDOT KOMn03UI{Ueu saiJaHHblx CPYHKI{UU. npH 3TOM V Ha3bI-
B310T npoMe1lCymo'lHbLM. apeYMeHmOM.
. B 06m.eM BMe nOH5ITHe KOMn03HUHH YHKUHH Onpe,lJ,eJISIeTCSI CJIe,lJ,YIO-
llI,HM 06pa30M. nYCTb 3MaHbI ,lJ,Be YHKUHH y=f(x), xE X, H x=cp(t),
t€ T, npHqeM Ka)!{,lJ,Oe 3HaqeHHe $YHKUHH x=cp (t) npHHaMe>KHT
MHO>KecTBY X. Tor,lJ,a no 3a,lJ,aHHOMY 3HaqeHHIO t MbI HaXO,lJ,HM 3HaqeHHe
x € X, a no 3HatJeHHIO x - 3HaqeHHe y. 3THM H onpe,lJ,eJISIeTCSI HOBaSI
<PYHKlJ,HH, CTaB5IlII.a51 Ka)K)J,OMY t E T B COOTBeTCTBHe HeKOTopoe y. Ee
Ha3hIBaIOT KOMnOSUI{Ueu cpYHKl{wl y=f(x) H x=cp(t) H 3anHCbIBaIOT B
BMe y=f[cp (t)J. t E T. HHorAa H306pa)!{a1OT 3TY $YHKUHIO TaK:
q> f
txy.
n p H M e p 1. nYCTb y=x'+ 1 H x=3t-4. 4T06bI nOJIyqHTb BbI-
p8)!{eHHe ,lJ,JISI KOMn03HUHH 3THX $YHKUHH, HMO B BbIpa)!{eHHH ,lJ,JIH y
3aMeHHTb x Ha 3t-4. nOJIyqaeM y=(3t-4)2+ 1.
n p HM e p 2. nYCTb y=Vi H x=-t 2 -1. B 3TOM cJIYtJae KOMno-
3HUHSI $YHKUHH He onp e,lJ,eJIeHa , TaK KaK Bce 3HaqeHHSI $YHKUHH
x=-t 2 -1 OTpHuaTe.rIbHbI, a $YHKUHSI y=V X onpe,lJ,eJIeHa JIHrnb ,llJISI
HeOTpHuaTeJIbHbIX 3HaqeHHH x.
YnpamUeHUJI
1. HaHJlHTe KOMn03MltMIO <j>YHKltHH:
a) y=x 3 H x=t2+4; 6) y=x2+4 H x=t 3 ;
B) y= (x8+ 1)2 H x= t 4 ; r) y=x4 H x= (t 3 + If.
2. CylltecTByeT JIH KOMn03HHH <j>yHKHii y= Y 4 x H x=8+t 2 ?
3. CyllteCTByeT JIM KOMn03HHH <j>yHKII.Hii y=8+x8 H x= Y 4-t?
4. BbIpa3HTe 06'beM Ky6a V 'Iepe3 nJIOllta.llb ero nosepxHoCTH S.
5. 06paTHaSi cJ>YHKHSI. ECJm HMO npOHTH nYTb B 100 KM, a nYT-
HHK ,lJ,BH)!{eTCH CO CKOpOCTblO 5 KM/'l, TO nYTb, KOTOpblH OCTaeTC5I
npORTH qepe3 t'l nOCJIe HalJaJIa .lI.BH)!{eHHSI, BbIpa>KaeTCSI TaK:
5=100-5t, t20 (1)
(ijepaBeHcTBo 0t20 nOKa3bIBaeT, qTO Bce nyTerneCTBHe 3aHHMaeT
20 'l). <I>opMYJIa (1) n03BOJIHeT HaRTH 5 no JII060MY t, npHHa,lJ,JIe>Kalll,eMY
OTpe3KY [0; 201.
l{JISI TOro tJTOObI pernHTb 06paTHYIO 3Maqy - HaHTH, CKOJIbKO Bpe-
MeHH npornJIO C HatJaJIa nYTernecTBHSI, ecJIH OCTaJIOCb npOHTH 5 KM, H8,/.lO
H3 paBeHCTBa (1) HaHTH 3HaqeHHe t:
100 - s
t=-r;-, 100. (2)
fOBOPSIT, qTO $YHKUHSI (2) o6paTHa .pYHKUHH (1).
lJ.8,/.lHM 06lI1.ee onpe,lJ,eJIeHHe 06pamnou CPYHKI{UU. nYCTb 4IYHKII.HSI
y=f(x) 3Mae-r HH'beKTHBHoe OT06pa:>KeHHe qHCJIOBOrO MHO)!{ecTBa X
B MHO>KecTBO eRCTBHTeJIbHbIX qHCeJI R (T. e. pa3JIHqHbIM 3HaqeHHHM x
147

COOTBe'fCTBYIOT pa3JIIItJifble 3HatJe-
HHH y). 0603HatJHM tJepe3 Y MHO-
IKeeTBO SHatJeHHH YHKU:HH y=
= f (X), X EX. T or,lJ,a ,lJ,JIH JlJo60ro
YoEY HaH,lJ,eTCH e,lJ,HHCTBeHHoe
3HatJeHHe Xo EX, TaKoe, tJTO
yo=f (xo). 3THM onpe,lJ,eJISJeTCH
OT06pa)!{eHHe Y Ha X, T. e. YHK-
a x UHH x=cp (y), y E Y. TaKYIO YHK-
UHIO Ha3bIBalOT o6pamlioiiiJ;lJI,
cpynK/iUU y = f (X), X EX. lh06bI
HaHTH BbIpa)!{eHl1e ,lJ,JI5I o6paTHoH
YHKUHH, Ha,lJ,O peIIIHTb ypaBHe-
HHe y=f (X) OTHOCHTeJIbHO X, 6epH
PHC. 94 JIHIIIb Te perneHHH, KOTopbIe npH-
HMJIe)!{aT MHO)!{eCTBY X. ECJIH )!{e
OT06pa)!{eHHe y=f (X), X EX, He HBJIHeTCH HH'beKTHBHbIM, TO 06paT-
HOH YHKUHH He CYlII.eeTBYeT, TaK KaK O,lJ,HOMY H TOMY :>Ke Yo E Y MorYT
COOTBeTCTBOBaTb pa3JIHtJHbIe 3HatJeHHH x.
n p H M e p. cI>YHKUHH y=x l , X E R, He HMeeT 06paTHOH YHKUHH,
TaK KaK, HanpHMep, 3HatJeHHHM x=4 H x=-4 COOTBeTCTBYeT OAHO H
TO)!{e 3HatJeHHe $YHKUHH: 4 1 =(-4)1= 16. n03TOMY no SHatJeHHIO y= 16
HeJIbSH 0,lJ,H03HatJHO Y3HaTb 3HatJeHHe x: MO)!{eT 6blTb, x=4, a MO)!{eT
6bITb, x=-4.
Ho eCJIH MbI B03bMeM YHKUHIO y=x 2 , X E R +, TO Tenepb Y:>Ke pa3-
JIH<::""IM 3HatJeHHHM y COOTBeTCTBYIOT pa3JIHtJHbIe 3HatJeHHH x, H nOTOMY
onpeAeJIeHa 06paTHaH YHKUHH. Ee 0603HatJaIOT x=VY' H Ha3bIBaIOT X
apucp.M.emUIleCKUM KBaopamnblM KOpneM H3 y. TaKHM 06pa30M, 8anHCb
x=Vy" 03HatJaeT, tJTO y=x 2 , npHtJeM x;;;::O, y;;;::O.
YopamBeBBJI
1. HaiillHTe <j>YHKIJ.HH, 06paTHble CJIellYI01II.HM <j>YHKIJ.HSlM:
a) y=2x+6, xER; 6) y-=-2x-8, xER+;
B) y=-x, -4<:x<:6; r) y= V 25---x2, O<.x<:5.
Ha'iepTHTe rpa<j>HKH HCXOllHblX H o6paTHbIX <j>YHKII.Hii.
2. OepHMeTp npSlMoyrOJIbHHKa paseH 20 M, a ero nJIOlIlallb pasHa S M 2 . 4eMY
paBHa AJlHH8 x OCHOBaHHSI npSlMoyrOJIbHHK8? OllH03H3'1eH JIH nOJIyqeHHblii OTBeT?
HaHAHTe Bbipa/KeHHe x 'Iepe3 S.
3. Ha pHcYHKe 94 H306p8/KeHbI rpa<j>HKH llBYX <j>YHKIJ.Hii. K3K8S1 H3 HHX HMeeT
06paTHYJO?
4. Bblpa3HTe AJlHHY x CTOpOHbI Ky6a Qepe3 nJI01Il.3llb S ero nOBepxHoCTH.
* 2. nOCTPOEHHE rPAClJHKOB ClJYHKD;HH
). nOCTpoeHHe rpa«l>HKa cno TO'IKaM». nOCKOJIbKY rpaHK CPYHK-
D.HH y=f (x), X EX, COCTOHT 06bJ1IHO H3 6ecKoHetJHoro MHO:>KecTBa
rotJeK, MbI He MO)!{eM H306pa3HTb ero «TOtJHO», a 1\!O)!{eM CAeJIaTb JIHllIb
9CKH3 TaKoro rpaHKa. 4aCTO CTpOHT rpaHK, OTbICKHBaH HecKOJIbKO
148

2. nOCTpoeHHe rpaHKoB ny-
TeM napaJIJIeJIbHOrO nepeHoca CHC-
TeMbi KOOpJl.HHaT. 4acTO MO:>KHO nocTpoHTb rpa$HK $YHKUHH, H3MeHHB
cHcTeMY KoopAHHaT H nocTpoHB B HOBOll CHCTeMe KOOp,lJ,HHaT y)!{e H3-
BecTHbIll rpaHK. HaqepTHM, HanpHMep, rpa$HK YHKUHH y=x 2 +3.
l{JIH STOro nepenHllleM paBeHCTBO y=x 2 +3 B BH,lJ,e y-3=x 2 H nOJIO)!{HM
X=x, Y=y-3. KaK 6bIJIO nOKa3aHO B n. 22 rJI. V, TaKoe npeo6pa30-
BaHHe KOOp,lJ,HHaT COOTBeTCTBYeT napaJIJIeJIbHOMY nepeHOCY ,lJ,eKapTOBon
CHCTeMbI KOOp,lJ,HHaT, npH KOTOpOM HatlaJIO KOOp,lJ,HHaT nepeXO,lJ,HT B
TOtlKY 0 1 (0; 3). B HOBO" CHCTeMe KOOp,lJ,HHaT nOJIYtlaeM $YHKUHIO Y=XI,
rpa$HKOM KOTOPO" CJIY)!{HT napa60JIa. TaKHM 06pa30M, tlT06hI nOCTpo-
HTb rpa$HK YHKUHH y=x 2 +3, HMO nepeHeeTH HatlaJIO KOOp,lJ,HHaT B
TOtlKY 0 1 (0; 3) H B HOBO" CHCTeMe KOOp,lJ,HHaT nOCTpOHTb napa60J1Y
Y =X2 (pHC. 95). 11Hatle MO)!{HO CKa3aTb, tlTO Ha,lJ,O nocTpoHTb napa60JIY
y=x 2 B HCXO,lJ,HOll CHCTeMe KOOp,lJ,HHaT H nepeHeeTH nOJIYtleHHbI" rpa$HK
BMeeTe C cHcTeMoll Koop,lJ,HHaT.
nOCTpOHM Tenepb rpa$HK
$YHKUHH y=(X-4)2. ECJIH IJOJIO-
:>KHTb X=x-4, Y=y, T. e. ne-
pCHeeTH HatlaJIO KOOp,lJ,HHaT B TOtl-
KY 0 1 (4; 0), TO nOJIYtlHM ypaBHe-
HHe Y=X2. 3HatlHT, HMO nepe-
HeeTH HaqaJIO KOOp,lJ,HHaT B TOtlKY
0 1 (4; 0) H nOCTpOHTb B 1I0BO" CHC-
TeMe KOOp,lJ,HHaT napa60JIY Y=X2,
HJIH, tlTO TO:>Ke caMoe, nOCTpOHTb
napa60J1Y y=x 2 H nepeHeCTH ee Ha
4 e.lI.HHHUbI BnpaBo (pHC. 96).
HaqepTIIM rpaHK YHKUHH
y= (X-2)2+3. 113 CKa3aI-IHOrO
HCHO, tlTO OH nOJIytlaeTcH
TOtleK, npHHa,lJ,.'Ie:>KalIl.IfX 3TOMY
rpa$HKY, H COe,lJ,HHHH IIX OT PYKH
,lJ,OCTaTOtlHO rJIaAKOH JIHHHe". npH
9TOM npe.ll.BapHTeJIbHO COCTaB.rIHIOT
Ta6JIHlI.Y 3HatleHHH: 3TOH $YHKUHH.
YupalKUeHBJI
OocTpofl're no TO'IK8M rpa<j>HKH CJIe-
AYlO1II.HX <j>YHKII.HH:
a) Y= (X-I)8;
6) y=x8-1;
B) y=x 3 -3x;
1
r) Y= x2+ 4 ;
x2-4
ll) Y = x2 + 4 ;
1
e) Y= x+8 '
-J -2 -1
1
2
3
x
-3 -Z -1 a 1
PHC. 95
2
;5
x
!J
y
y=(X-42
Y=X 2
6
5
4
6
5
J
2
1
-1 a
-J -2
1 2
2 X
6 x
PHC. 96
149

y


X
1 -2
-1 a 2/ , I 5

1 .3 * X
PHC. 97
Y Y
2


-6 - 2 -1 0 f x
-1
it -2
3 -3
2 -4


Puc. 98


-*-J-2-1 0 2 J If 5


PHC. 99


150


x


CJIe.lI.YlOlII.HM 06pa30M: nepeHOCHM
HaqaJIO KOop.IIJfHaT B TOqKY 0 1 (2; 3)
H B CHCTeMe KooPAHHaT X0 1 Y
CTpOHM napa60JIY Y=Xz (pHC. 97).
PaccMOTpHM Tenepb napaJI-
JIeJIbHbI" nepeHoc rpa(lmKOB B 06-
lII.eM BHAe. nYCTb 3MaH rpa$HK
$YHKUHH y=f (x) H Tpe6YeTc5I
nocrpoHTb rpa$HK $YHKUHH
y=f(x-a)+b. (1)
BMecTO y=f (x-a)+b MO:>KHO
HanHcaTb y-b=f (x-a). ECJIH no-
JIO:>KHTb { X =x-a,
Y=y-b, (2)
TO 3T0 ypaBHeHHe npHMeT BHA
Y=f(X). Ho paBeHcTBa (2) onH-
ChIBalOT napaJIJIeJIbHhI" nepeHoc
CHCTeMhI KOOp,lJ,HHaT, npH KOTO.
pOM TOqKa 0 (0; 0) nepeXOAHT
B TOtlKY A (a; b), a HanpaBJIe-
HH5I oce" KoopAHHaT OCTaIOT-
C5I HeH3MeHHbIMH. TaKHM 06pa30M,
HMO CAeJIaTb STOT napaJIJIeJIbHbIH
nepeHOC, nOCJIe qero B HOBO" CHC-
TeMe KoopAHHaT Hs06paSHTb rpa-
$HK $YHKUHII Y =f (X). I1Haqe
3T0 MOMHO C$opMYJIHpOBaTb TaK:
HMO B351Tb rpa
HK
YHKUHH
y=f(x) H HaATH ero 06pa3 npH na-
paJIJIeJIbHOM nepeHoce, nepeBO,lJ,51-
lII.eM TOqKY 0 (0; 0) B ToqKY 0 1 (a; b).
06palIl.aeM BHHMaHHe ql:\TaTe-
JIH, tlTO B ypaBHeHHH (1) nepeA a
CTOHT SHaK «MHHYC».
n p H M e p. I1s06pa311M rpa-
$HK $YHKUHH y= (X+4)3_6. 3ra
$YHKII.HH HMeeTBHAy=f (x-a)+b,
rAe f(x)=x 3 , a=-4 H b=-6.
3HatlHT, HMO nepeHeCTH HatlaJIO
KOOPAHHaT B TOtlKY 0 1 (-4; -6)
H B HOBOH CHCTeMe KoopAHHaT no-
crpOllTb rpa$HK Y=Xs (PHC. 98).


x


¥upaIKUeUBJI
Ha pHcYHKe 99 H306pa/KeH rpa<j>HK
<j>YHKlUfH y=f (x). DocrpoHre rpa<j>HKH
CJIellYJ01II.HX <j>YHK
HH:




3. rpaAj)HK KBaAPaTH'UlOH $YHK-
IJ,HH. PaccMoTpHM CHaIJaJIa Hec-
KOJIbKO npHMepoB. Ka}K)l.aSl OPJI.H-
HaTa rpa<pHKa <PYHKI..I.HH y=2x 2
BJI.Boe 60JIbllle cOOTBeTcTBYIOllI,eH
0pJI.HHarn rpa<pHKa<pYHKI..I.HHy=xl.
D03TOMY, qT06b1 nocTpoHTb rpa-
<pHK <PYHKI..I.HH y=2x 2 , HaJI.O Ka:>K-
JI.YIO opJI.HHaTY napa60JIbI y=x l
YBeJIHqHTb B JI.Ba pa3a (pHC. 100).
TOqHO TaK :>Ke, qT06b1 nOCTpoHTb
1
rpa<p HK <PYHKI..I.HH y="3X1, HMO
YMeHblllHTb Ka}!{JI.YIO OpJI.HHaTY na-
pa60JIbI y=X 2 B Tpn pa3a.
rpa<pHK <PYHKI..I.HH y=-X 2 no-
JIyqaerCSI H3 napa60JIbI y=X I ny-
TeM OCeBOH CHMMeTpHH OTHOCH-
TeJIbHO OCH a6cI..I.Hcc. A qT06bI
nOCTpoHTb rpa<pHK <PYHKI..I.HH
y=-2x 2 , HaJI.O OTpa3HTb rpa<pHK
<PYHKI..I.HH y=2x 2 OTHOCHTeJIbHO
OCH a6cI..I.HCC (pHC. 100).
DOKa}!{eM Tenepb, KaK CTpo-
HTCSI rpa<pHK 06llI,eH KBaJI.paTUq-
HOH <PYHKI..I.HH y=ax 2 +bx+c. CHa-
IJaJIa BbIHeceM 3a cK06KH K03<Jxlm-
IJ,HeHT a, nOTOM JI.OnOJIHHM Bblpa-
:>KeHHe B cK06Kax )1.0 nOJIHOrO KBaJI.paTa:
( be ) l ( b b 2 ) C b 2 ]
y=a x 2 +-x+- =a xZ+2-x+- +--- =
a a 2a 4a 2 a 4a 2
( b ) 2 4ac-b2
=a x+2i2 +
.
MbI nOJIYIJHJIH ypaBHeHHe BHJI.a y=a (X-a)I+
, rJI.e JI.JISI KpaTKOCTH na-
b 4ac-b 2
JIO:>KeHO a=- 2a H
="4a""' OrcIO)l.a cJIeJI.yer, IJTO rpa<pHK KBMpa-
THqHOH <PYHKI..I.JfH y=ax 2 +bx+c CTpOHTCSI CJIeJI.YIOllI,HM 06pa30M:
a) nepeHocSiT HaqaJIO KOOpJI.HHaT B TOIJKY 0 1 (a;
), rJI.e
_ b 4ac-b 2 .
a - _ A_
- 2a' t' - 4a '
6) B HOBOH cnCTeMe KOOpJI.HHaT CTpOSIT napa60JIY Y =Xz;
B) yMHO:>KaIOT Bce (HOBbIe) opJI.HHaTbI Ha lal H, ecnH a<O, OTpa}!{alOT
nOJlYIJeHHblH rpa<p HK OTHOCHTeJIbHO HOBO" OCH a6cI..I.HCC.


;1) y=f(x-2); r) y=f(x+3)-I;
6) y=f (x)-2;
a) y=f(x-I)+3; .l() y=f(x+I)+3.


If


8
7


!J=2X2


6


5


y=x'l


y= j X 2


x


y=-x 2


-6


-7
-8


y=-2x 2


Puc. 100


151




t'i
X
N
1\

y
10
9
8
Y=X 2
2 X
-5
-6
-7
-8
-9
2
y= - 2X
MO}KHO CJI.eJIaTb nocTpoeHHe
H B TaKoM nOpHJI.Ke:
a) nOCTpOHTb napa60JlY
y=x 2 ;
6) YMHO:>KHTb OpJI.HHaTbI
rpa<pHKa Ha lal H, eCJIH a<O,
oTpa3HTb rpa<pHK OTHOCHTeJIb-
HO OCH a6ClI,HCC;
B) nepeHeCTH nOJIyqeHHbIH
rpa<pHK TaK, qT06bI TOqKa 0 (0;0)
nepeWJIa B TOqKY 0 1 (a; P), rJI.e
b 4ac-b 2
a=- 2a ' P=----:ta.
D P H M e p. DOCTpOHM rpa-
<pHK <PYHKI..I,HH
y=-2x 2 +12x-16.
3aMeTHM, qTO
y=-2 (x 2 -6x+8)=
X =-2[(x 2 -6x+9)-9+8]=
=-2[(x-3)2-1J=
=-2 (X-3)2+2.
Tenepb flCHO, qTO HMO
nepeHeCTH HaqaJIO KOOpJI.HHaT
B TOqKY Oi (3; 2), B HOBOft CHC-
TeMe KOOpJI.HHaT nocTpoHTb na-
pa60JIY Y=X 2 , YMHO:>KHTb Bce
0pJI.HHaTbI 3TOH napa60JIbI Ha 2 H
OTpa3HTb nOJIyqeHHYIO KPHBYIO
OTHOCHTeJIbHO OCH OiX. MO:>KHO
TaK:>Ke nOCTpOHTb napa60JIY
y=x 2 , YMHO:>KHTb ee 0pJI.HHaTbI
Ha 2, oTpa3HTb nOJIyqeHHYIO
KPHBYIO OTHOCHTeJIbHO OCH a6c-
lI,HCC H CJI.BHHYTb eeTaK, 'IToOblToqKa 0(0; 0) nepeWJIa B TOqKY Od3; 2).
06a 9TH cnocoOa nOCTpoeHHfI H300pa}!{eHbI Ha pHCYHKe 101.
-4
-5
-6
-7
-8
N
+
N
.......
!<)
I
)(.
---
N
,
"
::n
-10
PHC. 101
YDpaacBeBBJI
nOCTpOHTC rpa<jJHKH CJJe.ll.YIOIII.HX <PYHKl\HH:
it) y-=x2-4x+6; 6) y=x2-3x+4;
8) y=2x2+8x+3; r) y=-2x2-4x+8.
4. fpa$HK AP06Ho-JIHHeHHoH $YHKI.{HH. cI>YHKI..I,HfI y= :; nOJIY-
'1aerCH npH JI.eJIeHHH JI.pyr Ha JI.pyra )l.BYX JIHHeHHblX $YHKlI,HH. Docno-
MY ee Ha3b1BaIOT OP06HO-IlUHeUHOii t/JyHKl{ueU. C qaCTHblMH CJIYIJaflMH
T3KHX <PYHKI..I,HH MbI Y:>Ke BCTpeIJaJIHCb. ECJIH c=O, a #0, TO JI.PoOHO-
U '" a + b -
JlHHeHHafi 't'YHKI..I,HfI npHHHMaer BHJI.Y=([X d' T. e. npeBpau.r.aeTCfl B JIH-
.52

HeHHYIO <PYHKlI.HIO (CM. n. 2  1). B cJIytJae KOrJI.a a=d=O, C:#=O,
JI.P06HO-J1HHeHHafi IfIYHKlI.HH npHHHMaer BHJI. y=!!..., rJI.e k=!?., T. e.
x C
CBOJI.HTCH K 06paTHoH npOnOpll.HOHaJ1bHOCTH (n. 3  1). ECJ1H C:#=O,
a b B
#0, HO ad-bc=O, TO c= d' 3TOM CJIyqae a=CA, b=dA, rJI.e
a b CAx+dA 
A= c=d' H nOTOMY y - cx+d = fl.. To me Cal\IOe 6YJI.eT, eCJIM
C:¥=O, d=O, ad-bc=O. Bo Bcex 3THX CJIytJaHX MbI 3HaeM, KaKOM
BHJI. HMeer rpalflHK IfIYHKlI.HH. DOKa:>KeM, qTO npH c*O, ad-bc=FO
rpalflHK JI.P06HO-JIHHeMHOH IfIYHKlI.HH nOJ1YtJaeTCH H3 rpalflHKa 06paTHOM
npOnOpll.HOHaJ1bHOCTH C nOMOblO napaJIJ1eJ1bHOrO nepeHoca. CHaqaJIa
paCCMOTpHM CJIeJI.yIOHH npHMep.
4x+6
n p H M e p 1. DOCTpOHM rpalflHK IfIYHKlI. HH Y= 2x+5 ' BblHeceM B
qHCJIHTeJ1e 3a CKooKY qHCJ10 4, a B 3HaMeHaTeJ1e -tJHCJIO 2, nOCJ1e tJero B
5
qHCJIHTeJIe npH6aBHM H BbIqTeM B cK06Kax "2:
4 ( x+ ) 2 [ ( x+ ) -+ J
_ 2 _ 2 2 2 -2 + 
y- ( 5 ) 5 - 5'
2 x+- x+- x+-
2 2 2
-2 5 X
TaKHM 06pa30M, y-2=--y;, HJ1H ecJIH nOJIO:>KHTb X+"2= , y-2=Y,
x+"2
-2 4x +6
TO Y =x' D03TOMY rpalflHK IfIYHKlI. HH y= 2x + 5 nOJIYtJaeTcfI H3 rpalflHKa
-2 .
IfIYHKlI.HH Y=- napaJIJ1eJ1bHblM Y
x
nepeHOCOM, KOTOpblH 3aJI.aercfI
5
$>pMYJIaMH X =X+"2' Y =y-2.
HHblMH CJIOBaMH, rpalflHK nOJIYIJeH
npH nepeHoce, KOTOpbIM TOqKY
0(0; 0) nepeBOJI.HT B TOtJKY
5
A (-"2; 2). 31'0T rpalflHK H306pa-
meH Ha pHcYHKe 102.
TaK :>Ke pelllaercH 3aJI.atJa B
ooeM CJIyqae. ECJIH a=O, C:¥=O,
TO IfIYHKlI.HH npHHHMaeT BHJI.
b
y= cx+d ' B 3TOM CJIYtJae JI.OCTa.
TOtJHO BblHeCTH 3a CKooKY B 3Ha-
MeHaTeJIe c:
b
y= d )
(; (x+c .
y
7
6
5
2 X
y=-x .3 4
X
1 2 ..-
.,"

,
I
I 2
I Y=--
I )(
153

d b
ECJIH nO.TJOmHTb X+C=X, y= Y H 0603HaIJHTb c IJepes k, nOJIYIJHM
k
<PYHK1.I. H IO y= X ' rpa<p HK KOTOpOH HaM H3BecreH, OrcIO)l.a CJIeJI.yer, IJTO
b
JI.JISI nOCTpoeHHSI rpa(lmKa <PYHK1.I.HH y= cx+d HMO CJI.eJIaTb napaJIJIeJIb-
HbIH nepeHOC Ha _.! e.II.HHH1.I. BJI.OJIb OCH aOC1.I.HCC, 8 nOTOM nOCTpOHTb;
c
. k b 0 0
rp&pHK ct>YHK1{HH y= X ' rJI.e k=c' DYCTb Tenepb a=/= H C =1= ,Tor.n,a
npe06pa3yeM <PYHKlI.HIO CJIe.II.YIOHM 06pa30M:
a ( x+ : ) a [ ( x+ : ) - : + : ]
y= ax+b _
cx+d- c ( x+ : ) c ( x+ : )
b- ad
=  + ( Cd =  + bC-ad d ) .
c x+c) c C S ( x+c
D d a bc-ad k T
OJIO:>KHM X=x+ -, Y=y-- H 0603HaIJHM  IJepes . OrJI.a
c c c
nOJIYIJHM <PYHK1.I.HIO Y= ; ,3HaIJHT, JI.JIfi nOCTpoeHHSI rpa<pHKa 3MaHHOH
<PYHK1.I.IlIl HaJI.O nepeHeCTH HaIJaJIO KOOpJI.IlHaT B TOqKY 0 1 ( -  ;  )
k
H nOCTpOHTb rpact>HK 06paTHoH npOnOp1.I.HO H aJIbHOCTH Y = X ' r)l.e
k= bc- ad
c 2 .
YnpamueUBR
nOCTpoiiTe rpa<jmKIi ,l\P06HO-JlIiHeiiHbIX <jJYHKl\liii:
1 . 4
a) Y= 2x-4 ; 6) Y -2x+6 ; B) Y= 2x- 1O;
x-I 3x-4 x+4
r) Y=-; .n) y=-; e) Y= 2x+3 '
x+ I 2x-2
 3. nOCJIEJJ;OBATEJILHOCTIl
1. lJHCJIOBbie nocneJl,OB8TeJIbHOCTH. BYJI.eM e:>KeJI.HeBHO 3anHCbIBaTb
TeMnepaTYPY B03.llYxa B 12 It JI.HSI, DOJIYIJaTCSI HeKOTopbIe IJHCJIa, MY-
He B onpeJI.eJIeHHOM. nOpflJI.Ke - CHaIJaJIa CerOJI.HfllllHflSI TeMnepaTypa,
nOTOM 3aBTpalllHfIfi H T. JI..
ECJIH B3S1Tb KYCOK paJI.HOaKTHBHoro BeecTBa, Macca KOToporo 3a
CYTKH BJI.BOe YMeHblllaercSI, H e:>KeJI.HeBHO 3anHCbIBaTb 3TY Maccy, TO
nOJIyqaTCfI qHCJIa, HJI.yrn.He JI.Pyr 3a JI.PyroM no onpe.II.eJIeHHOMY 3aKOHY,
HanpIlMep:
128, 64, 32, 16, 8, 4, 2, 1, 1/2,... .
(1)
154

B o6OHx cJIYIJaflx MO:>KHO CKa3aTb, qro Ka:>KJI.OMY HaTypaJIbHOMY IJHC-
JIY n - HOMepy JI.HfI Ha6JIIOJI.eHHH, nOCTaBJIeHO B COOTBerCTBHe onpe,n;e-
JIeHHOe qHCJIO (B nepBOM CJIyqae - TeMnepaTypa B03JI.yxa, BO BTOpOM -
MaCCa paJI.HOaKTHBHoro Bern.ecTBa; pa3YMeerCH, 3JI.ecb MhI OTBJIeKaeMQH
OT TOro. qTO JlI06afl CepHH Ha6JIIOJI.eHHH KOHeqHa, H CqHTaeM JI.aHHbIe ce-
pHH HaMlO)J.eHHH 6ecKOHeIJHbIMH). Ho TaKoe COOTBeTCTBHe ecTb He qTO
HHoeKaK CPYHKlI,HSI, 3aJI.aHHaSi Ha MHO:>KecTBe N HaTypaJIbHbIX tIHCe.n:
H npHHHMaIOw,afl 3HaIJeHUfi B MHO:>KecTBe R JI.eHCTBHTeJIbHbIX qHCM.
fIo)l.06Hble <PYHKI.I,HH Ha3bIBaIOT qHCJIOBbIMH nOCJIeJI.OBaTeJIbHOCTR.MH.
HTaK, MbI BBOJI.HM CJIe.JI.YIOrn.ee onpeJI.eJIeHHe:
lJUCA060U fWCJle006ame/lbHOCtnblO Ha3bIBalOT <PYHKlI,HIO y=f (n), sa-
)l.aHHYlO Ha MHO:>KecTBe N HaTypaJIbHbIX qHCeJI H npHHHMaIOYIO qHCJlO-
Bhle 3HaqeHHH.
06b1qHO qHCJIOBYIO nOCJIeJI.OBaTeJIbHOCTb 0603HaqalOT He f (n), a
a lt . . ., a,u . . ., HJIH, KOpOqe, (an). DpHMepaMH qHCJIOBbIX nOCJIeJI.OBa-
TeJIbHOCTeH Moryr CJIY:>KHTb:
a) nOCJIeJI.OBaTeJIbHOCTb BCex qerHblX HaTypaJIbRblX qHCeJI: 2, 4, 6,
8, ...;
6) nOCJIeJI.OBaTeJIbHOCTb Bcex npocTbIX qHCeJI: 2, 3, 5,7, 11, 13, ...;
B) nOCJleJI.OBaTeJIbHOCTb creneHeH qHCJIa 2: 2, 4, 8, 16, 32, ... .
.lI.JISI nOCJIeJI.OBaTeJIbHOCTeH a) H B) TaK :>Ke, KaK H Mfl nOCJIe.JI.OBaTeJIb-
HOCTH (1), cyrn.eCTByer BbIproKeHHe C nepeMeHHbIM n, KOTOpoe n03BOJIfier
no 33)1.aHHoMY 3HaIJeHHIO n HaHTH 3HaqeHHe an. HarrpHMep, JI.JIH nOCJIe-
,AOBaTeJIbHOCTH a) 3TO BbIpIDKeHHe HMeer BHJI. an =2n: ecJIH nOJIO-
:>KHTb n= 1, TO nOJIyqHM tIHCJIOBOe Bblpa:>KeHHe 2.1, 3HaqeHHe KOToporo
paBHo )l.BYM, T. e. HaqaJIbHoMY qJIeHY nOCJIeJI.OBaTeJIbHOCTH a); eCJIH no-
JIO:>KHTb n=4, TO nOJIyqHM 2.4, T. e. 8, a 3TO H eCTb qerBepTbIH IJJIeH no-
CJIe.JI.OBaTeJIbHOCTH a) H T. JI.. JI.JIH nOCJIeJI.OBaTeJIbHOCTH B) BblpmKeHHe
an qepe3 n HMeer BM an =2 n . OrMerHM, qTO Mfl nOCJIe.JI.OBaTeJIbHOCTH
6) TaKoro BbIpa:>KeHHSI He cyrn.ecTByer - Her <POPMYJIbI, BbIpa}KalOllI,eii
n-e npOCToe qHCJIO IJepe3 n.
ECJIH 3aJI.aHO BbIpIDKeHHe n-ro qJIeHa nOCJIe.JI.OBaTeJIbHOCTH (HJIH, KaK
rOBOpSlT, i!O 4.AeHa), TO JIerKO HaHTH 3HaqeHHe 3Toro 1JJIeHa npH JIIO-
60M HaTypaJIbHOM 3HaqeHHH n. HanpHMep, nepBbIMH nfiTblO qJIeHaMH
nOCJIe.JI.OBaTeJIbHOCTH C 06rn.HM qneHOM n 2 flBJIR.IOTCH qHCJIa }2, 2', 3 2 ,
4 2 ,5 2 , T. e. IJHCJIa 1,4,9, 16, 25. OJI.HaKO no 3aJI.aHHbIM nepBbIM qJIeHaM
nOCJIeJI.OBaTeJIbHOCTH HeB03MO}KHO OJI.H03HaqHO HaHTH Bblpa}KeHHe 06-
ero IJJIeHa 3TOH nOCJIeJI.OBareJIbHOCTH. HanpHMep, Te :>Ke qHCJIa 1,4, 9,
16, 25 nOJIYIJaTCfI, ecJIH nOJIO:>KHTb
an=n 2 + (n-l) (n-2)(n-3) (n-4) (n-5).
YupaIKBeBBB
J. 3arrHWHTe nepBhJe mlTb qJleHOB nOCJle.l\OBaTeJlbHOCTeii:
nl- 1 (_I)n
a) a,,=n s ; 6) a n = n2+ 1 ; B) an=;
r) a n = 1+(-1)"; .II) an=n [1 +(-I)n].
155

2. HaH,l\HTC O,l\HY H3 <jJopMYJI o6II1.ero '1J1CHa nOCJJC,l\OBaTeJJbHocTH, 3Hall HeCKOJlbKO
nepBblx '1J1CHOB:
a) 1,3,5,7,9, ...; 6) 3, 7, 11, 15, 19, ...; B) 3, 9, 27, 81, 243, ...,
1 1 1 1 1
r) 2,5,10.17,26, ...;.n) 2' 4' 8' 16 ' 32 '....
2. PeKyppeHTHbie nOCJIe)).OBaTeJIbHOCTH. He Bcer.zr.a nOCJIeJI.OBaTeJlb-
HOCTH 3aJI.aIOT, YKa3aB BbIpa:>KeHHe ee 06llI,ero IJJleHa. HHor)).a BMecTO
3TOro YKa3bIBaercH npaBHJIO, n03BOJIHIOllI,ee no nepBbIM n IJJleHaM nOCJIe-
JI.OBaTeJIbHOCTH HaXO)).HTb CJIe,n;YIOIIJ.HH 3a HHMH (n+ 1)-H IJJIeH 3TOH no-
CJleJI.OBaTeJIbHOCTH. .lI.JIH TaKHX nOCJIe,n;oBaTeJIbHOCTeH, KpOMe cpoPMYJlhI.
BbIpa:>KalOIIJ.eH tln+1 IJepe3 alt ..., an, HeOOXOJI.HMO YKa3aTb O)).HH HJIH He-
CKOJIbKO nepBblx IJJIeHOB. DpH BbIIJHCJIeHHH IJJIeHOB TaKHX nOCJIe)).OBa-
TeJIbHOCTeH MbI Ka:>K)l.bIH pa3 KaK 6bI B03Bp'allI,aeMCH Ha3a)).. n03TOMY MX
Ha3bIBalOT e038pamHblMU HJIM peKyppeHmHblMU nocJle008ameJlbHOCmflMu
(OT JIaTHHCKoro reeurrere - «B03BpallI,aTbcH»).
DpocTeHIIIHMH npHMepaMH peKyppeHTHbIx nOCJIe,n;oBaTeJIbHOCTeH
HBJIHIOTCH apmpMerHIJeeKaH H reOMerpHIJeeKaH nporpeeCHM. ApurPMe-
mUIleCKOU npo2peccuet1 Ha3bIBalOT IJHCJIOBYIO nOCJIeJI.OBaTeJIbHOCTb,
Ka:>K)l.bIH IJJIeH KOTOpOH (HaIJHHaH co BToporo) nOJIYIJaerCH H3 npeJI.IIIecr-
BYIOllI,erO eMY IJJIeHa npH6aBJIeHHeM OJI.Horo H TOro :>Ke IJHCJIa. 31'0 IJHCJIO
Ha3bIBalOT pasHocmblO npoepeccuu H 0603HaIJalOT d. TaKHM 06pa30M,
ecJIH ai, a 2 , ..., tln, ...- apmpMerHqeeKaH nporpeccHH, TO )).JIH JII060ro
11 BblIIOJIHHerCH paBeHCTBO tln+l =tln +d, HJIH HHaqe tln+1-an =d.
HanpHMep, IJerHbIe IJHCJIa 06pa3YIOT apmlJMerHIJeeKYIO nporpeeCHIO c
pa3HOCTblO 2.
DpH d>O Ka:>K)l.bIH CJIeJI.YIOIII.HH IJJIeH nporpeecHH 60JIbIIIe npe,n;bI,ll.Y-
llI,ero, 1. e. OHa MOHOTOHHO B03paCTaer; npH d<O nporpeccHH MOHOTOH-
HO y6bIBaeT, T. e. KIDKJI.bIH ee CJIeJI.YIOllI,HH IJJIeH MeHbIIIe npe.D.bUI.YllI,ero.
QT06bI 33)1.aTb apmpMerHIJeeKYIO nporpeeCHIO, H3)I.O, KpOMe ee pa3-
HOCTH, 3aJI.aTb nepBbIH IJJIeH 3TOH nporpeeCHH, T. e. al' Tor)).a BTOpOH
IJJIeH HaXOJI.IlTCH no cpopMYJIe a 2 =al+d, TperHH - no cpopMYJIe a 3 =
a 2 +d H T. JI.. OJI.HaKO TaKOH nYTb BbIIJHCJIeHHH He BnOJIHe y)).o6eH, TaK
KaK Tpe6yer 60JIbIIIOrO IJHCJIa CJIO:>KeHHH. D03TOMY BbIBe,n;eM cpoPMYJIY,
HenOCpeJI.CTBeHHO BbIpIDKalOllI,YIO n-H IJJIeH apml>MerHIJeeKOH nporpec-
CHU qepe3 n. JI.JIH 3Toro 3aMeTHM, IJTO
a 2 =al+d,
a 3 =a 2 +d=(al+d)+d=ai+2d,
a4=a3+d= (al+2d)+d=ai+3d.
H paBeHCTBa JI.eJIalOT eeTeeTBeHHblM npeJI.nOJIO:>KeHHe, IJTO )).JIH JII060ro
n BbInOJIHHerCH <popMYJIa
tln=a 1 + (n-l)d.
(1)
.!JOKaMeM ee no HHAYKl\HH. CnpaBe,l\JlHBOCTb paBeHcTBa (1) npH n= 1 O'lCBH.j\Ha,
TaK KaK npH 9TOM 3Ha'lCHHH 01+ (1-1)d=al' npCAnOJlOMUM TenCpb, qTO paBeHCTBO
(I) ,l\OKa3aHO AJlJI KaKOrO-TQ HaTypaJlbHOrO 3Ha'leHH!I k, T. e. ak=al+ (k-1)d. Tor.l(a
ak+1=ok+ d =[al+ (k-I)d!+d=a1+kd=Ul+[ (k+I)-I]d.
156

::no nOKa3bIBaCT, '1TO H 011+1 HaXO.llUTCSI no <jJopMYJle (1), eCJJu nOJlOJKHTb n=k+ 1.
TaKuM o6pa30M, <jJopMYJla (I) BepHa npH n= 1 U U3 ee UCTHHHOCTU npH n=k CJJell.yer,
'1TO OHa U<::THHHa H npu n=k+ 1. 3Ha'lHT, OHa BepHa rrpu BCCX HaTypaJIbHbIX sJta-
lJeHHSlX n.
PaCCMOTpHM Tenepb reOMerpHqecKYIO nporpeCCHIO. H3BeCTHa JIe-
re <;> TOM, qTO H306peraTeJIb IIIaXMaTHOH HrpbI nOTpe6oBaJI ce6e B
. ,,. ,b,. _
Harpa,lJY IIlUeHHII.Y H3 TaKoro pacqera: 3a nepBYIO KJIerKY IIIaXMaTHOH
JI.OCKH =- OJI.HO 3epHO, 3a BTOPYIO - JI.Ba, 3a TperblO - qeTbIpe, 3a qer-
BPTYIO - BoceMb 3epeH II T. JI.. HHbIMH CJIOBaMH, peKyppeHTHaH <PoP-
MYJIa 3JI.ecb TaKOBa:
lln+i=2an.
DOCJIeJI.OBaTeJIbHOCTb qHCeJI 1, 2, 4, 8, 16, ...- npHMep reoMerpH-
qecKOH nporpeccHH. B 06III.eM BH)l.e reoMerpHqecKYIO nporpeccHIO onpe-
lleJIHIOT TaK:
feoMempullecKOu npoepeccuetl Ha3bIBalOT HeHYJIeBYIO qHCJIOBYIO nO-
CJIeJI.OBaTeJIbHOCTb, K3:>K)I.bIH qJIeH KOTOpOH, HaqHHafi co BToporo, nOJIY-
qaercfI H3 npeJI.bIJI.YllI,ero nYTeM YMHO:>KeHHH Ha OJI.HO H TO :>Ke qHCJIO q.
Ha3bIBaeMoe 8H.aMeHameJleM npoepeccuu. I1HbIMH CJIOBaMH, peKyppeHT-
Hafl <popMYJIa .lI.JIfi reoMerpHqecKoH nporpeeCHH HMeer BIlJI.:
lln+i=llnq.
PaCCMOTpHM HeCKOJIbKO qaCTHbIX CJIYIJaeB:
a) q>l, npHIJeM ai>O. B 3TOM CJIYIJae Ka:>KJI.bIH CJIeJI.YIOIII.HH IJJIeH
reOMerpHqecKoH nporpeccHH 60JIbIIIe npeJI.bIJI.YllI,ero. DpHMepOM TaKoif
nporpeeCHH MO:>Ker CJIY:>KHTb nOCJIeJI.OBaTeJIbHOCTb I, 2, 4, 8, 16, ... .
6) q> I, npHIJeM a1<0. B 310M cJIyqae Bce qJIeHbI nporpeccHH OTPH-
II.areJIbHbI, HO HX MOJI.YJIH 06pa3YIOT B03paCTalOIII.YIO nOCJIeJI.OBaTeJIb-
HOCTb. DpHMepoM TaKoH nporpeccHH MO:>KeT CJIY:>KHTb nOCJIeJI.OBaTeJIb-
HOCTb -3, -6, -12, -24, ... .
B) O-::::q<I, npHIJeM al>O. B 310M CJIYIJae qJIeHbI nOCJIeJI.OBaTeJIb-
HOCTH nOJIO:>KHTeJIbHbI, HO 3HaqeHHH ee IJJIeHOB npH B03paCTaHHH YMeHb-
lIIalOTCfI H HeorpaHHqeHHO npH6JIH:>KaIOTCfI K HYJIIO. DpHMepOM TaKOH
nporpeccHH MO:>Ker CJIY:>KHTb nOCJIeJI.OBaTeJIbHOCTb 128, 64, 32, 16, 8,
4, 2, I, 1/2, 1/4, ... .
r) q= I. B 310M CJIYIJae Bce qJIeHbI nporpeCCHH OJI.HHaKOBbI, HanpH-
Mep: 6, 6, 6, 6, ... .
)I.) q =-1. B 310M CJIyIJae IJJIeHbI nporpeccHH Ha Ka:>KJI.OM lIIary JIHIIIb
MeHHIOT 3HaK, HanpHMep: 4, -4, 4, -4, 4, -4, ..: .
e) q<-l. B 310M CJIYIJae 3HaKH qJIeHOB nporpeccHH MeHflIOTCH, a
MO)l.YJIH IJJIeHOB B03paCTalOT, HanpHMep: 1, -2, 4, -8, 16, -32, ... .
}K) -1<q<O. B 3TOM CJIYIJae 3HaKH IJJIeHOB MeHflIOTCfI, a HX MO)l.Y-
IC_ I 1 1 1,
JIH npHUJIIDK8IOTCH K HYJIIO, HanpHMep: '-2' 4"' -"8' ... .
TOIJHO TaK }Ke, KaK .lI.JIfi apH<pMerHIJecKoH nporpeccHH, BbIBOJI.HT <PoP-
MYJIY 06ero IJJIeHa reoMerpHIJecKoH nporpeccHH:
lln =all(' -1 .
DpHBeJI.eM Tenepb npHMep nOCJIeJI.OBaTeJIbHOCTH, nOJIyqaeMOH no
60JIee CJIroKHOH peKyppeHTHOH <PopMYJIe, DYCTb K3:>K)I.blH qJIeH nocJIeJ,{o-
157

Ba'l'eJIbHOCTIl, HaqHHaSi C TpeTberO, paBeH CYMMe nBYX npe)l.blJI.)'IUIlX,
T. . e. nYCTb
an+ 2=an+1+an.
TorJI.a ecJlH B3HTb al=l, a:a=I, TO nOJIyqHM, qTO a s =a.+ai=I+1=2,
a,=a s +a2=2+1=3, a,=a,+a s =3+2=5 H T. JI.. TIOJIYIJaerCSl IIOCJIe-
JI.OBaTeJlbHOCTp 1lHce.n 1, 1, 2, 3, 5, 8, 13,21, ..., Ha3blBaeMblX 'l.UCAaMU
<PunOHattI/,U (4)1l60HaqqH - HTaJlbHHCKHH MaTeM.\iTHK HaqaJIa XIII B. 
paccMOTpeBWHA 33)1.aIJY, B KOTOpOH BCTperHJIIlCb 3TIl qHCJIa).
YnpamueUBlI
I. HanHWHTe nepBbIe weCTb 1JJIeHOB IJOCJJe.l(OBaTeJJbHOCTH. 3a.l(aHHOii peKyppeHT-
IIHM COOTHOlIIeHHeM an+l=na m eCJJH 1lJ.= I.
t. H8rrHWHTe rrepBble rrHTb 'lJJeHOB nOCJJeAOBSTeJlbHOCTH, S8JJ;8HHOii peKyppeHT-
IIblM COOTHOlIIeHHeM an+l=a-I, eCJJH al=2.
3. DOCJJe.l(OBaTeJJbHOCTb 3a.l(aHa peK)'ppeHTHbIM COOTHOmeHHeM an+2=2an+l-an'
HalIH1lIHTe ee rrepBbIC lIIeCTb 'IJleHOB, eCJJH al= I, Gjj=3.
4. .l!.oKa}KHTe, '1TO nOCJJe.l(OBaTeJJbHOCTb C 06ll1,HM '1J1eHOM a n =3 n y.nOBJlCTBOpSieT
peKyppeHTHoMY COOTHOlIIeHHIO an+:a=6an+1-9an'
3. 5ecKOuelfHO Oo.1IbWHe M OecKOHelfHo MaJlble noc.neJJ.OBaTeJIbHOCTH.
DOCJIeJI.oBaTeJIbHOCTb HaTypaJIbHbIX qHCeJI 1, 2, 3, ..., n, ... MOHOTOHHO
B03pacTaer, II ee qJIeHbI CTaHOBHTCH Bce 60JIbIIIe II 60JIbIIIe. ECJIH, Ha-
npIlMep, 33)1.aTb qIlCJIO 1000, TO HaIJHH8H C lOOl-ro HOMepa BCe 'lJIeHbI
nOCJIeJI.OBaTeJIbHOCTIl CTaHYT 60JIbIIIe 3TOrO qIlCJIa. A Ha'lIlHaH C
1000 001-ro HOMepa qJIeHbI 3TOH nOCJIeJI.OBaTeJIbHOCTH OKa:>KYTcH 60JIbIIIe
MIlJIJIHOHa. 11 Bo06llI,e, KaKoe 6b1 rpoMa)l.Hoe tJHCJIO M MbI HM B3HJIH.
oaHJI.erCH TaKoH HOMep, HalJHHafl C KOToporo Bce IlJIeHbI HameH fiOCJIeJI.O-
SaTeJIbHOCTIl OKaR<yrcH 60JIbIIIe M.
TeMIl :>Ke CBOHCTBaMIl 06JIaJI.aer nOCJIeJI.OBaTeJIbHOCTb 1, 4, 9, ...
..., n 2 ,... KBaJI.paTOBHaTypaJIbHbIX qIlCeJI. ECJIIl, HanpHMep, MbI 3a)l.aJI.HM
'1HCJIO I 000000, TO 1000 2 =1 000 000, a HallMH8H C W01-ro XlJIeH8 BbI-
nOJIHH€TCH HepaBeHCTBO n\!>1 000 000. TaKMe nOCJIeJI.OOaTeJIbHOCTH
oa3hlBalOT crpeMHllI,HMHCH K +00. rOBOpHT, IJTO nOCJIe.n.OBaTeJIbHOCTb
ai, a 2 , ..., an, ... cmpeMumcfl K +00, JIH )l.JIH JIro60ro M>O HaMH
OOMep N, HaqIlHaH C KOToporo BCe qJIeHbI nOCJIeJI.OBaTeJIbHOCTIl y.n.OBJleT-
BOpHIOT HepaBeHCTBY an>M. DpIl 3TOM He 06513aTeJIbHO, <n06bI qJIeHbI
nOCJIeJI.OBaTeJIbHOCTIl MOHOTOHHO B03paCTaJIIl. Iloc.ne;l{OBaT€JIbHOCTb
1,1,2,4,3,9,4,16..., B KOTOpOH BnepeMeJKKYH,lI.yr HaTypaJIbHbIe qRC-
JIa II HX KBa.II.PaTbI, He HBJIHerCH MOHOTOHHO B03pacrarom.eH:. Ho ORa
crpeMHTCH K + 00, TaK KaK ee qJIeHbI CTaHOBHTCH M OCTalOTCH BnpeJI.b
60JIbIIIe JII06oro HanepeJI. 33)1.aHHOro IJIlCJIa. H8npHMep, nOCJIe 2000-ro
1IJIeHa BbInOJIHHerCH HepaBeHcTBo an> 1000, 8 nOCJIe 2000 OOO-ro
'1JIeHa - HepaBeHCTBO an> 1 000 000.
ECJIIl nOMeHHTb 3HaKIl y Bcex qJIeHoB nOCJIeJI.OBaTeJIbHOCTH, cTpeMH-
eHcH K +00, nOJIyqIlTCH nOCJIe)I.()BaTeJIbHOCTb, KOTopaH CTpeMHTCH K
-00, HanpIlMep: -1, -4, -9, ..., _n 2 , ... . ECJIIl nOCJIeJI.OBaTeJIbHOCTb
crpeMHTcH K +00, TO nIllIIYT !im an=+oo, a ecJIIl OHa CTpeMIlTCH K
n..",
-00, TO nIlUIYT lirn a,.=-oo.
n-+GO
158

DOCJIeAOB8TeJIbHOCTb 1, -4,9, -16, 25, -36, ..., (_l)n-l n S , w
He CTpeMHTC5I HM K +00, BH K -00- ee 'LJIeHbI TO nOJIOJKHTeJIbHbI, TO
arpm.taTeJIbHbI. Ho ecJIH 3aMeHHTb Bee qJIeHhI 3TOH lJOCJre,l{OBaTeJIbHOCTH
HX MOJI.YJIHMH, TO nOJIyqHTCH nOCJIeJI.OBaTeJI.bHOCTb I, 4, 9, 16, 25,36, ...
..., K<YropaH CTpeMJlTCH K +00. B 3TOM CJIyqae rOB()pHT. 'ITO 3a)l.aHHaH
lIOCJIe,IJ.OBMeJIhHOCTb cmpeMumcJlK 6ecKoHeVWCmu, H lImnyT lim an = 00.
 ' n
TaKHe nOCJIeJI.OBaTeJIbHOCTH Ha3bIBalOT 6eCKOHettHO 60J1.bUlUMU. ¥IHbIM.H
CJIOBaMH, nOCJIeJI.OBaTeJIbHOCTb at, ..., l1n, ... 6eeKOHeqHO BeJIHKa B TOM
H TOJIbKO B TOM CJIyqae, KOr)l.a lim Ian 1=+00. Pa3YMeeTCH, eCJIH
n-+ GO
Jim l1n=+00 HJIH lim l1n=-00, TO (l1n)- 6ecKOHeqHO 60JIblllaH nOCJIe-
n.....oo n-+-oo
)l.OBaTeJIbHOCTb.
MbI 3HaeM, qTO qeM OOJIbllle 3HaMeHaTeJIb nOJIO}KHTeJIbHOH )l.P06H,
TeM MeHbllle 3HaIJeHHe ::)TOR )l.pOOH. npH )l.OCTaTOqHO 60JIblliOM 3HaqeHHH
3HaMeHaTeJISf  CTaHOBHTCSI CKOJIb yrO)l.HO MaJIOH (HanpHJ\lep, €CmI
n>1 000000, TO..!.. <0, (00001). OrclO)l.a CJIeJI.yeT, qTO eCAU lim an =00,
n n-+ 00
mrt; nocJteOOeameAbHocmb (:J 6ecKOHettHO MaAa. TOqHee rOBopH, 3TO
03aqaeT, qTO KaKoe 6b1 nOJIO}KHTeJIbHOe MaJIOe qHCJIO 8 MbI HH BbI6paJIH,
HaHJI.erCH HOMep N, HaqHHaH C KOToporo BblnOJIHHeTCH HepaBeHCTBO ..!.. <
an
<8. ECJIH OT6poCHTb YCJIOBHe, qTO 'iJIeHbl nOCJIe,IJ.OBaTeJIbHOCTH nOJIO-
:>KHTeJIbHbl, BMeCTO HepaBeHc.TBa 1.. <8 npHJI.erCSI nlICOlTb I ..!.. 1 <t.
an an
HTaK, BBeJI.eM CJIeJI.YIOllI,ee OnpeJI.eJIeHHe:
DOCJJeJI.OBaTeJIbHOCTb at, ct 2 , ..., an, ... HaSbIBaerCH 6eCKOHettH()
MaAOU, eCJIH JI.JIH JII060ro 8>0 HaiiJI.erCH HOMep N, HaqIlHaH C KOToporo
BbIIIOJIHHerCH HepaBeHCTBO Ian 1<8.
CnpaBeJI.JIHBa TeopeMa.
ECAU nOCAeooeameAbHocmb aI' a 2 , ..., l1n, ... 6eCKOHettHO eeAUKa, mo
1 I 1
nOCAed08ameAbHOCtnb -, -, ..., - .., 6ecKOHettHO AtaAa. 06pamHO, eCAU
a 1  an
aIt a2' ..., an, ... - 6eCKOHettHo MaAafl nOCAeooeameAbHocmb, mo nOCAe-
1 I 1
ooeameAbHocmb - , - , ..., -, ... 6eCKOHettHO eeAUKa (pa3YMeerCH, MbI
at a2 It"
CqHTaeM npH 3TOM, qTO Bee an OTJlHqHbI OT HYJIH).
I I I
HanpHMep, nOCJle,n:OBaTeJIbHOCTb 1, - '2 ' '3' - '4 ' ... 6eCKOHetJHO
MaJIa, TaK KaK nOCJIe)l.OBaTeJIbHOCTb 1, -2, 3, -4, ... 6ecKoHetJHo
OoJIblllaH.
DOCJle,n:OBaTeJIbHOCTb ai, a 2 , ..., l1n, ... Ha3bIBalOT 02paHutteHHor1,
ecJIH cyrn.eeTByer TaKoe qHCJIO M, qTO )l.JIH BCex n BbIIIOJIHHerCH Hepa-
BeHCTBO 1l1n IM .
6
HanpHMep, nOCJIeJI.OBaTeJIbHOCTb C oOllI,HM qJIeHOM l1n = 1 +n2 orpa-
6
HHqeHa, TaK KaK l+n2> 1 JI.JIH Bcex n, H nOToMY 1l1n 1< T =6. OrpaHHIJe-
Ha H nOCJle,n:OBaTeJIbHOCTb 1, -1, 1, -1, ...; Ian 1=1 JI.JIH Bcex n.
169

npH npOBepKe Toro, qTO 3aJI.aHHa nOCJIe,lJ.QBaTeJIbHOCTb 6ecKOHeqHO
Mani. nOJIb3YIOTC CJIeJI.YIOIII.HMH JIerKO .q;OKaSbIBaeMbIMH TeopeMaMH:
1. BCAu nOCAeoooameAbHocmu (an) H (n) 6eCIWHettl-lO MaAbl, mo uX
CYMi!-a (an + n) 6eCIWHNHO Ma.lUl.
2. BCAu nocAeoooameAbHocmb (an) 6eCIWHftltHO 1oUl.lUl, a nOCAeOO8Q.-
m'AbI-lOCf1lb (/2n) oepaHutteHa, mo nocAeoo8ameAbHocmb (anan) tJeCKOHe'lHeJ
MaAlt. B ttGcmk06mu, 6ecIWHNHo MaAO npoU38eiJeHue 08YX 6eC/WHNHO.
MlMbtX nocAeIJo6ameAbI-lOCmet1.
n nl+4
p H M e p 1. DOCJIeJI.OBaTeJIbHOCTb C 06HM 'IJIeHOM ---::r
n
6ecKoHeqHO M8JIa. B caMOM JI.eJIe, ee 06III.Hi% qJIeH MO:>KHO 3anHcaTb B BH)l.e
n l 4 I 4
an = n' + nt = Ii' + iii .
OrcIOJI.a BHJI.HO, 'ITO 3aJI.aHHaH nOCJIe)l.OBaTeJIbHOCTb BJIHerCH CYMMOA
6e6KoHeqHO MaJIblX nOCJIeJI.OBaTeJIbHOCTeii (} ) H () H nOTOMY 6ecKO-
HeQHO MaJIa.
n p H M e p 2. DOCJIeJI.OBaTeJIbHOCTb C 06llI,HM 'IJIeHOM n 2  9 6ec-
KOHeIJl1O MaJIa. B caMOM JI.eJIe. ee 06III.Hi% qJIeH MO:>KHO 3anHcaTb B BHJI.e
n I I
a = ' =--.
n n 2 ( i + :1 ) n i +
Ho nOCJIeJI.OBaTeJIbHOCTb () 6ecKoHeqHO MaJIa, a nOCJIeJI.OBaTeJIbHOCTb
(  ) OrpaHHqeHa, TaK KaK JI.JIH Bcex n BbmOJIHHerC HepaBeHCTBO
I+
 <1. 3HaqHT, 38)1,aHHa nOCJIe,lI.OBaTeJIbHOCTb 6ecKoHeqHO MaJIa.
1+#2
OrMerHM nOJIe3HOe YTep:>K)l.eHHe, n03BOJIHIOIII.ee cpa3Y YCTaHaBJIH-
BaTb JI.JIH MHorHX nOCJIeJI.OBaTeJIbHOCTeH, qTO OHH 6ecKOHetJH() MaJIbI:
J1CAU
 nk+...+ao
a = t .
n {n +... + b o
u k<;,l, mo nocAeOoeameAbHOCmb (an) 6eCIWHettHO MaJla.
HanpHMep, 6eCKOHe'IHO MaJIa nOCJIeJI.OBaTeJIbHOCTb C 06IIJ.HM qJIeHOM
3n l -4n+5
ex =
n 6n4+3n-9
Y upamnellHlI
L!.oKamHTe, '1TO rrOCJJC,l\OBaTeJJbHOCTH C 06W;HM '1JICHOM an 6eCKOHe'lHO MaJIbI, CCJJH:
1 1 I
a) a n = n 3 -f-B ; 6) a"= 3 n ; B) a n = 2 n +3 n ;
+4 1 n
r) a n = n 4 +16 ; A) a n = n 2 +2 n : e) an= IQ!"
160

4. npe.ll.eJI nOCJIe.ll.OBaTeJIbHOCTH. DOCJIeJI.OBaTeJIbHOCTb C 06llI,HM
n 2 +9
IJJIeHOM an = n 2 --I- 4 He HBJIHerCH 6ecKoHeIJHO MaJIoiL Ho ee 06Hj:j IJJIeH
MO:>KHO 3anHcaTb B BH,lI.0
5
an = 1 + n 1 +4 '
T. e. B BH.II.e CYMMbI IJHCJIa 1 H 6ecKoHeIJHO MaJIO" nOCJIe.ll.OBaTeJIbHOCTH
( n 2 _ 4 ) . D03TOMY npH .ll.OCTaTOqHO 60JIbIllHX 3HaqeHHHX HOMepa MO.ll.YJIb
5
«nonpaBKH» h 2 +4 CTaHOBHTCH OIJeHb MaJIbIM, a 3TO 03HaIJaer, qTO IJJIeHhI
nOCJIe.ll.OBaTeJIbHOCTH CTaHOBHTCH nOIJTH HOOTJIHIJHMbIMH OT IJHCJla 1.
B 3TOM cJIYIJae rOBopHT, l1TO npe.ll.eJI nOCJIe.ll.OBaTeJIbHOCTH paBeH 1, H
nHIllYT:
n 2 + 9
1im'= 1
n-+<t> n +4
(Jim - cOKpaeHHe JIaTHHCKoro CJIOBa limes - «npeJI.eJI», cpaBHHTe co
CJIOBOM «JIHMHT»).
Bo06e, rOBopHT, IJTO nOCAe008ameAbl-lOCmb a10 a2, " " Cln, ...
UMeem npeoeA a, H nHIllyr: Jim a,. =a, ecJlH nOCJIe.ll.OBaTeJIbHOCTb C 06-
n..<t>
II\HM IJJIeHOM an =a,.-a 6ecKoHeIJHO MaJIa. B IJaCTHOCTH, npe.ll.eJI JII060K
6eeKOHeqHO MaJIOH nOCJIe.ll.OBaTeJIbHOCTH ai, a 2 , ..., an, ... paBeH
HYJIIO, lim an =0.
n-+<t>
H3 CBOH:CTB 6ecKOHeIJHO MaJlbIX nOCJIC)J.oBaTeJlbHOCTeH: BbITeKalOT
CJIeJI.YIOHe CBOH:CTBa npe.ll.eJIOB, 06JIerIJaIOHe HX BbIIJHCJIeHHe:
1. ECAU lim Cln=a U lim bn=b, mo
n:c nCI)
lim(Cln+bn)=a+b II lim a,.bn=ab.
n--.. 00 n-,oo
2. ECAU lim Cln =a, lim b n =b, nputteM b=#=0 U Bce bn=l=O, mo
n('J.) n....:c
lim an =  .
n..<t> b n b
KpoMe TOro, OIJeBHJI.HO, qTO eCJIH nOCJIe.ll.OBaTeJIbHOCTb (Cln) nOCTOHH-
H8, T. e. Bce ee qJIeHbI paBHbI OJI.HOMY H TOMY R<e IJHCJIY, an=C, TO H
1im an=C.
n-+",
D p H M e p 1.
BbIIJHCJIHM npe.ll.eJI
. 3n 2 +2n-1
 4n2+6n+7 '
(2)
.lI.mI 3Toro pa3.11.emiM IJHCJIHTeJIb H 3HaMeHaTeJIb .ll.p06H Ha n H npHMeHHM
TeopeMbI 0 npeJI.eJIe IJaCTHOro, npOH3Be.ll.eHHH H CYMMbI:
6 Nt 702
161

3+.!
I ' 3n 2 + 2n 1 } . n n 2
1m - _ 1m
h-o-004n l +6n+7 - n-+oo 4+ + .!.-.
n n 2
lim ( 3+.!- )
= .. ( : )
Um 4+-n +1i2
n-+'"
2 1
3+ lim -- Um fi2 3
n-+oo n noo
6 7 =4'"
4 + Um Ii" + Um Iii"
n-+cx> n-+CI'J
Boo6e, eCAU 06w.uu lUleH noc/leOOeameAbl-lOCmU R8AJlemCfl Op06blO,
6 flucAumeAe U 3HaMeHame/le KOmopOll cmOflm MHOC!OttAeHbl om n, UMetOtqUe
oUHy U my :»ce CtTI/!tleHb, nw npeau. aTnOU nocAeOooame.llbHoctnu pll6eH. om-
HOUlI!HUIO Koat/xfJUl{ueH11W8 npu cmaptUUX tUleH.aX=
I ' aonk+... +ak ao
1m b 11 + + b - - b "
n-+oo on . . . k 0
HanpHMep, cpa3Y HaXO)l.HM, IJTO
, 4n--6n+ 1 4 I
 8n3+5nl+9- 8=2.
YnpamBeBBJI
BbIQUCJIUTe CJIe.n.YIOIII.ue npe.n.enbI:
. 4nl+1
a) lim "_2...1.. +8 ;
n-+aoiNJ. In
, n l +9
B) hm 4 + 16 ;
n-+oon
2 n -1 .
A) Hm 2n + I ;
n....'"
. 8n 4 -1
6) nm", 16114+1&3+ I ;
. n 3 +1.
r) hm 3n + 8 '
" -0- 00
. 3" - 2 n
e) Um 3" + 2 n '
"-0- '"
 4. llPE)J,EJI «I»YHK
1. BOSpaCTaHHe H y6b1BaHHe 4»YHHii. Ha pHCYHKe 103, a) H30opa-
)KeH rpacpHK CPYHKD:HH, BH,nHO, IJTO eeJIH TOqKa l1epeMeaercH B,nOJlh
9TOro rpHKa CJIeBa Hanpaso (T. e, B HanpaBJIeHHH BospacraHHSI x),
TO op,nHHaTa 9TOA TOIJKH Bee BpeMH yseJIHlJlfBaercH H TOIJK8 no,nHHMaercSi
BBepx. fOBOpHT, IJTO <PYHKD:HH y=f(x), OOJla,nalOIll,aH TaKHM CBoikTBOM,
B03pacraer Ha BceH IJHCJIOBOH OCH.. cf)YlJKD:IUI )Ke. rpa<pHK KOTOpOH H306-
pa:>KeH Ha pHcYHKe 103, 6), YObIBaer Ha BceH qHCJIOBOH OCH,
YTOIJHHM nOHHTHH B03pacraHHH H y6bIBaHHH CPYHKD:HH. nYCTh
<PYHKD:HH y= f (x) 3a)1.aHa Ha MHOJKeCTBe X. f OBOp HT, qTO OHa e03pacmaem
lIa 9TOM MHOR<eCTBe, ecJIH npH YBeJIHIJeHHH x 3HaqeHHe f (x) TOR<e YBeJIH-
1JHBaeTCH, T. e. H3 YCJIOBHH Xl<XI CJIe.n.yer: f(Xl)<f(X2). Mhl3HaeM, Ha-
npHMep, IJTO eeJIH O<Xl<X 1 , TO 4<. 3r0 3HtJHT, tJTOfj}YHKD:HH y=r
16i

B03paCTaer Ha JIYIJe [0; +00[. MO:>KHO )l.OKa-
aaTb, qTO OHa B03paCTaeT Ha BceH IJHCJIOBoA
OCH (pHC. 104).
AHaJIOfHqHO, rOBopHT, Iff 0 <PYHKnHH
y=f(x) y6bwJem Ha .MH03ICeCtn8e X, eeJIH )l.JIH
m06bIx ,Il.BYX IJHCeJI Xl H X2 3Toro MHO:>Kec'rBa.
TaKHX, 1J'f0 XI<X 2 , BbHIOJIHHeTCH HepaBeHCTBO
f(xl»f (x 2 ).
HanpHMep. <PYHKD:HSI y=x 2 YObIBaer npH
x<O.
HCCJIe)l.OBaHHe !f>YHKU,HH Ha B03paCTaHae H
YObIBaHHe MO:>KHO npOH3BO,Il.HTb Ha OCHOBe
CBoiicTB HepaBeHCTB. .lI.oKIDKeM CJIe,n;YIOUJ;ae
}'TBep:>KJI.eH HH :
1) ECAU lfJUHKl{UU y=f(x) U y=g(x) e03-
pacmalOm Ha MHO[)fCeCf1JJJe X, nw ux CYMMa
y=f(x)+g(x) m03ICe 803pacmaem Ha 9tnOM
MH(Y.JfCeCmee .
B caMOM lI.eJIe, B03bMeM )l.Ba IJHCJIa Xi E X H
XI EX TaKHe, IJTO XI<X2' no YCJIOBHIO HMeeM
f(xl)<f(X2) H g(XI)<.g(X 2 ). A TOr,Il.a no CBOit:-
CTBaM HepaBeHCTB nOJIYIJaeM, IJTO f(xd+
+g(xI)<f(x 2 )+g(X 2 ). zno H 3Ha'IHT, lITO
<PYHKD:HH y=f (x)+g (x) B03paCTaer Ha MHO-
:>KecTBe X.
2) ECAU q;yHKl{UJl y=f(x) B08pacmaem Ha
MHOOtCecmee X, molfJYHKlfWly=-f(x) y6bloaem
Ha amOM MH03ICecmee.
B caMOM ,Il.eJIe, npH Xi, x2EX, Xl<XI'
HMeeM f(xJ<f(xJ. A Tor,Il.a no CBOHCTBaM
HepaBeHcTBa -f(xd>-f(X2)' zno H 3HaIJHT,
lITO !f>YHKD:HH y=-f (x) YObIBaer Ha X.
3) ECAU lfJYHKl{UU U=f(x) U y=g(x) 803-
pacmatOm Ha MHot1lCeCtn8e X, npU'leM 8Ha'leHWl
amux lfJYHKl{UU MA03ICUme/lbHbl, mo U lfJYHK-
l{Wl y=f(x)g(x) e08pacmaem Ha X.
B caMOM AeJIe, eCJIH Xi. X2 EX, XI<XI'
TO O<f(XI)<f(X2) H'O<.g(X!)<g(x 2 ). A TOr)l.8
no CBOHCTBaM HepaBeHCTB O<f(XI)g(XI)<
<j(xl)g(xjJ)' aro a 3HatlHT, Iff 0 <PYHKnHSI
y=f(x)g(x) B03paCTaer Ha X.
4) ECAU tjJYfUCl{UU U=t (x) U y=g (x) 808-
pacmatOm H.a MH.(J:)lCecmoe X, npU'leM ux aHa-
qeHUJl omp , Ul{lltneAbHbt mo lfJYHKl{UJ1. y=
=f(x)g(x) y6bt8O£m Ha X.
.lI.oKa3bIBaercH TOIJHO TaK :>Ke.
6.
q)
J<
x
0)
PHC. 103
Y
B
7
6 !J=X 3
5
It
'J
2
1
2 X
-6
-7
-8
PRC. 104
183

5) ECAU tPYHK/1Ufl y=t(x) e03pal;maem HU X U COXpUHflem 3HaK HU X,
I
mo tPYHKl{UJl Y= f(x) y6bleaem l-ta MHO{)fCeCmee X.
B caMOM JI.eJIe, eCJIH XI<X2, TO O<f (xl)<f (X2) HJIH f (xi)<f (X2)<0.
I 1 I
A TOma f (Xl) > f (X2) . 3HaIJHT, !f>YHKD:HSI y= f (X) YObIBaer.
D p H M e p 1. HCCJIeJI.yeM Ha B03pacTaHHe H YObIBaHHe !f>YHKD:HIO
y=x 2 .
3Ty !f>YHKD:HIO MOR<HO sanHcaTb B BH)l.e npOHSBe)l.eHHH )l.BYX !f>YHKD:HH
y=x H y=x. Ho 3TH !f>YHKD:HH B03pacraIOT. DpH X>O OHH npHHHMalOT
nOJIOR<HTeJIbHbIe 3HaIJeHHH, a npH X<O - OTpHD:aTeJIbHbIe. 3HaIJHT, no
CBoHcTBaM 3) H 4) !f>YHKD:HH y=x' B03pacraer npH x>O H YObIBaer,
eCJIH x<O.
D p H M e p 2. HCCJIe)l.yeM Ha B03pacraHHe H YObIBaHHe !f>YHKD:HIO
4
11== x 2 + 1 .
MbI Y:>Ke SHaeM, IJTO !f>YHKD:HSI y=x' YObIBaer npH X<O H B03pacraer
npH X>O. Ho TOma H !f>YHKD:HH y=x 2 + 1 YObIBaeT npH x<O H Bospac-
Taer npH X>O. Bce SHaIJeHHH 310M !f>YHKD:HH nOJIOR<HTeJJbHbI. D03TOMY
4
B CHJIY YTBepR<)l.eHHSI 5) !f>YHKD:HH y= X2 + 1 B03paCTaer npH X<O H YObl-
Baer OpH X>O. rpa!f>HK 310M !f>YHKD:HH H300p3:>KeH Ha pHcYHKe 92.
YupamUeUBJJ
J. .D:oKamHTe, '1TO <\>YHKl\HH y=x'+3x2+7 B03paCTaCT Ha JlY1Je [0; + cor.
I
2. .D:oKamHTe, 1JTO <\>YHKl\HH y= 4+ x 4 y6blBaeT Ha JlY1Je [0; + cor.
3. .LI.oKamHTe, 'ITa eCJlH n - '1eTHOe rrOJlOmHTeJJbHOe '1HCJlO, TO <\>YHKl\HH y=x n
y6bIBaCT Ha JlY'Ie ]-00; 0] H B03paCTaeT Ha JlY'Ie [0; + 00[.
4. .D:oKamHTe,1JTO eCJlH n - He'lCTHOe rrOJlOmHTeJJbHOe '1HCJlO, TO <\>YHKl\HH y=x n
B03paCTaeT Ha BCeR '1HCJJOBOH OCH.
5. I1CCJIe.!l.yiiTe Ha B03paCTaHHe H y6bIBaHHe <\>YHKl\HIO y=  , x*O, a) npH
x
'ICTHOM n, 6) rrpH He'leTHOM n.
6. I1CCJIe.nyiiTe lIa BOgpaCTaHHe H y6bIBaHHe <\>YHKl\UU:
I
a) y= x 4 +3x 2 + 7 ' xER; 6) y=x 6 +5x 2 + 1, xER;
I 6 3
B) y= 27 . x> 3; r) y=-r+2" x> O.
x - x x
2. OrpaHH1feHHble H HeOrpaHH1feHHble <PYHKD:HH. rpa!f>HK !f>YHKD:HH
y=x 2 , 2::;;;;x::;;;;3, I.I.eJlHKOM nOMeaercSi B nOJIoce, OrpaHHIJeHHOH npSlMbI-
MH y=O H y=9, napaJIJleJJbHbIMH OCH aOCI.I.Hcc. TaKYIO !f>YHKD:HIO HaSbI-
BalOT OrpaHUIJeHHOH Ha OTpe3Ke [-2; 3], Ha J3CeM IJHCJIOBOH npHMoii
!f>YHKI.I.HSI y=x' He SlBJISIerCH OrpaHHIJeHHOH - KaKHe ObI npHMble, na-
paJlJIeJlbHbIe OCH a6CI.I.liCC, MbI HH npOBeJJH, Hall)l.yrcH TOqKH rpa!f>HKa,
KOTopble He JIeR<aT MeR<JI.Y 3THMH npHMbIMH. HeOrpaHHIJeHa H !f>YHKD:HH
I
Y= - Ha npOMe}KyTKe }O; 1], nOCKOJIbKY npu npU6J1H}KeHHH K TOqKe
X
x=O ee rpacpHK HeOrpaHHIJeHHO YJI.aJIHeTCH OTocH a6CI.I.HCC.
164

B 06llI,e1 BHJI.e OrpaHHqeHHOCTb H HeOrpaHHqeHHOCTb <PYHKU.HH onpe-
JI.eJIHIOTCH C,'JeJI.YIOHI\I 06pa30M.
<I>YHKU.HH y=f (x), x EX, Ha3bIBaeTCH oepaHUtteHHOii, eCJIH CYllI,eeT-
BYIOT TaKHe qHCJIa a H b, qTO JI.JIH Bcex x EX BbInOJIHHIOTCH HepaBeHCTBa
af (x)b (3TO 03Haqaer, qTO rpa<pHK 3TOH <PYHKU.HH U.e.nHKOM pacnOJIO-
:>KeH B nOJIOce, OrpaHH4eHHOH npHMbIMH y=a H y=b). ECJIH:>Ke JI.JIH JIIO-
ObIX a H b, a<b,' HaHJI.erCH TaKoe x EX, qTO f (x)<a HJIH f (x»b, TO
<!>YHKU.HIO y=f (x), x EX, Ha3bIBalOT HeoepaHUtteHHoU.
B YCJIOBHH af (x)b qHCJIa a H b MO:>KHO BbIopaTb npOTHBOnOJIO:>K-
HbIMH JI.Pyr JI.Pyry. HanpHMep, eeJIH -2f (x)5 JI..'1H Bcex x E X, TO TeM
60JIee BbIIIOJIHHeTCH HepaBeHCTBO -5f (x)5. Ho HepaBeHcTBo -c
f (x)c paBHOCHJIbHO HepaBeHCTBY If (x) Ic. n03TOMY <PYHKU.HH y=
=f (x), x EX, OrpaHHqeHa B TOM H TOJIbKO B TOM CJIyqae, Kor,na cyrn;eCT-
Byer qHCJIO C, TaKoe, qTO If (x) Ic, JI.JIH Bcex x EX.
4
n p H M e p 1. <I>YHKU.HH y= l+x2 , x E R, OrpaHHqeHa, TaK KaK.
u ° 4
C OJI.HOI1 CTOpOHbI, Bee ee 3HaqeHHH nOJIO:>KHTeJIbHbI, < l+x2 , a C ,npy-
rOH CTOpOHbI, 1+x21, H nOToMY lx2 4.
4
n p H M e p 2. <I>YHKU.HH y= x 2 -16 , x+ + 4, He HBJIHerCSI orpa-
HHqeHHOH - KOrJI.a x JI.OCTaTOqHO 6JIH3KO K 3HaqeHHHM -4 HJIH 4.
rpa<!>HK 3TOH <PYHKU.HH HeOrpaHHqeHHO YJI.aJIHerCH OT OCH a6cu.HCC.
YupamBeBBJI
I. AOKamHTe, '1TO <PYHKl\RH y=x 4 + 16 OrpaHR'IeHa Ha oTpe3Ke [0; 2]. OrpaHH.
'1eHa .'IR 9Ta <PYHKJJ.HH lIa Bcen '1RCJJOBoii OCR?
IT '" I . u
2. ,l-I,OKaJKHTe, '1TO 't'YHKl\RH Y = x 4 + 16 OrpaHR'Iella Ha BceH '1RCJJOBOH OCH.
1
3. .!JoKamRTe, '1TO <PYHKJJ.HJI Y=- 3 OrpaHR'IeHa Ha oTpe3Ke [I; 2], 110 He
x-
OrpaHH'IelIa I1a npOlCmYTKe ]1; 3[.
4. KaK!le H3 CJJe.nYIOIIl.HX <PYHKLlHH OrpaHH'IeHbI lIa Bceil tlHCJJOBOil DCH:
I I
a) y= x 2 +9 ; 6) y= x 2 -25 , x:;6 :1:5; B) y=x 2 +9;
+9 +9 x
r) y= x 2 + 25 ; ,l\) y= x2_ 25 ' x:;6 :1:5; e) y= x 2 + 25 ?
3. 6eCKOHe1JHO MaJIble (jJYHKl\HH. <I>YHKIJ.HH Y=(X-4)2 ooparn;aercSi
B HYJIb npH x=4. ECJIH B3HTb 3HaqeHHH aprYMeHTa, JI.OCTaTOqHO OJIH3-
KHe K qHCJIY 4, HM YJI.YT COOTBerCTBOBaTb CKOJIb yrOJI.HO MaJIbIe 3Haqe-
HHH !f>YHKU.HH. HanpHMep, eeJIH Ix-41<0,1, TO Ix-41 2 <0,01, a 3TO
H 3HaqHT, qTO (X-4)2<0,01. HepaBeHcTBo Ix-41<0,1 MO:>KHO nepe-
nHcaTb TaK: -0, 1 <x-4<0, 1 HJIH 3,9<x<4,1. MbI JI.OKa3aJIH, TaKHM
oopa30M, qTO JI.JIH Bcex TOqeK npoMemyrKa )3,9; 4,l[ BbIIIOJIHHerCH He-
paBeHCTBO (X-4)2<0,01. TOqHO TaK :>Ke JI.OKa3bIBaerCH, qTO ,nJIH Bcex
TOqeK npoMemYTKa ]3,999; 4,00l[ BbIIIOJIHHerCH HepaBeHcTBo (x-4)2<
<0,000001. A Ha npOMe:>KYTKe 13, 999999; 4, 000 001 [HMeeM (x-4)2<
<0,000000000001.
165

li .MbI BH)l.HM, qro K3Koe 6bJ 'IHCJIO 8 MbI HH
 33)1.3JIH (e=O,OI; 0,000001; 0,000 0000(0001),
a+-6 BCer)l.a HaH)l.eTCH npOMe1KYTOK C I.I:eHTpoM B
PRC. 1()5 TOQKe 4, Ha KOTOpoM BbInOJIHHe1CR nepaBeH-
CTBO (x-4)\I<e. npOMe:>KYTOK Ia; a+lH
C ll,eHTpoM a H33bIBaIOT oKpecnu.wcmblO m()I{.KIU a ,p3,lJ.HYca  (poc.
105). Mw JI.OKa3aJlH, T8KHM 06pa3oM, qTO .l\JISI JIlo60r0 e>O H3H.l.eTCH
OKpecrHOCTb TOIJKH 4, 8 KOTOpOH: BbIHOJIHHeTCH HepaBeHCTBO (x-4)'<
<e. fOBOIHI,., tfro I}IYHKnn y=(x-4)\1 6ecKOHettHO MaAa, KQra x CTpe-
MHTCH K qHCJlY 4. BMeCro CJlOB «X crpeMHTCH K a» MbI 6Y.I\eM rrl:lC3Th x-+a.
(x-4)\1
<l>YHKll,HH Y= z\l-7x+12 He m.teer npH x=4 onpeJI.eJIeHHOrO 3Ha'le-
HHH, nOCKOJlbKY npH no.ncraHOBKe BMecTO x tfHCJIa 4 H <iOCiIHTe.JIb H 3Ha-
MeHaTeJIb 06paalOTcj:I B HYJIh. Ho H B 3TOM cnyqae ecJIH 83HTb 3H.aQeuHH
x, JI.OCTaTOtIHO 6JIH3KHe K 'mcJIY 4, TO 3HaqeHHH nOR !f>YHKlI.HH 6y.nYT
6JIH3KH K HYJIIO. 3'1'0 BHJI.HO H3 CJleJI.YIOui 1aOJIHlI.bI:
o
o
a-15
I
Q
x I 3,9 I 3,w j 3,999 j 4, I  4,01 I 4,001
1 I I t I I I I
Y -g -'99 - 009 9 99 999
.lI.3)l.HM Tenepb 06ee onpeJI.eJIeHHe 6eeKOHeqHO MaJIOH <PYHKlI.HH.
<l>YHKlI.HH y=f (x) 6eCKoHeI(:HfJ MQ,Ml np" x-+a, eeJIH JI.JIH JII060ro e>O
MO:>KHO YK83aTb oKpeeTHoCTb TOIJKH a, 80 Bcex TOqK<:1X KOTOpOH: (KpOMe,
6bITb MO}KeT, caMoii: TOqKH all} BbWOJUHleTC5l HepaBeOCTBD IJ (x) I<e.
I1poCTeHIlIHM npHMepOM 6ecKOHeqHO Ma.rIOH CPYHKlI.HH npH x-+a
HBJIHerCH y=x-a. DpH 33)1.aHHOM e>O JI.OCTaTO'IHO B3HTb OKpeeTHOCTb
70'IKH a pa,ll,wyca £. B 3TOH OKpecTHOCTH \x-a 1<8, a 3TO H 3HaIJHT, qTO
If (x) I<e. BOJIee CJIO}KHble npHMepbI 6eCKOHeqHO MaJIbIX <PYHKlI.HH nOJIY-
lJalOTCH C nOMOUl.blO CJH:'JJ;YlOlli,HX YTBep)K).{eHHH, JI.OKa3aTe.JIbCTBO KOTO-
pbIX MbI onycKaeM.
1. CYMMa 08YX CPYI-lKl{UU, 6ecKOHettiiO MaAbIX, Koma x-+a, 6eC1COHettHO
MllAa npu x-+a.
2. ECAU CPYHK'lI,U,," G (x) 6eCKOHettHO MaAa npu x-+a, a cflyNKl{Ufl Y=
=/(x) OZpa.HWleHG 8 HeKOnwpo.u OK{JecmHocmu mOttKU a, mo ljJYHKl{Ufl y=
f (x)a. (x) 6£acOHeUHO J4[l.JJa npu x-+a.
3aMeTHM, qTO JIlOOaH CPYHKIUUI y=a. (x), OeeKOHelJHO MaJIaH lIpH x-+a,
orpaBH'IeH3 B HeKOTOpoo OKpecrHOCTH 3TOH TOIJKH (T'cIK K.aK B HeKOTopOH
OKpecTHOCTH BblfIOJUlCH Hep.a.BeHCTBO lex. (x) 1<t::). n03TOMY H3 YT-
BepeHH.H 2 cne.n.yer, IJTO np0U38eOeHlJ.e deyx r;fJYH1(,lfUU, 6eCKOHeqHO
MaJlblX npu x-+a, 6ecK.OHJeI.l.HO .N.ilIJ[) np.u x-+a.
1) UroB8 «KpOMe, 61.n1. MOB<eT, ClIJIIOi TO'lDi a» ,II06aB.l!eHIol II07QMY, '1TO B 9Toii
(x-4)\1
TO'IKe <!>YHKIUUI MO>ReT H He HMeTb 3f1a1Jelftl" (KaK He HMeJla q,YHKIIH y= x\l-7x+12
npH z=4).
166

Il P H M e p 1. TaK KaK <PYHKIJ,HH y=x-a 6eCKOHeqHO MaJIa npH
x-+a, TO Bce <PYHKIJ,HH BHJI.a y=(x-a)n, rJI.e n - HaTypaJIbHOe qHCJIO,
TaK}Ke 6eCKOHeqHO MaJIbI npH X-+a (3TH <PYHKIJ,HH HBJIHIOTCSI npOH3Be-
JI.eHH5I'MH 6eCKOHeqHO MaJIbIX <PYHKIJ,HM:). ()qeBHJI.HO, qTO Hepa1leHcTBo
Ix-ajn<e BbIIIOJIHHeTCH, eCJJH Ix-a 1< V"i, T. e. eeJIH a-6<x<a+6,
n/-
rJI.e 6=y e.
DpHMep
2. JII06aH <PYHKIJ,HH BHJI.a
y=A 1 (x-a)+.. .+An(x-a)n
(1)
6eCKOHeqHO MaJIa npH x-+a. B caMOM JI.eJIe, Bce npOH3Be)l.eHHH BHJI.a
A k(x-a)k 6eCKOHe'lHO MaJIbI, a TOrJI.a 6ecKOHetlHO Ma.na H CYMMa (1).
Il P H M e p 3. <I>YHKIJ,HH y= Vx=a 6ecKoHeQHo M8JI8 npH x-+a.
v -
B caMOM JI.eJIe, HepaBeHCTBO I x-al<B "Meer MecTO, eC.JIH Ix-aI<B 3 ,
T. e. eCJIH a-e 3 <x<a+e 3 .
x-a
Il p H M e p 4. ECJIH #0, TO <PYHKIJ,HH y=- 6eCKOHeIJHO MaJIa
ax
npH x-+a. 1JJIH JI.OKa3aTeJIbCTBa 3TOrO YTBep:>KJI.eHHH JI.OCTaTOIJHO nOKa-
 1 _
3aTb, qTO 't'YHKll.HH y= - OrpaHHqeHa B HeKOTOpOH OKpeCTHOCTH TOqKH
ax
a. B KaqeCTBe TaKOH OKpeCTHOCTH npH a>O MO:>KHO Bbl6paTb npOMe:>KY-
l a 3a [ - 2 1 2 2
TOK '2; '2 . B 3TOH oKpeCTHoCTH PlMeeM 3a2 < ax < a 2 , H nOTOMY 3a 2 <
121
< - < 2' A 3TO H 3HaqHT, qTO <PYHKIJ,HH y= - OrpaHHqeHa B JI.aHHOH
u a u
OKpeCTHOCTH. CJIyqaM: 0<0 pa36HpaercH TaK :>Ke.
)"upaacBeBDJI
1. Haii,/(HTe OKpeCTHOCTb 'IUCJIaa, B KOTOpoH BblUOJIUJlCTC.II HepaBeHCTBO Jx-aI 5 <
<0,00001.
2. J{oKaJKUTe, 'ITO <jJYHKJJ.UH y=3 (x-4?+5 (X-4)3 6ecKOHe'lHO MaJla rrpu x-+4.
3. ,QOK&JKIJ1'e, 'ITO !)JoymrnH1i y= V x-6-/- 7 (x-6)3 6ecKoHeQHO MaJla llJ}JI x-+6.
4. npe.ll,eJI 4JYHX1I B TOIiKe. <I>YHKl\HH y=x 2 + 1 He SiBJIHerCR Gee-
KOHeqHO MaJIOH npH x-+3 (HanpHMep, ec.nH x=3,Ol. TO 3HaqeHHe 3ToH
<PYHKIJ,HH paBHo 10,0601). Ho 3TY <PYHKlJ,H1O MO:>KHO 3anHcaTb TaK:
y = 10+ (x 2 -9) = 10+ (x-3)(x+ 3).
3JI.eeb CJIaraeMOe (x-3)(x+3) 6ecKOOeIJHO MaJIO npH x-+3. D03TOMY,
KOrJI.a x MaJIO 'OTJIHqaeTCH OT 3, 3HaqeHHe 33JI.aHHOH <PYHKIJ,HH MaJlO at:..
JlH'IaerCR ar 10. fooopsrr, tJTO neJI H .pYHKll.HH npH x-+3 pa8etI 10,
H nHlllYT lim (x i + 1)= 10.
x.... 3
Bo06e, eCJIH <PYHKIJ,HIO y=f(x) MO:>KHO npeJI.CTaBHTb B BHJI.e CYMMbI
qHCJIa h H CPYHKll.MH y=a (x), 6ecKOHe<IHQ M8JIOH npH x-+a, T. e. B BH)l.e
y=b+a (x), TO {{HCJIO b Ha3hIB3lOT npOOeJWM. 3TOM: <PYHKIJ,HH npH x-+d
H nHillYT
lim f(x)=lJ.
x....a
167

H3 CBOHCTB 6eCKOHeqHO laJIbIX CPYI-IKlJJiH BblTeKalOT CJ1eJI.YIOHe
cBoikTBa npeJI.eJIOB:
1. ECAU fjJYHKl{UU y=f (x) U y=g (x) UMelOm npeoeAbl npu x-+a, mo ux
CYMMa U npOU3eeoeHue UMelOm npeoeAbl npu x-a, npUtteM
!im If(x)+g(x)]=!im f(x)+lim g(x)
x....... a X-+D x-+a
u
!im f(X)g(x)=lim f(x)!im g(X).
x-+-a x....... a x-+a
Kop, oIJe npeOeA CYMMbl paeeH cyMMe npeoeAoe, a npeoeA npoU38eoeHUfl
paeeH npOU38eoeHUIO npeoeAoe.
2. ECAU fjJYHKl{UU y=f(x) U y=g(x) UMelOm npeoeAbl npu x-+a, npu-
'l£M emopou npeoeA OmAUttRH om HyAfl, mo
lim f (x)
!im f(x) =
x->-a g (x) lim g (x)'
x->-a
D P H M e p 1. BbIIJHCJIHM npe.n.eJI
, x 2 + IO
hm 2 2- 1 .
x.... 5 x
OIJeBMHO, IJTO !im x=5. D03TOMY no CCPOPMYJIHpOBaHHbIM BbIllie YT-
x-+5
Bep:>K,neHHHM HMeeM
11m (X2 + 10)
I ' X2+ 10 x-+5
1m ' -
2X2- 1 lim (2x 2 - I)
x->-5 x->-5
( Jim X ) 2+ 10
x-+5
2 ( lim X ) 2_1
x->-5
5 2 +10
2.5 2 -1
35 5
4'9=7'
ECJIH B TOIJKe a 3HaMeHaTeJIb ,np06Ho-paHOHaJIbHOH CPYHKHH 06pa-
m.aerCH B HYJIb, a ee IJHCJIHTeJIb OTJIHIJeH or HYJIH, TO no Mepe npHOJIH-
:>KeHHH x K a 3HaIJeHHH 4>YHKHH CTaHOBHTCH CKOJIb yrOJI.HO 60JIbIllHMH
no MOJI.YJIIO. rOBOpHT, IJTO 4>YHKHH 6eCKOHettHO eeAUKa, KOrJI.a x CTpe-
MHTCH K a, H nHIllYT lim f « x» =00. HanpHMep,
x-+a g x
. x 2 +7
hm 4 =00.
x->-2 X -
ECJIH :>Ke npH x=a B HYJIb oopalOTCH II IJHCJIHTeJIb H 3HaMeHaTeJIb,
TO HMO c,neJIaTb TO:>KJI.eeTBeHHoe npe06pa30BaHHe, COKpaTHB IJHCJIHTeJIb
H 3HaMeHaTeJIb ,npOOH Ha x-a.
x2-9
D p H M e p 2. BbIIJHCJIHM  x2-7x+ 12 ' 1JJIH 3Toro pa3JIO:>KHM
IJHCJIHTeJIb H 3HaMeHaTeJIb Ha MHO:>KHTeJIH H cOKpaTHM ,nP06b Ha x-3:
I . x2-9 { " (x-:-3)(x+3) { . x+3 3+3 6
1m 2 = 1m = I1I1-=-=- .
x-+3 x -7x+12 x->-3(x-3)(x-4) x->-3 x:-4 3-4
168

¥upamHeHUJI
BbI'IHCJJHTe rrpe.l(eJlbl:
. x 2 +4
a) lim 2 + 16 ;
%42 x
x 2 _1 . x 3 -8 . x 2 _1
6) Um 2 2 ; B) lim 2 6 + 8 ; r) 11m -.
x4Ix -3x+ x42x - x x4_1x3+1'
5. npe)l.eJI c1JYHKIJ,HH Ha 6ecKOHeIJHOcTH. MbI 3HaeM, qTO eeJIH x>O,
1
TO C B03pacraHHeM X 3HaIJeHHH !f>YHKIJ,HH y= - craHoBHTCH Bce MeHbIue
x .
H OHH npHOJIHR<aIOTCH K HYJIIO. 130JIee TOIJHO 3TO MOR<HO BbIpa3HTb TaK:
KaKoe ObI MaJIOe nOJIOR<HTeJIbHOe qHCJIO e MbI HH B3HJIH, HaHJI.erCH TaKoe
I I
3HaIJeHHe N, IJTO N <e, a TorJI.a npH x>N Bce 3HaIJeHHH !f>YHKIJ,HH y= X-
I
OK8:>KYTCH MeHbIlle e. fOBOpHT, IJTO !f>YHKIJ,HH y= - OeeKOHeIJHO MaJla
x .
npH x+oo (nHWYT lim J..=O). l3eeKoHeqHO MaJIa, KOrJI.a X CTpeMHTCH
X4+ a> X
K +00, H !f>YHKIJ,HH y=_J... Ho 3HaIJeHHH 3TOH !f>YHKIJ,HH OTpHIJ,aTeJIb-
x
HbI, H nOTOMY BMeCTO HepaBeHCTBa  <e HaJI.O nHcaTb I  I <e.
lJaJI.HM Tenepb OOllI,ee onpeJI.eJIeHHe OeeKOHeqHO MaJIOH npH x+oo
!f>YHKIJ,HH. Ha30BeM !f>YHKIJ,HIO y=f (X) 6eCKOHettHO MaAOU npH X, CTpeMH-
rn;eMCH K +00, eeJIH JI.JIH JIIOOoro nOJIOR<HTeJIbHOrO qHCJIa e HaHJI.eTCH
TaKoe N, IJTO If(x)l<e JI.JIH Bcex X, TaKHX, IJTO x>N. 3TO MOR<HO 3anH-
caTh TaK: .
Ve>O 3NVx>N If(x)l<e.
HarJIHJI.HbIM npHMepOM oeCKOHeqHO MaJIOH <PYHKIJ,HH MOR<er CJIYR<HTb
Macca pa.n.HoaKTHBHoro BeeCTBa KaK !f>YHKIJ,HH BpeMeHH. DYCTb 3a
K3:>K)I.ble CYTKH pacnaJI.aerCH nOJIOBHHa 3Toro BeeCTBa. TorJI.a, KaKoe ObI
MaJIOe qHCJIO e>O MbI HH 3a.n.aJIH, HacTynHT JI.eHb N, HaqHHaH C KOTOpO-
ro KOJIHqeCTBO 3Toro BeeCTBa CTaHer H OCTaHeTCH BnpeJI.b MeHbwe,
qeM e.
1
KaK MbI BHJI.eJIH BbIIlle, !f>YHKIJ,HH y= - oeCKOHeqHO MaJIa, KOrJI.a X
x
cTpeMHTcH K +00. OHa oeCKOHeqHO MaJIa H npH CTpeMJIeHHH X K -00:
I
eeJIH B3HTb, HanpHMep, x=-1 000000, TO - =-0,000001, a 3TO IJHC-
x
JIO MaJIO OTJIHqaerCH OT HYJIH. H qeM 60JIbIlle no MOJI.y.'11O OKaR<erCH OTPH-
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I I
- . A 310 H 3HaqHT, qTO !f>YHKIJ,HH Y= - oeCKOHeqHO MaJIa H npH X-+-OO.
x x
TaK KaK OHa 6eCKOHeqHO MaJIa H npH X+OO, H npH x-oo, TO
rOBopHT, qTO OHa 6eCKOHeqHO MaJIa, KOrJI.a X CTpeMHTCH K oeCKOHeqHOCTH
(He YTOqHHH 3HaKa 3TOH oeCKOHeqHocrH), T. e. npH x-+oo.
DpH npOBepKe Toro, qTO JI.aHHaH !f>YHKIJ,HH oeCKOHeqHO MaJIa npII
x-f-oo, HCnOJIb3YIOT CJIeJI.YIOlIJ,He TeopeMbI:
169

1. ECAU rjJYHKl{UU y=a. (x) U Y= (x) 6eaWHettHo J.taAbl npu x-++ 00 ,
mo U ux CYMMa y=a. (x)+ (x) 6eCKOHettHO MaAa, KOZi1a x-++oo.
2. ECAU rjJYHKl{Ufl y=a. (x) 6ecKOHe<tHo MaAa npu x-++oo, a rjJYHKl{Wl
y=f (x) oC!paHUtteHa Ha AYtte euoa fa; +00[, mo U rjJ!Jmr:qwr. y=f (x)a. (x)
6eCfWHettHO MaAa npu x-++ 00.
DOCKOJIbKY BCHKafl 6ecKOHeqHO MaJlaH YHKD:HH orpaHHqeua Ha He-
KOTOpOM JIYIJe (npH x>N HMeeM Ia. (x) I<), TO H3 yrBepeHHH 2 CJIe-
AyeT, IJTO npoU38eOeHue oeyx 6ecKoHettHO MaAblX rjJYHKl{UU 6ecKOHe<tHO
1Ul/ltJ .
3. ECAU rjJYHKl{Wl y=f(x) 6eCKOHettHO eeAUKG, KOC!oa x-++ oo . T. e.
itAfl AlO6oC!o M>O HaiiiJrmcfl maKoe N>O, Ilmo I f(x) I> M OM x>N,
I
mo 4JYHKJ1u.R, y= f(x) 6ecKOHettHO MaAa npu x-++ oo .
n p H M e p 1. MbI 3H1reM, 'ITO !f>YHKIUfH Y= J.. OecKoHe<IHO MaJTa
x
ImH x-+oo. TaK I(ax  =.l. .2.. '" 2.. (n P a3 ) TO "'''HKUHH Y = 2..
t' " xn X X X ' 't'J x n
TWKe 6ecKOHeQHO Ma.r.2- npH x-+oe.
n p It M e p 2. 4>YHKl\HH 2H,aa Y = ::: + . . . + :1 OecKoHeqHO Ma-
.na, Kor.u,a x-+oo.
x 2
D p H M e p 3. <l>YHKllHIO y= x 4 + 10 MO:>KHO 3anHcaTb B BH,ne Y=
1 I I
=2- 10 ,HOtPYHKIJ:HH g=2' 6ecKoHetJOO MaJIa npH X-+oo, a !f>YHK-
x 1+- x
x 4
I 10 I
nHH y= -W0rpaHHqeHa, TaK KaK 1+ x4 >I, H nOToMY-W<1.
1+ x4 1+ x4
x!-
3HaIJHT, !f>YHKWIH y= x4+ 10 6eCKOHqHO MaJIa npH X-+OO.
Cnpaae,ZJ.TIHBO cJIe,nYIOee 06lI.I,ee yrBepeHHe:
ECAU n>m u "m=F0. bn=FO, mo l/JgHJCl4W1.
amx m +... +ao
y=. b
bnx n + ' . . + 0
6ef.()He<!lfO Malia npu x-+oo.
, x 3 +6x-8
HanpHMep, !f>YHKllHH Y= x4+2xz+ 7 6ecKOHeQHO MaJla npH x-+oo,
2x 2 + 3
cl>YHKJUU1!J= xZ+ 1 Be HBJHleTCfl OecKOOeQHO Ma.ooH npH x-+oo (aa-
npHMep, eCJIIr x=1000, TO y= :: >2). Ho OHa HBJIHerCH CYMMOH
tJHCJla 2 H 6ecKoHeQHO MaJlOM YHKD:HH y= xII I :
2x3+3 r
y= 'xz+, =2+ xZ+1 .
n03TOMY npH 6o.rIbIIIHX 3HaIJeHHHX x rpaq,HK 3TOM !f>YHKD:HH nOIJTH CJIH-
1170

BaerC51 C np51MOH JIHHHeH y=2. fOBopm, IJTO Y
3Ta <PYHKII.H51 CTpeMHTCH K IJHCJIY 2 npH xoo,
H nmnYT
y=b
1 . 2X2+3
1m 2 I 2.
X-+oo x +
Boo6llI,e, rOBopHT, IJTO lim f(x)=b, eeJIH 0
X+CXI'
f(x)=b+a (x), r.n.e <PYHKII.HH y=a (x) DeeKo-
HetJHO MaJIa UpR x+oo. B 3TOM cJIyqae D:J»i
x+oo rpa<pHK <PYHKII.HH y=f (x) CTpeMHTC51 CJIHTbCJI c npflMOH: JIH-
HHeH: y=b (pHC. 106), T. e. paccToflHlle OT TOIJeK rpa<pHKa JI.O 3TOH
np51MOH: cTpeMHTcH K HYJIIO, Kor,n.a Tom<a rpaIrn'llfo y.,!\aJIH'eTCSI no
rpa<pHKY BnpaBo. AHaJIOrHIJHO onpeJI.eJIHeTC51 3anHCb !im f (x)=b.
x-+- UI
Ha30BeM acuMnmamou HeKOTOpOH: JIHHHH JII06ylO np51MYIO, o6JIa,lj;alO-
llI,YIO TeM CBOHCTBOM, qTO ,paCCT051HHe OT TO'leK JIHHHH )\0 3TOH npHMOH:
crpeMlITC5I K HYmo, iroI',lI;a :rotfKa oe3rpaHHtlHD Y;IJ;aJIReTC51 or Ha>I8JIa KG-
0pJI.HHaT. DpHMaH y=b HBJI51erCH ropH30HTaJIbHOH aCHMIITOTOI'i )l.JIH
rp8tPHKa <PYmrn;HH y=f(X)B TOM PI TOJIblW 'B TOM .c, Ror)l.a
lim f (x)=b 'HJIR lim f'(x='b. TaKHM oopa30M, 1JTOl5bI Hmfm TCIfY'l'I30H-
X-+ + 00 X.D)
T3JIbHhIe aCRMnT0ThI f,MHKa CPYHK1l.ltH lI,f (x), )l.OCT1llFOIJH0H&M>fH .ee
npe)l.eJIhI .tI,PH X+OO 'H npH x__-oo.
EcJIH npeJI.em.I cP.YHKJl;HH ll=f (x) npH X+ooH x-+--oo COBJi[a)l.aIOT
TO HX 06llI,Be 3HalieHHe Ha3blBaIOT npeik.n.O:M fJmaU ,fPJ!HiK1IJfUIil I1,pU ..\7-+00
.}I OD03HaqalOT Hm! (x. OpH BbJtIHCJIemr'H JIpe,D,eJIa c;p.YHiW:HH HCDOJIb-
x-+<t>
x
PHC. 106
SYIOT CJIeJI.YIOIII.He )'TBep:>KJI. eHHH :
a) EcRU Jim f(x)=a u lim g(X)=b, mo
j(-+ 00
x-+oo
Hm [f{x) +g(x] =fIl+b
%-+00
(npeiJeJl CYMMbl paiJ8H £.IJMMe npeOeJloo) ,u
Hm f (x,) II (X.) =.llb
x-+oo
(npeoeJlnpoU3lJBiJeHJJ.JL paseH .npoU886Oe1iU1O fltIJf!iJ£.aoe).
6) ECAU Urn f (x) =au limg (x }= b" npUIleM ,b=;i=O, ,mo
x......<J) X-+-ct:J
. f() a
!: g (x) = -:b .
"IXi' +1
n p H Me p 2. BbIIJHCJIHMnpe)l.eJI <P-YHK1J.HH 1J= '2+3 IIPH X-+OO.:
4+ lim . ( Il+ 3 ) 4+ Urn.!.
' I ' 4x3+1 I ,' ,x _ x-+oo . ] _ O¥-+OO x 3 4 2
1m 2 3+3 = 1m 3 ( 3 ) 3 '2=.
.x...."" X %-+00 2+ X 3 lim 2+ . x 3 - 2+ 1im -
x-+<x> X-+oo .;3
171

Bo06w.e, BepHO paBeHCTUO
I' anxn+...+ao an
r lJllx n +... +b o = b;; .
YnpamHeHHJI
1. .1JoKamHTe, 'ITO CJJe.ll.YIOIII.He <jJYHKl\HH 6eCKOHe'lHO MaJlbl rrpH x-+-co:
I x3-8x+15 ) _. ) - I
a) Y = -' 6) Y = ; B Y - V . -, r y - 3/
x 4 +8' x 6 +x-14 x l/x 2 +16
2. Bbl'lHCJlHTe rrpe.l(CJlbl:
. x 2 +'2x+3. . 6x 4 +2x 3 - ?
a) hm 2 2 + _ I ' 6) hm 3 4_ 7 + II .
x- '" X X r_ '" X X
2x 4 + I
3. Haii.l(HTe rrpH6.HmeHHoe 3Ha'leHHe <jJYHKl\HH Y = x 4 + 4 ,eCJlH x=2 356 811.
4. Haii.l(HTe ropH30HTaJIbHbIe aCHMrrTOTbI rpa<jJHKoB <jJYHKl\Hii:
3x-4 4x 2 -1 x 2 +6 x 3
a) Y= x+5 ; 6) Y= x2-f-6 ; B) Y= 4x2- I ; r) Y= 3x3'+ t .
6. HenpepblBHble 4JYHKQHH. 4T06bI 113MepHTb nJIOllI,aJI.b KBaJI.paTHOro
yqacTKa 3eMJII1, 113MepHIOT ero CTOPOHY, nOCJIe IJerOB03BOJI.HT nOJIYIJeH-
1I0e IJHCJIO B KBaJI.paT. Pa3YMeercH, H3MepeHHe BblIIOJIHHerCH C HeKOTO-
pon norpelllHocTblO, H3-3a IJero H 3HaIJeHHe nJIOaJI.H nOJIYIJaerCH JIHlllb
npH6JIHLKeHHbIM. Ho, IJeM JIYIJllle BblIIOJIHeHO H3MepeHHe JI.JIHHbI CTOpO-
1IbI, TeM TOqHee nOJIYIJeHHOe 3HaIJeHHe nJIOaJI.H. ECJIH 3apaHee 3aJI.aTb
lIe06xoJI.HMYIO TOIJHOCTb H3MepeHHH nJIOlHaJI.H, BCerJI.a MOLKHO yKa3aTb,
C KaKon CTeneHblO TOIJHOCTH cJIeJI.yer H3MepHTb JI.JIHHY CTOpOHbI, IJT06b1
AOCTHIJb 3TOH TOqHOCTH B H3MepeHHH nJIOaJI.H. 31'0 06'bHCHHerCH TeM,
IJTO npH MaJlbIX 113MeHeHHHX IJHCeJI HX KBaJI.paTbI TOLKe MaJIO MeHHIOTCH.
fOBOpHT, qTO nnOllI,a)J.b YIJaCTKa HenpepbIBHO 3aBHCHT OT JI.JIIlHbI ero
CTOpOHbI. XOTH HenpepbIBHaH 3aBHCHMOCTb BeJIHIJHH JI.pyr OT JI.pyra BCTpe-
IJaerCH IJaCTO, HHOrJI.a OHa He HMeer MeeTa. HanpHMep, eCJIH Ha HHTH
BHCHT rpY3, Bec KOToporo 6JIH30K K npeJI.eJIY npOIJHOCTH 3TOH HIlTH, TO
MaJIOe npH6aB.leHl1e K rpY3Y MO)Ker BbI3BaTb pa3pblB HHTH, a TOrJI.a BbI-
COTa, Ha KOTOpOH HaXOJI.HTCH rpy3, CKaIJKOO6pa3HO MeHHerCH. 4T06bI
OTpa3HTb TaKoe pa3JIHIJHe B nOBeJI.eHHH BeJIHIJHH, BBeJI.eM nOHHTl1e 0 He-
npepbIBHbIX H pa3pbIBHbIX 4JYHKHHHX. TOIJHee rOBopH, nOCKOJIbKY OJI.Ha
II Ta )Ke BeJIHIJI1Ha MOLKer B OJI.HHX yCJIOBHHX H3MeHHTbCH nJIaBHO, a B
JI.PYfl1X - CKaIJK006pa3HO, CJIeJI.yeT pa3JIHIJaTb TOIJKH HenpepblBHOCTH
H TOIJKH pa3pblBa OJI.HOH H TOH )Ke <PYHKHHH. TOIJKH HenpepblBHOCTH xa-
paKTepH3YlOTCH TeM, IJTO B HHX npH MaJIblX H3MeHeHHHX 3HaIJeHHn apry-
MeHTa 3HaIJeHHe <PYHKHHH MaJIO MeHHerCH, a TOIJKH pa3pbIBa TeM, IJTO B
HHX MaJlbIM H3MeHeHHHM 3HaIJeHHH apryMeHTa COOTBeTCTBYIOT 3HaIJH-
TeJIbHbIe 113MeHeHHH 3HaIJeHHH <PYHKHI1H.
OJI.HaKO TaKoe Onl1CaHIle pa3HHHbl Me}KJI.y TOqKaMI1 HenpepblBHOCTH
H TOqKaMI1 pa3pblBa He MOLKeT CIJHTaTbCH MaTeMaTHIJeeKHM onpeJI.eJIe-
HHeM HenpepbIBHOCTI1. HCnOJIb30BaHHbIe B 3TOM onHcaHIlH CJIOBa «MaJIbie
H3MeHE'HI1H», «3HaqI1Te.'JbHbIe H3MeHeHHH) BeCbMa paCnJIblBIJaTbl H He-
onpCJI.e.TJ.eIIHbI. HanpHMep, H3MeHeHHe Ha 1 MM MaJIO, eeJIH peIJb HJI.er 0
172

paJI.Hyce 3eMHOrO lllapa, H BeJIHKO, eCJIH pelJb IIJI.eT 06 H3roroBJIeHlIII
lllapHKonO.lUllHnHHKa. A 1 000 000 KM - 3HaIJIITeJIbHaH BeJIHIJHHa, eCJJII
pelJb H)l.er 0 pacCToHHHH OT 3eMJIH)l.0 JIYHbI (KOTopoe paBHO 384 TbIC. KM),
HO BeCbMa MaJI no cpaBHeHHIO c paccToHHHeM or COJIHUa )1.0 CHpHyca.
BOJIee TOIJHO nOHHTHe HenpepbIBHOCTH OnpeJI.eJIHerCH CJIeJI.YIOIII.HM
06pa30M.
<l>YHKUHH y=f (x) Ha3bIBaerCH Henpepb18HOU e mottKe a, eeJIH OHa
onpe)l.eJIeHa B 310H TOIJKe H eeJIH
!im f (x) = f (a).
x-+a
TaKHM 06pa30M, <PYHKUHH HenpepbIBHa B TOIJKe a, eeJIH ee npe,u,eJI B
3TOH TOIJKe cyecrByer H ero MOLKHO BbIIJHCJIHTb, nOJI.CTaBHB B 31Y <PYHK-
UHIO 3HaIJeHHe aprYMeHTa x=a.
TOIJKH, B KOTOpbIX HapYlllaerCH YCJIOBHe (1), Ha3bIBalOTCH mottlCllMU
pf13pbl8a <PYHKUHH. LJaIII.e Bcero pa3pbIB B03HHKaer HJIH B TOIJKaX, pa3)1.e-
JIHIOIII.HX YIJ8CTKH, Ha KOTOpbIX <PYHKUHH aa,n,aHa pa3JIHIJHbIMH 8HaJIHTH-
lJeeKHMH BbIp8LKeHHHMH, HJIH B TOIJKaX, r)l.e 3HaMeHaTeJIb 06paIII.aercft
B HYJIb. 31'0 CJIe,u,yer H3 TeopeM 0 HenpepblBHblx <PYHKUHHX.
1. ECAU cfJYHKijUU y=f(x) U y=g(x) Henpepb18Hbl e mOttKe a, mo U
tjJYHKl{UU y=f(x)+g(x) U y=f(x)g(x) Henpepbl8Hbl e moil JICe mofUCe.
2. ECAU cfJYHKU,UU y=f(x) U y=g(x) Henpepb18Hbl e mofUCe a, npUtteM
g (a)=FO, mo u tjJYHKU,Wl y= : ; Henpepbl8Ha e moil JlCe mottKe.
1-13 3THX YTBep:>KJI.eHHH BH)l.HO, IJTO eCJIH <PYHKUHH 3a,n,aHa BbIp3:>Ke-
HIIeM y= : i: ' npHIJeM <PYHKUH1i y=f (x) H y=g (x) HenpepbIBHbI, TO
pa3pbIBOB MO)KHO O)KHJI.aTb JIHlIlb B TOIJKaX, r)l.e 3HaMeHaTeJIb 06paIII.aer-
C5I B HYJIb.
D P H M e p 1. 1-13 TeopeM 0 npeJI.eJIax BbITeKaer, IJTO npe)l.eJI JIIO-
60ro MHoroIJJIeHa f(x)=bnx n +.. .+b o npH x-+a paBeH 3HaIJeHHIO 3TOro
MHOrOIJJIeHa B TOIJKe a, lim f(x)=f(a). 3TO 03HaIJaer, IJTO <PYHKUIlH BH,lla
x-+a
y=bnx n +.. .+b o HenpepbIBHa )l.JIH Bcex aHaIJeHHH aprYMeHTa x.
n p H M e p 2. <l>YHKUHH BH)l.a
b n x n + . . . + b o e R
Y X' ,
= cmxm+...+co '
HBJIHerCH IJaCTHbIM OT )l.eJIeHHH JI.BYX MHOrOIJJIeHOB )l.Pyr H8 )l.Pyra. TaK
KaK 06a MHOrOIJJIeHa - HenpepbIBHble <PYHKUHH, TO H <PYHKUHH (2)
HenpepbIBHa BO Bcex TOIJKaX, KpoMe TOIJeK, B KOTOpbIX 3HaMeHaTeJIb 06-
paIII.aerCH B HYJIb.
2+9
HanpHMep, IJT06bI HaihH TOIJKH pa3pbIB8 <PYHKUHH y= x26X+8 '
HaJI.O pelIlHTb ypaBHeHHe x 2 -6x+S=O. 1-13 HerO nOJIyqaeM Xi=2, x2=4.
TOqKH 2 H 4 HBJI5JIOTCH TOIJKaMH pa3pbIBa aaJI.aHHoH <PYHKUHH.
n p H M e p 3. <l>YHKUHH
{ X 2 + 1,
f(x)= x-I,
(x-I)2,
(2)
x<o,
0x2,
x>2,
173

3aJI.aHa pa3JIHIJHbIMH BbIpIDKeHHH-
MH Ha npOMeLKYTKaX )-00; 0(,
[0; 2) H 12; +00[. KorJI.a x CTpe-
MHTCH K 0, OCTaBaSiCb MeHblIle HY-
JIH, TO npe.z1E}1I f (X) paBeH
Ii m (x 2 + 1) """ 1. ECJIH LKe X CTpe-
x.... 0
MHTCH K 0, OCTaBaHCb 60JIblIle
HYJIH, TO npeJI.eJI f(x) paBeH
tim (x-l)=-1. 3TH npeJI.eJIbI
X-+O
OKa3aJIHCb pa3JIHIJHbIMH. D03TOMY
X <PYHHH y=f (X) pa3pbIBHa B
TOIJKe X  o H HMeeT B Heii «CKa-
tlOK» , paBHbIH -1-1=-2 (pHC.
107).
B TOIJKe LKe x= 2 3Ta <PYHKUHH
HenpepbIBH8, T8K KaK H lim (x-I) = I, H lim (x-l)2= 1.
x2 x-+2
!J
5
If
!/=fx-1;2
-2 -1
PHC. 107
¥upaHCUeURJI
1. Haii.l(HTe TO'IKH pa3pblBa cneAYIOIII.HX <j>YHKUHii:
4 x 2 + I x 2 + 6x + 7 x 3 + 9
a) Y=xz:::g; 6) Y= x 8 ; B) Y x2-7x+10 ; r) Y= x4-81 '
2. Haii,!\HTe TO'IKH pa3pblBa <j>YHKaHH
{ x2+7' x<-I,
g= 9+x. l..;;:x..;;:3,
x;8. x>3.
7: CsOHCTBa 4JYHKD.HH, HenpepblBHbIx Ha OTpe3Ke. <l>YHKUHIO
g=f (x) H83bIBalOT l-lenpepbl81-l0u Ha Ompe3Ke [a; bI, eeJIH OHa HenpepblBHa
BO BCex TOIJKaX 310ro OTpe3Ka. <l>YHKUHH, HenpepbIBHbIe Ha OTpe3Ke,
06JI3JI.aIOT p HJI.OM Ba)KI-TbIX CBoiicTB.
1. ECAU cpYI-lKl{Ufl y=f (x) nenpepbl8Ha Ha Ompe3Ke [a; bJ, mo cpeou
ee 31-f,attel-lUU I-la amOM Ompe3Ke cyw,ecmeYlOm l-lau6oAbwee U HaUMel-lbUlee.
2. ECAU cpYI-lKl{Ufl y=f (x) l-lenpepb18l-la I-la Ompe3Ke [a; b) U npu-
HUMaem I-la e20 KOI-ll{GX 3Hattel-lUfl Pa3AUttHblX 3HaKOe (HanpHMep,
f (a)<O, f (b»O), mo e KaKOu-mo mottKe amo2o Ompe3Ka OHa o6paw,aem-
CJI, 8 I-lYAb.
n p H M e p. .lI.oKaLKeM, IJTO ypaBHeHHe
x3-6x+3=O
HMeeT XOTSI ('jbI OJI.HH KopeHb Ha OTpe3Ke [2; 31.
B Cf.iMOM JI.eJIe, HenpepbIBHaH <PYHKUHH y=x3-6x+3 npHHHMaer Ha
JIeBOM KoHue 3TOro OTpe3Ka 3HaIJeHHe f(2)=23-6'2+3=-I, a Ha
npaBoM KOHue - 3HaIJeHHe f (3)=3 3 -6.3+3= 12. 3TH 3HaIJeHHH HMelOT
pa3JIHIJJ>Ie 3HaKH, n03TOMY <PYHKUHH 06paaerCH B HYJIb Ha OTpe3Ke
[2; 31. A 3TO H 3I-Iaq11T, IJTO ypaBHeHHe x 3 -6x+3=0 HMeeT XOTH 61,[
OJI.HH KopeHb Ha 3TOM OTpe3Ke.
174

YnpamBeBBH
I. Haii,qHTe HaH60Jlblllee H HaHMeHblllee 3Ha'leHHH <j>YHKII.HH y=/(X) Ha OTpC3Ke
[a; b), eCJlH:
8
a) y=x., a=-7, b=2; 6) y= X2+ 4 ' a=-4, b=4;
7 7
B) Y= x+3 ' a=O, b=4; r) Y= x+3 ' a=-4, b=4, x3.
2. lI.oKaJKHTe, 'ITO ypaBHeHHe x2-7x+ 1=0 HMeeT KOpHH Ha npOMeJKYTKax )0; II
H )6; 7[.
3. lI.OKIDKHTe, '1TO ypaBHeHHe x 8 -8x+2=0 HMeCT KOpHH Ha npoMeJKYTKax
)-3;-2[ H ]0; 1[.
 5. ;nIlCl)CI)EPEHAJI, llPOIl3BO,l(HAlI, IIHTEfPAJI
J. npHpaeHHe 4»YHKLI,HH. OO'beM Kyoa 5IBJI5IeTC5I <PYHKII.HeH or
AJIHHbI era CTOpOHbI, V=x 8 . ECJIH KYO CJI.eJIaH H3 MeraJIJIa, TO npH Harpe-
BaHHH JI.JIHHa CTOpOHhl Kyoa YBeJIHIJHBaerCH, a nOTOMY YBeJIHIJHBaeTCH "
ero 06'beM. ECJIH AJIHHa CTOpOHbI Ky6a HMeJIa 3HaqeHHe x H YBeJIHIJH-
JIaCb Ha h, TO OHa npHMer 3HaqeHHe x+h H 06'beM Ky6a CTaHeT paBeH
(X+h)8. 3HaIJHT, npH HarpeBaHHH 06'beM Ky6a YBeJIHIJHTCH Ha (X+h)3-
-x 8 . 3ry pa3HOCTb Ha3bIBaJOT npHpaw.eHHeM 06'beMa Ky6a, 8 IJHCJ]O h,
nOKa3bIBalOw.ee, Ha CKOJIbKO YBeJIHIJHJIaCb JI.JIHHa CTOpoHbl Kyoa, Ha3bl-
BalOT npHpaw.eHHeM STOn AJIHHbI. Booow.e roBOpH, 3TO Ha3BaHHe He-
Y)l.aIJHO, nOTOMY IJTO (HanpHMep, npH OXJIa:>K.IJ.eHHH Ky6a) JI.JIHHa CTOpOHhI
MOLKer YMeHbIIIHTbCH, a TorJI.a npHpaeHHe OK3}KeTCH OTpHII.aTeJIbHbIM.
D03TOMY 6bIJIO 6bI JIYIJille roBOpHTb He npHpeHHe, a H3MeHeHHe, HO
MbI 6YJI.eM npHJI.epLKHBaTbCH TpaJI.HII.HOHHOH TepMHHOJIOrHH.
B MaTeMaTHKe npupaU{l!nue KaKOH-HH6YJI.b BeJIHIJHHbI x 0003HaIJaIOT
tJ.x, rJI.e tJ. - nponHCH8H rpeIJeCKaH 6YKBa «JJ.eJJbTa», HanOMHHalOaH 0
JIaTHHCKOM CJIOBe differentia - «pa3HOCTb». TaKHM 06pa30M, HOBoe
3HaqeHHe BeJIHIJHHbI x paBHO x+Ax, T. e. cYMMe ee IlCX0JI.Horo SHaIJeHHJI
x H npHpeHHSI tJ.x. ECJIH y=f (x) - HeKOTopaH <PYHKIJ;HSI H X nOJIY-
qaeT npHpeHHe !1x, TO MeHHerCH H SHaIJeHHe <PYHKIJ;MH, B pe3YJIbTaTe
IJero OHa nOJIYIJaer HeKOTopoe npIlpaw.eHHe tJ.y. 4T06bI BbIIJIlCJIHTb 3TO
npHpaw.eHHe, HYLKHO:
a) HaHTH 3HaqeHHe <PYHKII.HH np" HCXOJI.HOM SHaIJeHHH aprYMeHT8,
T. e. y=f (x);
6) HaHTH HoBoe SHaIJeHHe aprYMeHTa, T. e. x+tJ.x;
B) HaHTH HOBoe SHaqeHHe <PYHKII.HH, T. e. f(x+tJ.x);
r) H3 HoBoro SHaIJeHHH <PYHKII.HH BbIIJecTb HCXO)I.Hoe, T. e. H8UTH
pa3HOCTb f (x+ tJ.x)-f (x).
HTaK,
tJ.y=f(x+tJ.x)-f(x). (1)
HafuI.eM, HanpHMep, npHpallleHHe <PYHKII.HH y=x l , ecJIH 3HaIJeHHe
aprYMeHTa, paBHoe 4, nOJIYIJHJIO npHpeHHe 0,1. DpH x=4 3HaIJeHHe
<PYHKII.HH paBHO 4 2 = 16. nOCJIe npHpaw.eHHSI SHaqeHHe apr)'MeHTa CTaJIO
4+0,1=4,1, a nOToMY HOBoe 3HaIJeHHe <PYHKII.HH paBHO 4,1 1 =16,81.
3HaIJIIT, npHpaw.eHHe <PYHKII.HH paBHo 16,81-16=0,81.
175

108
B 06llI,eM BHJI.e npHpaeHHe YHKIJJm
y=x 2 BbIrJI5IJI.HT TaK:
t\y=(x+ t\X)2_X 2 =X 2 +
+2x t\x+t\X 2 -x 2 =2x t\x+t\x 2 . (2)
3JI.eCb IJepe3 t\x 2 0603HalJeH KBaJI.paT t\x, T. e.
(t\X)2 (He HMO CMellIHBaTb 3TO 0603HaIJeHHe C
t\ (x 2 ) , nOKa3bIBalOllI,HM npHpallI,eHHe YHK-
lI.HH y=x 2 ).
ECJIH I}>YHKlI.HH y=f (x) B03paCTaer Ha OT-
pe3Ke [a; bI, TO Ha 310M OTpe3Ke 3HaKH t\y
H t\x COBnaJI.aIOT - npH YBeJIHIJeHHH x YBe-
JIHqHBaerCH H y, a npH YMeHbllIeHHH x
yMeHblllaerCH H y (pHC. 108, a). EC,lH )Ke
I}>YHKlI.HH y=f (x) y6bIBaeT Ha 3TOM OTpe3Ke,
TO B JII060H ero TOqKe 3HaKH t\x H t\y npo-
THBOnOJIOLKHbI (pHC. 108, 6).
YnpamUeUBJI
HaHHTe npHpaIIl.eHHe 4>YHKII.HH y=,r-4x+3, eCJJH:
a) x=l, 4x=O,I; 6) x=l, L\x=-O,I;
B) x=2, L\x=O, 1; r) x=2, L\x=-0,2.
2. )J,HepeHu.HaJI YHKU.HH. DpHpallI,eHHe YHKlI.HH y=x 3 HMeeT
BHJI.:
t\y=3x 2 t\x+3x t\X 2 +t\X 3 .
31'0 npHpaeHHe MO)KHO 3anHcaTb CJIeJI.YIOllI,HM 06pa30M:
t\y=3x 2 t\x+ (3x t\x+ t\x 2 )t\x.
MbI BHJI.HM, IJTO OHO COCTOHT H3 JI.BYX CJIaraeMbIX 3x 2 t\x H [3x t\x+
+(t\X)2) t\x. DepBoe H3 3THX CJIaraeMbIX nponOplI.HOHaJIbHO npHpallI,e-
"HIO apryMeHTa t\x. BTopoe )Ke CJIaraeMOe CJIO)KHee, 3aBHCHT OT t\x. Ho
"PH MaJIbIX 3HaIJeHHHX t\x OHO ropa3JI.0 MeHbllIe, IJeM 3x 2 t\x, nOTOMY qTO
HBJIHeTCH npOH3Be)l.eHHeM t\x Ha BbIpa)KeHHe 3x t\X+(t\X)2, KOTopoe
crpeMHTcH K HYJIIO npH t\xO. 31'0 HCHO BHJI.HO H3 CJIeJI.YIOllI,eH Ta6milI.bI
(MbI B3HJIH 3JI.ecb x= 1):
L\x I L\y I 3x 2 L\x I [3xL\X+(L\X)2]L\x
0,1 0,331 0,3 0,031
0,01 0,030301 0,03 0,000301
0,001 0,003003001 0,003 0,000003001
TaKHM 06pa30M, CJIaraeMOe 3x 2 t\x, nponOplI.HOHaJIbHOe t\x, HBJIHeT-
CH npH MaJIbIX 3HaIJeHHHX t\x «rJIaBHOH IJaCTblO» npHpallI,eHHH CPYHKlI.HH.
31'0 CJIaraeMoe Ha3bIBalOT oucpcpepeHl{Ua/lOM YHKlI.HH H 0603HaqalOT dy,
176

dy=3x 2 I1x. OHO 3aBHCHT He TOJIbKO OT I1x, HO HOT X.
HanpHMep, JI.J1H <PYHKll.HH y=x 3 npH x= 1 H I1x=O,1
JI.HqxpepeHll.HaJI paBeH 0,3, a npH x=2 H I1x=O,1 OH
paBeH 1,2. ECJIH }Ke x= 1 H I1x=O,Ol, TO JI.HqxpepeH-
ll.HaJI 3TOH <PYHKll.HH paBeH 0,03.
Bo06w.e, <PYHKll.HIO y=f (x) Ha3bIBalOT outjJtjJepeH-
llupyeMou npH 3aJI.aHHOM 3HaIJeHHH x, eCJIH ee npH-
paw.eHHe l1y=f (x+ I1x)-f (x) MO}KHO npeJI.CTaBHTb B
BHJI.e CYMMbI JI.BYX CJIaraeMbIX, nepBoe H3 KOTOpbIX
npOnOpll.HOHaJIbHO I1x, a BTopoe 6eCKOHeIJHO MaJIO no
cpaBHeHHIO C I1x. HHbIMH CJIOBaMH, <PYHKll.HH y=f (x) JI.H<p<pepeHll.Hpye-
Ma npH 3a,rr."aHHoM 3HaIJeHHH x, eeJIH l1y=A I1x+a I1x, rJI.e !im a=O.
,lX40
HanpHMep, JI.JIH <PYHKll.HH y=x 3 HMeeM: A =3x 2 H a=3x I1x+(l1x)2.
CJIaraeMOe A I1x Ha3bIBalOT ouq)(pepeHl{uaAoM tjJYHKl{UU H 0603HaIJalOT
dy. DpHpaeHHe aprYMeHTa Ha3bIBalOT outjJcpepeHl{UaAoJ,t apZYMeHtna
H 0603HaIJalOT dx. TaKHM 06pa30M, dx=l1x H dy=A dx. DpH 3TOM A 3a-
BHCHT OT x, TaK IJTO TOIJHee nHcaTb dy=A (x) dx.
D P H M e p. HaHJI.eM )l.H<P<f>epeHll.HaJI <PYHKll.HH y=x 2 . DpHpaeHHe
3TOM: <PYHKll.HH HMeeT BHJI.:
l1y=(x+l1x)2- x 2=2x I1x+(l1x)2.
CJIaraeMbIM, npOnOpll.HOHaJIbHbIM I1x, HBJIHeTCH 2x I1x. 31'0 CJIaraeMoe
H 5JBJIHerCH )l.H<p<pepeHll.HaJIOM 3aJI.aHHOM: <PYHKll.HH: dy=2x I1x=2x dx.
3aMerHM, IJTO <PopMy.rra JI.JIH JI.H<p<pepeHll.HaJIa <PYHKll.HH y=x 2 HMeeT
npocToH reOMeTpHqeCKHH CMbICJI. TaK KaK S=x 2 - nJIOllI,aJI.b KBaJI.paTa,
CTopOHa KOToporo HMeer JI.JIHHY x, TO I1S - nJIOaJI.b <PHrYPbI, 3a1lITpH-
XOBaHHOH Ha pHcYI-IKe 109. ,5ICHO, IJTO npH MaJIbIX 3HaqeHHHX I1x rJIaB-
HYIO IJaCTb 3TOM: nJIOaJI.H COCTaBJIHeT nJIOaJI.b JI.BYX npHMoyrOJIbHHKOB,
paBHaH 2x I1x, T. e. JI.H<P<PepeHll.HaJIY <PYHKll.HH S=x 2 . Bblpa}KeHHe }Ke
(l1x)2 - nJIOaJI.b KBaJI.paTHKa, KOTopaH 6eCKOHeIJHO MaJIa no cpaBHe-
HHIO C I1x.
YnpamBeBUJI
1. HaiiJLHTC JLH<jxpepeHl\!laJ1bI CMJLYIOIII.HX <!>YHKl\Hii:
a) y=x 3 +4; 6) y=4x 2 +6x-l; B) y=x4; r) y=2x 4 -x+I.
2. .1JaiiTe reoMeTpH'IeCKOe HCTOJIKOaaHHe cpOPMYJIe
d (x 3 )=3,r dx.
3. npOH3BOJI.HaSi. Pa3JI.eJIHB 06e IJaCTH paBeHCTBa
l1y=A (x) I1x+a I1x
Ha I1x, nOJIYIJHM:
t!y
t!x =A (x)+a.
Ho no onpeJI.eJIeHHIO )l.H<p<pepeHll.HaJIa !im a=O. D03TOMY
,lX-> 0
!im y = A (x).
.lx40 uX
x Llx
PHC. 109
(1)
177

TaKHM 06pa30M, K03<P<PHll.HeHT A (x) B paBeH-
CTBe (1) HBJIHerCH npeJI.eJlOM OTHOIlIeHHH npH-
p a w. e HH51 <PYHKll.HH K npHpaw.eHHIO aprYMeH-
Ta, KorJI.a npHpaw.eHHe aprYMeHTa cTpeMHTcH
K HYJIIO. 3TOT Ko3<P<PHIJ,HeHT Ha3bIBalOT npo-
l u880ih-wu CPYHKlluu y=f (x) npH 3aJI.aHHOM
3HaqeHHH x. Ero 0603HaIJaIOT f'(x). TaKHM
06p a30M ,
A (x) = f' (x) = Hrn  .
dx-+O
TaK KaK dy=A (x) dx, TO dy=f' (X) dx.
HanpHMep, JI..'JH <PYHKll.HH y=x 3 MbI HalliJIH, IJTO dy=3x 2 dx. 3HaIJHT,
npOH3BOJI.HaH 3TOH <PYHKIJ,HH paBHa 3x 2 . A JI.JI51 <PYHKll.HH y=x 2 npOM3BOJI.-
H851 paBHa 2x:
nOHHTHe npoM3BoJI.HOH BCTpeIJaerCH B caMbIX pa3JIHIJHbIX Bonpocax
MaTeMaTHKH. HanpHMep, C ero nOMow.bIO MOLKHO npoBo)l.HTb KaCaTeJIb-
HbIe K KpHBbIM. .lI.a.n.HM CHaIJaJIa 06w.ee onpeJI.eJIeHHe nOH51TH51 KaCaTeJIb-
HOR K KpHBoii:. HaIJepTHM KaKYIO-HH6YJI.b KPHBYIO H BbI6epeM Ha Heii
TOIJKY M (pHC. 110). npoBeJI.eM IJepea 3TY TOIJKY ceKYYIO MN. KorJI.a
TOIJKa N 6YJI.er npM6JIIDKaTbCH K TOIJKe M, TO
ceKyaH MN 6YJI.er nOBOpaIJHBaTbCH BOKPyr
TOIJKH M. ECJIH npH CTpeMJIeHHII TOIJKI1 N K
TOIJKe M ceKYllI,aH MN CTpeMHTCH K HeKOTO-
pOH npHMoH 1 (T. e. yroJI Me:>K,lJ.y npHMbIMH M N
H 1 CTpeMHTCH K HYJIIO), TO np5lMYIO 1 Ha3bI-
BaIOT Kfl[;ameAbHOU K aaiJaHHOu KpUOOU e mott-
ICe M. HTaK, KaCaTeJIbHoii: K KpHBOH B TOIJKe
M Ha3bIBalOT npeJI.eJIbHOe nOJIOLKeHHe ceKY-
w.eii MN, KOrJI.a paCCT051HHe MeLKJI.Y TOIJKa-
MH M H N CTpeMHTC51 K HYJIIO.
KaCaTeJIbHaH MOLKer HMerb HeeKOJIbKO
06HX roIJeK C 33JI.aHHoii: KpRBOii: (pHC. 111).
MmKer CJIyqHTbCH, IJTO KPHB8H B OKpeeTHOCTH
TOIJKH KacaHHH nepexoJI.HT C OJI.HOR CTOpOHbI
KaCaTeJlbHOH Ha JI.PyrylO (pHC. 112). B Tex
TOIJKaX, rJI.e KpHBaH MMeer OCTpHH MJIH H3-
JIOMbI (pHC. 113), K HeH HeJIb351 npoBecTH
KaCaTeJIbH YIO.
BbIBeJI.eM Tenepb ypaBHeHHe KaCaTeJIbHOii,
npOBeJI.eHHoii: K rpa<pHKY <PYHKll.HH y=f(x).
B03bMeM Ha 3TOM rpa<pHKe TOIJKY C a6cll.HC-
coii Xo. OpJI.HHaTa Yo 9Toii TOIJKH paBHa f (xo),
H nOTOMY ypaBHeHHe KaCaTeJIbHOH HMeeT BHJI.:
PHC. lIO
PHC. III
PHC. 112
PHC. 113
178
(2)
(x 3 )' =3x 2 , (x 2 )' =2x.
y-f (xo) =k Kac (x-xo).
(3)

HaM HaJI.O HaHTH yrJIOBOH Ko3$PHlI.HeHT Ka-
CaTeJIbHoH. .lI.ml 3Toro BOCnOJIb3yeMcH reM, Y
IJTO KaCaTeJIbHaH - npeJI.eJIbHOe nOJImKeHHe
ceKYIII.eH, H CHaIJaJIa HaHJI.eM yrJIOBOH Ko3$PH-
lI,HeHT ceKyeH M N. 1-13 P HCYHKa 114 BHJI.HO,
'ITO k CeK =  . Ho Kor)l.a tJ.x crpeMHTcH K HY-
JlIO, TOIJKa N cTpeMHTcH K TOIJKe M, ceKYIII.aH
crpE'MHTCH K KaCaTeJIbHOH: H yrJIOBOH K03<l>- 0
<pHII,UeHT ceKeH - K yrJIOBoMY Ko3$PnII,H-
eHTY KaCaTeJIbHoH. 3HaIJnr,
, . !1y
k Kac = lim k CeK = 1m ....--
b.x-+O b.x-+O oX
J\.
PIIC. 114
ECJIH <PYHKII,HH y=f (x) JI.H<p<pepeHII,HpyeMa B TOIJKe xo, TO 3TOT npe.lI.E'JI
paBeH npOH3BOJI.HOH 3TOH <PYHKII,HH B TOIJKe Xo, T. e. f' (x 0)' n03TOMY
k K8C =f' (xo) H ypaBHeHHe KaCaTeJIbHOH UMeeT BHJI.:
y-f (xo) =1' (xo)(x-xo). (4)
D P H M e p. DpOBeJI.eM KaCaTeJIbHYIO K rpa<pHKY <PYHKII,HH y=x 3
B TOIJKe C a6cII,HCCOH 2. .lI.JIH nOH TOIJKH xo=2, a nOToMY YO==23=8.
.lI.aJIee, l' (x) =3x z , U nOToMY l' (xo) =3 .2 z = 12. 3Ha'IHT, ypaBHeHue Ka-
CaTeJIbHOH UMeeT BHJI.: y-8=12(x-2) HJIH y-8=12x-24, T. e. y=
= 12x-16.
PaBeHcTBo k Kac = f' (xo) nOKa3bIBaer zeOMempUlteCKUU CMbLCA npoU38oo-
HOU: npoU38ooHafl CPYHKl{UU f (x) 8 mottKe Xo paBna YZA080MY KOBcpcpUl{ueH-
my KaCameAbHOU, np08eoeHHOU 8 mottKe C a6cl{uccoU Xo K zpacpUKY ifJYHK-
tfUU y=f(x).
YnpamBeBBJI
I. HaiiHTe npoH3BoHbIe <!>YHKUHii: a) y=x 4 ; 6) y=x3-x z . HaiiHTe HX 9Ha.
qeHHH, eCJJH x=4 H eCJJH x=-l.
2. HanHlIlHTe ypaBHeHHe KacaTeJJbHoii, npoooeHHoli K rpa<jJHKY <jJYHKUHII y=x?-
B TO'IKe C a6cUHccoii 5.
3. no PHCYHKy 115 HaiiHTe 3Ha'leHHH npoH3BoHOii <jJYHKUHH B TO'IKaX c a6c-
I1,HCCaMH 1, 2, 3.
4. MeXaHHQeCKHH CMblCJ1 npoH3BOJI.HOH. nOHHTHe npOH3BOJI.HOH
IJaCTO BCTpeIJaeTCH B <pH3HKe. DYCTb TOIJKa M JI.BH}KerCH no KOOpJI.HHaT-
HOH npHMoH. TorJI.a KOOpJI.HHaTa x 3TOH TOqKH B MOMeHT BpeMeHH t HB-
JlHerCH <pYHKII,HeH OT t, x=f (t). 3ra <PYHKII,HH
3Maer 3llIWH G.8UJlCeHUfl mOttKU. DYCTb B MOo Y
MeHT BpeMeHH t 1 KOOpJI.HHaTa TOIJKH paBHHJIaCh
Xi., a B MOMeHT BpeMeHH t z 3Ta KOOpJI.HHaTa 1
paBHa Xz. TorJI.a 3a npoMe*YToK BpeMeHH
[ll; tzJ TOIJKa npOlIlJIa nYTb XZ-Xlt H ee cpeo- 0 X
Xz - Xl 1
HJlJI. cIWpocmb paBHa - t t ' T. e. -
z- I
XZ-X1 PHC. 115
VCP=- t t .
z- I
179

ECJIH BMeCTO ti HamUIIeM t, BMeCTO Xi HanHllleM X, a pa3HOCTIl t 2 -ti H
X2-Xi 0603HaIJIlM COOTBerCTBeHHO t!t H t!x, TO BblpaLKeHHe JI.JIH CPeJI.HeH
CKOpOCTH npHMer BHJI.:
Ax
V cp = M .
O)l.HaKO 3a npOMeLKYTOK BpeMeHH tJ.t CKOpOCTb TOIJKH MeHSIerCSI.
LJT06bI Y3HaTb ee CKOpOCTb B MOMeHT BpeMeHH t, Ha)l.O 6paTb Bee MeHb-
WHe H MeHblllHe npOMeLKYTKH BpeMeHH H HCKaTb npeJI.eJI CKOpOCTH 8a
3TH npOMeLKYTKH BpeMeHH. HHbIMH CJIOBaMH, M2H08eHHOU CKOpOClTlb1O
T01lKH B MOMeHT BpeMeHH t Ha3bIBalOT npeJI.eJI ee CpeJI.HeH CKOpOCTH 8a
npOMeLKYTOK BpeMeHH [t; t+MJ, KOr)l.a tJ.t CTpeMHTCH K HYJIIO:
vJ(rR=lim V cp '
MO
Ho MbI YLKe 3HaeM, IJTO V cp =  . D03TOMY vJ(rR=lim  . 8r0 03HaQa-
MO
er, qTO MrHOBeHHaSi CKOpOCTb B MOMeHT BpeMeHH t paBHa 3HaIJeHHIO B
3TOT MOMeHT npOH3BOJI.HOH OT YHKlI.HH x=f (t):
VJ(rR=f' (t).
HanpHMep, paCCMOTpHM CB060)l.Hoe naJI.eHHe TeJIa. 3aKOH CBo6oJI.-
Horo nMeHHH JI.aercSi OPMYJIOH: s= g2 . Ho npOH3BOJI.HaSi 3TOH YHKlI.HH
( t2 ) '
paBHa gt,  =gt. 3Ha1lHT, CKOpOCTb TeJIa npH CB060JI.HOM nMeHHH
B MOMeHT BpeMeHH t Bblpa:>KaeTCH OPMYJIOH
v=gt.
5. OPMYJIbI JI.HcpcpepeHLI,HpOBaHHSI. MbI 3HaeM OPMYJIbI )l.JISI OTbIC-
KaHHSI npoH3BoJI.HbIX YHKlI.HH y=x 2 H y=x 3 . LJT06bI pellIaTb npaKTH1Ie-
CKHe 3aJI.aIJH, B KOTOpbIX BCTpe1laerCSI npOH3BOJI.H, aSi Ha)l.O YMerb )l.H-
4>epeHlI.HpOBaTb 60JIee IlIHpOKHH KJIaCC YHKlI.HH. B 3TOM nYHKTe MbI
BbIBeJI.eM OPMYJIbI JI.JISI JI.HqxpepeHIIHpoBaHHSI JII06blX aJIre6paIlIJeCKHX
)l.p06eH (B IJaCTHOCTIl, JII06bIX MHOrOqJIeHoB) H HeKoTopbIX JI.pyrux YHK'
lI.HH.
1. n pOU3eOOHafl nocmoflHHou paeHa H YAIO: C' = O.
B caMOM JI.eJIe, npH JII060M 3HaIJeHHH t!x npHpallI,eHHe YHKlI.HH
y=C paBHo HYJIIO (tJ.y=O), n03ToMY  =0 II, CJIeJI.oBaTeJIbHO, C' =0.
2. npoU3eOOHafl CPYHKlluU y=x paeHa eOUHUl{e: x' = 1. B caMOM )l.eJIe,
ecJIH y=x, TO tJ.y=tJ.x, a TorJI.a y =1, H nOToMY x'=!im 1=1.
,-,x dx...O
3. np0U3800HOJl CYM.Mbl oeyx CPYHKl{UlI paBHa CYMMe ux npoU380oHblX.
B caMOM )l.eJIe, nyCTb u=u (x), v=v (x) H y=u+v. .lI.aJI.HM x npHpa-
llI,eHHe tJ.x. Tor)l.a U H v nOJIY1IaT npHpallI,eHHSI t!u H t!v, a nOTOMY y no-
JIYIJHT npHpaeHHe tJ.y:
Y+ tJ.y= (u+ t!u)+ (v+ tJ.v).
D03TOMY
tJ.y=(U+ t!u)+{v+ t!u)-(u+u)= t!u+ t!v.
180

Ho TOrJI.a
4Y  6.q + 6.v
6.x - 6.x 6.x
II, TaK KaK npeJI.eJI CYMMbI paBeH CYMMe npeJI.eJIOB,
1 . 6.y \ . Au + I ' 6.v
1m -= 1m - 1m -.
Ax-+O 6.x Ax-+O 6.x Ax-+O 6.x
OOCKOJIbKY
Hm y = y' = (u + v)', Hm u = u',
Ax->O x Ax-+O x
\ ' 6.tI ,
1m T=V'
Ax -+ 0 X
TO
(u+v)' =u' +v'.
(1)
4. n p0U3800Hafl npoU38eoeHWl 08YX ljJYHKl{UU eblttUCAflemcfl no ljJop-
MYAe
(uv)' =uv' +u'v. (2)
B caMOM )l.eJIe, eCJIIl y=uv, u=u (x) H v=v (x), TO y+Lly=(u+Llu) 
X (v+Llv)=uv+uLlv+vLlu+ LluLlv=y+uLlv+vLlu+ LlvLlu. 3HaIJHT,
Lly=uLlv+vLlu+ LluLlv. OrclOJI.a
6.y = u 6.v + v 6.u + 6.u 6.v Llx.
6.x 6.x 6.x 6.x 6.x
Ho
1 . ( 6.V ) I ' 6.v ,
1m UT =u 1m T=uv.
{),.x -> 0 x Ax -+ 0 X
T \ . 6.u ,
OqHO TaK LKe )l.OKa3bIBaeTCH, qTO 1m v =vu , a
M-+O x
\ . ( 6.u 6.v A ) " 0 0
1m  uX = U . v. = .
Ax -+ 0 oX oX
3HaqIlT, (uv)' =uv' +u'v. opMYJIa (2) JI.OKa3aHa.
TotlHo TaK LKe JI.OKa3bIBaeTCH, IJTO
( .::. ) ' = vu' -uv'
V v 2
(3)
OrMeTHM qaCTHbIH CJIyqaH cpopMYJIbI (2). DYCTb v=C- nOCTOHHHaH
CPYHKlI.IlH. TaK KaK npOH3BOJI.HaH nocToHHHOH paBHa HYJIIO, C' =0 II
cpopMYJIa (2) npmlHMaeT BIl)l.:
(Cu)' =C' U+CU' =Cu'.
TaKHM 06pa30M, nocmoflHHblU MI-lOJ/CumeAb MOJ/CHO 8bll-leCmU 3a 3HaK npo-
U3800HOU.
HanpHMep,
(4x 3 )' =4 (x 3 )' =4 .3X2= 12x 2 .
.lI.oKa}!(eM paBeHcTBo (kx+b)' =k. B caMOM )l.eJIe, (kx+b)' = (kx)' +b' =
=k (x)' +b' =k .1 +O=k, Ero reOMerpHqeCKHH CMbICJI COCTOIlT B TOM,
qTO yrJIOBOH K03(pqmll.nCHT npHMOH y=kx+b B JII060H TOqKe paBeH k.
181

5. llAfl AI06020 HamypaAbHoeo 3HatteHUfl n UMeem Mecmo paeeHcmeo
(xn)'=nxn-l. (4)
B caMOM JI.eJIe, npH n= 1 9TO paBeHcTBo YLKe JI.OKa3aHO BbIIlle: (x)' =
= 1 'X 1 - 1 =.xO= 1. Dno.nOJKHM TefIepb, 'q:ro )l.JIJI HeK"C'ro'f'OI'O k JI.OKa-
saHO: (Xk)' =kX k - 1 . TG'fJI.a no 4>oPMYJIe (2) HMeeM:
(xk + 1)' = (xkx)' = (Xk)' X+Xk;K' =kXk-1X+xk .1=kxk+xk=(k+ l)x k .
3Ha'IHT,
lxt+J.)' = (k+ 1 )X It + t '-l,
a 9TO H eeTb paBeHcTBo (4) npH n=k+ 1.
TaKHM 06pa30M, paBeHCTBO (4) HC'NIMHO npH n= 1 H H3 ero HCTHHHO-
CTH npH n=k BbITeKaeT, 'ITO OHO BepHo II npH n=k+ 1. 3Ha'IHT, OHO
HMeeT MerTO )l.JIH Bcex HarypaJIbRbIX SHatJeIDlit n. HmrpHMep,
(x 7 )'=7x 8 , (xl)'=15x14.
M<»KHQ ,!lOKa3aTb, 'ITQ WopMYJla (4) aep.H.a He TO.nbKD ,Zi,JIB BCex HaTY-
paJIbHbIX SHa'IJeaMH n, HO Ii ,:lI,JISI .mowu n, KaK llOJIQ)KIITe.nbHd;,IX, Tal< H
OTpHIJ,aTeJIbHbIX, KaK lI.eJIbIX, TaK H HeneJ1bIK. TaK.
( 1 ) ' 1 I
- - I --1 d -- I
( 1r x ) '= x 2 =-x 2 =-x 2__
r . 2 II - 2 yi '
(  ) '=(x-)'=-5X--1=-5x-6=- 56 '
 x
BbIBe)l.eHHbIe cpopMYJIbI )l.PI<t$epeHlI.RpOBaHH'H 003BOJIHIOT HaXO)l.IITb
npOH3BOJI.HbIe JII06bIX aJIrropaII'IeeKtHX JqJoOeH. HanpIIMep, no 4>OPMY.ne
(3) HMeeM:
( X 2 _ 1 ) ' = (x 2 + I) (x 2 _1)'_(x 2 + I)' (x 2 _1)
x 2 + I (x2+1)2 .
Ho (x 2 -1)'=2x, (x 2 +1)'=2x, II nOToMY
( X 2 _ 1 ) ' = (x 2 + I) 2x-2x (x 2 _I,) = 4x
x 2 + I (x 2 + 1)2 (x2+ 1)2 .
YnpamOeOBJI
I. HaH,I\HTe npcm9BOJ1HbIe cnYHJI1UIX cPYJiKURB:
7
a) y=8x-6x2+ I; 6) g=6 vx- or' ;
2x' 6x 3 +4 V - ( 1r- V - )
B) Y --&¥" ; r) 1/= x 4xy X- x.
2. HarruwuTe ypaBHeHue KaCaTeJJbHOii K rpa<jJuKY <jJYHKJJ.UU y=5i'-x+6 ..
TO'IKe c a6cJJ.uccoii 3.
6. Heonpe.u.eJIeHHbIii HHTerpaJI. HapSlJI.Y C SaJI.a'IaMH, B KOTOpbIX
110 38)I,aHlIOH q,YHKII.HH H8,lI,G HaHTH ee npOPI:imO.IUIyJO, 'IaIOTCSI sa-
,IUPIH, B KOTOpbIX no S8,lI,aHmm fIpOII3BO)l.HOH Ha,ltO HaHTH rpYHKlI.lIlO,
KOTOpyIO ,!lIIqxpepeHlI.HpOB8JIH. 3Ty rpYHKlJ,H1O Ha3bIBalOT nep8006pa3Hotl
182

AlISI 3a.1J,aHHou. TromM 06paaoM, <PYHIill,Im y=F (x) HBJIHeTCH nepB006-
pa3HoH JI.JJH CPYHK1U1H y=f (x) B TOM H TOJIbKO B TOM CJIYIJae, KorAa
F'(x)=f (x). HanpHMep, <PYHKlI.HH y=x 3 HBJIHerCH nepBo06pa3HOR MH
cpYHKlum y=3x-, TaK KaK (x:3)' =3x 2 . KpOMe 9TOB cpYHKIJ.HIi, JII06aH
cpYHKlI.HH BH)l.8 g=x 3 +C, r)l.e' C - nOCTOHHHafl, HBJIHeTCH nepBo06p33-
HOR )l.JIH y=3x 2 , TaK KaK (x 3 +C)'=3x 2 .
MbI BH)l.IiM, IJTO, XOTSI K3)KJI.aSl CPYHKQH1I MOLKeT HMeTb JIHIIIb O)l.HY
npOH3aOJI.H)'JO, Y Hee MOLKer 6bITb 6eeKOHe<IHO MHoro nepBo06pa3HbIx.
Bce 3TH nepBo06pa3HbIe OTJIHIJalOTCH JI.Pyr OT JI.Pyra JIJllllb nOCTOHHHbIM
CJIaraeMbIM. I-1HbIMH CJIOBalllli, eeJUI y=F (x) - OJI.Ha H3 nepBo06pa3HbIX
)l.JIH cpYHKlI.HH y=f (x), TO Bce OCTaJIbHbIe nepBo06pa3Hble 9TOH cpYHKlI.HH
HMelOT BHJI. y=F (x)+C. COBoKynHOCTb BCeX nepBoo6pa3Hblx mJ q,yHK-
lI.HH y=f(x) Ha3bIBaIOT HeOnpede.l£FUlblM UH.me2fXUOM 9TOii .pYHKUHH If
0603HaIJalOT  f (x) dx. TaKHM 06pa30M,
 f (x) dx=F (x)+C,
I')te C - npOH3BOJIbHaH nOC'ro1lHH8H.
HanpHMep,
 3x'dx=x'+C.
Bo06llI,e, TaK KaK (x IJ + 1 )'=(n+ l)x-, TO
 (n+ l)x n dx=xn+:t+C. (1)
Ha tPopMYJI J(H<fxPepeHlJJlp<:JBaJIU BI>lJ'eUIOO' CJIyrooure CBOHcma
HeOIIpeAeJleHHOI'O HHI'erpana:
I. H eonpeaeAeHHbU4 UHtneeptlA om cy.MMbl i16yx tfJYH.1ClfUU patJeH cyM.Me
wtlTJeZfXlJl08 om 9mUX rPgHKlfuU:
 [f(x)+g(x)Jdx= J f(x)dx+  g(x)dx. (2)
2. flocmOflHHblU MHOJ/CumeAb MOJ/CHO eblHocumb 3a 3HGK UHmeepaAa:
S 'J..f (x) dx - 'A 3 f(x) dx.
OrcIOJI.a c.leJI.yer, IJTO qx>PMYJIY (1) MffiKHO nepeIIHCaTb TaK:
(n+l) S x"dx=x n +1+C
(3)
HJIH
5 xn+
xndx= n+l +c.
(4)
OrMerHM, IJTO
 dx=x+C.
D p H M e p. HaH)l.eM HHTerpaJI  (x-4x2+8) dx. HMeeM
S (X-4X2+8)dx= S x 5 dx-4 S x 2 dx+8 5 dx=  - 4;3 + 8x + c.
183

OTMerHM, qTO cpopMYJIa (4) HCTHHHa )l.JIH Bcex 3HaIJemIH n, KpoMe
n=-l. HanpHMep,
1
5 V - S .l. X2'+l 2..!. 2 V -
xdx= x 2 dx=+C=3 x 2 +C= '3 xS+C.
-+1
2
r dx S x-H I I
J x 3 = x-sdx= -3+1 +C=- '2 X - 2 + C =- 2x 2 + C ,
YopamUeUDJI
BbI'IHCJJHTe Heonpe.l(eJJeHHble HHTerpaJIbI:
a)  (3x-6x4+2x-7)dx; 6)  V-;dxj
) 5 dx . ) S X3+4 Vx+5 d
B x 8 ' r x 2 x.
7. OnpeJJ,eJleHHblH HHTerpaJI. H3-3a TOro, IJTO B BblpaLKeHHe Heonpe-
AeJIeHHOro HHTerpaJIa BXO)l.HT npOH3BOJIbHOe nOCTOHHHoe C, HeJIb3H
BaHTH 3HaIJeHHe 3Toro HHTerpaJIa npH 3a)l.DHBOM 3HaIJeHHH x. TeM He
MeHee MOLKHO HaHTH pa3HOCTb 3HaIJeHIiiLHHreI'paJIa B )l.aHHbIX TOIJKaX b
H a. B caMOM )l.eJIe, KaKOBO 6bI HH 6bIJIO C, HMeeM:
[F (b)+CJ-[F (a)+CJ=F (b)-F (a).
::>To paBeHcTBo nOKa3bIBaer, IJTO )l.JIH Bcex nepBo06pa3HbIX cpYHKIJ.HH
y=f (x) pa3HocTb HX 3HaIJeHHH B TOIJKaX b H a OJI.Ha H Ta LKe, OHa He 3a-
BHCHT OT BbI60pa C. D03TOMY pa3HOCTb 3HaIJeHHH nepBo06pa3HOH )l.JIH
YHKIJ.HH y=f (x) B TOIJKaX b H a Ha3bIBalOT onpeoeAeHHbLM UHmeepaAOJrt
om amou l/lYHKl{UU no ompe3KY [a; bJ. Onpe)l.eJIeHHbIH HHTerpaJI no OTpe3-
b
KY [a; b) 0603HaIJalOT: \ f (x) dx. TaKHM 06pa30M,
.J
a
b
 f (x)dx=F (b)-F (a),
(1)
a
r)l.e F (x) - nepBo06pa3HaH )l.JIH f (x). Pa3HOCTb F (b)-F (a) 0603HaIJa-
lOT F (x) I. HanpHMep, nepBo06pa3HoH )l.JIH cpYHKIJ.HH y=x 2 HBJIHerCH
 D
y=3 . 03TOMY
5 5
S XS 53 2 8 125-8
x 2 dx=3 =3-3==39.
2 2
OrMerHM HeKOTopble CBoHcTBa onpe)l.eJIeHHbIX HHTerpaJIOB. 1-13
cBoHcTB 1 H 2 HeOnpe)l.eJIeHHbIX HHTerpaJIOB BbITeKaeT, IJTO
b b b
[f(x)+g(x))dx= f(x)dx+  g(x)dx (2)
a a a
\84

H
b b
 Af (x) dX=A  f (x) dx.
a a
(3)
.lI.oKa}KeM, IJTO eCJIIl a < c < b, TO
b c b
 f(x)dx=  f(x)dx+  f(x)dx.
a a c
(4)
b c
B CaMOM )l.eJIe,  f (x) dx=F (b)-F (a);  f (x) dx=F (c)-F (a);
a
a
b
 f(x) dx=F (b)-F (c).
c
DO)l.CTaBJIHH 3TH 3HaIJeHHH B )l.OKa3bIBaeMoe paBeHCTBO, nO"1yqaeM TOLK-
)l.eCTBO:
F (b)-F (a)=[F (c)-F (a)]+[F (b)-F (c)L
.lI.aJIee OTMerHM, qTO
b a
 f (X) dx=-  f (X) dx.
a b
(5)
3r0 BbITeKaer Il3 COOTHOllIeHHH
b
 f (x) dx=F (b)-F (a),
a
a
 f (x) dx=F (a)-F (b).
b
YopamBeBDJI
BbI'IHCJJHTe CJle.l(YIOIII.He HHTerpaJlbl:
4 5 9 4
a) 5x3dx; 6) S(X 3 -4X+3)dX; B) SX 2VX dX; r) S X:x4 dX'
1 3 1 1

rJIaBa VII
fEOMETPHqECKHE TIPEOBPA30BAHHJI
 1. rpYIIIIA ITPEOBPA30BAHII:A:
1. npe06pa30BaHHe MHOJKeCTB. B3aHMHO-O)l.H03HaliHoe (6HeKTHBHoe)
OT06pa}KeHHe MHO}KeCTBa X Ha Ce6H Ha3bIBalOT npe06pa30BaHUeM fnOrO
MHO}KeeTBa. JlI060e npe06paaOBaHHe <p KOHeliHOrO MHO}KecTBa MO}KHO 3a-
JI.aTb Ta6mneH H3 )l.BYX CTpOK. B nepBOH CTpOKe Ta6JIHbI 3anHCbIBalOT
9JIeMeHTbI MHO)J{eeTBa X, a BO BTOpOH - 06paabI 3THX 9JIeMeHTOB npH
)l.aHHOM npe06paaOBaHHH. DOCKOJIbKY npe06pa30BaHHe <p BaaHMHO-OJl.-
H03HaqHO oTo6pa)J{aeT X Ha ce6H, TO BO BTOpOH CTpOKe TaK}Ke CTOHT
9JIeMeHTbI H3 X, HO, Boo6llI,e rOBopH, B HHOM nopH)l.Ke, qeM B nepBOH CTpO-
Ke. D03TOMY qHCJJO npe06paaOBaHHH MHO}KecTBa X H3 n 9JIeMeHTOB
paBHO qHC,1IY nepeCTaHOBOK 9JIeMeHTOB 3Toro MHO}KeeTBa, T. e.
Pn=n!:
HanpHMep, eeJIH X COCTOHT H3 Tpex 9JIeMeHTOB: X ={a; b; c}, TO
CYllI,eeTByer 3! =6 npe06paaoBaHHH 3Toro MHO}KeeTBa. OHH aaJI.aIOTCH
CJIe)l.YIOllI,HMH Ta6JIHuaMH:
(), (%), (k), (g), (), (g).
DpH npeo6pa30BaHHH, 3a)l.aHHOM Ta6JIHueii (r!:c), 9JIeMeIITbI a H b
MeHHIOTCH MecraMH, a 3J1eMeHT c OCTaerCH HenO.BiBH)f{HbIM.
TaK KaK npeo6pa30BaHHe <p R3aHMHO-O)l.HoaHaliHO OT06pa)J{aer MHO-
}KeeTBO X Ha Ce6H, TO JI.JIH JII06bIX )l.BYX MHO}KeeTB A H B H3 X BbIIIOJI-
HHIOTCH paBeHcTBa:
H
<p (A n B)=<p (A) n <p (B)
<p (A U B)=<p (A) U <p (B).
Dpe06pa30BaHHH KOHellHblX MHO}KeCTB MO}KHO Ha06pa}KaTb rpacpaMH.
.lI.JIH 9Toro HY}KHO H306paaHTb 9JIeMeHTbI MHO}KeCTBa TOliKaMH H npo-
BeeTH CTpeJIKH OT Ka}K)l.oro 9JIeMeHTa K era 06paa)i.. HanpHMep, npe-
06pa30BaHHe, a3JI.aBaeMoe Ta6JIHueH (), H306pa}KeHo rpacpoM Ha PH-
cYHKe 116, a). DpH 310M ecJIH KaKOH-HH6Y)l.b 9JIeMeHT HenO)l.BH}KeH npH
npeo6pa30BaHHH <p, TO COOTBerCTBYIOaH CTpeJIKa HallHHaerCH H KOH-
lIaerCH B OJI.HOH H TOH )J{e TOliKe (nerJIH), a eeJIH )l.Ba 9JIeMeHTa MeHSIIOTCH
186

MOCTBMM, TO H306p.aH<aIGT )l.BOHHYIO CTpeJIKY. HanpHMep, npH npoo6pa-
3OBatlllm, rp8tP Kc:rroporo K306p;a;>KeH Ela pOCYHKe 116, 6), 3JIeMeHT a
HernAABHJKeH, a 3JleMeHThI b Ii C MeHmGTCjf MecTaMII.
DOCKOJlbKY cP - B3aHMHO-O.l(H03Ha'lHOe oToOpameH!le MIIOJKCCTBa X Ha Croll,
rrpH rpacpH'lecKOlll l1306paJKeHHH 9TOro npeo6p330BamI'I H3 KaJKAoH TO'IKl'I YJblXQl!HT
O.l(Ha H TOJlbKO O.l(Ha CTpeJJK3 H J'I Ka)KJl.oii TO'lKe IroH'IaeTCJI O)I.H<I E 'ro.II1.iro Q,lI,J!Ia crpeJJ-
KI. Ec.oIIiIJIATB IIJ xal!lO-i-HH6YJ{b 'I'O'IKH a H HJ.t!l'tl no CTpeJJ.K ro B HOH HOIIl\OB
nY'll> K(}H'ItiTCR B roii >Ke T.o'lJre" H3 KOTOpOH MbI .BbIW.!U!. llo.nY'ilITCH 3aMXH)'TaIJ
JlIl1lH!I, KOropYID !la3b1BalDT op6umou U/lU /{UK/lOM TO'IKH a. Bce M!lOJKeCTBO TO'IeK
pMl5HB3eTCJI Ha norrapHO-HenepeceKa1OIII.lIeCJI l\HKJIbI (pHC. H7). 1..!:'RCJJO TO'IeK B JJ.HK-
.lie l'IB3b1J'1I11GT tU'Jf'MKaM IJ,HKJl3. Hanp.lIMep, l\HKJI ) JfMeCT 1'I0p1l.l\OK -4 (poc. 117).
Ooo6ylO poJIb HrpaeT npeo6pa3OBaHHe, npH KOTOp<Thl see 3.'IeHThI
MHOecua X HenO)I.BHiKHbl. TaKoe npe06pa30BaHHe H330BeM mO:JICOeI::m-
Hb/..M. HJIH e(JUHU'lH#rIlM npeo6pa3O&lHUeM fI 0603H.aJ.IHM ero E.
TaK KaK Hpe06pa3OBa.HHe (jJ MHOlKecTBB X HMHeTCH B3aHMHO-(),l\IW-
3Ha'lHbIM O1ro6pIDKeHHeM 9TOfO MOOlKecTBa Ha ce6H, TO H .o6paTH OTO-
6pa:>KeHHe <p-l B3aHMHO-O)l.H03Ha'lHO oT06pa:>KaeT X Ha ce6H (CM. n. 2
 4 rJI. 4). 31'0 oT06pa:>KeHg H83bmalOT u6pamHblM npeo6pasoeaHueM
)I.JIH <po TaKHM 06pa30M, <p-l(b)=a B TOM H TOJJbKO B TOM CJJyqae. KOrJI.a
<p (a)=b. ECJIH npeo6pa30BaHHe <p 3aJI.aHO Ta6JIHlI.en, TO Ta6JIHlI,Y 06paT-
HOro npe06pa30BaHHH (f)-l nOJIyqalOT, nepecTaBJHlliIMecTaMH CTpOKH
Ta6JllI[f.bI npe06pa30BaHHH <po HanpHMep, )l.JIH npe06pa30BaHHH C Ta6-
JIHlJ,eii () 06paTHbiM HBJIHerCH npe06pa30BaHHe, lIMelOllI,ee Ta()JIHlJ,y
( bCa ) E v 6 v
.alJc.' CJIH paCCTaBHTb B 3TOH Ta JIHlJ,e 3JIeMeHTbI nepBOH CTpOKH B no-
pH,l{Ke a, b, c, TO nOJIytIHM Ta6.1IID{y (:).
B c.nY'Iae, Kor)l.a upe06pa3OBlHHe 3a,!\.8.HO rpatPoM, rpap06paTlI<ro
npe06pa30BaHHH nOJIY'IaerCH nYTeM H3MeHeHHH HanpaBJIeHlIH Bcex
crpeJIOK. Dpn 3TOM KaJK)l,brH lJ,HKJI nepeFi:)I. C(lM B oeOH, HO 6Y.zI.e'f npo-
6eraTbCR B 06p8THOM H8npaBJIeHHH.
a.  c
Oa
b
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c
....
a)
5J
PJJc. 116
T e<{
id g
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h,
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na npe06pa30BaHlie \jJ, a nOTOM npe06pa30BaHHe <po B pe3YJIbTaTe KaLK-
,lI,bIH 3JIeMeHT a MHO}KeCTBa X CHaqaJIa nepeHJI.er B 9JIeMeHT b=\jJ(a), a
nOTOM 3JIeMeHT b nepeHJI.eT B 9JIeMeHT c=<p (b), HJIH, qTO TO }Ke ('aMoe, B
9JIeMeHT c=<p [\jJ (a)) MHO}KecTBa X. B pe3YJIbTaTe nOJIYIJHJIOCb oT06pa-
}KeHlie MHO}KeCTBa X B ce6H, a-+ <p [\jJ(a)J.
OqeBHJI.HO, qTO npH 3TOM OT06pa}KeHHH B Ka:>K)l.bIH 9JIeMeHT MHO}KecT-
Ba X nepeXO,ItHT OJI.HH II TOJIbKO OJI.HH 3JleMeHT 3Toro MHO}KeCTBa: TaK KaK
cp - npe06pa30BaHHe, TO B 9JIeMeHT c npH npe06pa30BaHHH <p MO}Ker
nepeHTH OJI.HH H TOJIbKO OJI.HH 9JIeMeHT b, a TaK KaK \jJ TaK}Ke HBJIHerCSI
npe06pa30BaHHeM, B 3JIeMeHT b npH npe06pa30BaHHH \jJ nepeXOJI.HT JIHWb
9JIeMeHT a; 3HaqUT, B 3.'"leMeHT c npH pe3y.'IbTHpYIOllI,eM npe06pa30BaHHH
nepeXOJI.HT JIHWb 3.'IeMeHT a. D03T01llY OTo6pa}KeHHe a-+ <p [\jJ (a)) HB-
JJHerCH npeo6p330BamIeM MHO}KecTBa X. Era ua3bIBalOT KOMn03Ul{Ueu
npeo6paso8aHUll \jJ Ii <p H 0603HaqalOT <p\jJ. TaKHM 06pa30M, npH npe06pa-
SOBaHHH <p\jJ Ka}K,ItbIH 3JIeMeHT a nepeXOJI.HT B 9JIeMeHT <p [\jJ (a)):
(<p\jJ) (a)=<p ['I: (a)].
OT06pa}KeHHe 3JIeMeHTa a npH KOMn03HlI.HH npe06pa30BaHHH '"
$ II'
H <p MO}KHO HarJIHJI.HO H306pa3HTb TaK: a-+ b-+ c.
D p Ii M e p. DYCTb npe06pa30BaHHe \jJ 33JI.aHO Ta6JIHlI.eH (g), a
npe06pa30BaHHe <p - Ta6JIHlI.eH (). HaiiJI.eM Ta6JIHlI.Y npe06pa30Ba-
HHH <p\jJ. .lI.JIH 3Toro 3aMeTHM, IJTO npH npe06pa30BaHHH \jJ 9JIeMeHT a
nepeXOJI.HT B 3JIeMeHT a, a npH npeoopa30BaHHH <p 9JIeMeHT a
nepexo,ItHT B b. 3HaqHT, <p [\jJ(a)J=b. TOqHO TaK }Ke HaXOJI.HM, qTO
cp [\jJ(b))=c H <p [\jJ(c)J=a. D09TOMY TaOJIHlI.a npe06pa30BaHHSI <p\jJ HMeer
BHJI. (). ECJIH npe06pa30BaHHH <p H \jJ MHO}KecTBa X 3aJI.aHbI CBOHMH
rpaq,aMH, TO rpaq, HX KOMn03HlI.HH <p\jJ MO}KHO nOJIyqHTb CJIeJI.YIOllI,HM
06pa30M: H3 Ka}K,ItOH TOqKH a E X npoBecTH CTpeJIKY rpaq,a npe06pa30-
BaHHSI \jJ. ECJIH aHa KOHqaeTCH B TOqKe b, TO H3 TOqKH b npOBeJI.eM CTpeJI-
KY rpaq,a <po KOHell. 3TOH CTpeJIKH H nOKa3bIBaer, KY,Ita nepeXOJI.HT TOqKa
a npH Pe3YJIbTHpYIOllI,eM npe06pa30B3HHH <p\jJ. 31'0 nOKa3aHO Ha PHCYH-
Ke 118, rJI.e CTpeJIKIi npeo6pa30BaHHSl1P H306pa}KeHbI TOqetIHbIMH JIHHliH-
MH, CTpeJIKH npeo6pa30BaHHH <p - WTpHXOBblMH JIHHHSlMH, a CTpeJlKH
npeo6pa30BaHHH <p\jJ - Cll.1IOWHbIMH JIHHHHMH.
KOMn03HlI.HH npe06pa30BaHHH HanOMHHa-
er Onepall.HIO YMHo)J{eHHH qHCeJI. TaK }Ke
KaK H YMHO}KeHHe liHCeJI, KOMn03HlI.HSI npe06-
pa30BaHHH oOJIaJI.aer cBoiicTBOM aCCOll.HaTHB-
HOCTH (cOqeraTeJIbHOcTH): JI.JIH JII06bIX Tpex
npe06pa30BaHHH <p, \jJ, 8 BbInOJIHSlerCH paBeH-
CTBO <p(\jJ8)=(<p\jJ)8. B caMOM JI.eJIe, eCJIH npH
npe06pa30B3HHH 8 3JIeMeHT a nepeXoJI.HT B
9JIeMeHT b, npe06pa30BaHHe \jJ nepeBOJI.HT b
b
C: '-
. a . f \
. e I
...e \_/
PHC. 118
188

B C, a npeot5pa30BaHHe q; nepeBO)l.HT C B d, TO
H q;(1j16), H (q;1j1)6 nepeBO)l.HT a B d (pHC. 119).
OJI.HaKO CBOHCTBO KOMMYTaTHBHOCTH JI.JIH
KOMn03HlI.HH npeot5pa30BaHHH Y}I{e He HMeer
MeCTa: Boo6llI,e rOBopH, qnp OTJIHqaerCH OT 1j1<p
(XOTH B HeKOTopbIX CJIyqaHX MO}l{er H COB-
na)l.aTb c 1j1<p). HanpHMep, ecJIH 1j1 33JI.aer-
CH Ta6JIHlI.eU (g), a <p-Tat5JIHlI.eH (), TO, KaK 6blJIO nOKa3a-
HO BbIllIe, npeoI5p a30BaHHe q;1j1 SaJI.aerCH Tat5JIHlI.eH (). B TO LKe BpeMSI
npeot5pa30BaHHe \jJ<p 33JI.aerCH Tat5JIHlI.eii (:b)' 3HaIJHT, B ot5llI,eM CJIyqae
1j1cp:#= <p1j1.
POJIb «e,lI.HHHlI.bIJ Hrpaer npH KOMn03HlI.HH npe06pa30BaHHii TO}l{JI.ecT-
BeHHoe npe0t5pa30BaHHe E: )l.JIH JII060ro npe0t5pa30BaHHH <p BbInOJIHHIOT-
CSI paBeHCTBa:
IJIB
a c <p d
._ 
PHC. 119
E<p=<pE=q;.
(1)
LJT06LI y6eJI.HTbCH B CnpaBeJI.JIHBOCTH paBeHcTBa (1), JI.OCTaTOqHO sa-
MerHTb, IJTO npH npeot5pa30BaHHH E Bce 9JIeMeHTbI Hen OJI.B H}I{HbI, a no-
TOMY Bce paBHO - nOCJIeJI.OBaTeJIbHO BbInOJIHIITb npeot5pa30BaHHH <p H
E HJIH JI.eJIaTb JIHllIb npeot5pa30BaHHe <po D09TOMY E<p=<p. PaBeHCTBO
<pE=<p JI.OKa3bIBaeTCH TOqHO TaK LKe.
BbInOJIHHM JI.Pyr 3a JI.pyroM npeot5pa30BaHHe <p H ot5paTHoe eMY
npe06pa30BaHHe <p-l. ECJIH npH nepBOM H3 HHX 9JIeMeHT a nepeiiJI.er B
9JIeMeHT b, TO npH BTOpOM 3JIeMeHT b 06paTHO nepeiiJI.er B a. D03TOMY
nOCJIe 060HX npeot5pa30BaHHH Bce 3JIeMeHTbI MHO}l{eCTBa X OCTaHYTcH Ha
MeCTe. TaKHM 06pa30M, <p-l<p HBJIHerCH TO:>K)l.eCTBeHHblM npeo6pa30Ba-
HHeM MHOLKeCTBa X, <p-l<p=E. TOqHO TaK LKe JI.OKa3bIBaercH paBeHcTBO
<p<p-l=E. I-1TaK,
<p-l<p=<p<p-l=E.
(2)
ECJIH nOBTopHTb k pa3 OJI.HO H TO }l{e npe06pa30BaHHe <p, nOJlYIJHM
HOBoe npe06pa30BaHHe, Ha3bIBaeMoe k-u cmeneHblO npeot5pa30BaHHH q;
H ot503HaIJaeMOe <pk. LJepe3 <p-k ot503HaIJalOT k-IO CTeneHb npeo6pa30Ba-
HHH <p-l, T. e. <p-k=(<p-l)k, a IJepe3 <po - npe06pa30BaHHe E. DpOCTaSi
npOBepKa nOKa3bIBaer, IJTO JI.eHCTBHH H3JI. CTeneHHMH npeot5pa30BaHHH
npOH3BOJI.HTCH no 06bIqHbIM npaBHJIaM:
<pk<p{ =<pk+ (,
(cpk){ =<pk{.
Ha30BeM HenYCTYIO COBOKYIIHOCTb 0 npe06pa30BaHHii MHOMeCTBa X
epynnou npe06p03OeaHuu, eCJIH:
a) BMeCTe C JIlOt5blMH JI.BYMfl npeo6pa30BaHHHMH <p H 1j1 COBOI<ynHOCTb
o COJI.epLKHT HX KOMn03HlI.HIO <p1j1,
15) BMeCTe C JII06blM npeot5pa30BaHHeM cp 9Ta cOBoKynHocTb COJI.ep.
LKHT H npeot5pa30BaHHe <p-l.
189

113 YC.J10BIIH a) II 6) BbITeKaer, qTO JII06aH rpynna npe06pa30BaHHH
CO)l.ep}I<HT TO}KJI.ecTBeHHoe npeo6pa30BaHHe E. B caMOM )l.eJIe, rpynna G
HenycTa H nOToMY COJI.ep}KHT XOTSI 6bI OJI.HO npe06pa30BaHHe cpo Ho TOr)l.a
no YCJJOBHIO 6) OHa COJI.ep}KHT H npe06pa30BaHHe cp-i, a CJIeJI.OBaTeJIbHO,
B CHJIY YCJIOBHH a) H npe06pa30BaHHe cpcp-i, T. e. E.
Bce npeo6pa30BaHHSI MHO}KecTBa X o6pa3YIOT rpynny npeo6pa30Ba-
HHH fHoro MHO}KecTBa. DOKa:>KeM, qTO npe06pa30BaHIfH E=(),
cp=(:), \jJ=() MHO}KecTBa X={a; b; c} 06pa3YIOTrpynny. .lI.ml
9I'oro COCTaBHM «Ta6JIHlI,Y YMHO}KeHHSI», nOKa3bIBalOllI,YIO pe3YJIbTaT KOM-
 MH' I n03HlI,HH 9THX npeo6pa30BaHHH:
E (p \jJ 1-13 9TOH Ta6JIHlI,bI BH)l.HO, 'lT0
1 MH. npeo6pa30BaHHe cp 06paTHo "',
npHIJeM KOMn03HlI,HSI JII06bIX )l.BYX
H3 npe06pa30BaHHH E, cp, '" SlB-
JIHerCSI O)l.HHM H3 9I'HX npe06pa-
30BaHHH.
E
rp
\jJ
E
rp
\jJ
rp


cp
YupamHeHBH
t. HaiiHTe o6pa3 9J1eMellTa e npH npe06pa30BaHHH, H306pameHHoM Ha pllcYHKe
117. KaKoB nopSloK l\HKJla, nOpOJK,l\eHHoro 9THM 9JJeMcHTOM?
2. Haii.l(HTe npeoopa30BaHHSI cp\jJ H 'i'qI, eCJJH:
a) rp=(), ;=(Z);
6) rp=(f:d). =().
3. Haii.l(HTe 3-10 CTeneHb rrpeo6pa30BaHHSI rp, 3aaHHoro Ta6J1Hl\eii C:t/:/z).
4. Haii.l(HTC npeo6pa30BaHIISI, 06paTllbIe npeo6pa30BaHHSlM:
( abcde ) , ( abCde )
a) ecbda; 0) cdeab'
5*. lloK8mHTe, '1TO ecJIH 9JJeMeHT b npHHa.l(JlCmHT JJ.HKJlY 9JJCMeHTa a, TO H 9JIe-
IreHT a npHHaJleJKHT l\HKJlY 9JJeMeHTa b.
6*. .!JoKamHTc, qTO ecJlH 9JJCMeHT b npHHa)l,Jlem!lT l\HKJlY 9J1eMCHTa a, a 9J1eMCHT
c - l\HKJlY 9J1CMCHTa b, TO C np!lHa.l(Jlem!lT l\HKJlY 9J1eMCHTa a.
7. .!JoKamHTe, '!TO cOBoKynHoCTb npeo6pa30BaHHH
E=(), (f=(), =(), e=(:C;:)
MHOJKCCTBa X= {a; b; c; d} 06pa3YCT rpynny. CocTaBbTe Ta6JIHl\Y YMllomeHRSI .l(JJSI
9TOii rpynnbl.
8. HaiiHTe Bce nepccTaHOBK!I 6YKB a, b, c, d, np!l KOTOpbIX BblpamellHe ac+bd
nepeXO!lT B TOm.l(CCTBCHHO paBHoe eMY BbIpamcHHe. 06pa3YIOT JI!I 9TH ncpecTaHoBKH
rpynrry?
2. feoMeTpH'IeCKKe npe06pasoBaHKJI. Ha30BeM eeoMempU'leCKUM
npe06po30eaHueM JII060e B3aHMHO-O)l.H03HaqHoe ar06pa:>KeHHe f MHO}KecT-
Ba II TOqeK nJIOCKOCTH Ha ce6f1 (paccMaTpHBalOT H reoMerpHqecKHe ope-.
06pa30BaHHH Bcero npoCTpaHcTBa). TaK KaK MHOLKecTBO II 6ecKoHeqHO,
\1Ja.

reOMefpHIJecKHe npe06pa3OBaHHH He.JI.b3H 3MaBaTb Ta6JIIlaMH. qacTo
HX 33.l(31OT 4>oPMYJIaMH, Bblpa:>Kmom.HMH KOOp,AHHaTbI x' H y' TOqKIl f (M)
1Jepes KOOP.llHH3ThI x H Y TOIJKH M. B 06m.eM BHJI.e 3TH 4'OPMYJIbI IIHUJYT
TaK:
{ x: =cp (X, y),
y ='i' (X, y).
HanpHMep, paseHCTBa
{ X' =x+y-3,
y' =2x+3y+4 (2)
33,ltam reoMeTpHIJecKoe npeoopasoBaHHe nJIOCKOCTH. qTOObJ: HaiiTH 06-
pa3 TOIJKH A (-2; 4) IIPH npeo6pa30BaHHII (2), HaJI.O 38MeHHTb B 3THX
p8BeHCTBax X H Y IJHCJlaMH -2 H 4:
x' =-2+4-3=-1,
y' =2 (-2)+3.4+4= 12.
(1)
3HaIJHT, TOqKa A (-2; 4) nepeXOJI.UT B TOqKY B (-I; 12). lJT06bI HaMTH
cf>oPMYJlbI 06paTHoro npe06pa30BaHHH, HMO penlHTb ypaBHeHHH (2)
OTHOCnreJIbHO X H y. MbI nOJIyqaeM, qTO
{ X = 3x'-y' + 13, (3)
y=-2x' +y' -10.
HanpHMep, Y3H, aeM KaKaSi TOqKa nepeXOJI.HT npH npe06pa30B8HHH
(2) B TOqKY N (-5; 4). JJ:JIH 3Toro nOJIOLKHM B <popMYJIaX (3) x' =-5,
y'=4:
x=3 (-5)-4+ 13=-6,
y=-2(-5)+4-10=4.
3Haqnr, B To<IKY N (-5; 4) nepeXO)l.HT TOqKa M (-6; 4).
<l>YHKlI,HIl, 3MalOIle npe06pa30BaHHe nJIOCKOCTH (I), HeJIb3J1 BbI-
6HpaTb npOH3BOJIbHO. OHII JI.OJILKHbI YJI.OBJIefBOpHTb YCJIOBIlIO, qTO OT06-
paLKeHHe (x; y)-+(q:> (x, y); 'i' (x, y» B3aHMHO-0JI.H03HaqHO. 31'0 3Haqnr,
IJTO OHII )l.OJI:>KHbI 6f:,m, ODpe.!l.eJleHbI )l.JlJI BCex nap (x; y) H, C )Q)yroii CTO-
pOHbl, )I.JIH .mo60H napbl IJUCM (a'; b') .n.omKHa HaHTOCb e,IUIHCTBeHHaH
napa qIlCeJI (a; b), TaKaH, qTO
q:> (a, b)=a', 'i' (a, b)= b'.
HanpHMep, «PYHKHH
{ x:= v x+y,
y = v x y
He 3a)l.aIOT npe06pa30BaHHH nJIOCKOCTH, TaK KaK Onpe)l.eJJeHbI JIUWb npH
YCJlOBHH, IJTO x+yO H x-yO. He 33.l(aJOT npe06paaoBaHHH ITJIOCKD-
CTH H 4'YHKlI,HH
{ X' =x'+y',
y' =x'-y",
WI

TaK KaK TOqKH A (a, b) H B (-a, -b) HMelOT
O)l.HH H TOT Me 06pa3 M (a 2 +b 2 , a 2 -b 2 ).
3r0T Me 06pa3 HMelOT TO'lKH C (a, -b) H
D (-a, -b). 3Ha'lHT,3TH <l>YHKII.HH He 3Ma-
lOT B3aHMHO-O)l.H03Ira'lHOrO oT06paMeHHSI
nJIOCKOCTH Ha ce6H.
DpH reOMeTpH'IeCKOM npe06pa30BaHHH q>
KaLK)l.aH reOMeTpHqeCKaH <l>Hrypa <D nepexo-
JI.HT B reOMeTpH'IecKYIO <l>Hrypy <D', Ha3bIBae-
MYIO ee 06pa30M npH npeo6pa30BaHHH q>.
<l>Hrypa <D' COCTOHT H3 TO'leK BH)l.a <p (A), r)l.e A - TO'lKa <l>HrYPbI <D.
HanjmMep, npH CHMMerpHH OTHOCHTeJIbHO npHMoH 1 cpHrypa <D nepe-
XOJI.HT B CHMMerpH'IHYIO eM: <l>HryPY (pHC. 120).
MOMer CJIY'lHTbCH, 'ITO reOMerpH'IecKoe npe06pa30BaHHe <p nepeBo-
)l.HT cpHrypy <D B ce6H, T. e. 'ITO <p (<D)=<D. DpH 3TOM He 06H3aTeJIbHO
KaMJI.aH TO'lKa cpHrypbI <D OCTaerCH HenO)l.BHMHOM: npH npe06pa30BaHHH
<po HanpHMep, npH CHMMerpHH OTHOCHTeJIbHO npHMOM:, npOXO)l.HIlI,eM: 'Ie-
pe3 II.eHTp OKPYMHOCTH, 3Ta OKPYMHOCTb nepeXOJI.HT B Ce6H. Ho Heno-
)l.BHMHbIMH OCTalOTCH JIHWb TO'lKH nepeCe'leHHH npHMoii H OKPYMHOCTH.
OcraJIbHble Me TO'lKH OKPYMHOCTH nepeXO)l.HT B CHMMerpH'IHbIe HM TO'l-
KH, JIe}KaIlI,He Ha Toii Me OKPYMHOCTH. .
P).
"47 ( ) $(
l
PHC. 120
¥npamaeaHJI
I. npe06pa30BaHHe nepeBOJl.HT TO'IKY M (x; y) B TO'IKY M' (x'; y'), rJ.{e
Ix'=-x+l,
\y'=-y+3.
a) HaiiJl.HTe KOOpJl.HHaTbI 06pa30B CJJeJl.YIOIII.HX TO'leK:
Md4; 5); M 2 (-4; -5); M 3 (-2; 7); M 4 (7; -2).
6) HaiiJl.HTe KOOpJl.HHaTbI rrpo06pa30B CJJeJl.YIOIII.HX TO'IeK:
Nd2; 4); N 2 (-3; 8); N 3 (4; -6); N 4 (-8; -9).
2. HaiiJl.HTe 06pa3 KBaJl.paTa ABCD, rJl.e A (3; 3), B (-3; 3); C (3; -3); D (-3;-3)
rrpH rrpeo6pa30BaHHH
{ X'=X-4
y'=y+2.
3. Ha'lepTHTe TpeyrOJlbHHK OBC, 3HaJl KoopJl.HHaTbI ero BeplllHH 0 (0; 0); B (5; 5);
C (0; 5). HaiiJl.HTe 06pa3bI 0', B', C' TO'IeK 0, B, C, npHMeHHB CJJeJl.YIOIII.He rrpeo6pa-
3OBaIlHJI, H nocTpoiiTe Bce rroJlyqeHHble TpeyroJlbHHKH Ha OJl.HOM qepTe}Ke, 3aKpacHB
HX B pa3Hble l\BeTa:
jx'=-x, { X'=X+2, Ix'=x-4, { X'=X-4,
a) \y'=y; 6) y'=y+3; B) ly'=y+2; r) y'=-y-6.
4. Ha'lepTHTe TpeyroJlbHHK OBC H3 npeJl.bIJl.YIIl.erO ynpamHeHHJI H HaiiJl.HTe 06.
pa3bI 0', B', C' TO'IeK 0, B, C npH CJJeJl.ylOIII.HX npeo6pa30BaHHJlx:
f x'=x-.i. y f x,=.i.x- y
5 5' 5 5'
a) ) 4 3 6») 3 4
 Y'=Sx+sY; { Y'=Sx+sy.
5. reoMeTpHqeCKOe npeo6pa30BaHHe 3aJl.aHO <jJopMYJlaMH
I x'=3x-5y+2,
\y'=2x+y-7,
192

HaiilXHre 06pa3 TOqKH M (1; 8). Haii.nHTe rrpoo6pa3 TOqKH N (-5; 2).
6. reoMeTpHqCCKOC npeo6pa30BaHHc 3alXaHO opMYJlaMH
I x'=x-2y+4,
\y'=2x+y-5.
Haii.nHTC 06pa3 KBa.l(paTa C BcplllHHaMH A (I; 3), B (I; 6), C(4; 6); D (4; 3). 51BJ1HeTCH
JlH 9TOT o6pa3 KBa.zq> aTOM?
7. Haii.nHTC npeo6pa30BaHHe, o6paTHoc rrpeo6pa30BaHHIO
{ X'=2x+3 y -1,
y'=x-2y+5.
8. Haii.nHTC KOMrr03HJJ.HH Ip1jJ H 'i'<P npe06pa30BaHHii
<p: { X;=X-3 Y +4,
Y =2x-y-1
'1" I x'=2x+2y-3,
. ly'=x+3y+ I.
H
9. * 5!BJlJlIOTCH JlH npeo6pa30BaHHHMH oT06pamcHHH, 3a.l(aHHble <jJoPMYJlaMli:
{ x'=x2+3 y2 ,
a) y'=x2-4y2;
6) I x'=3x+3y-I,
\y'=x+y+2?
10. * npOBCpbTC, '1TO '1CTblpC npeo6pa30BaHHJI, 3a.naBaCMblC <jJopMYJlaMIi
E.lx'=x,
'\y'=y,
<P o I x'=-x,
. \y'=y,
.1)' I x'=x, f). J x'=-x,
. \y'=-y, . \y'=-y,
06pa3YIOT rpynny.
11. * npOBCpbTC, '1TO npeo6pa30BaHHJI, 3a.naBaCMble <jJopMYJlaMH
E.I x'=x, <p : { x',=-y, .h: { x'=-x,
. \y'=y, y =x, 'I' y'=-y,
He 06pa3YlOT rpynny.
* 2. rpYllllA llEPEMEEHlln llJIOCKOCTH
II EE llOrpYllllbI
1. rpynna nepeMem.eHHH nnOCKOCTH. Ha30BeM reoMerpHqecKoe
npe06pa30BaHHe <p nepeMew,eHueM nAOCKocmu (HJIH, KOpOqe, nepeMew,e-
HueM), ecJIH OHO He H3MeHHeT paCCTOHHHH Me:>KJI.Y TOqKaMH nJIOCKOCTH.
I1HbIMH CJIOBaMH, <p HBJIHeTCH nepeMeIII.eHHeM nJIOCKOCTH, eCJIH JI.JIH JIIO-
6bIX JI.BYX TOqeK nJIOCKOCTH A H B BbIIIOJIHHerCH paBeHCTBO lA' B'I =
= lAB I, rJI.e A' - 06pa3 TOqKH A, a B' - 06pa3 TOqKH B npH 3TOM ne-
peMeIII.eHHH, A' =<p (A), B' =<p (B).
BOJIbIllHHCTBO reoMeTpHqeCKIlX <tmryp H HX CBOHCTB OnpeJI.eJIHIOTCSI C
nOMOllI.blO nOHSITHSI paccToHHHH MeLKJI.Y TOqKaMH. HanpHMep, OKPYLK-
HOCTb - 310 MHOLKecTBO TOqeK, HMelOIII.HX OJI.HHaKOBbIe paccToHHHH )1.0
)l.aHHoH TOqKH 0 (ueHTpa OKPYLKHOCTH), OTpe30K AB - MHOLKecTBO
TOqeK M TaKHX, qTO lAM! + 1M B 1 = IA B I, H T. JI.. T aK KaK nepeMeIII.e-
HHSI nJIOCKOCTH coxpaHHIOT paccToHHHH MeMJI.Y TOqKaMH, TO OHH nepe-
BOJI.HT OKPYMHOCTH B OKPYiKIIOCTIl C TeM Me paJI.HYCOM, OTpe3KH B OTpe3-
7 NI 'iu_
193

KH TOM w:e JI,:mHbI II T. JI.. D03Tm1Y, qT06bI H3HTH 06p33 OKPYW:HOCTI1 C
U.eHTpOM 0 II p3JI.HYCOM R npn 33JI.aHHOM nepeM€rn,eHHH, JI.OCTaTOqHO Hali-
TII 06P33 0 1 ee lI,eHTpa II nOCTpOHTb OKPYLKHOCTb C lI,eHTpOM 0 1 II P3JI.H-
YCOM R. A qT06bI nOCTpoIlTb 06pa3 OTpe3Ka AB, JI.OCTaToqHO HaRTH 06-
pa3bI A' II B' ero kOHlI,OB II nocrpoHTb oTpe30K A' B' .
JJ:oKa)l{eM, qTO c080KynHocmb Bcex nepeMeU{€Huu nAOCKocmu o6pCl3yem
cpynny, T. e. COJI.epIKHT BMeCTe C JII06bIMH JI.BYMH nepeMeeHHHMH IIX
KOMn03IllI,HIO, a BMeCTe C Ka)l{JI.bIM nepeMeeHHeM - 06paTHoe eMY npe-
06p330B3HHe. B caMOM JI.eJIe, nycTb <p Il'i' - JI.Ba nepeMeIII.eHHH nJIOCKO-
CTII. TaK KaK OHH He H3MeHHIOT paCCTOHHHR Me)l{JI.Y TOqK3MH nJIOCKOCTH,
TO II IlX KOMn03HlI,HH OCTaBJIHeT 3TH paCCTOHHHH HeH3MeHHhlMIl. 3HaqHT,
KOMn03HlI,HH JI.BYX nepeMeIII.eHHR HBJIHerCH nepeMeeHHeM. JJ:3JIee HCHO,
qTO eCJIH <p He H3MeHHer paCCTOHHHH Me:>KJI.Y TOqKaMH nJIOCKOCTH, TO H
06paTHoe npe06pa30BaHHe <p-1 OCTaBJIHer IIX HeH3MeHHbIMH. 3H3qHT,
eCJIH <p - nepeMellI,eHHe n.1OCKOCTIl, TO H <p-l TO)l{e HB.TIHeTCH nepe!e-
III.eHHeM. TaKHM 06pa30M, nepeMeU{eHUJl nAOCKOcmu 06pCl3YlOm cpynny.
Ha30BeM nJIOCKYIO <tmrypy <D' KOHzpyaHmHou nJIocKoii qmrype <D,
eCJIIl cecTByer nepeMeIII.eHHe <p, KOTOpoe OT06paIKaer <D H3 <D', T. e.
TaKoe, QTO <p (<D)=<D'. ECJIIl qmrYP3 <D' KOHrpY3HTH3 <pHrype <D, TO nH-
WYT: <D  <D'. JJ:OKa)l{eM, QTO omHOUJeHue KOHzpyaHml-locmu t/Jucyp o6.,1Q-
oaem C80ucmeaMU pet/JAeKCUBHOCmu, CUMMempU'iHxmu U mpaH..'1UmUB-
Hocmu.
B caMOM .neJJe, Jl106aSl <jJHrypa 11> caMa ce6e KOHrpY9HTHa, TaK KaK TOm)(eCTBeHHoe
npe06pa30BaHHe nJlOCKOCTH nepeBO,l\HT ee B Ce6S1, a To)!{,!(ecTBeHHoe npe06pa30BaHHe
HBJlSleTCSI nepeMeIIl.eHHeM. I1TaK, 11>11>, H nOTOMY OTHoweHHe KOHrpY9HTHOCTH pe-
tj!JleKCHBHo.
.L!.oKam:eM Tenepb, '1TO 9TO OTHoweHHe CHMMCTpH'IHO, T. e. E3 11>11>' BblTeKaeT
11)' 11>. B caMOM ,l\eJle, ecJlH II) e=11> , , TO CYIII.eCTBieT nepeMeIIl.eHHe qJ TaKoe, '1TO IjJ (<<1»=
=11>'. Ho TOr,l\a 06paTHoe npeo6pa30BaHHe Ip- oT06pamaeT 11>' Ha 11>, 1p-1 (11)')=11>,
npH'IeM 1p-1 TaKme SlBJlSleTCSI nepeMeIIl.eHHeM. n09TOMY 11>'11>. 3Ha'lHT, H3 YC.10B!!JI
1I)11>' CJJe,l\yeT 11>'11>.
HaKoHeu, ,l\OKameM TpaH3HTHBHOCTb OTHoweHHSI KOHrpY9HTHOCTH, T. e. nOKameM,
'1TO eCJJH 11>11>' H 11>' 11>", TO 11>11>". Ho YCJJoBHe 11>e=11>' 03Ha'laCT cYIIl.eCTBOBaHne
TaKoro nepeMeIIl.eHHIl1jJ, '1TO 1jJ()=I1>', a H3 YCJIOBHSlI1>'" CJJe.nyCT CYIII.ecTIJOBaHHe
TaKoro nepellleIIl.eHHJI Ip, '1TO Ip (11)')=11>''. Ho TOr,l\a W TaKm:e HBJlHeTCH nepeMeIIl.eHHCM,
npH'IeM Ip'IjJ oT06pamaeT 11> Ha 11>": (1p\jJ) (11))=Ip[1\J (11)}]=q) (11)')=11)". 3Ha'lHT, 11>2:"'«1>",
H OOTOMY OTOOUJeHHe KOHrpY3HTHOCTH TpaH3HTHBHO.
DOCKOJIbKY OTHOIIIeHHe KOHrpY3HTHOCTH pe<PJIeKCHBHo, CHMMeTpHQ-
HO n TpaH3HTHBHO, OHO HBJIHeTCH OTHOIIIeHHeM 3KBHBaJIeHTHOCTH H 3a-
.n3er pa36HeHHe cOBoKynHOCTH BCeX reoMerpHQecKHX <pIlryp Ha K.TJ3CCbI
9KBHBaJIeHTHOCTH.
n.nMeTOM WKOJIbHOH reOMerpHIl HBJISIerCSI B OCHOBHOM H3YQeHIle
CBOHCTB <pIlryp, He Il3MeHHIOIII.IlXCH npIl nepeMeIII.eHHSlX, T. e. CBOHCTB,
06III.HX BceM KOHrpY3HTHbIM Me)l{JI.Y co6oH <PHrYP3M. I1HbIMIl CJIOBaMH,
)l.JlSI reoMeTpHH He HMeer 3HaQeHIlH, B K3KOM Mecre nJIOCKOCTIl H306pa)l{e-
Ha JI.3HH3fl <pHrypa, KaK OHa nOBepHYTa H T. JI.., 3 B3)1{HbI JIHWb cpopMa H
pa3Mepbl 9TOll <PHrYPbI.
B )l.3JIbHeiiureM MbI 6YJI.eM IJacTO nOJIb30BaTbCSI TpeMSI H3BecTHbIMH H3
WKOJIbHOrO Kypca yrBep)l{JI.eHHSlMIl, YK33bIB31OIII.HMIl JI.OCTaTOIJHbIe yc-
JJOBHfl KOHrpY3HTHOCTH TpeyrOJIbHHKOB:
194

D tj
""
f "'.
I r ",.

f
Cl
PHC. 121
1) ECAU IAB!=IA'B'I, IBC!=IB'C'I, IACI=\A'C'I, mo mpeY20Ab"
HUKU ABC U A' B' C' KOH2pyaHmHbl.
...............  .........--
2) ECAU IABI=IA'B'I, !ACI=IA'C'I H BAC=B'A'C' (Qepe3 HAC
0603HaQaeTcH BeJIHqHH3 yrJI3 BAC), mo mpeY20AbHUKU ABC U A' B'C'
KOH2pyaHmHbl.
............... ........-- ............... ...............
3) ECAU !AB\=IA'B'I, BAC=B'A'C' u ABC=A'B'C', mo mpc.
YcOAbHllKU ABC U A' B'C' KOHcP!J:JHtnHbl.
DpB 9TOM, eCAU mpeycoAbHUKU ABC H A' B' C' He paBH06eopeHHbl, 1710
808cex mpex CAyttaflX cyw,ecmByem AU/ub OOHO nepe.J.tew,elltll' cp, mOKOe,4.mO
<p(A)=A', <p(B)=B', <p(C)=C'.
YnpamliellDJI
. I. nOCTpoiiTe paBHocTopOHHHii TpeyroJlbHHK ABC, .l\IIHHa CTOpOHbI KOToporo
paBHa 4 CM. Hail.l\HTC oupa3 3Toro Tpeyro.1bHHKa npH nepeMCIII.eHHH, fiepeaOJ,!Lle.1
TO'IKY A B TO'lKY A', a TO'lKY B B TO'IKY B'. O,H03HatIl!0 JlH pellleHlle 9Toii 3a.L(a'lH?
npOR3BOJlbHO JlH MomHO BblUpaTb TO'IKH A' H B'?
2. YKaJKHTe Ha pIlcYHKe 121 napbl KonrppHTHbIX <jmryp.
3. * LI,oKamRTe, '1TO npc06pa30BaHIISI
a ) j x:=y" (, ) { x:=-y, ) J x'=x+a,
\y =x: y =x; B \y'=y+b
SlBJJ5UOTCSI nepeMeU\eHIIJlMH n.OCKOCTH.
4. LI:oKaiKHTe, '1TO <jmrypa, KonrpY9HTHaJl pOM6y, JlBJlJleTCJI PO!60M.
5. .!JoKamllTc, '1TO eCJJH TO'lKH A, 13 H C JlCiKaT Ha O.L(HOli npJllOii, npH'IeM TO'II;:J
B JlemllT MemJlY A H C, H eCJlII A', B', C' - 06pa3bl 9THX TO'leK npR nepeMeIIl.eHHH q\,
TO B' JJemllT Ha oTpe3Ke A' C' .
6. .!JoKamHTe, '1TO eCJlR TO'IKH A H B Jle)!{aT no pa3Hble CTOpOHbl OT npSlMoi'! I
H A', B', l' - 06pa3b1 9THX TO'IeK H npJllOii npH nepeMeIIl.eHHIlIp, TO A' H B' Jle)j;,
no pa3Hble CropOHbI l'.
7. .!JoKamHTe, '1TO eCJlH yr.1b1 AOB H BOC HMelOT 06myIO CTopony OC H He m.le!c;;
HHblX 06U\RX TO'leK, a yr JlbI A' 0' B' H B' 0' C' - HX 06pa3b1 npH nepeMeIIl.eHlI1I (I).
TO 9TH yr.1b1 HMelOT 06myIO CTOPOHY 0' B' H He HMelOT HHblX 06II1.HX TO'leK.'
2. OceBaH CHMMeTpuH. TOqKa A' nJIOCKOCTH n H3SbIB3ercSi CUMfttl'-
mpUttHOU mottKe A omHocumeAbHO npRJtoU l, ecJIH OTpe30K AA' (pHC. 122)
7* 193

l
rreprreHJI.IlKYJlHpeH 3TOH npmWH II ero CepeJI.IlH3
JleMIlT H3 Heii. TOqKIl rrpHMoii I CqIlT3IOTC51 CHM-
A' MerpHIJHbIMIl C3MIl cece OTHOCIlTeJIbHO 3TOii rrpH-
MOH.
Ha30BeM oceBou cUM'Mempueu omHocumeAbHO
1 reOMeTpIlqeCKOe npeo()p330B3HIle S/, npIl KOTO-
pOM K3MJI.3H TOqK3 nJIOCKOCTH II nepeXOJI.IlT B
TOqKY, CIlMMeTpIlqHYIO CHell OTHOCIlTeJIbHO l.
DpHMYIO 1 H33bIBalOT OCblO CUMMempuu JI.JIH S/.
OIJeB, HJI.HO qTO Bce TOqKH OCH CHMMerpHIl OCT3-
npH npe06pa30BaHHH Sp T.
A
8=8'
Puc. 122
lOTCH HenoJI.BIlMHbIMH
(A E l) => S/(A)=A.
JJ:OK3MeM HeKOTopble CBoiicTB3 oceBOH CIlMMeTpHH.
a) ECAU S/(A)=A', mo S/(A')=A.
B caMOM JI.eJle, p3BeHCTBa S/(A)=A' H S/(A')=A 03Haq31OT OJI.HO
H TO Me :.....- OTpe30K AA' nepneHJI.IlKYJlHpeH npHMoii 1 H JI.eJIHTCH elO no-
n0JI3M (HJIIl A JIeMIlT H3 npHMoll l).
1-13 a) cJIeJI.yer, qTO S/2(A)=S/(S/(A))=S{(A')=A. T3KHM 06pa30M,
npeoopa30B3HIle S OCT3BJlHeT Bce TOqKIl nJlOCKOCTIl HenOJI.BIlMHbIMH,
T. e. HBJlHeTCH TOMJI.eCTBeHHbIM npe06p330B3HHeM E, Sl=E. Do-JI.PY-
rOMY 3TO MO}KHO C<POPMYJlIlpOB3Tb T3K: oceBafl cUMMempufl S/ C3M3 ce6e
06p3TH3, T. e. Si1=S/.
6) ECAU S/(A)=A' U M - mottKa ocu cUMMempuu, mo mOttKa M
paeHoyoaAeHa om mOtteK A U A' U YZOA MeJ/Coy npflMblMU 1 U MA KOH-
2pyaHmeH YZAY MeJ/Coy npflMblMu I U MA' (pIlC. 123).
B C3MOM JI.eJle, nycTb B - CepeJI.IlH3 oTpe3K3 AA'. Do onpeJI.eJIeHHIO
CHMMeTpIlIl IlMeeM IABI= IBA'I, npIlqeM yrJIbI MBA H MBA' npHMble.
TaK K3K, KpoMe TOro, TpeyrOJIbHIlKIl M BA II M BA' IlMelOT 06III.YIO CTO-
POHY MB, TO OHII KOHrpY3HTHbI, 6MBA  6MBA'. OrclOJI.a H CJIe-
AyeT, IJTO IAMI=IA'MI H A'MB=A'AfB.
B) Oceeafl CUMMempUfl R8AflemCJl nepeMeU{eHueM nAOCKOcmu.
B caMOM JI.eJle, nYCTb S/(A)=A' H S/(B)=B'. B03bMeM JII06ylO TOIJ-
KY M H3 OCH cIlMMeTpIlIl I (pIlC. 124). 1-13 YTBepMJI.eHIlH 6) cJleJI.yer, qTO
 /""-- 
IAMI=IA'MI, IBMI=IB'MI, AMT=A'MT II BMT=B'MT. Ho
TorJI.a AM'B=A'MB' II, CJle,n:OB3TeJlbHO, TpeyroJlbHIlKHAMB II A'MB'
T
s
HC. 1:23
Puc. 124
196

1/ Y
!1 (-X;Ij)
---
M(x;y)
--I
I
a
x
I
.
M'(X;-yJ
PHC. 125
PHC. 126
KOHrpY3HTHbI. n03TOMY lA' B ' 1= lAB I. TaKHM 06pasoM, paCCTOSlHHe
Me:>KJI.Y 06pa3aMH TOtIeK A H B paBHO paccToSlHHIO Me:>KJI.Y 3THMH TOtI-
KaMH, a 3TO H 03HatIaer, tITO S{ - nepeMern.eHHe nJIOCKOCTH.
ECJIH npH cHMMerpHH s.{ <tmrypa <D nepexOJI.HT caMa B COO5l, TO ee
Ha3bIBalOT CUMMempUttHOU OTHOCHTeJIbHO np5lMOH l, a 1 Ha3bIBalOT OCblO
CUMMempuu <t>HrYPbI <D. TaKaH <t>Hrypa COCTOHT H3 JI.BYX nOJIOBHH, nepe-
XOJI.51rn.HX JI.pyr B JI.pyra npH CHMMerpHH S{ (pHC. 125). cI>HrYPbI Moryr
HMeTb HecKOJIbKO oceR cHMMerpHH. .lI.oK3:>KeM, HanpHMep, tffO /ltOOlJH.
npRMOJI. l, npoxOOJUlll1R Ilepe3 l{eHmp 0 oKpyicH()cmu <D; R8AflemcR OCblO
uMMempuu amou OKpYJICHocmu. B caMOM JI.eJIe, nYCTb A - TOtIKa OK-
PYLKHOCTH H S{(A)=A I. TaK KaK u:eHTp OKPYLKHOCTH 0 JIeLKHT Ha IIp5l-
MOR l, TO lOA 1= lOA' I. A 3TO it 03HatIaer, tITO A' HaX0JI.HTC5I Ha TOM LKe
paCCTOHHHH OT u:eHTpa OKPYLKHOCTH, tITO H A, T. e. JIe:>KHT Ha TOR LKe
OKPYLKHOCTH <D. 3HatIHT, S{(<D)=<D.
113 CBoHCTBa 6) oceBoH cHMMerpHH BbITeKaer, tITO OCblO cUMMempuu
ye/la R8AflemcR npfl.MO,J/" Ha KOmopOU /leJICum eeo 6ucGeKmpuca, a OCblO
CUM.M.empUU Ompe3Ka - npflMaJ/., npoxOORUIpJ/. Ilepe3 eeo cepeouHY u nep-
nenauKljAflpHaR eMY. J1.pyeou OCbIO cUMMempuu Ompe3Ka R8AflemcJi npR-
MaR, Ha KOmopou OH /leJ/Cum.
ECJIH Ha nJIOCKOCTH BbI6paHa CHCTeMa KOOpJI.HHaT, TO TOtIKa M',
CHMMerpHtIHaSi TOtIKe M (x; y) OTHOCHTeJIbHO OCH a6cU:HCC, HMeeT KOOp-
JI.HHaTbI x H -y, a TOtIKa M", cHMMerpHtIHa51 TOtIKe M OTHOCHTeJIbHO
OcH OpJI.HHaT, HMeer KOOpJI.HHaTbI -x H y (pHC. 126). 31'0 3HatIHT, tITO
cHMMerpHH OTHOCHTeJIbHO OCH a6cU:HCC 3aJI.aerC5I <p<>pMYJIaMH
{ X' =x,
y'=-y,
a cHMMeTpHH OTHOCHTeJIbHO OCH 0pJI.HHaT - <t>opMYJIaMH
{z: ;X,
YupamaeaHJI
1. YKam:HTC OCH CHMMeTpHH <jJHryp, H306pIDKeHHblX Ha pHcYHKe 127.
2. KaKHe H3 3arJlaBHblX 6YKB pyccKoro an<jJaBHTa (pHC. 128) HMelOT XOTSI 61i
OAHY OCb CHMMCTpHH? KaKHe H3 HHX HMelOT HeCKOJlbKO oceH CHMMCTpHH?
197

0 O} 0
C 
e) 0)
PHC. 127
3. H a JlHcre 6YMarH H306pIDKeH yroJl A 0 B. C nOMO-
IUblO rreperHUaHHJI ,noro JlHCTa, He rrpHeHJIJI reoMeTpH'Iec-
KHX HHCTpYMeHTOB Haii.ll.HTe 6HcceKTpncy 9TOro yr.la.
4. H a JlHCTe 6YMarH H306pameHbI TPH TO'lKH A, B H C.
C nOMOIIl.blO nepenl6aHHJI 9Toro JlHCTa HaH)(HTe:
a) UeHTp oKpymHocTH, OIJI!CaHHOii BOKpyr Tpeyro,1b-
HHKa ABC;
6) eHTp oKpymHocTH, BnHcaHHoli B TpeyrOJlbHHK ABC.
5. KaKHe rrpJlMble nepeXO)(JlT B ce6J1 rrpH cHMMeTpHH
OTHOCHTeJJbHO rrpJlMoii l?
6. KaIrne OKPYJKHOCTn nepeXO)(SlT B ce6J1 rrpH cHMMeTpHB
OTHOCHTeJlbHO npJlMOii l?
7. KaKne rrpJlMbie nepexO)(HT npn oceBOii cHMMeTpHH B
napaJlJle.'lbHble HM rrpSiMbIe?
8. nOCTpOHTe TpeyroJlbHHK ABC TaKoii, '1TO IBCI=7 CM,
CAI=6 CM, IABI=5 CM. Ha CTOpoae AB OTMCTbTe TO'll{)' X
Ha paCCTOJlHHH 1 CM OT TO'lKH B, lIa cTOpoHe AC - TO'IKY Y
Ha paccTOHHHH 3 CM OT TO'lKH C. nocTpoiiTe TpeyrOJlbHHK,
cHMMeTpH'IHblH TpeyroJlbHHKY ABC OTHOCHTeJlbHO OCH XV.
9. nepe'lepTHTe <jJHrypbl, H306pameHHbIe Ha pHcYHKe 129, H nocTpoiiTe CHMMeT-
pH'IHble HM <jJltrypbl OTHOCHTeJJbHO OCH I.
lD. .lI.oKamHTe, '1TO 06'be)(HHeHHe <jmrYPbl <II C CHMMeTpH'IHOA eA OTHOCIlTeJJbllO
npSlMoii I <jmrypoii <II' cnMMeTpH'IHO OTHOCHTeJlbHO 9T011 npSlMoii.
II. .lI.oKamHTe, '1TO nepece'leHHe <jJHrYPbl <II C cHMMeTpH'IHOH eH OTHOCHTeJJbHO
npSlMoH I <jJHrypoH <II' CHMMeTpH'IHO OTHOCHTeJlbHO 9TOii rrpSiMOH.
12. .lI.oKaJoKHTe, '1TO eCJlH )(Be oKpymHocTH HMelOT paBHble pa,/J,HYCbI, TO OBH CHM'
MeTpUlJHbI OTHOCHTe.lbHO HeKoTOpoii npSlMoiL
13. KaKHe H3 nepe'lHCJleHHbIX HHJKe <jmryp HMelOT OCH cHMMeTpHJI: pa3HocTO-
pOHHHii TpeyroJlbHHK; paBHo60lJHaSi TparreUHSI; paBHocTopOHHHii rpeyro.%Hlm; na.
A68rAE
>K3V1K/\
MHOnp[
TYcpXll.4
WLlLb bbl
3[)
PHC. 128
L
o
I
0)
19&
L
L
1;J
0) I
tJ)
PHC. 129

pa,Me.10rpaMM; pOM6; KBa.npaT; nOJlYKpyr, CeKTOp; CCrMeIlT; Kpyr C rrpOBe,l\eHHbIMR
K IIC1.1Y 113 TOlJKH KaCaTeJJhllblMH; Kpyr C npOSe,'\eHHblMH ,'\3YMH 1J3ZIIMHO rreprrell,l\H-
KYJlHpHbI1H JHaMCTpaMH; <jJHrypa, COCTOSlIIl.aSl 113 ;'\BYX KaCalOIII.HXCSI KOHrpY9HTHbiX
oKpymHoCTeH? KaKHe H3 9TH X <jJllryp HMelOT 60Jlee O.l(IIOii OCH cHMMeTpHH?
14. CKOJlbKO oceii cHMMeTpHH HMeeT <jJHrypa, 06pa30BaHllan:
a) .l(BYMSI rrapanJIeJJbHblMH npnMblMH; 6) ,l\BYMSI rreprreH,l\HKYJlnpHblMH rrpSlMblMH?
15.. HatlepTHTe TpeyroJlbHHK ABC C BepWHHaMH A (3; I), B (-I; 4), C (0; -5).
nOCTpoHTe TpeyrOJlbHHKH, CHMMeTpH'IlIbIe C ABC OTHOCHTeJJbllO
a) OCH a6cJJ.Hcc, 6) OCH Op.l(HHaT, B) 6HcceKTpHCbI rrepBoro H TpeTbero KOOp.nH-
HaTHbIX yrJloB.
16. .1JoKamHTe, '1TO cHIMeTpHSI OTIIOCHTeJJbHO rrpnMoii y=x BblpamaCTcSI <jJop-
MYJlaMH
{ x;=y,
y=x.
17. * .1JoKmKHTe, '1TO rpa<jJIIK cHMMeTpH'IHOrO OTlIoweHHH Mem,l\y '1HCJJaMH x H Y
cHMMeTpHlJeH OTIIOCHTeJJbHO npSlMoii y=x.
18. nocTpoiiTe TpeyrOJlbHHK, cHMMeTpH'IHblii TpeyroJlbHHKY ABC OTHOCHTeJJbHO
ero cpe.iiHell JlIIHHH DE.
19. nocTpoihe OKppKHOCTb C UeHTpOM 0 (1; 2) H pa.l(HYCOM 5 H npoae,l\HTe rrpSlMYIO
y=x-2. nocTpoflTe OKPYJKIIOCTb, CHMMeTpH'IHYIO .l(aHHOH OTHOCHTeJlbHO 9TOii rrpm'loii.
20. nocTpoDTe napaJlJleJJbHbIe npSlMbIc I H m, HaXO.l(SlIII.Hecn Ha paccTOnHHIi 3 CM
APyr OT .l(pyra. Ha'lepTHTe TpeyrOJlbHHK, He nepeceKalOIII.HiicSi C 9THMH rrpSlMblMH,
H c,l\e.1afITe Clla'laJla oceaylO CHMMCTpHIO S{, a no TOM Sm.
3. napaJIJIeJIbHhlH nepeHOC. DpOBeJI.eM Ha nJIOCKOCTH JI.Be napaJI-
JIeJIbHbIe npHMble 1 H m; BbIIIOJIHIlM CHaIJaJIa oceBYIO cHMMeTpIlIO S{, a
nOTOfI,1 Sm. DYCTb npIl nepBOH Il3 3THX CHMMeTpHH TOIJKa A nepeXOJI.IlT
B TOIJKY A', a npIl BTOpOH TOIJKa A' nepeXOJI.HT B TOIJKY A". TorJI.a npH
nepeMellI,eHHH S mS { TOIJKa A nepeXOJI.HT B TOqKY A". T aK KaK oTpe3KH
AA' H A'A" nepneHJI.HKYJIHpHbI npHMbIM 1 H m, TO II OTpe30K AA" nep-
neHJI.ll:KYJIHpeH HM. DpH 3TOM TaK KaK IABI=IBA'I H IA'CI=ICA"I
(pIlC. 130), TO JI.JIIlHa OTpe3Ka AA" BJI.BOe 60JIbIIIe paCCTOHHHH d MeLK.n:y
npHMbIMII 1 H m, IAA"/=2d.
i\1bI JI.OKa3aJIH CJIeJI.YIOllI,ee YTBepiKJI.eHIle:
Ec,w flpRMble 1 /J m napaAAeAbHbl, mo OAfl AI060ii mottK/J A flAOClwcmu
ompe30K AA", coeOuHfllOUI,UU mottKY A c ee 06pa30M npu npeo6pa308aHuu
S mS I' nepneHOUKlj/lflpeH. amUM npflMblM U eeO OAUHa eoeoe 60Abwe pac-
cmOflHWl ftteJ/Coy npflMbtMU 1 U m.
31'0 )TBepiK.il.eHHe MOiKHO YTOIJHHTb, paCCMaTpIlBaH nanpaBJIeHHble
OTpe3KH. Ha30BeM OTpe30K AB HanpaeAeHHblAt, ecJIIl YKa3aHO, KaKYIO
H3 TOqeK A, B HaJI.O CIJHTaTb nepBoH, a KaKYIO - BTOpOH. HanpaBJIeH-
--r
HbIH bTpe30K 0603HaIJHM AB (pIlC. 131). OH A"
3aJI.aeTCH ynopHJI.OqeHHOf! napoH TotJeK (A; B).
HanpaB.'IeHHbIH oTpe30K BA Ha3bIBaeTCH npo- m C
mU80nO;lO;)ItHblM HanpaBJIeHHoMY oTpe3KY AB.
1-13 pHcYHKa 130 BIlJI.HO, Iff 0 HanpaB.'IeHIIH
. --r--+
OTpe3KOB AA" II BC OJI.HHaKOBbI. 31'0 3HaqHT,
--r
Iff 0 OTpe30K AA" IIMeeT TO iKe HanpaBJ'Iem!e,
Iff 0 H 06III.HH nepneHJI.HKYJIHp BC K npHMblM
A'
8
fA
PHC. 130
199

A
.


AB


8
-..


PI1C. 131


A1 8 !!nJ


PHC. 133


A A' B B'
.
. . =--.
0)
PHC. 132


--+
1 H m H B)l.Boe )l.JIHHHee ero, npHIJeM HanpaBJIeHHe OTpe3Ka BC HJI.er OT
TOIJKH, JIeLKa
eH Ha npHMoH t, K TOIJKe, JIeLKallI,eii Ha npHMoii m.


--r --r
OTpe3KH AA ' H BB', He JlemaIIl.He Ha O.l(Hoii npSlMoii, O.l(HHaKOBo HanpaBJIeHbI
B TOM H TOJlbKO TOM CJlY'Iae, Kor.l(a JlY'IH AA' H BB ' napaJlJlCJlbHbI H JlemaT no O.l(ny
--r --r
CTOpoHY OT npSlMoii AB (pHC. 132, a). ECJJH me AA' H BB ' JlemaT na o.l(Hoii npSlMoii,
TO OHH O.l(HHaKOBO HanpaBJleHbI B TOM H TOJlbKO TOM CJJY'Iae, KOr.l(a O.l(HH H3 JlY'Ieii AA I,
BB' CO.l(eQmHT BTOpoii (pHC. 132, 6).
Ha30BeM reoMeTpHIJeCKOe npe06pa30BaHHe, npH KOTOpOM Bce TOIJKH
nJIOCKOCTH nepeMellI,alOTCH B OJI.HOM H TOM LKe HanpaBJIeHHH Ha OJI.HO H
TO LKe paCCTOHHHe (pHC. 133), napaAAeAbHblM nepeHOCOM HJIH eeKmopOM
->
(TaKoe npe06pa30BaHHe 0603HaIJalOT a). PaccToHHHe MeLK)l.Y TOqKaMH
H HX 06pa3aMH Ha3bIBalOT OAUHOU eeKmopa, a HanpaBJIeHHe oTpe3KoB
-> ->
AA ' - HanpaBAeHUeM eeKmopa. .lI.JIHHY BeKTopa a 0603HaIJaIOT lal.
.lI.oKa3amroe BbIwe YTBepLK)l.eHHe 03 Haqaer , qTO KOAtn03Ul{Wl CUM-
Mempuil. omHOCUmeAbl-:O napaAAeAbHblx npflMblX t U m R8AflemCfl napaJl-
lteAbHblM nepeHocoM e HanpaBAeHUU, nepneHOUKYAflPHOM amUM npflMblM
u uOYW,UM om I K m, /-ta paccmoflHue eoeoe 6oAbwee, tteM paccmoflHue
MeJ/Coy amUMU npfl.MbIAttl.
CnpaBeJI.JIHBO H 06paTlioe YTBepLKJI.eHIIe: AI060ii napaJl/leAbHblu nepe-
->
HOC a MOJ/CHO npeocmaeum/) e p,uoe KOMn03Ul{UU CUMMempuu OmHOCUmeAb-
HO oeyx napaAAe.1b/-tbIX np.<i.MbIX. .lI.JIH 3Toro )l.OCTaTOqHO B3HTb JII06bIe
npHMbIe, nepneJlJI.HK)'nHpJlbIe HanpaBJIeHHIO nepeHoca H HaXOJI.H
HecSi
->
Ha B)l.BOe MeHbweM paCCTOHHI1H, qeM )l.JII1Ha BeKTopa a, H COOTBerCTBYIO-
llI.HM 06pa30M Bbl6paTb nOpHJI.OK CHMMeTpHii t H m.
1-13 onpe,.'I,eJIeHHH napaJIJleJIbHOrO nepeHoca CJIe)l.yeT, qTO OH noJI-
HOCTblO onpeJI.e.TJHeTCH 33.aaHlfeM 06pa3a KaKoii-HH6YJI.b TOqKH: eCJIH npH
napaJIJIeJIbHOM nepeHoce ;; TOllKa A nepeXOJI.HT B TOqKY A I, TO Bce OC-
TaJIbHbIe TOqKH nepe
H:'UL3IOTCH B TOM }!{e HanpaBJIeHHH H Ha TO LKe caMoe


200




paCCTOHHHe. I103TOMY, tITOObI nOCTpOHTb 06pa3 II:
TO'IKH B npH TaKOM nepeHOCe, Ha.l(O npOBeCTH

'Iepe3 Hee JIY'I, HMelOw.HH TO}Ke HanpaBJIeHHe, A
'ITO H JIY'I AA', H OTJIO>KHTb Ha HeM OTpe30K

BB', HMelOw.HH TY >Ke)l.JIHHY, 'ITO AA' (pH
. 134). 8 1
I1ocKOJIbKY napaJIJIeJIbHbIH nepeHoc a O)l.HO-
3Ha'IHO Onpe)l.eJIHeTCH 3a)1.aHHeM 06pa3a A' HeKO- B
TOpOH TOIIKH A, MbI 6Y)l.eM )l.JIH KpaTKOCTH BMe- PHC. 134
CTO «napaJIJIeJIbHbIH nepeHoc, npH KOTOpOM TO'I- __
Ka A nepeXO)l.HT B TO'IKY A'» rOBopHTb «napaJIJIeJIbHbIH nepeHOC AA'».
Pa3YMeeTCH, Ha.l(O CJIe)l.HTb 3a TeM, 'IT06bI He cnYTaTb 3TOT napaJIJIeJIb-
--
HbIH nepeHOC C HanpaBJIeHHbIM OTpe3KOM AA'.
-+
ECJIH TO'IKa A nepeXO.l(HT npH napaJIJIeJIbHOM nepeHoce a B TO'IKY A',
a TO'IKa B JIe}KHT Ha npHMoH AA', TO H ee 06pa3 B' JIe}KHT Ha TOH >Ke
npHMOH - Be)l.b HanpaBJIeHHH AA' H BB' O.l(lIHaKOBbI. B 3TOM cJIYllae
npHMaH AB nepeXO.l(HT caMa B ce6H. TaKHM o6pa30M, napG/l/le/lbHb111
....
nepeHoc a nepee08um e ce6fl eee npflMble, uMelOl-/1ue mo JlCe HanpaeAeHUe,
....
'-Imo U a.
I1pe)l.nOJIO}KHM Tenepb, 'ITO TO'IKa B He JIe}KHT Ha npHMOH AA' H
....
liTO B' - ee 06pa3 npH nepeHoce a, oT06pa}KalOIl1.eM A B A'. TaK KaK
HanpaBJIeHHH JIY'IeH AA' H BB' O)l.HHaKOBbI, TO TO'IKH A' H B' JIe}KaT lIO
O.l(HY CTOPOHY OT npHMoH AB (pliC. 135). I10 TY }Ke CTOPOHY OT (AB)
JIe}KaT H Bce TO'IKH npHMoH A' B'. OTCIO)l.a CJIe.l(yeT, 'ITO o6pa3 A' B' npJl-
MOU AB npu napG/l/le/lbHOM nepeHoce napa/l/le/leH Bmou npflMou. B caMOM
.lI.eJIe, eCJIH 6bI npHMbIe AB H A' B' nepeceKaJIHCb B KaKoH-HH6Y.l(b TO'IKe
C, TO npHMaH A' B' HMeJIa 6bI TO'IKH no 06e CTOpOHbI OT AB H He MOrJIa
6bI JIe}KaTb no O.l(HY CTOPOHY OT Hee.
-+
HTaK, eC/lU [A' B') - o6pa3 [AB) npu napa/l/le/lbHOM nepeHoce a,
-- --
mo lA' B'I= IABI u [A' B'JiI[AB]. I1pu BmOM ompe3KU A' B' U AB UMeIOm
08UHaK08ble Hanpae/leHUfl. 31'0 CJIe)l.YeT H3 TOro, 'ITO B CHJIY napaJIJIeJIb-
HOCTH npHMbIX AA' H BB' 3TH OTpe3KH JIe}KaT no O)l.HY CTOPOHY OT npS!-
MOH AA'.
.iloKa3aHHoe YTBeplK.l(eHHe MOIKHO CcpopMYJII!pOBaTb CJIe.l(YIOIl1.HM 06-
--+ --+
pa30M: eC/lU ompe3KU AA' U BB' UMeIOm 08uHaKoeylO 8/lUHY U 08HO U mo
--+ --+
3ICe Hanpae/leHUe, mo U ompe3KU AB H A' B' o6/la8alOm meM JlCe ceoucm-
80M. OrclO)l.a BbITeKaeT CJIe)l.YlOIl1.ee He06xo.l(HMOe
H )l.OCTaTO'IHOe YCJIOBHe )l.JIH TOro, 'IT06bI reOMeT-
pH'IeCKOe npe06pa30BaHHe <p 6bIJIO napaJIJIeJIb-
HbIM nepeHOCOM:
r eOMempU'-IecKoe npe06pa3oeaHue <p fle/lflem-
iffl napa/l/le/lbHbIM nepeHOCOM e mOM U mO/lbKO e
mOM C/l!lfae, eC/lU 06pa3 /l106oco Hanpae/leHHOcO
ompe3Ka UMeem my JlCe 8/lUHY U mo JlCe Hanpae-
/leHUE, limo U Bmom ompe30K. PHC. 135


201




B caMOM )l.eJIe, MbI Y>Ke BH)l.e-
JIl!, 'ITO npH napaJIJIeJIbHOM nepe-
Hoce )l.JIHHa H HanpaBJIeHHe 01'-
--+ ---+
pe3Ka AB H era 06pa3a A' B'
COBna)l.alOT. IIYCTb, 06paTHo, )l.JIH
 --+
JIr060ra OTpe3Ka AB era 06pa3
---+
X A' B' HMeeT TY >Ke )l.JIHHY H TO >Ke
HalIpaBJIeHHe, 1fT0 H 3TOT OTpe30K.
Tor)l.a no )l.OKa3aHHOMY BbIIlle YT-
Bep}K)l.eHMlO .31'0 }Ke CnpaBe)l.JIHBO
--+ --+
H )l.JIH HanpaBJIeHHbIX OTpe3KOB AA' H BB'. A 3T0 H 3Ha'IHT, 'ITOnpH
npe06pa30BaHHH 'P OTpe3KH, coe)l.lIHHlOIl1.He TO'IKH nJIOCKOCTH C HX 06pa-
3aMH, HM€IOT O)l.HHaKOBbIe )J,JUlHbI H O)l.HHaKOBbie HanpaBJIeHHH, T. e.
cp - napaJIJIeJIbHbIH nepeHoc.
--+ ---+
OrMeTlIM, 'ITO eCJIH OTpe3KH AB H A' B' HMelOT O.l(UHaKOBbIe )l.JIHHbI H
--+ ---+
O)l.HHaKOBbIe HanpaBJIeHHH, TO A B nepeXO.l(HT B A' B' npli napaJIJIeJIbHOM
nepeHoce, oT06palKalOIl1.eM TO'IKY A B TO'IKY A'.
HaH,1I.eM Koop)l.HHaTbI 06pa3a M' TO'lKH M (x; y) npli napaJIJIeJIbHOM
nepeHoce . I1YCTb npH .3TOM nepeHoce Ha'IaJIO KOOp)l.HHaT 0 (0; 0) nepe-
XO)l.HT B TO'IKY 0' (a; b). H3 pHcYHKa 136 BH.l(HO, 'ITO Koop)l.HHaTbi x'
H y' TO'IKlI M' BbIpalKalOTcH 'Iepe3 'IHCJIa a, b 1I KOOp)l.HHaTbI x, y TO'IKH
M no cpopMYJIaM
!J . M'(x;y'J
---a---x-1
,--
, - OYa.;b) )
y ___x;y} b
I!J X I
lJ
,
-,
X'
PHC. 136
{ x'=x+a,
y' =y+b. (1)
MOIKHO )l.OKa3aTb, 'ITO .3TlI cpopMYJIbI OCTalOTCH cnpaBe.n.,7IHBbIMH npH JIlO-
6bIX 3HaKax KOOp)l.HHaT.
II p H M e p. I1pH napa.nJIeJIbHOM nepeHoce KOOp)l.HHaT TO'IKa
A (3; -4) nepeIll.lla B TO'IKY A' (7; 2). B KaKYIO TO'IKY nepeIll.'la TO'IKa
B (5; -8)?
CHa'IaJIa Hai1.l(eM KOOp.l(HHaTbI TO'IKH, B KOTOpylO nepeIllJIO Ha'IaJIO
KOOp)l.lIHaT 0 (0; 0). .ilJIH :OHoro B cpopMYJIbI (1) nO)l.CTaBliM BMeCTO x KO-
Op)l.HHaTbI TO'IKH A, a BMeCTO x' H y' - KOOp.l(HHaTbI ee o6pa3a, T. e.
TO'IKH A'. MbI nOJIY[lHM, 'ITO
7 =3+a, 2=-4+b, H lIOTOMY
a=4, b=6. 3Ha'IHT, cpopMYJIbI KO-
'\ Op)l.HHaT 06pa3a npH )l.aHHOM na-
h paJIJIeJIbHOM nepeHoce HMelOT CJIe-
JJ:YIOIl1.HH BH)I.:
{ x'=x+4,
y' =y+6.
1 7
a/
f
:.
g/
/'
d
..
e
..
A Tenepb nO)l.CTaBHM B 3TH cf>oPMY-
JIbI BMecTO x H y KOOp)l.HHaTbi TOq.
PHC. 137
1D2

Kli B. I10JIY'IaeM x' =5-1-4=9, y' =-8+6=-2.
3HaQHT, TOQKa B (5; -8) nepeXO.l(HT B TO'IKY
B'(9; -2).
YopamoeOBH
Dn C

A 8
J. Cpe.l\H HanpaBJleHHblX OTpe3KOB, H306paJKeHHblX
H3 pHcYHKe 137, HaH.l\HTe napbl OTpe3KOB, HMelOlI\HX
O.l\HHaKOBble ,lI,J1HHbI H HanpaBJIeHHH, a TaKJKe napbl oTpe3-
KOB, HMelOlI\HX O.l\HHaKOBbie MHHbi H npoTHBOrr0J10JKHble
HanpaBJIeHHH. ITocTpOHTe .l\JIH 3'I'Oro MHOJKeCTBa OTpe3KOB
rPa4> OTHOlIIeHHH «oTpe3KH O.l\HHaKOBO HanpaBJ1eHbI H HMelOT paBHble .l\J1HHbI». 51B-
J1HeTCH J1H 3'I'O OTHoweHHe 3KBHBaJIeHTHOCTbIO?
---+ ---+ ---+ --+- ---+ --+
2. ITocTpOllTe HanpaBJJeHHbie oTpe3KH AB, CD, EF, OF, AC, BD, r.l\e A (-1; 2),
B(O; 5), C(3; -7), D(4; -4), £(3; 2), F(2; -1), 0(4; 0). ECTb JlH cpe.l\H H!IX OT-
pe3KH, HMelOlI\He O.ll.HHaKOBble MHHbi H HanpaBJIeHHH? O.l\HHaKOBble .l\J1HHbI H npoTH-
BOn0J10JKHbJe HanpaBJIeHHH?
3. ITpH napaJ1J1e.JIbHOM nepeHoce TO'IKa A nepeXO.l\HT B TO'lKY A' (pHC. 138). HaH-
)J,HTe 06pa3 KBMpaTa ABCD npH 3'I'OM nepeHm:e.
4. ITpoBe.l\HTe napaJJJle.JIbHble npHMble 1 H m Ha paccTOHHHH 3 CM ,aPyr OT .l\pyra.
Ha'iepTHTe rrapaJIJ1e.JIOrpaMM ABCD no O.l\HY CTOPOHY OT npHMoii 1 H Hai'I.l\HTe ero
o6pa3b1 npH oceBoii CHMMeTpHH Sl H napaJ1J1eJ1bHOM nepeHoce SmSI' ITpOBepbTe,
COBnMalOT J1H 06pa3b1 3'I'Oro napaJ1J1e.JIOrpaMMa npH napaJIJ1eJ1bHblX nepeHocax SlS"
H SmSI'
--+ --+
5. LLOK3JKHTe, '1TO ecJ1H oTpe3KH AB H CD HMelOT O.l\HHaKOBbie MHllbi H HanpaB-
JleHHH, TO cepe.l\HHbI oTpe3KoB AD H BC COBnM3JOT. LLOKaJKHTe, 'ITO, 06paTHo, ecJIJI
. --+--+
Cepe.l\HHbI oTpe3KoB AD H BC COBna.l\3JOT, TO OTpe3KH AB H CD HMelOT O.l\HHaKOBble
,lI,J1HHbI H HanpaBJ1eHHH.
---+ --+
6. HaqepmTe TpeyrOJ1bHHK ABC H ,ll,Ba HanpaBJ1eHHbIX OTpe3Ka MN H PQ.
. ---+--+
ITepeHeCHTe TpeyroJ1bHHK ABC CHa'l3JJa H BeKTOp MN, a nOTOM Ha neKTOp PQ.
--+ ---+
ITOCJIe .:noro nepeHeCHTe TpeyrOJ1bHHK CH'13JJa Ha aeKTOp PQ, a HOTOM Ha BeKrop MN.
Cp3BHHTe pe3YJ1bTaTbi o6oHx nepeHocoB.
7. q)Hrypa $' nOJ1Y'lHJIaCb H3 (ImrYPbl $ C nOMOlI\blO napaJ1.eJ1bHOrO nepeHoca
....
a. KaK nOJIy<IHTb \f!urypy $ H3 \f!Hrypbl $'?
8. CTOpolThl AB B CD npnroyroJIbHHKa ABCD napaJIJ1eJ1bHbI npHMoii 1. ITyCTlt
A' B'C' D' -06pa3 3'I'Oro npHMoyrOJ1bHHKa npH CHMMeTpHH Sl' LLOKaJKHTe, '1TO A' B'C'D'
MOJKHO nOJ1Y'lHTb H3 ABCD H C rrOMOlliblO nap3JJJ1eJ1bHOrO rrepeHoca. 8 KaK)'1O
TO'IKY nepeH,AeT npH 3'roM nepeHOCe TO'lKa A?
. ....
9. KaKHe npHMb!e nepexo)l.HT raMH B ce6H npH napaJIJleJlbHOM nepeHoce a? KaKJ!e
OKPYJKHOC1'B nepexo)J,n caMH B cOOH npH 3TOM nepeHoce?
10. KaKHe napa.nJIeJlbHbJe nepeHOCbI nepeBO)I.JIT caMY B ceuJI npHM)"1O l?
11. TpeyroJ1bHHK D£K nOJ1Y'IeH H3 TpeyrOJ1bHHKa ABC napaJ1J1eJ1bHblM nepe-
HOCOM, npH'IeM TO'IKa A nepeXO.ll.HT B m'lKY D. ,aoKaJKHTe, '1m Me}UfaHbI AM H DB
nap3JJJ1eJJbHbI.
12. ,a:m TpeyroJlbHHK ABC. ITocTpoiiTe TpeyrOJ1bHHKH, no.rry'laeMbIe H3 Hero
--+ --+ --+
nap3JLlleJIbHhlMH nepeHocaMH AB, BC, CA.
13. ITyCTb A - TO'IKa Ha OKPYJKHOCTH H 0 - neInp 3TOH OKpYJKHOCTH. ITepeHe-
CHTe oXPYXalocn. TaK, Qr06h1 1t)lfKa 0 nepewJ1a B TO'IKY A.
14. ITpH napaJIJ1e.JIbHOM nepenoce TO'IKa A (3; 4) nepeIllJl3 B TO'iKY A' (-1; 2).
Ha1lepTHTe TpeyroJ1bHHK, B KOTO'pbJii nepeJJreJl TpeYrOJIbHHK BCD: B (I; 6), C (5; 0),
D (-3; 1).
15. Ha'iepTHTe npmroyrOJ1bHnK C BeplIIHRaMH A (I; 6), B (5; 6), C (5; 2), D (I; 2).
IJepette<:JFre era napaJJ.l!eJIbHO TaK, '1r06bJ neH'Tp n1H1Moyr0J1bHHKa nonaJ1 B TOIJKY
£(-1; 2).
PHC. 138


4.' OnepaHH HaJJ. napaJIJIeJlbHblMH nepe-
8" HocaMH. TaK KaK Ka}K.l(bIH napaJIJIe.nbIIbI i1
nepeHoc - nepeMeIl1.eHHe nJIOCKOCTH, TO Ii
KOMn03Ul{Ufl 8eyx napa/l/le/lbHblX nepeHOCOB
JU3/lflemCfl nepeMeUlf!HueM. .iloKa}KeM, 'ITO Bma
KOMn03Ul{Ufl mO:J/Ce napa/l/le/lbHblll nepeHOC,
T. e. 'ITO OHa nepeMeIl1.aeT Bce TO'IKH nJIOC-
KOCTH B O.l(HOM H TOM >Ke HanpaBJIeHHH Ha
O.l(HO H TO >Ke paCCTOHHHe. B03bMeM TO'I-
KH A H B. IIYCTb npH napaJIJIeJIbHOM ne-
....
peHoce a OHH nepeXO.l(HT COOTBeTCTBeHHO B TO'IKH A' H B', a npH na-
....
paJIJIeJIbHOM nepeHoce b TO'IKH A I H B ' nepeXO.l(HT B TO'IKH A" H B"
-- ---
(pHC. 139). Tor.l(a, KaK 6blJIO nOKa3aHO B n. 3, oTpe3KH AB H A' B ' HMe-
--+-
lOT O)l.HHaKOBbIe .l(JIHHbI H HanpaBJIeHHH, paBHO KaK H OTpe3KH A' B' H
-.,.,.+ -- ---
A"B". II03TOMY OTpe3KH AB H A"B" TaK>Ke HMelOT O.l(HHaKOBble .l(JIHHbl H
HanpaBJIeHHH. TaKHM 06pa30M, npH KOMn03HlLHH napaJIJIeJIbHbIX nepe-
-... -... -+ ---+
HOCOB a H b Ka>K)l.bIH OTpe30K AB nepeXO.l(liT B OTpe30K A"B", liMelOIl1.lIH
TY }Ke .l(JIliHY li TO >Ke HanpaBJIeHlie. OTCIO.l(a B CliJIY He06xo.l(liMOrO li .l(0-
CTaTO'IHOrO npli3HaKa napaJIJIeJIbHOrO nepeHoca CJle.l(yeT, 'ITO 3Ta KOM-
n03lilLliH caMa HBJIHeTCH napaJIJIeJIbHbIM nepeHOCOM. Mbl Y}Ke rOBOpliJIli,
'lTO napaJIJIe.nbHbIe nepeHOCbI Ha3blBalOT liHa'Ie BeKTopaMH. BeKTOp,
.... ....
COOTBeTCTBYIOIl1.lIH KOMn03HlLHH napaJIJIe.nbHbIX nepeHOCOB a li b, Ha3bI-
-... -... -... -...
BalOT CYMMOIl eeKmopoe a li b li 0603Ha'IalOT a+b.
.... ....
4T06bI onpe.l(eJIliTb BeKTOp a+b, .l(OCTaTO'IHO 3HaTb 06pa3 XOTH 6bI
OJI.HOH TO'IKH A. .ilJIH 3TOro CHa'IaJIa HaH.l(eM 06pa3 B 3TOH TO'IKli npli
....
napa.n.neJIbHOM nepeHoce a, a nOTOM 06pa3 TO'IKli B npli napaJIJIe.nbHOM
.... .... ....
nepeHoce b. Tor.l(a npli oT06pa>KeHlili a+b TO'IKa A nepeli)l.eT B TO'IKY D
(pHC. 140).
TO'IKa C HBJIHeTCH BepIllliHoH napaJIJIe.norpaMMa ABCD co cTopoHa-
Mli AB H BD, a OTpe30K AD - .l(lIarOHaJIblO TOro napaJIJIe.norpaMMa.
n03TOMY CJIO}KeHHe BeKTopoB onlIcaHHbIM BbIIlle cnoco60M Ha3bIBalOT
CJIO}KeHlieM no «npaBliJIY napaJIJIe.norpaMMa». ECJlli CJIO}KliTb TeM }Ke
.... ....
cnOC060M BeKTopbI b li a, TO nOJIY'IHTCH TOT >Ke caMbIH pe3YJIbTaT, 'ITO li
.... ....
npli CJIO}KeHlili a li b - CHa'IaJIa TO'IKa A nepeH)l.eT B TO'IKY C, a nOTOM
B TO'IKa C - B TO'IKY D. II0CKOJIbKY napaJI-
JIeJIbHbIM nepeHoc 0.l(H03Ha'IHO onpe.l(eJIHeTCH
06pa30M O.l(HOH TO'IKli, a napaJIJIeJIbHbIe ne-
-+-+- -+-+
D peHocbI a+b li b+a oT06pa}KalOT TO'IKY A B
-+ -+ -+ -+
O)l.HY li TY }Ke TO'IKY C, TO a+b=b+a. 31'0
3Ha'IliT, 'ITO KOMn03Ul{UR napa/l/le/lbHblX nepe-
Hocoe KOMMymamueHa.
a+h
A
A'
PHC. 139
A
C
PHC. 140
204

I1pe06pa30BaHHe,' 06paTHOe
-> .
napaJIJIe.nbHOMY nepeHOCY a, caMO
HBJIHeTCH napaJIJIeJIbHbIM nepeHO-
COM. I1pH .3TOM npeo6pa30BaHHH .
Bce TO'IKH nepeMew.alOTCH Ha TO
>Ke paCCTOHHHe, 'ITO H npH nepe-
-+
Hoce a, HO B npOTHBOnOJIO>KHOM
HanpaBJIeHHH. 31'0T napaJIJIeJIb-
-+
HbIH nepeHOC 0603Ha'Ia1OT - a.
-+ -+
KOMn03HlLHeH nepeHOCOB a H -a HBJIHeTCH TO}K)l.eCTBelfHoe npe06pa-
30BaHHe E. Ih06bI coxpaHHTb e)l.HH006pa3He 0603Ha'IeHHH, 6Y)l.eM BMec-
-+
TO E nHcaTb 0 H Ha3bIBaTb TO>K)l.eCTBeHHoe npe06pa30BaHlie HY/leeblM
eeKmopOM. HTaK,
...
a
-
a
..
.
..
.
..:
l1a (i\< 0)
0)
.
),a (11)0)
a)
PHC. 141
-+ -+ -+--+--+
a+(-a)=(-a)+a=O.
II0CKOJIbKY KOMn03HlLHH TO}K)l.eCTBeHHOro npe06pa30BaHHH C JIlO6bIM
reOMeTpH'IeCKHM npeo6pa30BaHHeM 'P OCTaBJIHeT 'P He1I3MeHHbIM, TO )l.JIH
-+
JIlO60ro BeKTopa a BbInOJIHHlOTCH paBeHcTBa
-+ -+- -+- -+ -+-
a+O=O+a=a.
TaKHM 06pa30M, He TOJIbKO KOMn03Hu.lIH napaJIJIeJIbHbIX nepeHOCOB
HBJIHeTCH napaJIJIeJIbHbIM nepeHOCOM, HO H npe06pa30BaHHe, 06paTHoe
napaJ1JIeJIbHOMY nepeHOCY, TaK}Ke HBJIHeTCH napaJIJIeJIbHbIM nepeHOCOM.
31'0 03Ha'IaeT, 'ITO coeoKynHocmb napa/l/le/lbHbtx nepeHocoe n/lOCKOCmU 06-
pa3yem epynny. KaK MbI BH)l.eJIH BbIIlle, KOMn03Hu.HH napaJIJIeJIbHbIX ne-
peHOCOB 06JIa)l.aeT CBOHCTBOM KOMMYTaTHBHOCTH. 3Ha'IHT, zpynna napa/l-
/le/lbHblX nepeHocoe KOMMymamueHa.
Onpe)l.eJIHM Tenepb onepau.HlO YMHO}KeHHH BeKTopa Ha 'IHCJlO. I1YCTb
A - nOJIO}KHTeJIbHOe 'IHCJlO. Ha30BeM npoU3ee8eHueM ttUC/la A Ha 8eKmOp
-+ -+ -+
a HOBbIH BeKTop 'Aa, HMelOw.HH TO >Ke HanpaBJIeHHe, 'ITO H BeKTOp a,
-+ -+ -+
npH'IeM )l.JIHHa 'Aa nOJIY'IaeTCH H3 )l.JIHHbI a YMHO}KeHHeM Ha A, 1'Aa1=
-+ -+
=Ala I. ECJlH >Ke A<O, TO 'Iepe3 'Aa 0603Ha'IalOT BeKTOp, HanpaBJIeHHe KO-
-+
Toporo npOTHBOnOJIO}KHO HanpaBJIeHHlO BeKTopa a, a )l.JIHHa nOJIY'IaeTCH.
H3 )l.JIHHbI -; YMHO>KeHHeM Ha IAI (pHC. 141, a, 6). HaKoHeu., npOH3Be)l.e-
-+- -+- -+- 
HHeM 'IHCJIa 0 Ha BeKTop a Ha30BeM HYJIeBOH BeKTop 0, O'a=O.
MO>KHO )l.OKa3aTb CJle)l.YIOIl1.He CBoHcTBa onepau.HH YMHO}KeHHH BeK-
Topa Ha 'IHCJlO:
-+
a) 8M /l106btX ttUGe/l A U  U /1106020 eeKmopa a ebtnO/lHflemcfl paeeH-
cmeo
-+ -+
A (a) = (A)a;
-+
6) 8/lfl /l106btX ttuce/l A U  U .11106020 eeKmopa a 8btnO/lHflemcfl paeeH-
cmeo
-> -+ -+
(A+  )a='Aa+ a;
20:)'

C
A 8
PHC. 142
B) 8/lfl /l106O 'lUC/la A U /II06blX 6eKmOp08
; u b BbtnO/lflflemCfl paeeflcmeo
-+- -+- -+--+-
A (a+b)=Aa+Ab
-+
r) 8/lfl .11106020 eeK11lopa a ebllW/lflflemCfl
paeeHcmeo
-+ -+
(-I)a=-a.
¥opamaeBBH
-+
I. npH I1apaJ1J1eJ1bHOM I1epeHOCe a TO'lKa A I1epeXOHT B TO'lKY H, a npH I1apaJI-
-+
JIeJlbHOM nepeHoce b OHa nepeXO,.!J;HT B TO'lKY C (pHC. 142). nepe'lepTHTe 3TOT PHCYHOK
B -rerpMb H HaH.l\HTe TO'lItH, B KOTOpbIe I1epexo,!{HT TO'lKa A npH CJle"Y:OlQHX I1apaJI-
JIe.JIbHbIX ncpeHocax:
--:P -. -. -+--+-
a) -a; 6) 2b; B) -2b; r) a+b;
-+- -+- -+- -+-
}J,) a-3b; e) -2a+4b.
2. YrrpOCTHTe BbIpaJKeHHe
-+-+ -+ -+--. -+--+-
2 (a-3b)+5 (a+6b-c)-6 (a+5,5b-c).
-+ -+ -+
KaK .l\aJ1ee ynpocTHTCH 3TO BbIpaJKeHHe, eCJ1H c=a+b?
-+
3. PeW!lTe cJ1eYIOee ypaBHeHHe OTHOCHTeJ1bHO BeKTOpa x H nocTpoHTe 3TOT
-+ -+
BeKrop, Bbl6paB npOH3BOJ1bHO BeKTOpbI a H b:
-+--+ -+ -+ -+-+
(2x+a)- (4x+b-3a)=3a-2b.
5. nOBOpOTbI nJlOCKOCTH. I1pOBe)l.eM Ha n.'lOCKOCTH II )l.Be npHMbIe 1
H m, nepeCeKalOIll.HeCH B HeKoTopoH TO'IKe 0, H C)l.eJIaeM CHa'IaJIa oceBYIO
CHMMeTpHIO SI' a nOTOM oceBYIO CHMMeTpHIO Sm. TaK KaK H SI' H Sm -
nepeMeIll.eHHH, TO H KOMno3Hu.HH 'I)J=SmSI 3THX CHMMeTpHM TO>Ke HB-
.nHeTCH nepeMeIll.eHHeM nJIOCKOCTH. TIpH 3TOM H npH SI' H npH Sm TO'IKa
o OCTaeTCH HenO)l.BH}KHOH, a nOTOM)' OHa HenO)l.BH>KHa H npH nepeMeIll.e-
HIm <po 3Ha'IHT, JII060H JIY'I, Ha'IHHalOIll.HH:CH B TO'IKe 0, nepeH)l.eT B JIY'I,
Ha'IHHalOIll.IIHCH B TOM )Ke TO'IKe.
HaMJ{eM yroJI Me}K)l.Y JIY'IOM OA Hero 06pa30M npH nepeMeIll.eHlfH
'tjJ=SmS{, I1pH CHMMeTpMH S{ JIY'I OA nepeXO)l.HT B JIY'I OA', a npH CHM-
MeTpHH Sm Y'I OA' nepeXO)l.HT B JIY'I OA". I1pH 3TOM 6HcceKTpHCOH yrJIa
A 'OA HBJIHeTCH npHMaH 1, a 61IcceKTpncoH
yrJIa A'OA" - npHMaH m (pHC. 143). OrClO-
.u.a HCHO, 'ITO BeJIH'IHHa yrJIa ADA" BJU300
60JIbIlle BeJIli'IHHbI yrJIa Me>K)l.y npHMbIMH 1
............ .....-..-
H m, T. e. AOA"=2 (lOrn). I1pH 3TOM KaK
yrOJI AOA", TaK H yrOJI 10m OTC'IHTbIBalOTCH
npoTHB 'IaCOBOH CTpeJIKH.
HTaK, npH nepeMeIll.eHHH <p=SmS{ TO'IKa
o OCTaeTCH HenoJ{BH}KHOH, a Ka>K)l.bIH JIyq OA,
Ha'IHHalOw.HH:CH B 3ToH TO'IKe, nepexo)l.HT B
............
PHC. .'IY'I OA", TaKoi'!, QTO AOA"=2<p, r)l.e <p-
6.

yrOJI Me)K)l.y npHMbIMH 1 H m. TaKoe nepeMeIl1.€HlI€ Ha3bIBalOT 11080pO-
mOM n/lOClwcmu 80KPYZ mOttKu 0 Ha yrOJI 2fjJ. TO'IKY 0, HenOJJ,B!DKHYIO
npH nOBopOTe, Ha3hIBaIOT l{eHmpOJt n080poma, a yrOJI a MelK.l(Y JIY'IaMlI,
Ha'IHHalOIl1.HMlICH B TO'IKe 0 H lIX 06pa3aMH,- YZ/lOM n080poma. fJOBO-
pOT Ha yrOJI ex. BOKpyr TO'IKlI 0 0003Ha'IalOT R3.
TaK KaK npH nOBopOTe R't; He 1I3MeHHIOTCH paCCTOHHHH Me}K)l.y TO'I-
KaMli nJIOCKOCTH, a caMa TO'IKa 0 OCTaeTCH HenO)l.BlI}KHOH, TO )l.JIH JIIOOOH
TO'IKlI A nJlOCKOCTli II lIMeeM lOA 1= lOA' I, r)l.e A' =Rg(A).
MbI )l.OKa3aJIlI, 'ITO Ko.Mn03Ul{ueu 88YX oceeblX cUMMempUll omHOCU-
me/lbHO nepeCeKaIOU{UXCfl npflMbIX 1 U m fl8/lRemcfl n08opom n/lOCKocmu.
ECJIH 3aMeHHTb npHMbIe ill m )l.pyrHMH npH1'.fbIMli p 1I q, nepeceKalOllLH-
MliCH B TOH .tKe TO'IKe 0 H 06pa3YIDmHMK TOT )Ke yrOJI ex. (O'l'CqlITblBaeMbIH
01' p K q), TO nOJIY'IHTCH nOBopOT BOKpyr TOH }Ke caMOH TO'IKH Ha TOT }Ke
caMbIH yrOJI, T. e.
SmSz=SqSp.
JII060H nOBopOT BOKpyr TO'IKlI 0 MO)KHO npe)l.CTaBHTb B BH)];e KOMno-
31I1I.1II! .l(BYX oceBbIX CliMMeTpliH. JJ.JIH 3Toro )l.OCTaTO'IHO npOBeCTH 'Iepe3
ex
3T)' TO'IKY np5IMbIe ill m, 06pa3YIOIl1.lIe yrOJI 2' r)l.e a - yrOJl nOBopOTa,
H C)l.eJIaTb CliMMeTpHH Sz H Sm.
JJ.OKa:meM, 'IT{) noeOpombl 80KPYZ </JUKCUpoBaH1iOU mOttKU 0 06pa3YIOm
zpynny. JJ.ml3Toro HaH,!J,eM KOMn03HlLlIIO Rg R3 nOBopOTOB R3 1I Rg. IIpH
OOoliX nOBopOTax TO'IKa 0 OCTaeTCH HenO.l(BlilKHOH, a nOTOMY OHa He-
nO)l.BH}KHa H npH nepeMeIII;eHlIli RBR'b. JJ.aJIee, ecJIli JIY'I OA npH nOBO.
pOTe R3 nepeXO)l.HT B JIY'I OA', a JIyq OA' rIPH nOBOpoTe RS nepexo)l.HT
B JIY'I OA", TO npH nep€MeIII;eHHH RB JIY'I OA nepeXO)l.lIT B JIy'I OA"
....---........---.... ....---....
(pliC. 144). IIpli 3TOM AOA'=ex., A'OA"=, a TIOTOMY AOA"=ex.+p.
I1TaK, npli nepeMeIl1.eHlIli RR3 TO'IKa 0 OCTaeTCH Heno.l(BlI)KHOH, a
Bce JIY'IlI, BbIXO)l.SlUJ.lIe H3 3TOH TOIJKH, nOBOpa'IHBalOTCH Ha O)l.HH H T<>T JIre
yrOJI a+p. 3HatlJIT, Rf,Rg - nOBOpOT BOKpyr TO'IKH 0 Ha yroJI a+J!.
R  R a.- R a.+
o (J- 0 .
TaK KaK a+=p+a, TO +Il=Rg+a., H nOTOMY R=R3R. Ta-
KliM 06pa30M, KOMf1.()3Ul.fUfl 1W80pomOO OOKpyz oiJHoU u mot/. e mOttKU 0
06/laiJaem CBoUcmooM KOMMymamUIJHOCmu.
MbI ,lJ.OK83aJIH, 'ITO KOMn03RlLHH OOBOpoTOB BOKpyr TOIfKlI 0 CROBa
HBJIHeTCH IlOBOpoTOM BOKpyr TOM }K€ T()!{KH. OrMeTHM .rr.aJlee, 'ITa npll no-
BopOTe Rb 80 Bce JIyqH nepexo)l.JIT C3MlI B Ce6H.
TaK KaK, KpoMe TOro, He H3MeHSIIOTC paCCTOH-
HHH TOqex nJIOCKOCTH 01' lLeHTpa 0 nOBopOTa,
TO R3Jo nepeBO)l.lIT KIDK)l.YIO TO'IKY nJIOCKOCTli B
Ce6H, T. e. HBJIHeTCH TO)K)l.eCTBeHHbIM npe06pa-
30BaHHeM (paBHO KaK H IlOBOpOT R'b Ha 0°). 3Ha-
'fHT,
Rb 80 =R'b=E.
4T06bI )l.OKa3aTb, lITO cOBoKynHocTb nOBOpo-
TOB BOKpyr TO'IKH 0 06pa3yeT rpynny, OCTa.'IOCb
a
A
PHC. 144
-

IJ HaHTH )l.JIH KalK)l.OrO nOBOpOTa R5 06paTHbIH eMY
nOBOpOT. VIM HBJIHeTCH RbBO-a.. B CaMOM )l.eJIe,
Rg. R'bBO-a.=R'bBO=E.
I1YCTb Ha nJIOCKOCTH 3a)l.aHbI )l.Ba KOHrpY3HT-
HbIX OTpe3Ka AB H CD, IABI=ICDI. ECJlH OHH
napaJIJIeJIbHbI, npH'IeM HanpaBJIeHHH OT A K B
HOT C K D COBna)l.aIOT, TO Cyw.ecTByeT napan o
JIeJIbHbIH nepeHOC, npH KOTOpOM TO'IKa A nepe-
XO)l.HT B TO'IKY C, a B - B TO'IKY D (pHC. 145).
B cJIY'Iae lKe, KOr)l.a 3TH YCJIOBHH He BbInOJIHHIOTo
--+ --+
CH (TO ecTb JIH60 OTpe3KH AB H CD He napaJIJIeJIb-
HbI, JIH60 OHM: napaJIJIeJIbHbI, HO HMelOT npOTH-
BonOJIOIKHbIe HanpaBJIeHHH), HaH)l.eTCH nOBopOT
nJIOCKOCTH, npH KOTOpOM TO'IKa A nepeXO)l.HT B
m TO'IKY C, a TO'IKa B - B TO'IKY D. ::nOT nOBopOT
CTpOHT CJle)l.YIOIl1.HM 06pa30M. I1poBe)l.eM OCb cHM-
MeTpHH 1 TO'IeK A H C (pHC. 146) H C)l.eJIaeM
oceBYIO cHMMeTpHIO Sl' I1YCTb SI(B)=F. TaK
KaK SI(A)=C, TO IABI= ICFI. A Tenepb npo-
Be)l.eM npHMYIO m, )l.eJIHIl1.YIO nOnOJIaM yroJI FCD
H C)l.eJIaeM oceBYIO CHMMeTpHIO Sm. IIpH 3TOH: CHM-
MeTPHH JIY'I CF nepeXO)l.HT B JIyq CD, a TaK KaK
ICFI=IABI=ICDI H Sm(C)=C, TO Sm(F)=D.
SmS I(A)=Sm[S I(A)]=Sm(C)=C
SmSl(B) =Sm[S I(B)] =Sm(F)=D.
3HaqHT, nepeMeIl1.eHHe SmSI oT06paIKaeT TO'IKY A B TO'IKY C, a TO'IKY
--+ --+
B - B TO'IKY D. H3 Toro, 'ITO OTpe3KH AB H CD He HBJIHIOTCH napaJI-
JIeJIbHbIMH H O)l.HHaKOBO HanpaBJIeHHbIMH, JIerKO CJle)l.yeT, 'ITO OCH CHM-
MeTPHH 1 H m nepeceKalOTCH B HeKOTOpOH: TO'IKe 0, a nOTOMY SmSl-no-
BOpOT nJIOCKOCTH BOKpyr 3TOH: TO'IKH. YTBeplK)l.eHHe )l.OKa3aHO.
MbI paCCMOTpe.JIH TPH BH.lI.a nepeMeru:eHHii: oceBYIO CHMMeTpHIO, napaJIJ1e.JIbHbIii
nepeHOC H nOBopOT. CYru:eCTByeT eru:e O.l\HH BH.l\ - nepeHOCHQJI CUMMemplm. TaK Ha-
o S .... ....
SbIBaIOT KOMI103H1.{HIO oceBOH CHMMeTpHH I H I1apaJIJ1e.JIbHOrO I1epeHOCa a4= 0 B
HaI1paBJIeHHH, napaJIJ1e.JIbHOM OCH 1. MOJKHO .l\OKa3aTb, '!TO 3THM HCqepnbIBaeTcII
COBOKYI1HOCTb nepeMeru:eHHii: m060e I1epeMeru:eHHe I1J10CKOCTH IIBJ1l1eTCII HJ1H I1apaJI'
.IIe.JIbHblM I1epeHOCoM, HJ1H I10BOpOTOM, HJ1H oceBOii cHMMeTpHeii, HJ1H I1epeHOCHoii
CHMMeTpHeii (T eo p eM a III a J1 II).
A
PHC. 145
n
PHC. 146
Ho TOr)l.a HMeeM
"
B
%COW
a) 0) 8) z)
208

e) )K)
fV'\
.a)
PHC. 147

)"npamBeBBJI
1. Ha pHcYHKax 147a) - JK) H8ii.l\HTe <PHrypbl, no-
.nY'laeMbie .l\pyr H3 .l\pyra nOBopoTaMH. Haii.l\HTe .IJ.Jl1I !miX
<pllryp lleHTpbl H yrJlbl nOBOpoTa.
2. ITepepHcyiiTe B TeTPMb<PHrypy«1>, H306p3JKeHH)'IO
H8 pHcYHKe 148. H306pa3HTe H8 TOM JKe '1epTeJKe <pHry-
py, I10JlY'laeMYIO H3 <PHrYPbl «1> I10BOpOTOM Ha 90° BOKpyr
TO'IKH O. OTMeTbTe WTPHXOBKOii 06ylO '1aCTb 06eHX <PHfYP,
IToBepHHTe <pHrypy 11> BOKpyr TO'IKH 0 Ha 270°.
3. <I>Hrypa «1>' I10JlY'laeTcH H3 <PHrYPbl 11> I10BOpOTOM Ha
yroJ1 a no '1acoBoii CTpe.JIKe BOKpyr TO'IKH O. KaKHM I1epe-
MeeHHeM nOJ1Y'laeTcH <pHrypa «1> H3 <PHrypbl «1>'?
4. KaKHe TO'IKH nepeXO.l\HT caMH B Croll I1PH I10BOpOTe
f;IOKpyr TO'IKH 0 Ha yroJi a? KaKHe I1pHMble I1epeXO.l\IIT
CBMH B CroH I1pH 9TOM I1peo6pa30BaHHH?
5. KaKHe <PHrYPbl I1epeXo.l\IIT caMH B Croll I1PH Bce-
B03MOJKHblX I10BOpOTax BOKpyr TO'IKH O?
6. * Haii.l\HTe Bce I10BOpOTbl, I1epeBO.l\IIHe KBMpaT«1>
B KBMpaT «1>' (pHC. 149). Haii.l\HTe.IJ.Jl1I KaJK.IJ.oro H3 9THX
nOBOpoTOB yroJi H lleHTp I10BOpoTa.
7. nYCTb 11> H «1>' - .l\Be KOHrpY3HTHbie OKPYJKHOCTH.
Haii.l\HTe Bce nOBopoTbI, nepeBO.l\HHe O.l\HY H3 HHX B
pyrylO.
8. Ha'lepTHTe KBMpaT co CTOpOHOii 3 CM H KBM-
paT, I10J1Y'laeMbllI: H3 .l\aHHOrO I10BOpOTOM Ha 45° BOKpyr
MHTpa. KaKall <pHrypa HBJ1HeTCII 06'be.l\HHeHHeM 9THX
KBMpaTOB? KaKall <pHrypa IIBJIlleTClI HX I1epece'leHHeM?
9. LL3H I1pllMoyroJlbHHK ABCD. ITocTpoiiTe I1J>IIMO-
yroJlbHHK, I1oJlY'laeMblii H3 Hero I10BOpOTOM Ha 45 BOK-
pyr lleHTpa. HMeeT JlH 06'be.l\HHeHHe 9THX I1pIlMoyroJlb-
kKOB OCH cHMMeTpHH? HMeeT JlH OHO lleHTp CHMMeT-
pHH?
10. LLaH I1pHMoyroJlbHHK ABCD, TaKoii, '1TO IABI=
=8 CM H IBCI=4 CM. ITocTpoiiTe I1J>IIMoyrOJlbHHK, nOJlY-
tlaeMblii H3 Hero nOBopOTOM Ha 90 BOKpyr Cepe.l\HHbI M
l:TopOHbI AB. HMeeT JlH 06'be.l\HHeHHe 3THX I1pIlMoyroJlb-
HHKOB OCb CHMMeTpHH?
. 11. Ha'lepTHTe B TeTpa.l\H <pHrypy «1> (pHC. 150) H C.l\e-
.naiiTe CHa'laJIa I10BOpOT 3TOii <PHrypbl Ha yroJ1 45° BOKpyr
10IJKH 0, a I10TOM I10BepHHTe I10JlY'lHBIlIYIOCII <pHrypy Ha
80° BOKpyr TOii JKe TO'IKH. KaKHM O.l\HHM I1peo6pa30Ba-
fIHeM MOJKHO 3aMeHHTb 3TH I1peo6pa30BaHHII? BblI10JlHH-
te Te JKe I10BOpOTbI B 06paTHOM I10pll.l\Ke. CoBI1aJI JlH
pe3YJlbTaT C I10JlY'l'eHHblM paHee?
12. KaKHM I10BOpOTOM MOJKHO 3aMeHHTb I10CJ1e.l\OBa-
Te.IIbHble nOBOpoTbI BOKpyr TO'IKH 0 Ha 45°, 30°,60°, 75°?
13. Ha'lepTHTe B TeTPMH <pHrypy«1> (pHC. 151). CHao
qaJIa C.l\e.naiiTe oceBYIO CHMMeTpHIO OTHOCHTe.JIbHO npllMoii
'I, a I10TOM I10BOpOT Ha 45° BOKpyr TO'IKH O. IToTOM BbI-
tJOJlHHTe Te JKe I1Pe<?6pa30BaHHII B 06paTHoM I10pH.l\Ke.
CoBI13J1H J1H pe3YJIbTaTbI?
14. Haii.l\HTe 06pa3 TO'IKH M (3; 4) I1PH I10BOpoTax
BOKpyr Ha'laJIa KOop.l\HHaT Ha yrJlbl: 30°, 135,° 240°, 270°.
15. Haii.l\HTe TO'IKy,-I1epeXO.l\H)'IO B TO'IKY M (4; -3)
npH I10BOpOTe Ha yroJi 60° BOKpyr Ha'l3J1a KOop.l\HHaT.
Pac. 148
(/J
cp'
PHC. 149
PHC. 150
II
PHC. 151
209

16. HaH,zU!Te 06pa3 KBa)J,paTa ABCD: A (0; 0),
B (3; 0), C (3; 3), D (0; 3) npH nOBopoTe lIa 45°
BOKpyr TO'IKH A.
A ' 17.. ITOCJIe}40BaTeJlbHO BblnOJlHeHbI TPH npe06-
pa30BaHHSI: OCeB3SI CHMMeTpHSI O'l'HOCB'teJIbHO npSlMoH
I, npoxo,zlHeii: "Iepe3 TO'IKY 0, IIOfIOpoT Ha yroJi 45 Q
80Kpyr mllKH 0 H CHOBa OceB3SI CHJlMeTpHH OTlJO-
CH'reJIbHO 1. K3KHM O.ll.HHM npe06pa30B3HHeM MOJKH<>
SaMeHHTb 9TH TPH npe06pa30BaHHH?
6. l.(eHTpaJ1bHaH H nOBopoTHasr CHMMeT-
pHH. I10BOpOT Ha yrOJI 180 0 BOKpyr TOQKH
o nepenO)).RT KmK)).blH JIY'I C HaqBJIOM B 3TOH
TO'IKe B npoTliBOnOJIO}KHblH eMY JIY'I (1'. e. B JIY'I, JIe*aw.HH Ha TO"}Ke
npHMOH, HO lI)l.YIl1.lIH B I1poTHBOnOJIO>KHOM HanpaBJIeHHI'I). B pe3YJIbT3re
Ka}K)l.aH TQIIKa A, OTJIlI'IHaH 01' 0, nepeH)teT B TaKYIO TO'IKy A', 'ITO TOqKa
o HBJIHeTCH C€pe)l.lIHOH OTpe3Ka AA'. EYJ{eM rOBOplITb, tJTO TOQKH A 1I
A', TaKlIe, 'ITO 0 -cepe)l.lIHa OTpe3Ka AA', CUMMempU'-IHbl omHocume,1bHO
O. TaKHM 06pa30M, npH nOBopOTe Ha 180" BOKpyr TO'IKH 0 caMa 3Ta
TO'IKa OCTaeTCH HenO)l.BlI}KHOH, a Bce OCTaJIbHbIe TO'lKH nJWCKOCTli nepe-
XO)l.HT B TOqKH, CliMMeTpHqHble C HlIMH OTHOCHTeJIbHO TMKH 0 (pliC. 152).
fi03TOMY IIOBOpOT Ha 180 0 BOKpyr mqKK 0 Ha3blBalOT TaKiKe l{eHmp(1/ltJ-
HOt1 cUMMempuet1 OTHOClITeJIbHO 3TOH TO'IKlI.
I1plI lLeHTpaJIbHOH CHMMeTpliH OTHOCHTeJIbHO TO'IKlI 0 Ka)KJ{aH nJIO-
CKaH cplirypa <D nepeXo)I,HT B cplirypy (1)'; rOBOpHT, 'ITO qmrypa ([)' CUM-
MempUttHa <D OmHOCUme.J/.bHO nwt.KU O. ECJiH 4>Hrypa <D nepexoJJ.lIT npu
u.eHTpaJ'IbHOH cliMMeTpHH OTHOClITeJIhHO 0 caMa B ce6H, ee lIa3bIBalOT
cUMMempUttHot1 omHocumeAblW amot1 f1lOIlfCU, a TO'IKY 0 - l{eHmpo/1t CUM-
J.,empuu )l.aHHOH cplIrYPbI<D. HanpliMep. 11.eHTp OKpy2l<HOCTH HBJIHeTCH ee
u.eHTpoM CHMMeTplIlI, u.eHTpOM CliMMeTplili napaJIJIeJIOrpaMMa HB.1HeTCfl
TO'IK nepeCe'IeHlIH ero )l.HarOHaJIefi H T. .
.lI.oKa}KeM HeKOTOpbl€ cBoikTBa u.eH1'pBJIbHOH CliMMeTpHli.
a) EC/lU mO'-lKa A' CUMMempU'-IHa mO'4Ke A OmHOCUme.!l.bl«J 0, mo -
Ka A CUMMempU'-Ifla A' omHOCUme/lbHO O.
B caMOM )l.eJIe, 06a YTBep:H<)l.eHHH 03HaIJaIOT, qTO T()!IKa 0 - cepe)l.H-
Ha OTpe3Ka AA'.
1-13 YTBep)K)l.eHlIH a) BblTeKaeT, 11m IJOC.1Ie )l.BYKpaTHoro BbJnOJ1HeH1I5I
u.eHTpa.nbHoH CIIMMeTpnH OTHOClITeJIbHO TO'IKH 0 Bce T01JKlI nJIOCKOCTH
B03BpaIl1.aIOTCH HB 1:ICXO)l.Hble MeCTa, T. e. 'ITO
lWMfW3Ul{Wl. f4eHmpaJlbHot1 ClI.MJtefllPuu C ctlMoU
coOOii. Jl8/lRemCR mo:JJCiJeCI1l6e:HfUJUt n.pe06paooea-
Hue.M .
6) n pu z;emnpaAbHOt1 CUM..MemPUU omHOCU-
X tne/lbHO mOtt1CU 0 npflMaR, npoxOO1fUJ,l1Jl flepe3 0,
nepex08um 8 ce6fl, a npflMaR., H£ npo.xoiJfl.U¥lJI
fJ£pe3fJtny mOttK.fl.-8 napaA.lleAbHlJ1O eU npflMYIO.
B caWM ,B.eJIe, npHMUI 1, npoxO){jlIIJ,aH 1Je-
pe3 KY O. COCTOI'IT H3 .1J.BYX JlY'ieR OA H OA'
C H'IaJlOM B 3T08 'J'01IKe, KOTOpL1e MeHHIOTCSI
A
B
PHC. 152
y
A'(-x;-!l)
Pm:. 153
210

MeCTaMH npH u.eHTpaJIbHOH ClfMMeTpHH OTHQCHTeJIbHO O. I10OMY rrpHMaH
1 nepexo,I1T C3Ma B ce6H. B03bMeM Terrepb rrpHMYIO m, He rrpOXO)l.HIIJ,YIO
'Iepe3 TO'IKY 0, H 0603Ha'IHM 'Iepe3 1 napaJIJIeJIbHYlO eH npHMYIO, npo-
XO.II.HW.YIO 'Iepe3 O. I1pH lLeHTpaJlbHOH CHMMeTpHH, K3K 11 npH JIIo6oM
nepeMeIl1.eHHH, napaJIJIeJIbHbIe npHMbIe nepeXO)l.HT B napaJIJIeJIbHble.
A TaK KaK npHMaH 1 nepeXo)l.HT B ce6H, TO npHMaH m nepeXO)l.HT B
npHMYIO m', napaJIJIeJIbHYIO 1, a TeM caMbIM napaJIJIeJIbHYIO H m. 3H3-
'IHT, 06pa3 npHMOH m npH 3TOH cHMMeTpH1I napaJIJIeJIeH m.
B) EC/lU mO'iIW A UMeem KOOpaUHambl x U y, a mO'iIW A' CUMMem-
pU'iHa eu omHOCUme/lbHO Halla/la KOOpaUHam, mo KOOpaUHambl mo'iKU A'
pa8H.b1 -x U -y (pHC. 153). 3H3'IHT, lLeHTpaJIbHaH cHMMeTpHH OTHO-
CHTe.nbHO Ha'IaJIa Koop)l.HHaT 3a)l.aeTCH cf>opMYJIaMH
{ x: =-x,
y=-y.
Bo.'lee 06Il1.HM, 'IeM lLeHTpaJIbHaH CHMMeTPH'IHOCTb, HBJIHeTCH nOHH-
THe nOBopOTHOH CHMMeTpHH. rOBOpHT, 'ITO cpHrypa <I> 06JIa)I.T nooopom-
u u a 3600
HOU CUMMempueu nopJ/ Ka n, ec.nH npH nOBopOTe Ha yrOJI - OHa nepe-
n
XO)l.HT caMa B COOH. HanpHMep, npaBHJIbHbIH TpeyrOJIbHHK 06JIa)l.aeT no-
BOpOTHOH CHMMeTpHeH nOpH)l.Ka 3 - npH nOBopOTe BOKpyr u.eHTpa Tpe-
360 0
yrOJIbHHKa Ha yrOJI 1200=3 3TOT TpeyrOJIbHHK nepeXO)l.l1T B COOH.
KBa)l.paT 06JIa.l(aeT nOBopoTHOH cHMMeTpHeH nopH.l(Ka 4 - OH rrepexo-
J(HT B ce6H npH nOBopOTe BOKpyr lLeHTpa Ha 90°= 30° . Boo6w.e, .11106011
npa8U/lbHblU n-yeO/lbHUK 06/looaem n080pomHOu cUMMempueu nOpflaKa fl.
nHTI1KOHe'lHaH 3Be3)1.a 06JIa)l.aeT nOBopoTHOH cHMMeTpHeH rrOpH)l.Ka 5.
IJ.eHTpa,1bHaH CHMMeTpH'IHOCTb - 31'0 nOBopOTHaH CHMMeTpHH nopH)I.-
Ka 2.
EC/lU q;uzypa <I> UMeem aee ocu CUMMempuu I U m, yeO/l MeJlcay Komo-
180 0 6 -;:I u u a
pblMtl pa8eH -, mo OHa 0 /luvaem n080pomHOU CUMMempueu nopfl Ka n.
n
31'0 CJIeJ{yeT H3 Toro, 'ITO npeo6pa30BaHI1e SISm HB.'lHeTCH rroBopoTOM
360 0
Ha yrOJI - H nepeBo)l.HT cpHryp y <I> B COOH.
n
¥upamHeHBH
1. H aii,[\HTe l\eHTpbl CHMMeTpHH <PH'
ryp, H306paJKeHHblX Ha pHcYHKe 154.
2. KaKHe H3 6YKB, H306paJKeHHblX
Ha pHcYHKe 128, l\eHTpaJ1bHO-CHMMeTpH'I-
mz?
3. 0603Ha'lHM '1epe3 A MHOJKeCTBO
93fJlaBHblX 6YKB, HMeIOW.HX BepTHK3Jlb-
HYJO OCb cHMMerpHH, '1epe3 B - MHO-
JKeCTBO 3arJlaBHblX 6YKB, HMelOw.HX ro-
pH30HTaJ1bHYIO OCb CHMMeTpHH, H '1epe3
C - MHOJKeCTBO 3arJ1aBHbIX 6YKB, HMelO-
Ul,HX l\eHTp cHMMeTpHH. Haii.l\HTe MHO-
ji!:eCTBa:
AnS, (AnS)' nC, (A UB)nC'.
DQZ
008
PHC. 154
21\

4. '.KsKile npSlMbIe<nepexO.ll.SlT B CeOSi npH cHMMeTpHH C u.eHTpoM O?
5. KaKHe OKPYJKHOCTH nepeXO.ll.SlT B ceOSi npH cHMMeTpHH C U.eHTpOM O?
6. LLOK3JKHTe, 'I1'O J1lO6bIe JlBa KOHrpY3HTHbiX H napaJ1J1e.JIbHbIX oTpe3Ka U.eHTpaJIb-
Ho-cHMMeTpH'IHbI. .
7. LLOKaJKHTe, 'I1'O JIlO6bIe .lI.Be KOHrpy3HTHbie OKPYJKHOCTH U.eHTpaJIbHO-CHMMeT-
pH'IHbI.. .. .
8.. HaH.lI.HTe U.eHTp CHMMeTpHH <j>Hry, pbl o6pa30BaHHoH JlBYMSI KaCalO\lJ.HMHCSI
KOH rpY3HTHblMH OKPYJKHOCTSlMH.
9. LLOK3JKHTe, 'I1'O KOMn03Hu.HSI JlBYX oceBbiX cHMMeTpHii OTHOCHTe.JIbHO neprreHJlH-
KYJ1l1pHbIX npSlMblx SlBJ1S1eTCI1 u.eHTpaJIbHOH CHMMeTpHeii OTHOCHTe.JIbHO TO'IKH nepe-
Ce'IeHHSI 3THX npSlMblx.
. 10, ):(oKaJKHTe, 'I1'O nepece'IeHHe <j>HrYPbl cD C <j>HrYPoH cD', cHMMeTpH'IHOH eft
OTHOCHTe.JIbHO 1'O'IKH 0, HMeeT U.eHTp cHMMeTpHH O. LLOKaJKHTe aHaJIOrH'IHOe YTBepJK-
JleHHe npo 06'be,[\HHeHHe <j>Hryp cD H cD' .
II. HaHJlHTe U.eHTpbI cHMMeTpHH <j>Hrypbl, 06p33oBaHHoii JlBYMI1 napaJ1J1eJ1bHblMH
JIHHHSlMH.
7. rpynna CaMOCOBMew.eHHH <!JHrypbJ. 60PJl,IOpbl H OpHaMeHTbI'
TO>K)I,ecTBeHHoe lIpe06pa30BaHlie nJIOCKOCTH nepeBO)l,HT KIDK)l.YIO cpHry-
py B ce6H. J{JIH HeKOTopbIX cpHryp CYW.eCTBYIOT H HHbIe nepeMeIll.eHHH,
cOBMew.alOw.He )l.aHHYIO cpHrypy C C060H. HanpHMep, paBH060'IHaH Tpa-
nelLHH .cOBMeIl1.aeTCH C co60H lIPH oceBOH CHMMeTpHH OTHOCHTeJIbHO npH-
MOM, COe)l.HHHIOw.eu Cepe)l,HHbI ee oCHoBaHHu, a napaJIJIeJIOrpaMM - lIPH
lLeHTpaJIbHOH cHMMeTpHH OTHOCHTeJIbHO ero lLeHTpa. J{JIH lIpHMoyrOJIb-
HHKa eCTb 'IeTbIpe TaKHX nepeMeIll.eHHH: TO}KJ{ecTBeHHoe npe06pa30Ba-
HHe, )l.Be oceBble CHMMeTpHH H lLeHTpaJIbHaH CHMMeTpHH. TaKHe }Ke 'Ie-
TbIpe rtepeMeIl1.eHHH COBMeIl1.alOT C co60u pOM6. TOJIbKO )l.JIH lIpHMOyroJIb-
HHKa OHMH cHMMeTpHH CJlY>KaT lIpHMble, COe)l,HHHIOw.He Cepe)l.HHbI npo-
THBOnOJIO>KHblX CTOpOH, a )l,JIH pOM6a - ero J{HarOHaJIH. KBaJ{paT CO'Ie-
TaeT B ce6e cHMMeTpHIO pOM6a H npHMoyroJIbHHKa H cOBoKynHocTb ero
caMocoBMew.eHHH COCTOHT H3 BOCbMH 3JIeMeHTOB: lIOBOpOTOB BOKpyr
u.eHTpa Ha yrJIbI 0°, 90°, 180°, 270° H cHMMeTpHH OTHOCHTeJIbHO oceH l,
m m, p, q (pHC. 155). Y OKPY}KHOCTH 6eCKOHe'IHO
MHOro caMoCOBMeIl1.eHHM-oHa lIepeXOJ{HT B ce-
6H lIPH JII060M lIOBopOTe BOKpyr lLeHTpa H
lIPH JII060H oceBOH CHMMeTpHH OTHOCHTeJIbHO
l lIpHMOH, npOXO)l.HIl1.eH 'Iepe3 ee lLeHTp. J{JIH
lIOJIOCbI (pHC. 156) COBOKYlIHOCTb caMOCOBMe-
Il1.eHHH COCTOHT H3 nepeHocoB B)l.OJIb OCH l,
oceBblX CHMMeTpHH OTHOCHTeJIbHO lIpHMbIX l
H m H KOMlI03HlLHH 3THX lIapaJIJIeJIbHbIX lIepe-
HOCOB H oceBblX cHMMeTpHH (B 'IaCTHOCTH-
cHMMeTpHH OTHOCHTe.nbHO Bcex TO'IeK OCH l
H cHMMeTpHu OTHOCHTeJIbHO Bcex lIpHMbIX,
napaJIJIeJIbHblX OCH m).
J{JIH. JII060H cpHrypbI COBOKYlIHOCTb ee
caMOcoBMew.eHHH 06pa3yeT rpYlIlIY npe06pa-
30BaHHH, Ha3b1BaeMYIO epynnou CaMOC08MeU{e-
IiUU 3TOH cpHrYPbI. 3TO CJIe)l.yeT H3 Toro, 'lTO
eCJIH lIpe06pa30BaHHH fjJ H '\jJ lIepeBoJ{HT cpHry-
py <1> B ce6H, TO H HX KOMn03Hu.HH qn\J nepeBO-
)l,HT <1> B <1>. TOtIHO TaK }Ke eCJIH fjJ lIepeBo)l.HT
PHC. 155
m
PHC. 156
212

<D B ce:)ll, TO TeM}Ke CBOHCTBOM 06JIa)l.aeT H
npe06pa30naHHe rp-l. fpyrrna CaMOCOBMeIl1.e-
HI1H cpl1rYPbI XapaKTepl13yeT ee CI1MMeTpl1q-
HOCTb.
fpynnbI CaMOCOBMeIll.eHHM KOHeqHbIX cpH-
ryp COCTOHT JIHIllb 113 BpaIl1.eHl1ll H OCeBbiX
CHMMeTpHH. 4eM 06IlllIpHee 3Ta rpynna, TeM
CHMMeTpl1QHee )l.aHHaH cpHrypa (HanpHMep,
KBa)l.paT CI1MMeTpl1QHee npOH3BOJIbHOrO npH-
MoyrOJIbHHKa, TaK KaK BCe CaMOCOBMeIl1.eHHH
npHMoyrOJIbHI1Ka HBJIHlOTCH H CaMOCOBMeIl1.e-
HHHMH KBa)l.paTa, HO y KBa)l.paTa ecTb eIl1.e
'IeTbipe )l.onOJIHHTeJIbHbIX CaMOCOBMeIl1.eHHH -
BpameHHH BOKpyr lLeHTpa Ha 90 H 270°, a TaK-
>Ke oceBbIe CHMMeTpHH OTHOCHTeJIbHO )l.HarOHa-
JIeM) .
MHome }KHBbIe opraHH3MbI 06JIa)l.alOT TOll
HJIH HHOM CHMMeTpl1eM, H rpynnbI HX caMOCOB-
MeIl1.eHI1M COBna)l.alOT C rpynnaMI1 caMOCOBMe-
Il1.eHHM npaBHJIbHbIX MHoroyrOJIbHHKOB. Han-
pHMep, y opraHH3Ma, H306pa}KeHHoro Ha PH-
cYHKe 157, Ta }Ke cHMMeTpHH, 'ITO y KBa)l.pa-
Ta, aHa pHcYHKe 158 H306pa}KeH opraHH3M,
HMelOIl1.HH TY }Ke cHMMeTpHIO, QTO H npaBHJIb-
HbIll IlleCTHyrOJIbHHK.
<l>HrypbI, 06JIa)l.alOIl1.He nOBopoTHbIMH CHM-
MeTPHHMH, MO}KHO nOJIYQHTb, nepem6aH JIHCT
6YMam H npope3aH ero HO}KHHlLaMH. HanpH-
Mep, eCJIH B3HTb KBa)l.paTHbIM JIHCT nanHpocHoM 6YMam, CJIO}KHTb era no
)l.HaroHaJIH, nOTOM CJlO}KHTb nOJIYQHBIllHMCH paBH06e)l.peHHbIH TpeyrOJIb-
HHK no BbiCOTe, a HOBbill TpeyrOJIbHHK eIl1.e pa3 CJlO}KHTb nOnOJIaM H C)l.e-
JIaTb pa3pe3, npOpe3alOIl1.I1H Bce CJIOH 6YMam, TO nOJIYQHTCH cpl1rypa,
HMelOIl1.aH TY }Ke CHMMeTpHIO, 'ITO H KBa)l.paT (pHC. 159). TaKI1M nYTeM
nOJIY'IaroycH cpHrypbI, Ha3bIBaeMbIe p03emKaMu. 3aMeQaTe.nbHbIe npHMepbl
cHMMeTpHH, COBna)l.alOIl1.ell C cHMMeTpHeM npaBHJIbHOrO IllecTHyrOJIb-
HHKa, )l.alOT CHe}KHHKH (pHC. 160).
BecKoHe'IHbIe cpHrYPbI MorYT COBMeIl1.aTbCH C co60H He TOJIbKO np"
nOBopOTax H CHMMeTpI1HX, HO H npH napaJIJIeJIbHbIX nepeHocax. Ha-
npHMep, KBa)l.paTHaH ceTKa (pHC. 161) COBMeIl1.aeTCH C C060M npH nepe-
HOcax BH)l.a m-;'+nb, r)l.e m H n - lLeJIbIe QHCJIa. HHbIMH CJIOBa-
MH, eCJIH MbI CMeCTHM 3TY ceTKY Ha m KBa)l.paToB no rOpl130HTaJIH (Bne-
pe.n. HJIH Ha3a)1.) H Ha n KBa)l.paToB no BepTHKaJIH (BBepx HJIH BHH3), TO
CHOBa nOJIY'IHM TY }Ke caMYIO ceTKY.
4T06bI nOJIY'IHTb 6eCKOHeQHYlO B O)l.HOM HanpaBJIeHHH cpl1rypy, rpyn-
na caMOCOBMeIl1.eHHM KOTOpOM CO)l.ep}KHT napaJIJIeJIbHbIe nepeHOCbI, no-
CTynalOT CJIe)l.YIOIl1.HM 06pa30M. PHCym KaKYIO-HH6Y)l.b cpHrypy <1> H BbI-
6HpalOT BeKTOp ;, I10CJIe 3Toro nepeHOCHT cpHrypy <1> Ha Bce BeKTopbI BI1-
-:'"'" -'-- .
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PHC. 157
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PHC. 159
PHC. 160
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PHC. 161
)l.a na, r)l.e n - lLe.noe lJHCJIO. TaKlIM 06pa-
30M nOJIYlJaIOT, HanpliMep, 60 pJ{!OPbI IIa CTe-
Hax (pliC. 162). I1plI 3TOM ecJHI cpl1rypa <1>
caMa HMeJIa TY HJIH HHYIO CHMMeTpHIO, TO no-
JIYlJeHHbIM 60p)l.IOp MO}KeT 06JIa)l.aTb )l.OnOJIHH-
TeJIbHOM CliMMeTplIeti:: HanpHMep, 60 p)l.lOp ,
H306pa}KeHHbIM Ha pl1CYHKe 163, nepeXO)l.HT
B ce6H He TOJIbKO npH lIapaJIJIe.lbHbIX nepeHO-
....
cax Ha BeKTOpbI BH)l.a na, HO 1I lIPH oceBoH
CHMMeTpHH OTHOCHTeJIbHO lIpHMOH t. Bop)I.IDp,
H306pIDKeHHblM Ha pHcYHKe 164, lIepeXO)l.HT B
ce651 11 npH u.eHTpaJIbHbIX cHMMeTplIHx OTHO-
CHTeJIbHO lLeHTpoB p03eTOK, a H306pa}KeH-
Hbln Ha pucYHKe 165 - npH cHMeTpliH OT-
HOClITeJIbHO npHMbIx t, m 1I lIM aHaJIOrl1'IHbIM.
O'IeHb cHMMeTplI'IeH 6op)l.lOp, 1I306pa)KeHHbIH
Ha pHcYHKe 166. Bcero cyw,em18yem ceJttb
munoe cuMMempuii 6opalOpoo.
3HalJHTeJIbHO 6o.'lbIlle THnOB cHMMeTpl1H
MO}KeT 6bITb y HeorpaHlI'IeHHbIX cpHryp, npo-
CTHpalOIII.HXCH B )l.BYX HanpaBJIeHHHx. TaKHe
cpHrYPbl Ha3bIBalOT opHaMeHtnaMu. HeKoTopbIe
113 HHX H306pameHbI Ha pHcYHKe 167.
B KpHCTaJIJIOrpacplIl1 Ba}KHeMwylO pOJTb
mpaeT H3YQeHHe rpYlIlI caMOCOBMeIll.eHlIH
KpliCTaJIJIu'!€.CKHX peIlleTOK. Hx nepelJHCJlHJI
H3BecTHbIH PYCCKHH Y'IeHbIU E. C. <l>e)l.opoB, H
nOTOMY ax Ha3bIBalOT fjJeaOp08CKJ1MU C!pynnaMu.
YnpamoeOIlM
I. LLml KaJK.lI.oii H3 4JHryp, H306pmKeHHblx Ha
pHcYHKax 127 H 154, HaH.lI.HTe rpynnbl camCOEMelll,enHH.
2. HaH.lI.HTe rpYImbi CaMOCOBMeH.\CIIHii 6op,ll.1OpoB,
H306paJKeHHblX Ha pHcYHKe 168.
3. HaH.lI.HTe rpynIlbi CaMOCOBMem,eIlHii opHaMeH-
1'OB, H306p3JKeHHblX Ha pHcYHKe 169.
4. Hall.l\HTe rpynnb1 CaMOCOEMeIlI.eHHH Ky6a H
npaBHJ1bHOrO TeTp,a3.l1.p,a.
214
PHC. 162

PHC. 163
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 3. rEOMETPIIQECInIE llPEOBPA30BAHII1I
llOOBlIlI II CiKATlIlI
t. rOMoTeTHH. BbI6epeM Ha lIJIOCKOCTH
TO'IKY 0 H 3a)l.a)l.HM lIOJIO}KHTeJIbHOe 'IHc.nO k.
r oMomemueu C u,eftmpOM 0 U Koaf/Jf/Juu,ueft- k >0 6 i1
mOM k Ha3bIBaIOT reOMeTpH'IeCKOe npeo6pa-
30BaHHe, lIPH KOTOpOM TO'IKa 0 HenO)l.BH}K-
Ha, a Ka}K)l.aH OTJ1H'IHaH OT 0 TO'IKa M nJIOC-
KOCTH lIepexo)l.IiT B TaKYIO TO'IKY M' (pHC. 170), 'ITO M' JIeIKHT Ha
JIY'Ie OM, npH'IeM 10M' I=k 10M I. Ka}K)l.aH cpHrypa <D nJIOCKOCTH
nepeXo)l.HT npH rOMOTeTHH B )l.pyrylO cpHrypy <D'. rOBOpHT, 'ITO cpHry-
pa <D' eOMOmemU'ifta <D (c u.eHTpoM rOMOTeTHH 0 H K03cpcpHu.HeHToM k).
Ec.nH k<l, TO paCCTOHHHH OT Bcex TO'IeK nJIOCKOCTH )1.0 TO'IKH 0
YMeHbIllaIOTCH, a ecJIH k>1 - YBeJIH'IHBaIOTCH. I103TOMY HHOr)l.a rOMO-
TeTHIO C u.eHTpOM 0 H I<03cpcpHu.HeHTOM k, MeHbIlllIM I, Ha3bIBaIOT cJ/Camu-
eM K mO'iKe 0, a npH k>1 - paCmfl:JICeHUeM n/lOCKOCmu om mO'iKU O.
ECJIH k= I, TO Bce TO'IKH nJIOCKOCTH OCTalOTCH HenO)l.BH}KHbIMH. HHbIMH
CJIOBaMH, eOAtomemUfl C Koaf/JrpUl{ueftmOAt 1 R8/lRemCfl mOJ/coeCm8eftHblM
npeo6pa300aftUeM n/lOCKocmu.
C)l.eJIaeM )l.Ba npe06pa30BaHHH rOMOTeTHH C O)l.HIIM JJ TeM )Ke u.eHTpoM
o H K03cpcpHu.HeHTaMH k l H k 2 COOTBeTCTBeHHO. I1pH nepBoM npe06pa30Ba-
HHH TO'IKa M n.'lOCKOCTH nepeXO)l.HT B TaKYIO TO'IKY M' JIY'la OM, 'ITO
10M' l=klIOM I, a npH BTOpOM TO'llKa M' nepeXO}I.HT B TO'lKY M", JIe-
}KaIl1.YIO Ha TOM )Ke JIY'Ie, TaKYIO, 'ITO IOM"I=k 2 10M'I. Ho TOr)l.a
IOM"I=k 1 k 2 10MI. 3TO 3Ha'IHT, 'ITO KOMn03Hu.HeH 3THX )l.BYX npeo6pa-
30BaHHH HBJIHeTCH rOMOTeTHH C TeM lKe u.eHTpOM 0 H K03cpcpHu.HeHTO
k 1 k 2 : npu KOAtn03UU,UU eOAwmemuu C 06W,UM l{eHmpoM UX Koarp:jmu,ueft111.bl
nepeMftO:JICalOmcfl. .
JJ.aJIee, 06paTHbIM npe06pa30BaHHeM K rOMOTeTHH C u.eIHpoM 0 H KO-
3cpcpHu.HeHTOM k HBJIHeTCH rOMOTeTHH C TeM }Ke u.eHTpOM H K03CPCPHu.HeH-
I B ' u u
TOM T' caMOM )l.eJIe, KOMn03Hu.HeI1 3THX )l.BYX rOMOTeTHH HBJIHeTCH
J
rOMOTeTHH C TeM }Ke u.eHTpOM 0 H K03cpcpHu.HeHTOM k. Ii = I. Ho TaKaH
rOMOTeTHH HBJIHeTCH TO}K)l.ecTBeHHbIM npeo6pa30BaImeM. 3TO H 3Ha'lHT,
'ITO BTopaH rOMOTeTHH 06paTHa nepBOH. haK, C080KfjnftOCmb eOMomemuu
c u,eHmpOM 0 It nO/lOJ/CUn1.e/lbHblMU Koarpf/Juu,ueftmaMU o6pa3yem epyn-
ny.
PaCCMOTpHM Tenepb 60JIee IllHPOKYIO rpynny lIpeo6pa30BaHHH, lIPH-
COe,1J.HHHB rOMOTeTHH C OTpHu.aTe.lbHbIMH K03cpcpHu.HeHTaMH. Ha30BeM
eOMomemueu C l{eftmpo_'d 0 U Koarprpuu,ueflmo.M k<O reOMeTpH'IeCKOe
lIpe06pa30BaHHe, lIPH KOTOpOM TO'IKa 0 HenO)l.BH}KHa, a KaIK,llaH OTJIH'l-
HaH 01' Hee TO'IKa M nepeXO)l.HT B TaKYIO TO'I- I M I l : JkII OM I
KY M', 'ITO: a
a) TO'IKH M, 0, M' JIe)KaT Ha o)l.Hoi npH- 1
MOH, lIpHLIeM TO'IKa 0 HaXo)l.HTcH MeIK,lly T01j- M'
KaMH M H M';
,.
10M I=k IOMI
"
M
PHC. 170
I<. O
o M
PHC. 171
217

PHC. 172
6) BbIIIO,'1HHeTCH paBeHCTBO jOM'I=/kIIOMI (pHC. 171).
TIpH rOMOTeTHH C u.eHTpOM 0 H K03cpcp.iu.HeHToM -1 KalK-
,llaH TO'IKa M lIepeXO)l.HT B TO'IKY M', cHMMeTpHl/HYIO C M
OTHOCHTeJIbHO TO'IKH O. YIHbIMH CJIOBaMH, 2QMOmemtlfl C
l{eftmpOM 0 U Koaq;q;Ul{UeftmOM -1 M/lJlenu:.>l l{eftmpaAbftOU
cUMMempueu C l{eHmpoM O.
JlerKo lIOKa3aTb, 'ITO Ka:JICaylO eOMomemUiO C l{eftmpOM 0
U ompul{ame/lbHblM Koaq;q;ul{ueHmoM k MO:JICftO npeacmaeumb
e euae KOMno3ul{uu cUMMempuu C l{eHmpo./It 0 U eOMomemuu
C l{eftmpOM 0 u K09ljJq;Ul{UeHmOM 1 kl.
¥opamHeHHH
1. f1epepHcyiiTe B TeTpa.Qb H306p3JKeHHYlo Ha pHcYHKe 172 <lmrypy cD. 1130-
opa3HTe <lmrypbI, rOMOTeTH'IHble <pHrype 111 C L\eHTpoM 0 H K03<1><1>HL\HeHT3MH rOMOTeTHH
1 1
2' -2'
2. <l>Hrypa 111' - 06pa3 qmrypbI 111 npH rOMOTeTHH C Ko3(jxjJHL\HeHTOM 3 H L\eHTpOM
O. C nOMOllJ:blO K3KOH rOMOTeTHH nOJIY'laeTcH <pHrypa 111 H3 <PHrYPbI Ill'?
3. <l>Hrypa 111' nOJJY'laeTCH H3 <PHrYPbI 111 rOMOTeTHeH C LleHTpoM 0 H K03(jxjJHL\H-
eHTOM rOMOTeTHH k. C nOMOllJ:blO KaKoH rOMOTeTHH nOJJY'l3eTcH H3 <PHrypbI 111 <pHrypa
111", cHMMeTpH'IH3H 111' OTHOCHTe.JIbHO TO'lKH O?
4. <l>Hrypbllll' H Ill" rOMOTeTH'IHbI <pHrype 111 OTHOCHTe.JIbHO TO'lKH 0 C K03<l><l>HLlH-
eHT3MH rOMOTeTHH k 1 H k 2 . K.aKa5! rOMOTeTHH rrpeo6pa3yeT <pHrypy 111' B <pHrypy Ill"?
5. OTpe3KH AB H CD rrapaJIJIe.JIbHbI )J.pyr .l\pyry, HO He JIeJKaT Ha O.l\HOH npHMOH.
LLOK3JKHTe, 'ITO eCJIH HX )J.JJHHbI p33JIH'IHbI, TO CYllJ:ecTBYIOT )J.Be rOMOTeTHH, nepeBO-
)J.HIl(He OTpe30K AB BCD. CKOJIbKO T3KHX rOMoTeTHH, eCJIH )J.JIHHbI oTpe3KoB paBHbl?
6. Ha'iepTHTe OKPYJKHOCTb C L\eHTpOM 0 H nOCTpOHTe rOMoTeTlI'lHYIO eH OKPYJK-
HOCTb, eCJIH: 3) LleHTpOM rOMOTeTHH HBJIHeTCH TO'lK3 0 H K03<1><1>HLlHeHT rOMOTeTHH
1
p3BeH 2; 6) LleHTp rOMOTeTHH .1eJKHT Ha OKPYJKHOCTH H Ko3(jxjJHLlHeHT rOMOTeTHH
p3BeH 2.
7. K.aKHe rOIOTeTHII nepeBo)J.HT B Ce6H:
a) OCTpbIH yroJI,
6) <pnrypy, 06p330B3HHYlO )J.BYMH nepeCeK3IOllJ:HMHCH npHMbIMH,
B) nOJIynJIOCKOCTb?
2. fOMoTeTHR " npe06pa30BamfR nO)l,068R. ECJIH K03<pCPHu.HeHT
rOMOTeTHH OT.'lH'IeH 01' 1 HJIH -1, TO paCCTOHHHH TOl/eK lI.l0CKOCTH OT
lLeHTpa rOMOTeTHH MeHHIOTCH B I k I pa3. I103TOMY, eCJIH k OT.1H'IHO OT 1
"JIH -1, rOMOTeTHH He HB.1HeTCH lIepeMeIl1.eHHeM - lIP" 3TOM lIpeo6pa-
30BaHHH coxpaHHeTCH JIHIllb cpopMa cpHrypbI, HO He ee pa3MepbI - XOTH
OKPY}KHOCTH lIepeXOJlHT B OKPYIKHOCTH, HO HX pa)l.HYCbI lIPH 3TOM Me-
HHlOTCH, MeHHeTCH )l.JIHHa CTOpOHbI KBa)l.paTa H T. )I..
I1peo6pa30BaHHH, lIPH KOTOpbIX COXpaHHeTCH <p<>pMa cpHrYPbI, XOTH
" MeHHIOTCH ee pa3MepbI, Ha3bIBalOTCH lIpeo6pa30BaHHHMH nO)l.06HH.
DOJIee TOl/HO 31'0 lIOHHTHe Onpe)l.eJIHeTCH CJle)l.YIOIl1.HM 06pa30M:
. I1pe06pa30BaHHe lIJIOCKOCTH cp Ha3bIBaeTCH npe06pG.308O.HueM noo06Ufl
C Koaqxpuu,UeftmOM k, r)l.e k>O, ecJIH OHO H3MeHHeT Bce paCCTOHHHjf
Me}K)l.Y TO'IKaMH lIJIOCKOCTH B k pa3. HHbIMH CJlOBaMH, eCJIH npH npe06-
pa30BaHHH lIO)l.06HH cp TO'IKa A lIepeXO)l.HT B .TO'IKY A', a TO'IKa B-
B TO'IKY B', TO CnpaBe)l..'1HBO paBeHCTBO IA'B'/=kIABI. OrclO)I,a CJIe-
2J8

)l.yeT, 'ITO JII060e nepeMeIll.eHUe HBJIHeTCH npe06pa30BaHlieM nO)l.061IH C
K03cpcplIu.I1eHTOM 1 - )l.JIH nepeMeIl1.eHHH cnpaBe)l.JIlIBO paBeHCTBO
IA'B'I=IABI.
ECnlI C)l.e.naTb npe06pa30BaHI1e nO)l.06HH C K03cpcpHu.lIeHTOM k i , a
nOTOM npeo6pa30BaHlie nO)l.061IH C K03cf>cf>lIlLHeHTOM k 2 , TO B pe3YJIbTaTe
nOJIY'IHTCH npe06pa30BaHHe nO)l.061IH C K03cpcpHlLlIeHTOM k 1 k 2 . B caMOM
)l.eJIe, ecJIH np" nepBOM npe06pa30BaHHlI TO'IKlI A H B rrepeIllJIH B TO'IKH
A I H B', a npli BTOpO!\! TO'IKlI A' H B' nepeXO)l.HT B TO'IKH A" H B W ,
TO lIMelOT MeCTO paBeHCTBa IA'B'I=k1IABI 1I /A"B"I=k 2 IA'B'I. H3
HHX BbITeKaeT, 'ITO IA "B"I=k 1 k 2 IA' B' I. 31'0 1I 03Ha'IaeT, 'ITO pe3YJIb-
THpYIOIll.ee npe06pa30BaHlie MeHHeT Bce paCCTOHHHH B k 1 k 2 pa3. I1pe06-
pa30BaHlIe, 06panIOe npe06pa30BaHliIO nO)l.OOliH C K03cf>cf>HlLlIeHTOM k, .
1
HBclHe1CH npeo6pa30BaHHeM nOJ{061IH C Ko3cf>cf>l1u.l1eHTOM T' B caMOM
)l.eJIe, eC.'1lI IA'B'j=kIABI, TO IABI=+IA'B'I. OmoJ{a Cne)l.yeT, 'ITO
npe06pa30BaHH nO)l.06HH oopa3YIOT rpynny. Ee Ha3bIBalOT epynnou
noa06wz.
EcnlI npe06pa30BaHlie nO)l.061IH qJ npeo6pa3yeT qmrypy <1> B cplirypy
<1>', TO rOBopHT, 'ITO cpHrypa <1>' no806Ha cpHrype <1>. HanpliMep, 11306pa-
)KeHlIe nOJIH Ha nJIaHe nO)l.06Ho caMoMY nOJIIO (K03cpcplIu.lIeHT nO)l.o6JfB
Ha3bIBaeTCH npli 3TOM Macmma60M n/laHa). OrHoIlleHlIe nO)l.06HH 0603-
Ha'IalOT TaK: <1>,...-<1>'. OHO 06JIa)l.aeT cBoiicTBaMli pecf>.neKclIBHocTH, ClIM-
MeTpH'IHOCTIi 1I TpaH3HTHBHOCTli. HHbIMH CJIOB8MH, JII06aH cplirypa no-
)l.OOHa caMOH ce6e, <1>,...-<1>'; ecJIli <1>,...-<1>', TO <1>' ,...-<D, a ecnlI <1>......,<1>' H
!D' ,...-<1>". TO <1>,...-<1>". 31'HM onpe.n.eJIHeTCH pa36HeHlIe MHO}KeCTBa Bcex
qmryp Ha KJIaCCbI nO)l.06HbIX )l.pyr )l.pyry cpHryp.
H3 onpe)l.eneHlIH npe06pa30BaHliH nO)l.OOHH BbITeK8eT, B 'IaCTHOCTH.
'ITO npli TaKOM npe06pa30BaHHH OKPY}KHOCTb nepexo)l.HT B OKPYIKHOCTb.
B caMOM )l.eJIe, nycTb <1> - OKPY:H<HOCTb paJ{lIyca R C u.eHTpOM 0 H
<1>' - ee 06pa3 npH npeoopa30BaHlili nO)l.OOHH qJ, HMelOIll.eM I<03cpcplIlLlIeHT
k. 0603Ha'IHM 'Iepe3 0' 06pa3 lLeHTpa 0 npli npe06pa30BaHIII1 qJ. I1YCTb
TO'IKa A' HB.'lHeTCH oopa3OM TO'IKH A OKPY:H<HOCTII (D. Tor)l.a 10' A' 1=
=k lOA I, H n03TOMY 10' A' l=kR. HTaK, MbI )l.OKa3aJllI, 'ITO .'l106aH TO'lKa
A' 06pa3a OKPY:H<HOCTII <1> HaXO)l.HTCH Ha pacCTOHHHH kR )1.0 TO'IKH 0',
T. e. <1>'- OKPY)KHOCTb p8)1.Hyca kR C lLeHTpOM 0'.
rOMomemufl C KOWUtfueHmOAt k>O fl8/lflefnCfl npeo6pa306aHueM noiJo-
6UJl C me.M 3ICe K03!/JfjJUtfUl!HnwM. I1OCKOJIbKY npH npe06pa30BaHHHx
nO)l.OOHlI OKPY)KHOCTH nepeXO)l.HT B OKPY:H<HOCTlI, OTpe3KH - B OTpe3KH,
TJ>eyroJIbHHKH - B nO)l.06HbIe HM TpeyrOJIbHlIKH H T. )I.., TO H m06all
rOMOTeTHH npe06pa3yeT OKPYIKHOCTb B OKPY:H<HOCTb, OTpe30K -B OTpe-
3OK, Tpeyro.1JbHHK - B nO)l.06HbIH eMY TpeyronbHlIK H T. )I.. To :H<e
C3Moe BepHo II )J.JIH rOMOTeTHH C OTpHu.aTeJIbHbIM Ko3cf>cf>lIu.HeHToM, no-
CKOJIbKY TaK3H rOMOTeTHH HBJIHeTCH lIpOH3Be)l.eHlIeM rOMOTeTlili C no-
JlO:H<HTeJIbHblM Ko3cf>cf>lIlLHeHToM H CHMMeTpHH OTHOClITeJIbHO lLeHTpa O.
Mo:H<HO )l.OK83aTb, qTO /lJ06oe npe06pG.308aHUI! C Kowu/{ueHmoM noiJo-
OUR k fl8.tlJ/RfnCfl KOMfW3UtfueU HeKOmopou eOM,omemuu C meM {)ICe Ko-
3tfxPuI4UiHm(J.M u nepe.MeU¥!HWl.
2)1)

YnpamHeHBR
1. <I>Hrypa <D' rrO.l\06Ha tpHrype <D C K09qxjmLlHeHTOM rrO.l\06HH k 1 , a qJHrypa <D*
nO.ll.06Ha qJHrype <D C Ko9<jxjJHLlHeH1'OM nO.l\06HH k 2 . LLOKaJKHTe, '11'0 cjJHrypa <D" nO.l\06Ha
cjJHrype (}:I'. I(aKoB Ko9<jxjJHII.HeHT nO.ll.06HH cjJHrypbI (}:I* OTHOCHTeJIbHO cjJHrYPbI <D'?
2. LLOK3JKHTe, '11'0 .lI.Ba KpyroBbIx CeK1'Opa nO.ll.06HbI 1'Or.ll.a H 1'OJlbKO 1'Or.ll.a, KorJ{a
HX II.eHTpaJIbHbIe yrJIbI paBHbI.
3. LLOK3JKHTe, '11'0 .lI.Ba KpyroBbIx cerMeHTa nO.l\06HbI 1'Or.ll.a H TOJIbKO 1'Or.l\a, KorJ{a
HX II.eHTpaJIbHbIe yrJIbI PaBHbI.
4. * LLOK3JKHTe, '11'0 napa60JIa y=x 2 nepexo.ll.HT npH rOMOTeTHH OTHOCHTeJIbHO
. x 2
HaqaJI Koop.ll.HHaT B napaooJIY Y=7(' rJ{e k - Ko9<jxjJHII.HeHT rOMOTeTHH.
5. * LLOKalKHTe, '11'0 .lI.Ba KpyroBbIx KOJIbIl.a, 06pa30BaHHbIx napaMH KOHII.eHTpH-
qecKHX OKPYJKHOCTeH, nO.ll.06HbI 1'Or.ll.a H 1'OJIbKO 1'OrJ{a, Kor.ll.a OTHoweHHe P3.l\HYCOB
OKpYJKHocreH y OOOHX KOJleIl. O.ll.HO H TO JKe.
3. npe06paaoBaHHe C:H<aTHH K npHMoH. Onpe)l.eJIHM eIIJ,e O)l.HO reo-
MeTpH'IecKoe lIpe06pa30BaHHe lIJIOCKOCTH, He HBJIHIOIIJ,eecH nepeMeIIJ,e-
HHeM. J{JIH 3TOro lIpOBe)l.eM lIPHMYIO I H 3a)l.a)l.HM lIOJIO}KHTeJIbHOe 'IHCJIO
k. Ka}K)l.OH TO'IKe lIpHMOH I lIOCTaBHM B COOTBeTCTBHe 3TY }Ke TO'IKY, a
KIDK)I.OH TO'IKe A, He JIe}KaIIJ,eH Ha lIPHMOH,- TaKYIO TO'IKY A', 'ITO:
a) TO'IKa A' JIe}KHT Ha lIeplIeH)l.HKYJIHpe K lIpHMOH I, lIpOXO)l.HIIJ,eM
'Iepe3 TO'IKY A, H HaXO)l.HTCH C TOM }Ke CTOpOHbI OT lIpHMOH I, 'ITO H TO'I-
Ka A;
6) OTHOIlleHHe paccToHHHH TO'IeK A' H A OT lIpHMOH I paBHO k, T. e.
lA' PI=kIAPJ, r)l.e P - OCHOBaHHe lIeplIeH)l.HKYJIHpa K lIpHMOH I,
npOXO)l.HIl1.eM 'Iepe3 TO'IKY A (pHC. 173).
H3 3TOro OlIpe)l.eJIeHHH BH)l.HO, 'ITO lIPH k< 1 TOlJKa A' 6JIH}Ke K npH-
MOH I, 'IeM TO'IKa A, a lIPH k> 1 - )l.aJIbIlle OT I, 'IeM A. Mbi 6Y)l.eM Ha-
3b1BaTb 3TO lIpe06pa30BaHHe C{)fCamUeM n/lOCKOCmu K lIpHMOH I C KH-
u.HeHTOM k (XOTH lIPH k> 1 ero JIY'IIlle 6bIJIO
6bI Ha3bIBaTb He C}KaTHeM, a paCIllHpeHHeM
nJIOCKOCTH).
PHCYHOK 174 lIOKa3bIBaeT, 'ITO lIpeo6-
pa30BaHHe C}KaTHH MeHHeT He TOJIbKO pa3-
MepbI, HO H CPOPMY cpHryp. HalIpHMep, OK-
PY}KHOCTb, u.eHTp KOTOpOH JIe}KHT Ha lIpHMOM
I, lIPH 3TOM lIpe06pa30BaHHH C}KHMaeTCH K
3TOH lIpHMOH H lIepeXO)l.HT B OBaJIOBH)l.HYIO
JIHHHIO. 3Ty JIHHHIO Ha3bIBalOT 8/l/lUnCOM. OHa
HrpaeT Ba}KHYIO pOJIb B aCTpOHOMHH - Bce
lIJIaHeTbI ]J:BH}KyTCH BOKpyr COJIHu.a no 3JI-
JIHlIcaM.
MO}KHO )l.OKa3aTb, 'ITO npu npe06paao-
8aliuu C{)fCamUfl K npflMOU /l106afl npflMafl /ilJ
n/lOCKOCmu nepexooum 8 npflMYIO. IIpH 3TOM
napaA.Ile/lblible npflMble nepexOaflm 8 napa/l-
/le/lblible npflMble. B 'IaCTHOCTH, JII060H napaJI-
JIeJIOrpaMM lIPH TaKOM C}KaTHH lIepeXO)l.JIT B
napaJIJIeJIOrpaMM. BTO >Ke BpeMH KBa)l.paT
A
I
A
P
/PA'/=I<.IPAI
PHC. 173
A
PHC. 174
220

MOlKeT npH TaKOM C}KaTHH nepeHTH B napaJIJIeJIOrpaMM co CTOpOHa-
MH pa3JIH'IHOH )l.JIHHbI H yrJIaMH pa3JIH'IHOH Be.'lH'IHHbI.
YnpamHeHBH
I. TIocTpoihe OTpe30K H HerrapaJIJIeJJbHYIO eMY npHMYIO I. TIoCTpoHTe OTpe30K,
rrOJIY'IaeMblH H3 .l\aHHOrO CJKaTHeM K rrpHMOH 1 C Ko3<jxpHl(HeHTaMH 112, -1/2, 2, -2.
2. TIoCTpoHTe OKpYJKHOCTb pa.l\Hyca 3 CM. TIpOBe.l\HTe ee ropH30HTaJIbHbIH.l\HaMeTp.
TIocTpoHTe <PHrYPbI, nOJIY'IaeMbIe CJKaTHeM OKpYJKHOCTH K .3TOMY .l\HaMeTpy C KO.3Cf>-
<pHl(HeHTaMH 112, I, 3, 116.
3. TIoCTpoHre KBa.l\paT cO CTOpOHOH 4 CM H rrocTpoHTe <PHrYPbI, nOJIY'IaeMbIe B3
Hero CJKaTHeM K O.l\HOH H3 CTOpOH C Ko3<jxpHl(HeHTaMH 112, -112, 2, -2.
4. TIoCTpoHTe KBa.l\paT co CTOpOHOH 4 CM H HaH.l\HTe <PHrypbI, nOJIytlaeMbIe B3
Hero CJKaTHeM K O.l\HOH H3 .l\HarOHaJIeH C Ko3<jxpHl(HeHTaMH 1/2 H 2.
5. KaKHe TO'IKH nepeXO.l\HT B CroH npH CJKaTHH K npHMoH l?
6. KaKHe npHMbIe nepeXO.l\HT B CroH npH CJKaTHH K npHMoH l?
7. KaKHe npHMbIe nepeXO.l\HT B napaJIJIeJJbHbIe HM npHMbIe npH CJKaTHH K npHMoii: l?
8. TIpH CJKaTHH KOCH aGCl(HCC TO'IKa M (-I; 6) nepewJIa B TO'IKY K (-I; 2).
4eMY paBeH Ko3<jxpHl(HeHT CJKaTHH? B KaKYIO <pHryPY 11>' nepeH.lI.eT npH .3TOM CJKaTBB
KBlI,l\paT C BepWHHaMH A (I; 3), B (7; 3), C (7; 9), D (I; 9)?
9. 4eM HBJIHeTCH CJKaTHe K rrpHMoH 1 C Ko3<jxpHl(HeHTOM -I?
10. LLOKaJKHTe, '1TO KOMn03Hl(HH CJKaTHH K OCHM a6cl(HcC H 0p.l\HHaT C O.ll.HBM
H TeM JKe KO.3<p<pHl(HeHTOM k HBJIHeTCH rOMoTeTHeii: OTHOCHTe.JIbHO Ha'laJIa KOOPAH-
HaT C KO.3<p<pHl(HeHTOM k.

rJIaBaVIII
AJIrEBPAUQECHHE OIIEPAlflI II AJIrE6Pbl
 t. AJIfEBPAIPIECKHE OllEPAIUUI If AJIrE6PhI
1. 06rn,ee nOHRTife aJire6paIf"ecKoH onepau.HH. B lllKO.'le H3Y'IaIOT
pa3JHi'lHbIe onepau.HH Hall lIHCJIaMH: CJIOiKeHHe, BbI'IIiTaHHe, YMHOlKeHife
H )l.eJIeHHe. I1pIf BbIIIO.'lHeHIIH KalKllOH H3 3THX Onepau.HH no 3aJ{aHHbIM
)l.BYM 'IHCJIaM HaXOJ(5IT pe3YJIbTaT onepau.m, KOTOpbIH TaKlKe HBJIHeTCH
'IHCJIOM. HalIpHMep, lIPH CJIO}KeHHH no 'IHCJIaM 5 H 9 HaXO)l.HT HX CYMMY
14, a npH YMHoiKeHIfH no TeM lKe 'IHCJIaM HaXOJ{HT IIX npoH3Be.n.eHHe 45.
Pa3HocTb 'IHCe,'] 5 11 9 B MHOJKeCTBe HaTypaJlbHblX 'IHCeJl He onpeJ{eJleHa;
eCJIH )j{e onepau.II5I I3b:'IlITaHH5I BbmO.'lH5IeTC5I 13 MHOIKCCTBE' Bcex ILeJlbIX
'I IfCeJI , TO pa3HOCTb 5-9 onpeJ{eJIeHa H paBHa -4. HaKoHeu., )l.eJIellHe
5 : 9 M01KHO BblIIOJlHHTb B MHOlKeCTBe pau.HoHaJIbHbIX 'IHCeJI, r,'J.e nOJIY-
'IaeTCH OTBeT 5/9, H Ht'Jlh3H BbmOJlHHTb B MHO/KeCTBe HaTypaJIbHbIX 'IH-
CeJI.
KaIK)l.aH H3 onepau.IfH HaJ{ 'IHCJIaMH I1MeeT CBOH CBOHCTBa. Han;JH-
Mep, CYMMa He 3aBI!CHT 01' rropHJ{Ka CJlaraeMbIX: 5+9=9+5. B TO }Ke
npeMH pa3HOCTb MeHHeTCH, eC.1H rrm1eH5ITb MeCTaMI! Y:\leHbIllaeMoe H
BbI'IHTaeMoe: 9-5=4, a 5-9=-4.
Onepau.HH MOiKHO npml3BO)l.HTb He TOJJbKO. Ha)l. 'IHCJlaMH, 110 H Ha)l.
HHbIMH 06'beKTaMH. MbI y}Ke BCTpe'IaJlHCb C onepau.HHMH nepeCe'IeHHH
H 06'be.:I.HHE'HHH MHO/KCCTB, l\IB'bIOHKILHH H !WHbIOHKu.HH BbICKa.<bI13aHHH,
KOMlI03Hu.HH lIpe06pa30BaHHH II 1'. )I.. MHorue [\3 HHX 06J1a)l.aJIH noxo-
LKHMI! CBOHCTBaMH. HanpHMep, YKa3aHHbIe BbIIlle onepau.IiH Ha)l. MHO-
}KeCTBaMH H BbICKa3bIBaHIIHMH 6bIJIH KOMMYTaTHBHbI, T. e. pe3YJlbTaT
He MeH5IJlCH lIPH nepeCTaHOBKe MHO/KeCTB IIJ1H BbICKa3bIBaHHH:, HaJ{ 1<01'0-
pbIMH BbInOJIHHJIaCb 3Ta OlIepau.nH, KOMlI03nu.IfH lIpe06pa30BaHHH ac-
cou.HaTlIBlIa If T. )I.. B 3TOH rJIaBe MbI H3Y'IHM 06III.ee lIOHHTHe aJIre6paIf-
'IecKOH onepau.HH H paccMoTpHM HeKoropbIe CBoHcTBa, 06III.He )l.JIH MHO-
rIfX aJIre6paH'IeCKHX cnepau.uH.
Bo Bcex paccMOTpeHHbIx BbIIlle npHMepax onepau.Hii MbI Hl\IeJIH )l.eJIO
C HeKoTopbIM MHO}KeCTBOM X (MHolKecTBoM 'IHCe.n, BblcKa3bIBaHHH, lIpe-
06pa30BaHHH If T. J{.). I1pHBbIIIOJIHeHHH onepau.HH no )l.BYM 3JIeMeHTaM
3Toro MHO>KeCTBa HaXO)l.51T TpeTHH 3J1eMeHT Toro >Ke MHO>KecTBa (iiO
MYM 3a)l.aHHbIM 'IHCJIaM HaXO)l.HT If X CYMMY, lIO )l.BYM 3a)l.aHblM BbI-
CKa3bIBaHHHM - HX KOH'hIOHKlLHIO If T. )I..). I1pH 3TOM OTBeT, Bo06III.e ro-
BOpH, 3aBHCHT OT lIopHJ{Ka 3THX 3JIeMeHTOB (KaK, HalIpHMep, lIPH one-
222

paUHH BbI'IliTaHIHt tJHCe-JI). L{a)l.HM Tenepb 06m.ee onpe)l.eJIeHHe aJIreopaB-
'IeCKOH onepalLHH.
AJlze6pau'I£CKOU onepal{ueu B MH02KecTBe X Ha3b1BaIOT OT06pa:H<eHHe
(x; y}-+z, CTaBHm.ee B COOTBeTCTBHe JII060H YlIOQH)l.O'IeHHOH lIape 3JIe-
MeHTOB (x; Y) 3Toro MHO>KeCTBa TpeTHH 3JIeMeHT Z Toro >Ke MHO>KecTBa.
BbIIlle MbI Ha3BaJIH MHO>KecTBO X X X YlIopH)I.O'IeHHbIX lIap (x; y).
r)l.e x EX, y EX, )l.eKapTOBbIM lIpOH3Be)l.eHHeM X Ha X. 3Ha'IHT, aJlre6pa-
HIJeCKaH OlIepau.HH B MHO>KeCTBe X HBI1HeTCH oT06paJKeHHeM X X X B
X. 3JIeMeHT x Ha3b1BalOT nep80u KOMnOHeHmOU onepalLHH, 3JIeMeHT
y - ee 8TTWpoU KOMnonenTTWu, a 3JIeMeHT Z - pe3yAbmaTTWM onepal{UIl-
B MaTeMaTHKe paCCMaTpHBalOT H 60JIee 06m.ee lIOHHTHe aJIre6paH'IeCKoH
onepalLHH, npH KOTOpOM oT06pIDKaIOT B X )l.eKapTOBO npOH3Be)l.eutre
BH)l.a Xx. . . xX. TaKYIO onepalLHIO Ha3b1BaroT n-apnou. HanpInlep,
'----v---"
n paa
onepau.HH, npn KOTOpOH 3a)1.aHHbIM nHTH 'IHCJIaM alt a 2 , a3, a 4 , as CTa-
BHTCH B COOTBeTCTBHe lIX CYMMa al+a 2 +a 3 +a 4 +aS, 5IBJIHeTCH 5-apHoH.
tIT06bI oc06o BbI)l.eJIHTb H3Y'IaeMbIii HaMH cJIYIJafI, OT06pa}KeHHe X X X B
X Ha3blEalOT 6unaplioil aJITe6paHQeCKOH onepalLlIefI (OT JIaTHHcKoro CJlO-
Ba bis, 'iTO 03HaQaeT «)l.BIDK.1U>D». qacTo Y)l.aeTCH cBecrH n-apHYIO aJIre-
6paHQecKYlO olIepau.HIO K 6HHapHblM; HanpHMep, CJI02KeHHe n5iTH 111IceJl
MO}KHO 3aMeHHTb nOCJIe)l.OBaTeJIbHbIM CJIO>KeHHeM )l.BYX I1HCeJI:
al +a 2+a 3 +a. +as=( «(al +a2) +a3) +a 4 )+as.
C onHcaHHoH TO'IKH 3peuBSI mo6oe OT06pa:H<eHlfe MHO>KecTBa X B X
cJIe)l.YeT Ha3BaTb YHapHOU aJIrropaHIJecKoH Onepau.HeH (OT JIaTHHCKOCO
CJIOBa unus - «O)l.HH»).
I1pHBe.n.eM HeKOTOpbIe npHMepbl aJIrropaH'IeCKHX onepalLHH.
I1 p H M e p 1. Cn:OJKeHBe HBJI5IeTCH aJIre6p3llQOCKOH onepalLlIeH B
MHO>KeCTBe HaTypaJIbHbIX 'lHceJI, lIOCKOJIbKY cYMMa H3TYPaJIbHblX qyeeJI
HBJIHeTCSI HaTypaJIbHbIM qHC.IIOM. HHblMH CJIOB3MH, onepaIl.YH CJIO>KeHHg
CTaBHT B COOTBeTCTBHe mo6oii nape (x; y) mrrypaJlbHbIX qHceJI HaTYpa.1Ib-
Hoe 'IHCJIO, KOTOpoe 0603HaIJalOT x+y.
I1 p H M e p 2. CJIOJKeuHe SfBJIHeTCH aJIre6paHtteCKoii oIIepalLHeH Jf B
MHO}KOCTBe Z lLeJlbIX qBeen: cYMMa )I.BYx II.eJ1blX 'IHCeJI HBJUleTCH lLeJILlM
'IHCJlOM. B MHO>KecTBe qeTHblX qHCe.7I CJlO1KeHHe - aJIre6paH'IecKaH
onepau.lIH, a B MHO>KecTBe HeIJeTHbIX IJHCeJI CJIO>KeHHe Y>Ke He HBJIHeTC1I
aJIre6paHqecKOH onepanuei - CYMMa )l.BYX 'IeTRbIX IJHCeJI qeTHa, a
CYMMa )l.BYX HeQeTHbIX 'IHCeJI Y>Ke He HBJISIeTCSI He'IeTHhlM qHC.'lOM.
I1 p H M e p 3. YMHOJKeHHe - aJIre6paB'IecKaH onepalLHH B MHO-
>KecTBe N HarypaJIbHbIX 'lHceJI, lIOCROJIbKY npoH3Be.a.elme )l.BYX HaT)'-
paJIbllliX qHCM - 83TYPaJIbHoe 'IHCJlO. B MHO>KecTBe 'IeTHbIX 'IHce.JI
YMHOJKeHHe TO}Ke aJIre6paJflIecKaH onepalLIDf (npoH3Be.n.eHHe 'IeTHIn
'IHCeJI qeTHo). TO'IHO TaK }Ke YMHO>KeHHe HBJIHeTCH aJIre6paH'IecKoH OIle-
palLHeii H B MHO>KeCTBe HetleTHblX 'IHCeA', TaR K3K npOH3Be)l.eHHe ABYX
tJe1leTRbIX 'JYCM HetJeTHO.
n pH M e p 4. Bbl'lHTaHHe He HBJIHeTCH aJIre6paHIJeCKOH onepa-
lLneH B MHC»KeCTBe HaTYPaJIbHbIX IJnce.n: H3 HaTypaJIbHOrO 'InCJla a
123

MO>KHO BbI'IeCTb HaTypanbHoe tlHCJlO b JIHIllb npH yCJIOBHIi, 'ITO a>b.
B MHO}KeCTBe Z lLeJIbiX 'IHCeJI BbI'IJITaHHe Y>Ke HBJIHeTCH anre6paH'Ie-
CKOU OlIepalLHeH: pa3HOCTb m06bIx )l.BYX lLeJIbiX 'IHCeJI - lLeJIoe 'IHCJIO
(HanpHMep, 7-15=-8). BbI'IHTaHHe - aJIre6paH'IecKaH olIepalLHH H
B MHO)KeCTBe Bcex 'IeTHbiX 'IHCeJI (nOJIO>KJITeJIbHbIX, OTpHlLaTeJIbHbIX
H HYJIH), HO He HBJIHeTCH aJIre6paH'IecKoH OlIepalLHeH B MHO}KeCTBe
He'IeTHbiX 'IHCeJI (pa3HOCTb )l.BYX He'IeTHbIX tlHCeJI 'IeTHa).
II p H M e p 5. B03Be)l.eHHe B CTelIeHb, lIPH KOTOpOM lIape 'IHCeJI
(a; b) CTaBHTCH B COOTBeTCTBHe 'IHCJlO a h , HBJIHeTCH aJIre6paH'IecKoH OlIe-
palLHeu B MHO>KecTBe N HaTypaJIbHbIX tlHCeJI: ecJIH a H b - HaTypaJIbHbIe
tlHCJIa, TO a h - TO}Ke HaTypaJIbHOe 'IHCJIO (HalIpHMep, 2 4 = 16). B MHO-
>KeCTBe lLeJIbiX 'IHCeJI B03Be)l.eHHe B CTelIeHb y}Ke He HBJIHeTCH aJIre6pa-
H'IecKOH OlIepau.Heu. HalIpHMep, nape 'IHCeJI 2 H -4 CTaBHTCH B COOT-
BeTCTBHe 'IHCJIO 2-4, T. e. ;4 = :6 , a 3TO 'IHCJlO HelLeJIOe.
I1 p H M e p 6. Ka>K)l.bIM )l.BYM HaTypaJIbHbIM 'IHCJIa!\! a H b MO}KHO
nOCTaBHTb B COOTBeTCTBHe HX HaH60JIbIIIHU 06W.HU )l.eJIHTeJIb D (a, b)
H HaHMeHbIllee 06w.ee KpaTHoe K (a, b). TaK KaK H D (a, b), H K (a, b) -
HaTypaJIbHbIe 'IHCJIa, TO 3THM OlIpe.n.eJIeHbI )l.Be aJIre6paH'IecKHe Ollepa-
lLHH B N. 4T06bI coxpaHHTb e)l.HHCTBO 0603Ha'IeHHH, 6Y)l.eM Ha lIpOTH)j{e-
HHH 3TOH rJIaBbi lIHcaTb aDb BMeCTO D (a, b) H aKb BMeCTO K (a, b). Ta-
KHM 06pa30M, TelIepb D H K - 3HaKH anre6paH'IecKHx olIepau.HH.
I1 p H M e p 7. 0603Ha'IHM 'Iepe3 X COBOKYlIHOCTb Bcex lIO)l.MHO-
>KecTB HeKoToporo MHO>KecTBa U. ECJIH A e:: V H Be:: V, TO An B -
nO)l.MHO}KeCTBO U. 3Ha'IHT, lIepece'IeHHe MHO>KecTB HBJIHeTCH aJIre6pa-
H'IecKOH olIepalLHeu B X. TO'IHO TaK}Ke A U Be:: V, H lIOTOMY 06'be)l.HHe-
HHe MHO>KeCTB - aJIre6paH'IeCKaH olIepalLHH B X. J{aJIee, eCJIH A e:: V,
TO A'- lIO)l.MHO}KecTBO V, T. e. A' E X. 3T0 3Ha'IHT, 'ITO olIepalLHH,
CTaBHw.aH B COOTBeTCTBHe KIDK)I.OMY lIO)l.MHO}KecTBY A B V ero )l.OlIOJI-
HeHHe A', HBJIHeTCH aJIre6paH'IeCKOH olIepau.HeH. B OTJIH'IHe OT paccMoT-
peHHbIx BbIIlle olIepalLHH 3)1.eCb HMeeTCH JIHIllb O)l.Ha KOMlIOHeHTa - caMO
MHO>KecTBO A. I103TOMY OlIepal(HH A-A' HBJIHeTCH He 6HHapHOH, a
YHapHOH.
I1 p H M e p 8. KOH'blOHKlLHH, )l.H3'blOHKiLHH H HMlIJIHKalLHH BbICKa-
3bIBaHHH - 6HHapHbIe aJIre6paH'IecKHe olIepau.HH, a OTpHlLaHHe BbICKa-
3bIBaHHH - YHapHaH aJIre6paH'IecKaH olIepau.HH B MHO>KeCTBe Bcex
BbICKa3bIBaHHM.
I1 p H M e p 9. KOMlI03HlLHH npeo6pa30BaHHu HBJIHeTCH aJIre6paH-
'IecKOH olIepau.Heu B cOBoKynHocTH Bcex npe06pa30BaHHH )l.aHHOrO
MHO}KecTBa (lIpH KOMlI03HlLHH )l.BYX lIpe06pa30BaHHH MHO>KeCTBa lIOJIY-
'IaeTCH CHOBa npe06pa30BaHHe Toro }Ke MHO}KecTBa). OHa HBJIHeTCH aJI-
re6paH'IecKoH onepalLHeH H B MHO}KeCTBe Bcex nepeMew.eHHH lIJIOCKOCTH.
O)l.HaKO B MHO}KeCTBe oceBbIX CHMMeTpHH KOMn03HlLHH lIpe06pa30BaHHH
He HBJIHeTCH aJIre6p aH'IeCKOU OlIepalLHeH - KOMlI03HlLHH oceBbiX CHM-
MeTPHH He HBJIHeTCH oceBOU CHMMeTpHeu.
I1pHBe)l.eHHble npHMepbI nOKa3bIBalOT, 'ITO, XOTH nOHHTHe aJIre6paH-
'IecKOH OlIepalLHH OXBaTbIBaeT lLeJIbIH pH)I. pa3Ho06pa3HbIX CJIY'IaeB,
HH BbI'IHTaHHe, HH )l.eJIeHHe HaTypaJIbHbIX 'IHCeJI He HBJIHlOTCH aJIre6pa-
224

H'IeCKHMH onepau.H5iMH. l11'06bI BKJIIO'IHTb B paCCMOTpeHHe H TaKHe
Onepau.HH, HaJ{O BBeCTH 6o"'lee IllHpOKOe nOH5iTHe. I1MeHHo, MbI paCCMOT-
pHM H CJlY'Ian, Kor)l.a He BeeM napaM (x; y) CTaBHTCH B COOTBeTCTBHe
.9JIeMeHT Z.
l{acmU'-IIiOU a/l2e6paUlfeCKOtl onepaqueu B lIfHO}KeCTBe X Ha3bIBaeTCH
oT06pa}KeHHe HeKOToporo nOJ{MHO}KeCTBa Y )l.eKapTOBa npOH3Be)l.eHHH
XxX B X.
I1HblMH CJIOBaMH, B !\IHO}KeeTBe X 3aJ{aHa 'IaCTH'IHaH aJIre6paH'Ie-
CKaH onepalLHH, eCJIH H e K 0 l' 0 P bI M napaM (x; y), x EX, y EX,
nOCTaBJIeH B COOTBeTCTBHe 0J{H03Ha'IHO onpe)l.eJIeHHbIH 3JIeMeHT Z EX.
MHO>KeCTBO Y nap (x; y), KOTOpbIM COOTBeTCTByeT 3JIeMeHT z, Ha3bI-
BalOT 06/lacmblo onpeae/leliUfl 'IaCTH'IHOH aJIre6paH'IeCKOH onepalLHH.
BbI'IHTaHHe H J{eJIeHHe B MHO}KeCTBe HaTypaJIbHbIX 'IHCeJI HBJIHIOTCH
1j:aeTH'IHbIMH aJIre6paH'IeCKHMH onepau.HHMH. 4aCTH'IHaH aJIre6paH-
'IeCKaH onepalLHH MO}KeT OKa3aTbCH H nycmou, T. e. TaKoH, 'ITO HH
O.u.HOH nape (x; y) He eOOTBeTCTByeT 3JIeMeHT Z. I1pHMepaMH nYCTbIX 'Ia-
CTH'IHbIX aJIre6paH'IeCKHX onepalLHH HBJIHIOTCH CJIO}KeHHe B MHO}KeCTBe
He'IeTHbiX 'IHCeJI (HH OJ{HOH nape He'IeTHbIX 'IHCeJI He COOTBeTcTByeT He-
'IeTHaH CYMMa), KOMn03Hu.HH oceBbIX CHMMeTpHH H T. )I..
I1YCTb B MHO}KeCTBe X 3a.u.aHa aJIre6paH'IeCKaH OlIepalLHH H A - lIO.n;-
MHO}KeCTBO X. B03bMeM lIapy 3JIeMeHTOB (x; y), lIpHHa)l.JIe}Kaw.Hx A.
3ToH nape COOTBeTCTByeT HeKoTopbIH 3JIeMeHT Z H3 X. Bo06Il1.e rOBopH,
OH MO}KeT He npHHa)l.JIe}KaTb nO)l.MHO}KeCTBY A. ECJIH )l.JIH JII060H lIapll
(x; y) 3JIeMeHTOB H3 A COOTBeTCTBYIOIl1.HH 3JIeMeHT Z TO}Ke lIpHHa)l.JIe>KHT
A, TO rOBopHT, 'ITO lIO)l.MHO}KeCTBO A 3aMKliymo OTHOCHTeJIbHO )l.aHHOii
aJIre6paH'IeCKOH olIepau.HH.
HalIpHMep, lIO)l.MHO}KecTBO 'IeTHbIX 'IHCeJI 3aMKHYTO OTHOCHTeJIbHO
onepalLHH CJIO>KeHHH H YMHO}KeHHH B MHO}KeCTBe HaTypaJIbHbIX 'IHCeJI:
H cYMMa, H lIpoH3Be)l.eHHe 'IeTHbIX 'IHCeJI 'IeTHO. MHO}KeCTBO }Ke He'IeT-
HbIX 'IHCeJI 3aMKHYTo OTHOCHTeJIbHO olIepalLHH YMHO}KeHHH (lIpoH3Be-
)l.eHHe He'IeTHbIX 'IHCeJI He'IeTHO), HO He3aMKHYTo OTHOCHTeJIbHO one-
palLHH CJIO}KeHHH.
ECJIH nO)l.MHO}KeCTBO A 3aMKHYTO OTHOCHTeJIbHO HeKoTopoH aJIre6pa-
H'IecKOH olIepalLHH, TO, paCCMaTpHBaH ee JIHIllb Ha 3TOM lIO)l.MHO}KeCTBe.
nOJIY'IaeM aJIre6paH'IeCKYIO OlIepalLHIO B A.
¥npamHeHBlI
1. OTHOCHTeJIbHO KaKHX orrepaUHH: (CJIOJKeHHH, Bbl'lHTaHHH, YMHOJKeHHH, .l\e.JIeHHH)
SaMKHYTbl CJIe.l\YIOLUHe MIIOJKeCTBa: a) HaTypaJIbJIbIX 'IHCeJI; 6) He'leTHblX 'IHCeJl;
B) rrOJIOJKHTeJIbHbIX paUHOHaJIbHbIX 'IHCe.JI; r) {OJ, .ll) {O; I}, e) {3n+ IlnEZ}?
2. LLJI5! KaKHX rrap {*, X} HCTHHHO BbICKaSbIBaHHe «* 5!BJI5!eTC5! 3J1re6paH'IecKoii
onepaUHen B X»:
a) *- CJIOJKeHHe, X = Z _ ; 6) * - .l\e.JIeHHe, X = R + ; B) * - .l\eJJeHHe, X = Z; r) *-
)'MHOJKeHHe, X - MHO)KeCTBO UeJJblX 'IHCe.JI BH.l\a 3k; .l1) * - .l\eJJeHHe, X=R_;
e) *-D, X - MHOJKeCTBO 'IeTHbIX 'IHCe.JI; JK) * -K, X - MHOJKeCTBO He'leTHblX 'IHCeJI?
3. Haii.l\HTe 06JIaCTH orrpe.l\e.JIeHHH CJ1e.l\yroLUHx 'IaCTH'IHbIX 3J1re6paH'IeCKHX
onepaUHH:: a) Bbl'lHTaHHe B N; 6) .l\eJIeHHe B N; B) Bbl'lHTaHHe B MHOJKeCTBe nOJIOJKH-
Te.JIbHbIX 'mce.JI; r) H3BJIeqeHHe KOpH5! B MHOJKeCTBe N: (a; b)-+Vb:
8 N. 7.J

4. B KaKHX H3 CJIeJ\YlOllI.HX coBOKynHocreH reoMeTpH1.J.ecKHx rrpeo6pa30BaHHH
KOMn03HUHH HBJlHeTCH 3J1rOOpaH'IecKoH onepauHefi: a) napaJIJIeJJbHble rrepeHoCbl;
6) nOBOpoTbI BOKpyr 3a.ll.aHHoH T01.J.KH; B) oceBble CHMMeTpHH; r) ueHTpa.1bHb!e CHM-
MeTpHH?
2. Anre6pbl. B n. 1 MhI n03HaKOMHJIHCb C pa3JIH'IHbIMH aJIre6paH'Ie-
CKHMH onepalLHSlMH. K8}K)I.aSl H3 3THX onepalLHll HMeJIa CBoe 0603Haqe-
HHe: CJIo:meHHe 0603Ha'IaJIocb 3HaKOM «+», BbIqHTaHHe - 3HaKOM «-»,
)l.eJIeHHe - 3HaKOM« : », nepece'IeHHe MHo:meCTB - 3HaKOM «n », HX
.06'be.ll.HHeHHe - 3HaKOM «U» H T. )I.. 06bI'IHO 3HaK a.nre6paH'IecKoH
onepalLHH CTaBHTCH Me}K)l.Y KOMnOHeHTaMH. O)l.HaKO CMbICJI 3Toro 3HaKa
OTJIHQeH OT 3HaKa OTHOIlleHHH. KOTOpbIll TO>Ke CTaBHTCH Me}K)l.Y )l.BYiH
3JIeMeHTaMH MHO>KecTBa. HanpHMep, allb 03Ha'IaeT, 'ITO npHMaH a na-
paJIJIeJIbHa npHMoH b, a a; b 03Ha'IaeT, 'ITO 'IHCJIO a )l.eJ1HTCH Ha qHCJIO
b. HH 0 KaKOH TpeTbell npHMoH HJIH TpeTbeM 'IHC.le 3)1.eCb HeT pe'IH.
Kor)l.a >Ke Mbi nHIlleM a+b, TO nO)l.pa3YMeBaeM TpeTbe 'IHCJIO C - CYMMY
'IHCeJI a H b.
Mbi HaqHeM H3Y'IaTb 06Il1.He cBollCTBa aJIre6paH'IeCKHX onepalLHH.
I103TOMY )l.JIH HX 0603Ha'IeHHH HeJIb3H HcnO.'lb30BaTb 3HaKOMbIe HaM
3HaKH +, -, n, U H T. )1..- HHa'Ie MbI CBH3bIBaJIH 6bI COOTBeTCTBYIC-
llLue cBoHCTBa JIHIllb C TeMH onepalLHHMH, KOTopbIe 0603HaqalOTCH 3TH-
MH 3HaKaMH. BY)l.eM 0603Haqa'Fb aJIre6paH'IeeKHe H qaCTHqHbIe a.lre6pa-
H'IecKHe onepalLHH YCJIOBHbIMH 3HaKaMH, TaKHMH, KaK *, T, 0 H T. .n.
J1HbIMH CJIOBaMH, 3JIeMeHT C - pe3YJIbTaT onepalLHfI )l.JIH 3,leMeHTOB
(a; b) - 6Y)l.eT 0603Ha'IaTbCH )l.JIH O)l.HOll onepaUHH a*b, )l.JIH J{Pyroll
a ° b H T. )I..
MHO>KeCTBO, B KOTOpOM 3a)l.aHbI aJIre6paHqeCKHe onepalLHH, Ha3bIBa-
eTCH aAZe6poii. I1pH 3a)l.aHHH aJIre6pbI Ha)l.O YKa3aTb H MHO}KeCTBO X,
H paccMaTpHBaeMbIe B HeM aJIre6paH'IeCKUe onepalLHH. ECJIH B MHOlKe-
TBe X 3a)l.aHbI onepaUHH * H 0, TO COOTBeTCTBYIOIl1.YIO aJIre6py 6YJ{eM
0603Ha'IaTb (X; *, 0). HanpHMep, (N; +) 0603HaqaeT MHO}KeCTBO Ha-
rypaJIbHbIX 'IHCeJI, B KOTOpOM paCCMaTpHBaeTCH JIHIllb aJIre6paHQecKaH
onepalLHH CJIO>KeHHH, a (N; .) - TO >Ke MHOJKeCTBO, B KOTOpOM pac-
CMaTpHBaeTCH aJIre6paH'IecKaH onepalLHH YMHO>KeHHH. 3TH )l.Be aJIre6-
pbI C'IHTalOTCH pa3JIH'IHbIMH, npH'IeM OHH OTJIHQalOTCH H 01' aJIre6pbI
(N; +, .) (MHO}KeCTBO HaTypaJIbHbIX 'IHCeJI, B KOTOpOM paCCMaTpHBa-
lOTCH H CJIO>KeHHe, H YMHO}KeHHe).
MO>KeT CJIY'IHTbCH, 'ITO )l.Be aJIre6pbI OTJIHQalOTCH )l.pyr OT J{pyra
BHeIIIHe, a no CYTH )l.eJIa OHH COBna)l.alOT. TaKHe aJIre6pbI, Ha3blBalOT
U30MOpQ;HblMU. I1pe}K)l.e 'IeM )l.aBaTb TO'IHOe onpe)l.eJIeHHe nOHHTHlO H30-
MopcpH3Ma aJIre6p, paccMOTpHM CJIe.n.YlOIlLHll npHMep. 0603Ha'IHM Qepe3
A MHO>KeCTBO, COCTOHllLee H3 'IHCeJI BH)l.a 2 n , r)l.e n - HaTypa.'lbHoe
qHCJlO, T. e. A={2; 4; 8; 16; 32; . . .}. TaK KaK 2 n .2 m =2 n + m , TO npo-
H3Be)l.eHHe )l.Byx 'IHCeJI 3TOro MHO}KecTBa CHOBa npHHa)l.JIe}KHT 3TOMY
>Ke MHo:meCTBY. J1HbIMH CJIOBaMH, YMHO}KeHHe HBJ1HeTCH aJIre6pallQe-
CKOll onepalLHell B A. C )l.pyroH CTOpOHbI. MbI 3HaeM, 'ITO CJIO}KeHHe HB-
JIHeTCH aJIre6paH'IecKoH onepauHeH B MHO>KeCTBe N HarypaJIbHbIX 'IHCe.lI.
TaKRM 06pa30M, HMeeM )l.Be aJIre6pbI - {N; +} II {A; .}. Ha nepBbIH
B3rJIH.lI. OHH COBepIlleHHO paSJIH'IHbI: MHO>KeCTBa N H A He COBna)l.aIOT,
226

npH4ervr B N 33JJ.aHa 011epau.1U1 CJIO>KeH, IHI a B A - YMHO>KeHflH. Ho
nOCTaBHM B COOTBeTCTBHe Ka}K)l.OMY HaTypaJIbHOMY 'IHCJIY n 'IHCJlO. 2 n .
3TuM onpe)l.eJIHeTCfI B3aHMHO-O)l.H03Ha'IHOe OT06pa>KeHfle cp MHO:meCTB3
N Ha A:cp (n)=2 n . OHO npe06pa3YeT onepaUfllO CJIO}KeHHH HaTypaJIbHhIX
'IHCeJI B cnepaUfllO YMHO)KeHHH 'IflceJI H3 A. B caMOM )l.eJIe,
cp (n+m)=2n+m=2n .2m=cp (n) cp (m).
HHblMU C.10B, a:\1H 6e3pa3JIn'IHO, 'ITO MbI C)l.eJIaeM: CJlO}KHM 'lflCJIa n
o m H nOTOM Hafl;J,e!\! 06pa3 cp (n+m) fiX CYMMbI flJIO CHa'Iana AaH)l.eM
06pa3oI cp.(n) Ii cp (m) 3THX 'I HCeJI , a nOTOM nepeMHO>KHM H nOJIyqHM
cp (n) cp (nz).
Tenepb J{3L\H:vI oOIl1.ee Ol1pe.n.eJIeHHe H30MopcpH3Ma )l.ByX aJIre6p. J{ml
npOCTOTbI paCCMOTpHM aJIre6pbI C O!l.HOH aJIre6paH'IecKoH onepaUHeH.
Ha30BeM aJIre6pbI (X; * ) H (A; 0 ) ll3OMOpQ;HblMU, ecJIH CYIl1.ecTBYeT B3a-
OMHO-0J,H03Ha'IHOe oT06pa}KeHHe cp MHO>KecTBa X Ha MHO>KecTBO A,
TaKoe, 'ITO cp (x*y)=cp (x) 0 cp (y). B 3TOM CJIY'Iae 06paTHoe OT06pa:meHoe
cp-l 06.'la.'I.aeT TeM CBOHCTBOM, 'ITO
cp-l(a 0 b) =cp-l (a)*cp-l (b).
B C-1Y'1ae, Kor;!.a B X H A orrpe.l\e.JIeHO HecKOJIbKO aJIre6paH'IeCKHX onepauHA,
.lI.OJIJKHO 6b1Tb YKa3aHo H B3aaMHO-Q.lI.H03Ha'lHoe COOTBeTCTBHe MeJK.lI.Y onepauHSlMH.
HanpHMep, ame6pbl (X; *, T) H (A; 0, f) H30MOP<PHbI, eCJIH cymeCTBye-rTaJroeB3a-
HMHO-0)\H03Ha'1HOe oT06paJKeHHe <p MHOJKeCTBa X Ha MHOJKeCTBO A, 'ITO cp (ny)=
=<p (x)o<p (y) H <p (xTy)=<p (x)r<p (y).
IlpHMepoM H30MOP<PHbIX a.'Ire6p C }J.BYMSI orrepaUHSlMH MorYT CJIYJKHTb (X; n, U)
H (X; U, n), r.l\e X - cOBoKyrrHoCTb Bcex rrO.ll.MHOJKeCTB HeKoTOporo MHOJKeCTlla U.
ECJ111 rrOCT3BHTb B COOTBeTCTBlle KaJK}J.OMY rrO.l\MHOJKeCTBY Ac::U ero )\OrrOJfHeHHe A'
B U, T. e. rrOJIOJKIITb <p (A)=A', TO 6Y.l\eM HMeTb
<p (A n B)= (A nB)'=A' UB'=<p (A) U<P (B)
<p(A UB)= (A UB)'=A' nB'=<p(A)n<p(B).
II
A 9TO 03Ha'lae-r, 'ITO npH oT06p3JKeHHH <p onepauHH U H n MeHlllOTCIi MecTaMII,
II nOTOMY aJIfe6pbl (X; U. n> H (X; n. U) H30MOP<PHbI.
B n. 1 93 rJI. V MbI BBeJIB nOHHTHH 'IHCJIOBOrO BbIpIDKeHHSI. TaM
IIIJIa pe'lb 0 BbIpIDKeHflHx, B KOTophIe BXO)l.HJIH 'IHCJIa, H Ha)l. .9THMH qHC-
naMfI BbIIIOJIHHJIHCb apHcpMeTH'IecKHe olIepaUHH CJIO}KeHflH, BbIqHTaHHSI,
YMHO>KeHHH H )l.eJIeHHH. TO'IHO TaK }Ke MO}KHO OlIpe)l.eJIHTb nOHSITHe
BbIpa>KeHflH H 6YKBeHHoroBbIpIDKeHHSI JJ.JIH JIroOoH aJIre6phI. HanpH-
Mep, nycTb X - cOBoKynHocTb BceX nO)l.MHO}KeCTB HeKOToporo MHO-
>KecTBa U H A, B, C, D - nO)l.MHO}KeCTBa H3 3TOH cOBoKynHoC1'H. Tor.na
(A nB)U (CnD), (A uBuC)nD
H T..1I:. - BbIpIDKeHHSI B aJIre6pe (X; n, U). ToqHO TaK}Ke (4 K6)D54-
BhIpIDKeHfle B aJIre6pe (N; K, D).
L(JIH H3Y'IeHHH aJIre6p cYIII.eCTBeHHYlO pOJIb Hrp3IOT npaBHJIa TO}K-
)l.eCTBeHHOro npe06pa30BaHHH 6YKBeHHhIx BbIpIDKeHHH. 3TH npaBHna
CBH3aHbI co CBOHCTBaMH COOTBeTCTByroIII.HX aJIre6paHqecKHx onepanlfii,
KOTopbIe MhI paCCMOTpHM B CJIYlOW.eM naparpacpe.
8-
'01

YnpamllellOfi
1. 060311aLJHM LJCpC3 X MHOJKeCTBO LJeTHbIX tIHCeJI. LLOKaJKHTe, qTO oT06paJKeHHe
41' (a)=2a HBJIHeTCH H30MOp<pHbIM oT06paJKeHHeM aJIre6pbr (N; +) Ha aJIre6py (X; +).
51BJI5!eTcH JIH 3Ta oTo6paJKeHHe H30MOp<pH3MOM MeJK.l\Y aJIre6paMH (N; .) H (X; .)?
2. 1130MOp<pHbI JIH ameopbI (N; +) H (N; .)? A (N; +) H (N; +, .)?
3. LLOKaJKHTe, 'ITO oT06paJKCHHe (f! (x)= Jgx HBJIHeTCH H30MOp<pH3MOM MCJK.l\Y aJI-
re6paMH (R +; .) H (R; :_), r.l\c R +- MHOJKeCTBO rrO.OJKHTeJIbHbIX .l\eiIcTBHTeJIbHbIX
mceJI.
4. * 0603HaQHM Qepe3 X MHO>KeCTBO HaTypaJIbHblX .l\e.JIHTeJIeH QHCJIa 60 H nOJIO-
60
)KHM <p (a) = -. LLOKaJKHTe, QTO <p - H30MOp<pHoe oTo6paJKeHHe (X; D) Ha (X; K).
a
ti. I1MeeT JIH BbIpaJKeHHC (21-25).3+20 3HaQeHHe B aJIreope (N; +, -, .)? I1MeeT
JlH B Toii JKe a,1re6pe 311aQerIHe BbrpaJKeHHe 21.3+20-25. 3?
6 TIYCTb A={I' 2' 3' 4' 5' 6} B={3' 4' 5' 6' 7' 8} C={-2'-I' 0'1' 2}
Haii.l\Te 3!IaQeHlI!! .l\YW:HX' BblpaJKeHHft: 'a) , (i u b n'B; 6) (A"B) U iBA);
B) (AUC)n (BUC).
7. TIYCTb A HCTHHHO, B JIOJKHO, C HCTHHHO. Haii.l\HTe 3Ha'1eHH5! cJIe.l\YIOll(HX
BblpaJKeHHii:
a) (A I\B)vC; 6) (A ) B)VS
-:) (B .....;) A)I\C; r) (AC)I\B.
8. Haii.l\HTe 3HaQerme BblpaJKeHH5! (A",,-B) U (B",,-A), eCJ1H:
a) A= {a, b, c, d}, B= {c, d, e, f};
6) A= {I, 2, 3, 4}. B= {2, 3, 4, 8};
B) A= {I, 3, 5, 7}, B= {2, 4, 6, 8}.
9. Haii.l\HTe 3HaQeHHe BblpaJKeHHH (A ) B) V C, eCJIH:
a) A HCTHHHO, B HCTHHHO, C JIOJKHO; 6) A JIOJKHO, B HCTHHHO, C HCTHHHO;
B) A HCi'HHHO, B JIOJKHO, C JIOJKHO.
10. Haii.l\HTe 3HaQeHHe BblpaJKeHHH (aDb)Kc, eCJIH: a) a=60, b=72, c=20;
6) a=81. b=54, c=12.
S 2. CBOnCTBA AJIrE6PAHQECKIIX OllEPAU,Hfl:
1. ACCOIJ,HaTHBHbIe anre6paHI.JeCKHe OlIepalJ,HH. I13Y'IeHHe BOlIpoca
o TO}K)l.eCTBeHHbIX lIpe06pa30BaHHHx BbIpa>KeHHH MbI Ha'IHeM C BbIpa>Ke-
HUH, CO)l.ep>KaIl1.HX JIHIllb O)l.HY aJIre6paH'IeCKYIO olIepau.HIO. 0603Ha'IHM
!ny orrepau.HIO *. Ba>KHeHIllHM H3 CBOHCTB aJIre6paH'IeCKHX orrepaUHH,
HClIOJIb3yeMbIx lIPH lIpe06pa30BaHHH BbIp, a>KeHHH HBJIHeTCH CBOHCTBO
accOl{uamU8HOCmu. OHO COCTOHT B TOM, 'ITO )l.JIH JII06bIX TpeX 3JIeMeHTOB
a, b, C H3 X BblIIOJIHHeTCH paBeHCTBO
a* (b.c) = (a*b)*c.
ECJIH orrepau.HH * 06JIa.rr.aeT CBOHCTBOM accou.HaTHBHOCTH, TO BbIpa-
:meHHH, co.rr.ep>KaIl1.He JIHIllb 3TY orrepau.HIO, MOIKHO 3arrHCbIBaTb, ony-
CKaH CK06KH. 3T0 BbiTeKaeT H3 CJIe)l.YIOIl1.ero YTBep}K)l.eHHH, )l.OKa3aTeJIb-
CTBO KOToporo Mb(OlIYCKaeM:
EC/lU a/lee6pautteCKafl onepal{Ufl * 8 AtHOJICeCmee X accOl{UamU8Ha, a
(XI; . . .; X n ) - KOpmeJIC a/lUHbl n, COCma8/lei-lHblU U3 8/leMeHmoa .l>tHOJICe-
cm8a X, mo 8ce 8blpaJICeHUfl, nOAyttaeMble ll3 maKoeo Kopme"ca nymeM
paccmaH08KU CK060K U npUMeHeHUfl onepa/,{uu ., UMeIOm oaJ-tO U mo Jfce
8flatteflue (31'0 03Ha'IaeT, 'ITO nOCJIe BbJIIOJIHeHHH .rr.eHCTBHH B lIopH)l.Ke,
YKa3aHHOM cK06KaMH, Mbi BO Bcex CJIY'IaHX no.'1Y'IliM O)l.HO H TO >Ke 3Ha'Ie-
HHe BbIpa>KeHHH).
228

HanplI!\Iep, eCJIl-! onepau.HH * accou.HaTHBHa, TO BCe BbIpalKeHHH
(a*b)* (c*d) , ((a*b):;,c)*d, (a* (b*c))*d, a* (b* (c;:..d)) , a*((b.c)*d)
HMelOT O)l.HO H TO >Ke 3Ha'IeHHe.
06Il1.ee 3Ha'IeHHe BbIpalKeHHH, nOJIY'IaeMbIX OnHCaHlIbIM BbIllie cno-
060M H3 KOpTe>Ka (Xl; . . .; X n ), 0603Ha'Ia1OT XI*' . .*X n . HanpHMep,
nHlllYT He (3+9)+(4+ 7), a 3+9+4+7. 3T0 3Ha'IeHHe MOIKIIO BbI'IHC-
JIHTb, HanpHMep, CKJIa)l.bIBaH 'IHCJIa CJIeBa HanpaBo:
3+9+4+7= 12+4+7= 16+7=23.
I1pHBe)l.eM npHMepbI accou.HaTHBHbIX H Heacc0u.HaTHBHbIX onepau.HH.
I1 p H M e p 1. CJIO>KeHHe HaTypaJIbHbIX 'IHCeJI - aCCOUHaTHBHaH
onepau.HH. Accou.HaTHBHbI H onepaUHH CJIOlKeHHH paUHOHaJIbHbIX 'IH-
ceJI, )l.eHCTBHTeJIbHbIX 'IHCeJI H T. )I.. I103TOMY CYMMY HeCKOJIbKHX 'IHCe..'1
MO>KHO nHcaTb 6e3 cK060K.
I1 p H M e p 2. YMHO>KeHHe HaTypaJIbHbIX 'IHCeJI aCCOu.HaTHBHO,
paBHO KaK H YMHO>KeHHe pau.HOHaJIbHbIX, )l.eHCTBHTeJIbHbIX 'IHCeJI Ii T. )I..
I103TOMY npOH3Be)l.eHHe HeCKOJIbKHX 'IHCeJI TO>Ke nHIllYT 6e3 CK060K.
I1 P H M e p 3. Onepau.HH nepece'IeHHH H 06'be)l.HHeHHH MHO>KeCTB
06JIa)l.alOT CBOHCTBOM accou.HaTHBHOCTH (CM. c. 32 H 34). I103TOMY, Ha-
npHMep, nepeCe'IeHHe MHO>KeCTB (A n B) n (C n D) MO>KHO 3anHcaTb B
BH)l.e A n B n C n D.
I1 P H M e p 4. Accou.HaTHBHbI H onepau.HH )l.H3'bIOHKU.HH H KOH'blOHK-
U.HH BbIcKa3bIBaHHH (CM. c. 17 H 19). I103TOMY nHlllYT TaK:
A!\B!\C!\D, A Y ByCyD, mIYCKaH CK06KH.
I1 P H M e p 5. Onepau.HH KOMn03Hu.HH npe06pa30BaHHH MHO}KeCT-
Ba X accou.HaTHBHa (CM. c. 188).
I1 P H M e p 6. Onepau.HH BbI'IHTaHHH B MHO>KeCTBe u.eJIbIX 'IHCeJI
He HBJIHeTCH accou.I1aTHBHoii: eCJIH a, b, c - u.eJIbIe 'IHCJIa, TO npll
C+O HMeeM a- (b-c)=I= (a-b)-c. He HBJIHeTCH accOu.HaTHBHoH H one-
pau.HH )l.eJIeHHH B MHO>KeCTBe nOJIO>KHTeJIbHbIX 'IHCeJI: eCJIH c+ 1, TO
a : (b : c)=I= (a : b) : c.
L(JIH accou.HaTHBHbIX onepaUHH HMeeT CMbICJI nOHHTHe cmenenu aile-
Menma. B03bMeM KaKoH-HH6Y)l.b 3JIeMeHT a H KOpTe>K )l.JIHHbI n, Bce
KOOp)l.HHaTbI KOToporo paBHbI a, T. e. KOpTe>K (a; . . .; a) (n pa3). 3ToMY
KopTelKY cooTBeTcTByeT 3JIeMeHT H3 X, paBHbIH a*. . .*a. Ero o603Ha-
'IalOT an H Ha3bIBalOT n-H cmenenblO aileMenma a HanpHMep, a 3 =a*a*a.
I1pH n= 1 nOJIaraIOT a 1 =a.
B c.'lY'Iae Kor)l.a *- Onepau.HH )'MHO>KeHHH, BBe)l.eHHOe nOHHTHe
CTeneHH COBna)l.aeT C 06bI'lHbIM: an=a. . .a (n pa3). ECJlH >Ke * - one-
pau.HH CJIOlKeHHH, TO 0603Ha'IeHHe an 03Ha'IaeT a+. . .+a. 06bIQHO B
3TOM CJIY'Iae BMeCTO an nHlllYT na H Ha3bIBalOT na KpamnblM QHCJIa a,
a He CTeneHblO :)Toro 'IHCJIa.
H3 aCCOu.HaTHBHOCTH onepaUHH * BbITeKalOT CJIe.llYlOIII.He npaBHJIa
)l.eHCTBHH Ha)l. CTeneHHMH:
I) (an)*(am)=a n + m ,
2) (an)m=a nm .
229

"boobl )1.0K33aTb CBOH:CTBO 1), 3aMeTHM, 'ITO
a"*am=(a*. . .*a)*(a*. . .*a).
.... ......1
" p"3 m paa
Ho B CHJIY 3CCOUHaTHBHOCTH onepaUHH *, 31'0 Bblpa}KeHHe paBHO a*.
. . .*a(n+m p33), T. e. a n + m . 3Ha'IHT, a"*am=a,,+m. CBOHCTBO 2))1.0-
Ka3bIBaeTCH aHaJIOrH'IHO.
B cJIY'Iae KorJ{a * - onepau.uH CJIO}KeHHH, nO.1Y'IaeM H3 CBOHCTB
1) H 2) CJIeJ{YIOIl1.He npaBHJIa J{eHCT!HH HaJ{ KpaTHbIMH:
1) na+nb=n (a+b),
2) m (na) = (mn) a.
OlIepaUHH nepece'IeHH5I MHO}KeCTB aCCOUHaTHBHa. OJ{HaKo )l.JI5I 3TOH:
onepaUHH HeUeJIeco06pa3Ho rOBopHTb 0 CTeneHH: J{JIH JII060ro MHO}KecT-
Ba A HMeeM A n A =A, H rrOTOMY J1I06aH «CTerreHbY> MHO}KeCTBa A COBna.
)1.aeT C 3THM MHO}KecTBOM. Heu.eJIecoo6pa3Ho rOBopHTb 0 CTeneHH H J{JIH
onepaUHH 06'be.n.HHeHHH MHO}Kt'CTB.
¥lIpa»CHeHBJI
J. KaKHe R3 CJ1e.l\YI01l(HX Onepall.HH B MHOJKecTBe Z lI.eJlblX qHCe.JI aCCOll.HaTHBHbI:
a) CJ10JKeHHe; 6) BbIqHTaHHe; B) YMHOJKeHHe; r) Onepall.HH a*b=3a-2b?
2. KaKHe H3 CJ1e.l\YI01l(HX onepall.Hii B MHOJKeCTBe N HaTypaJIbHbIX qHCe,l acco-
nHaTHBHbI: a) CJ10JKeHHe; 6) BbIqHTaHHe; B) YMHOJKeHHe; r) .l\e.JIeHHe; .l\) aDb; e) aKb;
:IK) a*b=a b ?
3. LLOKaJKHTe, qTO eCJJH onepall.HH * B MJlOJKecTBe X aCCOL\HaTHBHa, H <p - Hoo-
Mopqmae Or06paJKeHHe CHCTeMbI (X; *) Ha CHCTeMY (Y; 0), TO 0 - aCCOL\HaTHBHaH
OnepaL\RH B Y.
4. 0603HaqHM qepe3 0 Onepall.HIO KOMn03HL\HH reoMeTpHqeCKHX rrpe06pa30-
BaHHH, a qepe3 <p - nOBOpoT Ha 60° BOKpyr roqKH O. 4TO 03HaqalOT npe06pa30Ba-
BHIl , <f, <p', <t, <p6, <p7? KaKoe H3 3THX rrpeo6pa30BaHHH HBJlHeTCH TOJK.l\eCTBeHHblM
npe06paooBaHHeM? B03bMHTe TOqKy M, OTJlHqllYIO OT TOqKH 0, H HaH.l\HTe ee 06p33bI-
npH rrpeo6pa30BaHHHx <p, q;2, <p3, <p4, q;", (fi6. BepwHHaMH KaKoH qmrYPbI 5!BJlHIOTCH
nOJlyqHBwHecH roqKH?
-+
5. 0603HaqHM Qepe3 <p npeo6pa30BallHe napaJlJleJIbllOrO nepelloca (BeKTOpa) a.
1JTO 03HaQaeT npeo6pa30BallHe <p"? B03bMHTe TOQKY M H lIaH.l\HTe ee o6pa3bI npH npe-
06pa30BaHBllx <p-3, <p-2, <p_l, <po, <po <p2, <p3.
2. KOMMYTaTHBHocTb. MbI BII)l.eJIH, 'ITO accou.HaTIIBHOCTb a.lre6paH-
'lecKOH: olIepauHH * n03BOJIHeT 3anHCbIBaTb 6e3 cK060K Bce BbIpa}KeHHH,
cOJ{ep}KaIII.He JIHIllb 3TY onepaUHIO. Ho nepeCTaBJIHTb BXO)l.HIII.He B 3TO
Bblpa}KeHHe 3JIeMeHTbI, Boo6III.e rOBopH, HeJIb3H, H nOTOMY CJIe.1I.YeT
C'lHTaTb BblpIDKeHHH a*b H b*a pa3JIH'lHblMH. JIHIllb B cJIY'Iae, Kor)l.a
onepauHH * He TOJIbKO aCCOUHaTHBHa, HO H KOMMYTaTHBHa, B03MO>KHO
)J.aJIbHeHIllee YlIpOw.eHHe BbIpIDKeHHH.
AJIre6paH'IecKYIO onepaUHIO * Ha3bIBalOT KOMMymamuSfwu, eCJIH
pe3YJIbTaT ee lIpHMeHeHHH He 3aBHCHT OT lIopSlJ{Ka KOMTIOHeHT, T. e.
ecJIH )l.JI5I JIlO6bIX J{BYX 3JIeMeHTOB a H b H3 X BbITIOJIHSleTCH paBeHCTBO
a*b=b*a.
IIpHMepaMH KOMMYTaTHBHblX onepaUHH MorYT CJIY>KHTb CJIO}KeHHe H
YMHO>KeHHe HaTypaJIbHblX 'IHCeJI, nOCKOJIbKY J{JlH JII06bIX HaTypaJIbHbIX
D)

4HCe,'l m II n BbInOJIHHIOTCH paBeHCTBa:
m+n=n+m,
mn=nm.
3TH paBeHCTBa CnpaBe,l1,JIHBbI He TOJIbKO ,l1,JIH HaTypaJIbHbIX 'IHCeJI, HO
H ,l1,JIH JIlO6bIX )l.eikTBHTeJIbHbIX 'IHCeJI: B MHOJKecTBe R )l.eikTBHTeJIbHhIX
'IHCeJI Onepau.HH CJlO}KeHHH H YMHO}KeHlIH TO}Ke KOMMYTaTlIBHbI.
I1plJlllepOM HeKOMMYTaTlIBHOH Onepau.lIH MO}KeT CJIY:H<lITb BbI'IliTaHlie
B MHO}KeCTBe Z U.eJIbIX 'IlICeJI: ecJIH a=l=b, TO a-b=i=b-a. HeKOMMYTaTHB-
HO H ,l1,eJIeHlIe B MHO}KecTBe nOJIO}KlITMbHblX pau.HOHaJIbHblX 'IHCeJI:
ecJIH a=j=b, TO a : b=i=b : a.
Onepau.HH KOMn03Hu.HH npe06pa30BaHHH MHO}KecTBa X, Bo06rn.e
roBOpH, HeKOMMYTaTHBHa (CM. c. 189). Ho KOMn031Iu.HH rrapaJIJIeJIbHblX
rrepeHocoB KOMMYTaTHBHa, paBHO KaK H KOMn03Hu.HH nOBopOTOB BOKpyr
TO'IKlI (CM. c. 204 H 207).
ECJIH onepau.HH * KOMMYTaTHBHa H accou.HaTHBHa, TO JIlO60e BbIpa-
>KeHHe, cO,l1,ep:maw.ee JIHIllb 3TY onepalLHlo, MO}KHO rrpe.n.cTaBHTb B BH)l.e
  J{
a l *. . .*ak , r,l1,e 3JIeMeHTbI ai, . . . , ali rronapHO pa3JIH'IHbI. JIH TaKoro
rrpe06pa30BaHHH )l.OCTaTO'IHO, BOCrrOJIb30BaBIIIHCb aCCOlLHaTHBHOCTbIO
H KOMMYTaTHBHOCTblO, rrepecraBHTb 3JIeMeHTbI, BXO)l.HIl1.He B 3T0 BbIpa-
>KeHHe, TaK, 1JTOObI paBHbIe ,l1,Pyr )l.pyry 3JIeMeHTbI IIIJIH rrO,l1,pH)I.. Ha-
npHMep,
((al*a 2)* (al *al) )*( al *a 2) =al *a 2*al*al *a l *a 2 = al *al *al *al*a 2*a 2 =.
LI:oKaJKeM CJIe)l.YlOrn.ee CBOHCTBO CTeneHeH, HMelOIl1.ee MeCTO, ecJIli
onepalLHH * KOMMYTaTHBHa H aCCOlLHaTHBHa:
an*b n = (a*b)n.
B caMOM )l.eJIe, rro orrpe.n.eJIeHHlO
(a*b)n=(a*b)*. . .*(a*b)=a*b*. . .*a*b.
 "'
(1)
n pas
I1epecTaBJIHH 3JIeMeHTbI, MO>KeM aanHC8Tb 3TO BblpIDKeHHe B BH)l.e
(a*. . .*a)*(b*. . .*b)=an*b n .
"'- "' ..... .
n pas n pas
3Ha'IHT, (a*b)n=an*b n .
ECJIH * - onepau.HH YMHO}KeHHH 'I HCeJI , TO H3 paBeHCTBa (1) CJIe-
)l.YeT, 'ITO anb n = (ab)n. ECJIH JKe * - orrepalLHH CJIO>KeHHH, TO paBeHCTBO
(I) 03Ha'IaeT, 'ITO na+nb=n (a+b).
¥upaarseSBR
I. KaKHe H3 CJle.l\yrom:HX onepaUHH B MHO>KeCTBe Z KOMM)'TaTHBHbl: a) CJlO>KeHHe;
6) BblIJHTaHHe; B) YMHOJKeHHe; r) onepaUHSI tl*b=3a+2b?
2. 06JIa.l\3eT JIH cBOikTBOM KOMMYTaTHBHoCTH: a) onepaUHSI BhI'IHTaHHSI MHO-
>KeCTB; 6) onepaUHSI AAB= (A"B)U (B"A)?
231

3. KaKHe V,3 cJIe,ll.YIOIlI,v,x orrepaUYH B MHOJKeCTBe N H3TypaJIbHbIX '1HCeJI KOIllMY-
TaTHBHbI: a) CJ10JKeHHe; 6) Y!HOJKeHHe; B) aDb; r) aKb;,lI.) a*b=a b ; e) a*b=3a+2b?
4. KaKHe H3 CJIeJJ:YIOIlI,HX Orrepal\HH Ha,ll. Pb!CKa3blBaHH5IMH KOMMYTaTHBHbI:
a) KOH'bIOHKl\H5I; 6) .llH3'blOflKLlV,5!; B) HMrrJIHKal\H5!; r) 3KBHBaJIeHl\H>I?
5. 0603Ha'lHM '1epe3 cp napaJIJIeJIbHbIH rrepeHOC Ha I e,ll.IIHHl\Y B HarrpaBJIeHHH
OCH a6Cl\HCC, a '1epe3 1jJ - rrapMJJeJIbHbII! nepeHoc Ha 2 e.ll.HHHl\bI B HarrpaBJIeHHH
OCH op.l\HHaT. B KaKYIO TO'lKY rrepeXO.l\HT TO'IK3 0 rrpH rrpe06pa30BaHHH If'l\:? a rrpH
rrpe06pa30BaHHH 1jJcp? 06'b5lCHHTe rrOJIY'IeHHbIH pe3YJIbTaT.
6. 0603Ha'lHM '1epe3 cp rrapaJIJIeJJbHbIH rrepeHoc Ha 2 e.l\HHHl\bI B HarrpaBJIeHHH
OCH a6Cl\Hcc. a lJepe3 1jJ - rroBopoT Ha 90 0 BOKpyr HatJaJIa KOOp.l\HHaT O. H aH.l\HTe
o6pa3bI TO'lKH 0 rrpH rrpe06pa30BaHH5Ix cp1jJ H 1jJcp. BepHo JIH, '1TO <p'IjJ= 'l\JIf?
3. .nuCTpu6YTUBHOCTb. J{o CHX nop Mbl paccMaTpliBaJIH BbIpa}KeHHH,
co.n.ep}KaIl1.l1e JIHIllb O)l.HY aJIre6paU'IecKYlO onepau.HlO, H HaY'IHJIHCb
npe06pa30BbIBaTb TaKHe BbIpa}KeHHH npli YCJIOBlIH, 'ITO onepau.HH ac.
cou.HaTHBHa H KOMMYTaTlIBHa. Tenepb paccMoTpliM BbIpa}KeHHH, cO)l.ep.
:maIl1.l1e )l.Be onepau.lIli. 0603Ha'IHM 3TH onepaUlili * 1I o. ECJIH a, b, c, d-
KaKlie-TO 3JIeMeHTbI MHO)J{eCTBa X, B KOTOpOM OHH 3a)l.aHbI, TO MbI MO}KeM
COCTaBlITb, HanpliMep, T3KlIe BbIpa}KeHlIH, KaJ{ (a*b)o (c*d). 3T1I BbIpa-
:meHlIH Y)l.aeTCH npeo6pa30B3Tb, eCJIli Onep31UilI * 1I 0 CBH3aHbI )l.Pyr C
.n.pyroM HeKOTopbIMli COOTHOIlleHlIHMlI.
4aIl1.e Bcero paCCMaTpHB3lOT napbl onepaUlIH, CBH3aHHbIe )l.pyr C
.n.pyroM )l.lICTPH6YTlIBHOCTblO. Ha30BeM onepaUlilO 0 aucmpu6ymueHou
OTHOCHTeJIbHO onepau.HH *, eCJIH )l.JIH JIlO6bIX Tpex 3JIeMeHTOB a, b, c
MHO}KCCTBa X BbinOJIHHlOTCH paBeHCTBa:
ao (b*c)=(aob)*(aoc)
(I)
Ii
(b*c)oa= (boa)*(boc).
(2)
B CJIY'Iae KOr)l.a * H 0 - 'IaCTH'IHbIe aJIre6palI'IeCKHe onepau.HH, paBeH-
CTBO (I) nOHHMaeTCH CJIe)l.YlOIl1.lIM 06pa30M: ecJIli XOTH 6bI O)l.HO H3 BbI-
pa}KeHHH a 0 (b * c) 1I (a 0 b) * (aoc) HMeeT CMblCJI, TO HMeeT CMbICJI
U BTOpoe BbIpa}KeHHe, npI1'IeM 3Ha'IeHHH 3THX BbIpa}KeHlIH paBHbI
.n.pyr )l.pyry.
ECJIH Bb!!JOJIH5IeTC5I JIHWb O,ll.HO H3 paBeHCTB (I), (2), TO rOBOp5lT 0 (Jucmpu6ymuB-
Hocmu CAeBa (COOTBeTCTBeHHO cnpaBa). Pa3YMeeTC5I, B CJIY'Iae KOr,ll.a 0 - KOMMYTa-
THBHa51 Orrepal\H5I, MbI HMeeM aob=boa. aoc=coa, ao (b*c)= (b*c)oa, a rrOTOMY B
9TOM CJIY'Iae He Ha.l\O pa3JIHIJaTb JIeBYIO H rrpaBYIO .l\HCTPH6YTHBHOCTH.
TIpHBe.l\eM rrPHMep Orrepal\HH, .l\HCTPIl6YTHBHOH crrpaBa. HO He .l\HCTPH6YTHBHOH
CJIeBa. TIOJIOJKHM B MHOJKeCTBe HaTypaJIbHbIX '1HCeJI aob=a b . Tor.l\a HMeeM:
(bc)oa= (bc)a= bac;a= (boa) (coa).
3HaIJHT, Orrepal\H5I o.ll.HCTPH6YTHBHa crrpaBa OTHOCHTeJIbHO YMHOJKeHH5I. Ho a 0 (bc)=
=ab c , B TO BpeM5I KaK (aob) (aoc)=abac=ab+c, H rroToMY .9Ta Orrepal\H5I He 51B-
JI5IeTC5I .lI.HCTPH6YTHBHOH CJ1eBa OTHOCHTe.JIbHO YMIJOJKeHH5I.
Onepau.HH YMHO}KeHHH HaTypaJIbHbIX 'IlICeJI )l.IICTPH6YTHBHa OTHo-
CUTeJIbHO onepau.HH CJIO}KeHHH: )l.JIH JIlO6bIX Tpex HaTypaJIbHbIX 'IUCM
a, b, C HMeeT MeCTO paBeHCTBO
a (b+c)=ab+ac.
232

Onepau.lIH YMHO}KeHlIH HaTypa,'!bHbIX 'IlICeJI )l.lICTP1I6YTlIBHa 1I OTHOCH-
TeJIbHO Orrepau.lIH BbI'IliTaHlIH: eCJIli b>c, TO
a (b-c) = ab-ac.
Onepau.HH CJIO}KeHlIH He HB.'lHeTCH )l.lICTp1I6ynlBHoM OTHOCHTeJIbHO
onepau.HH YMHo:meHHH, nOCKOJIbKY a+bc=F (a+b) (a+c).
ECJlH onepau.HH 0 )l.HCTP1I6YTlIBHa OTHOCHTeJIbHO onepau.HH *,
npHIJeM 06e onepau.HH accou.lIaTlIBHbI, TO B JIlO60M BbIpaJKeHHH, CO)l.ep-
:maIl1.eM JIHIllb3TH )l.Be onepau.lIH, MO}KHO paCKpblTb Bce CK06KH. HanpH-
Mep, MbI 3HaeM, IJTO YMHO}KeHlIe )l.eHCTBHTeJIbHbIX IJHCeJI )l.HCTPH6YTHB-
HO OTHOCHTeJIbHO CJIO}KeHHH. I103TOMY
(a+b) (c+d)=a (c+d)+b (c+d).
HCnOJIb3YH eIl1.e pa3 )l.HCTPH6YTHBHOCTb YMHO}KeHHH H accOu.IfaTHB-
HOCTb CJIO}KeHHH, nOJIY'IaeM, IJTO
(a+b) (c+d)=ac+ad+bc+bd. (3)
I1plI BbIBO)J,e 3TOH CPOPMYJIbI MbI He HCnOJIb30BaJIli lIHbIX CBOHCTB onepa-
U.HH YMHO}KeHlIH 1I CJIO}KeHlIH, KpOMe )J,lICTP1I6YTlIBHOCTli YMHO}KeHHH
OTHOClITeJIbHO CJIO}KeHlIH H accou.lIanlBHOCTli CJIO}KeHlIH. I103TOMY, eCJIH
* H 0 - JIlO6bIe )l.Be aJIre6palI'IeCKlie onepau.HH, TaKHe, IJTO * accou.Ha-
THBHa H 0 )l.HCTPH6YTHBHa OTHOCHTeJIbHO *, TO HMeeT MeCTO paBeHCTBO
(a*b) o (c*d) = (aoc)*(aod)*(boc)*(bod). (4)
H3npHMep, nYCTb 0 - orrepallH5I rrepeCe'leHH5I MHOJKeCTB, 3 * - orrepallHSI
06'be.l\HHeHH5I MHOJKeCTB. TaK KaK orrepallH5I rrepeCe'leHH5I MHOJKeCTB .l\HCTPH6YTHBHa
OTHOCHTeJIbHO onep311HH HX 06'be.l\HHCHH5I, a orrepallH5I 06'be.l\HHeHH5I MHOJKeCTB ac-
COllH3THBH3, TO 33MeH5I5I B cpopMYJIe (4) 0 Ha n, 3 * H3 U, rrOJIY'IaeM:
(AUB)n (CUD)= (AnC)U (AnD)U (BnC)U (BnD). (5)
Ho Mbi 3HaeM, '1TO H orrep311H5I 06'be.l\HHeHIl51 MHOJKeCTB .l\HCTPH6YTHBH3 OTHOCHTe.JIbHO
orrep311HH HX nepeCe'leHH5I, rrpH'IeM orrepallH5I rrepeCe'leHH5I 3CCOllH3THBH8 (eM. c. 32 H
34). D03TOMY MOJKHO B <f;opMYJle (4) 3aMCIIHTb 0 Ha U, 8 * Ha n H rrOJIY'IHTb, '1TO
(A nB)U (Cn D )= (A UC) n (A UD)n (BUC) n (B UD). (6)
TO'lHO TaK JKe H3 <f;OPMYJIbJ (4) BbIBO,l\HM COOTHOWeHHe .l\JI5I BbICK83bIB8HHi!:
(AVB)A (CVD)= (AAC)V (AAD)V (BAC)V (BAD). (7)
MbI BH.l\HM, '1TO 0.l\H8 <f;opIYJIa (4), rrpH BblBO.l\e KOTOpOH 6bIJIH HCnOJIb30B8HH
JIHWb .l\HCTPH6YTHBHOCTb 0 OTIIOCHTeJIbHO * H 8ccOllHaTHBHOCTb orrep811HH *, .l\aJIa
lleJIbIH p5!.l\ paBeHcTB, KaC810111HXC5I KOHKpeTHblx 8JIre6p8H'IeCKHX onep811HH. ECJlH
6bI Mbi He BocrrOJIb30B8JIHCb 06111HMH CBOHCTBaMH 8JIre6p8H'IeCKHX onepallHH, K8JK.l\)'IO
H3 cpOPMYJI (5) - (7) npHWJIOCb 6bI ,l\OKa3bIB8Tb OT.l\eJIbHO.
PaCKpblTHe CK060K n03BOJIHeT npHBeCTH BbIpa}KeHHH, CO)l.ep}KaIl1.He
)l.Be onepau.HH 0 H *, K CTaH)l.apTHOMY BH)l.Y. I1YCTb 06e 3TH onepau.HH
aCCou.HaTHBHbI, npH'IeM 0 )l.lICTPH6YTHBHO OTHOCHTeJIbHO *. Ha30BeM
«O)l.HO'IJIeHOM» BbIpa}KeHHe BH)l.a alo. . .oa n , a «MHOrOIJJIeHOM» BbIpa-
}KeHlIe BlI)l.a A 1 *. . .*A,,, r)l.e A J , . . ., A k - O)l.HOIJJIeHbI. MOJKHO )l.OKa-
3aTb, 'ITO JIlO60e BbIpaIKeHHe, cO)l.epIKaIl1.ee JIHIllb onepau.HH 0 If *.
nOCJIe pacKpbITHH CK060K npeBpaIl1.aeTCH B «MHOrO'IJIeH». .lJ.aJIbHeMIllee
ynpOIl1.eHHe B03MO}KHO npH YCJIOBlIH, 'ITO onepau.HH 0 H * KOMMYTaTHB-
233

Hbl. Tor)l.a JII060ii «O)l.HOqJIeH» MOJKHO 3anlicaTb KaK npOH3Be)l.eHlIe
creneHeH pa3JIH'IHbIX 3JIeMeHTOB, a B «MHOrO'I,'1eHax» nplIBecni nO)l.06Hble
tjJIeHbI.
B 'IaCTHOCTlI, JII060e 6YKBeHHoe BbIp aJKeH lie, CO)l.epJKaIl1.ee JIlIlllb
onepau.lIli CJIOJKeHHH H YMHO}KeHlIH, MOJKHO npe)l.CTaBlITb B BlI)l.e CYMMbI
O)l.HO'IJIeHOB. IIpH 3TOM, nOJIb3YHCb KOMMYTaTHBHOCTbIO YMHOJKeHlIH,
MOJKHO Ka>K)l.bIH O)l.HO'lJIeH npe)l.CTaBHTb B BH)l.e npOH3Be)l.eHHH CTeneHeH.
I1plIMeHHH TaKlie npeo6pa30BaHliH K Bblpa>KeHHHM (a+b)2 H (a+b)3,
nOJIyqaeM H3BecTHbie 1I3 IllKOJIbI <PoPMYJIbI cOKpaw.eHHoro YMHOJKeHHH:
(a+W=a 2 +2ab+b 2
H
(a+b)3=a 3 +3a 2 b+3ab 2 +b 3 .
ORH HBJIHIOTCH qaCTHblMH CJlY'IaHMH 06Il1.eH <PoPMYJIbI, Ha3b1BaeMOH <fIop-
MYJIOH 6UHOMa HblOmOHa. 3Ta <PoPMYJIa lIMeeT BlI)I.:
(a+b)n=an+CAan-1b+. . .+Can-kbk+. . .+b n ,
f.!l.e 'Iepe3 C 0603Ha'IeHo 'IHCJlO COqeTaHliH H3 n 3JIeMeHTOB no k (CM.
c. 48).
Do.l\06HO TOMY KaK Qx>pMYJIa (3) OK33aJ1aCb 'IaCTHbiM CJIy'!aeM o6m.eii Qx>PMYJIbi
(4), Qx>pMY.1Ja 6HHoMa HblOTOHa - '!acTHblii CJIy'!aii 06m.eii Qx>PMYJIbI, KOTOpaH BepHa
AJlH JII06blX I<OMMYTaTHBHblx H aCCOLlHaTHBHblX orrepaLlHii 0 H *. TaKHX, '1TO 0 AH-
CTpH6YTHBHO OTHOCHTe.1IbHO *.
¥opa*HeHHII
I. LLJIH KaKHX rrap (*. 0) aJlre6paH'!ecKHx orrepaLlHii HCTHHHO BblCKa3b1BaHHe
dJrrepaLlHH 0 .l\HCTPH6YTHBHa OTHOCHTeJJbHO onepaLlHH *»:
a) 0 - AeJIeHHe B MHOJKeCTBe nOJIOJKHTeJIbHblX '1HCM, * - CJIOJKeHHe;
6) 0 - CJIOJKeHHe B R, * - YMHOJKeHHe B R;
B) 0 - 06'be.l\HHeHHe MHOJKeCTB, * - nepece'!eHHe MHOJKeCTB;
r) 0 - rrepece'!eHHe MHOJKeCTB, * - 06'be.l\HHeHHe MHOJKeCTB;
.l\) 0 - CJIOJKeUHe LleJJbIX '!HCe.JI, * - BbI'!HTaHHe;
e) 0 - YMHOJKeHHe Lle.JIblX '!HCM, * - Bbl'!HTaHHe;
JK) 0 - .l\H3'bIOHKLlHH BbICKa3blBaHHii, * - KOH'bIOHKUHH;
3) 0 - HMrr.1JHKaLlHH BblCKa3blBaHHii, * - KOH'bIOHKUHH?
2. LLHCTPH6YTHBHa JIH orrepaUHH B03Be.l\eHHH B creneHb OTHOCHTMbHO CJIOJKeHHH?
3. * YrrpocTHTe BblpaJKeHHe
(aoa)* (boc)", (aoc) ,
eCJIH H3, BecTHO '1TO onepaUHH * H 0 aCCOUHaTHBHbI H KOMMYTaTHBHbI, a onepaUHH *
AHCTPH6YTHBH3 OTHOCHTe.JIbHO onepaUHH 0, npH'IeM .IJ.JUI JII06oro xEX HMeeM: x*x=x
H xox=x.
4. COKpaTHMocTb. EC.'IH a, x H y - HaTypaJIbHble 'IlICJla H a+x=
=a+y, TO OTClO)l.a MOJKHO C)l.eJIaTb BbIBO)l., 'ITO x=y. TO'IHO TaK JKe H3
Toro, 'ITO ax=ay, r)l.e a, x, y - HaTYPaJIbHble qHCJla, BbiTeKaeT paBeH-
CTBO x=y. Ho eCJIli Mbi 3HaeM JIHIllb, 1JT0 a, x H y - UeJIbIe qHCJla, TO
H3 paBeHCTBa ax=ay HeJIb3H 6e3oroBOpO'IHO C)l.eJIaTb BblBO)l., 1JT0 x=y;
MO}KeT CJI, Y'IlITbCH qTO a=O, a TOr)l.a npli JII06b1X x H .II lIMeeM ax=
=ay=O. BBe)l.eM c.ne)l.YIOw.ee onpe)l.eJIeHue.
234

AJIre6paH'IecKaH Onepau.I!H * B [ 
MHO>KeCTBe X Ha3blBaeTCH COKpamU-
MOll, ecJIH H3 YCJIOBHH a*x=a*y CJIe-A V X
.nYeT, lITO X=y, H H3 YCJIOBliH x*a=
= y*a CJle.nYeT X=y.
OnepaUHH CJIO>KeHHH HaTypaJIb- PHC. 175
HblX 'IHCeJI, KaK MbI BlI)l.eJIH BbIIIJe,
COKpaTHMa: H H3 a+x=a+y, H H3 x+a=y+a BbiTeKaeT paBeHcTBO
X=y. Onepau.HH YMHoIKeHlIH HaTypaJIbHbIX 'IHCeJI cOKpaTHMa, paBHo
KaK H onepau.HH YMHoIKeHHH B MHoIKecTBe 'IHCeJI, OTJIH'IHbIX OT HYJIH.
B MHO>KecTBe IKe Z Bcex u.eJIbIX 'IHCeJI 3Ta onepau.HH He HBJIHeTCH COKpa-
THMOH - H3 O.x=O. y He BbITeI{aeT, 'ITO x=y.
npHMepOM HecoKpaTHMOH aJIre6paH'IeCKOH onepau.IIH MOIKeT CJIY-
)J{HTb onepau.HH nepeCe'IeHHH MHOIKecTB; H3 Toro, 'ITO A n X =A nY,
HeJIb3H CJ{eJIaTb BblBO)l.a 0 paBeHCTBe MHOIKeCTB X H Y (pHC. 175).
TO'IHO TaK >Ke H3 Toro 'ITO A U X =A U Y, HeJIb3H, Bo06Il1.e rOBopH, BbI-
BeCTH, 'ITO X=Y.
YnpamHeHHII
1. KaKHe H3 CJ1e.l\YIOll\HX aJIre6p3H'IeCKHX oneparU!l1 COKpaTHMbI: a) BbI'IHTaJme
B MHOJKeCTBe Z; 6) .l\e.JIeIlHe B MHOJKeCme R +; B) onepallHH aDb H aKb B MlrOJKeCTBe
H3TyPaJIbHblX '1HCeJI; r) orrep311H5! (a; b)-+3a+2b B MHO)\{eCTBe lleJIblX '11:ceJI?
2. DYCTb orrep311H5! * COKp3THM3. KaKoii BblBO.l\ MOJKHO C.l\eJI3Tb 113 p3BeHcna
a*nb=a*y*b?
5. 06paTHbie onepaUUH. Onepau.HH BbI'IHTaHHH 06paTHa onepau.HH
CJIOIKeHHH, a onepau.HH )l.eJ1eHHH - onepau.HH YMHO)!{eHHH. 3TO 3Ha'IHT,
'ITO H3 b+x=a CJIe)l.yeT x=a-b, a H3 bx=a BbITeKaeT x=a : b. Onepa-
UHH BbI'IHTaHHH H )l.eJIeHHH HMelOT 06Il1.He CBoHcTBa; HanpHMep, paBeH-
CTBa a- (b+c)=a-b-c H a: (bc)=a : b : c aHaJIOrlI'IHbI.
OrMeTHM, 'ITO XOTH onepau.HH CJlO>KeHHH H YMHO>KeHHH onpeJ{eJIeHbI
Ha BceM MHO>KeCTBe HaTypaJIbHbIX 'IHCeJI (cYMMa H np01I3Be)l.eHJle .1110-
6bIX )l.BYX HaTypaJIbHbIX 'IlICeJI - HaTypaJIbHble 'IHCJla) , 06paTHbie HM
onepau.HH BbI'IHTaHHH H .neJIeHHH HBJIHIOTCH lIHWb 'IaCTH'IHbIMH a,'1re6pa-
lI'IeCKHMH onepau.HHMlI - pa3HocTb a-b onpeJ{eJIeHa JIHWb npll YCJIo-
BHH a>b, a 'IaCTHOe a: b - npH YCJIOBHH, 'ITO a Hau.e.no J{eJIHTCH
Ha b.
B 3TOM nYHKTe MbI onpe)l.eJIHM )I..'lH KaIKJ{oH cOKpaTliMoH H KOMMY-
TaTHBHoH aJIre6palI'IeCKOH onepau.HH * 06paTHYIO eH 'IaCTII'IHYIO aJIreo-
paH'IecKYlO onepau.HlO T H BbIBe)l.eM 06Il1.He cBoHCTBa TaKHX OnepaII.HH,
H3 KOTOpbIX, B 'IaCTHOCTlI, 6Y)l.YT BbITeKaTb CBoHcTBa onepmulH BbI'IU-
TaHHH H )l.eJIeHHH.
OTMeTHM, '1TO 06paTHble onep311HH MOJKHO orrpe.l\eJISlTb H .l\JISI HeKOMMYT3THBHblX
8JIre6paH'IeCKHX onepallHii, HO B 3TOM CJIY'Iae npHWJJOCb 6b1 p33JIH'I3Tb JIeBble H npa.
Bble 06p3THble orrep311HH (::no BHAHO XOTSI 6b1 H3 Toro, '1TO HeH3BeCTHOC J;.CJIHMOe Hm.yT
C nOM01ll,b1O orrep311HH YMHOJKeHH5!, 3 HeH3BecTHblH .l\eJIHTeJIb - C rrOMOUJ.bIO onepallHIt
.l\e,lCIIH5!; 31I3'1HT, .l\JI5! HeKOMMYT3THBIloii orrepallHH ».eJIeHH5! eCTb .lI.BC 06paTHblc
orrepallBH - YMHOJKeHHe H ,/J.CJJeHHC).
235

HTaK, rrYCTb .j(- - COKpaTHMaH H KOMMYTaTHBHaH aJIre6palI'IeCKaH
orrepau.HH Ha MHOIKeCTBe X. 0603Ha'IliM 'Iepe3 Y MHO)"eCTBO nap (a; b),<
a EX, bE X, J{JIH KOTOpbIX HaHJ{eTcH TaKoe x EX, 'ITO b*x=a. .1J.JIH Ka}K-
AOH TaKOH napbI 3JIeMeHT x 0J{H03Ha'IHO onpeJ{eJIeH, TaK KaK eCJIH b'l:x=a
H b*y=a, TO B CHJIY COKpaTHMOCTH orrepau.HlI * HMeeM x=y. 31'0 3Ha-
lIHT, 'ITO, conOCTaBJIHH Ka}K)J,oH nape (a; b) H3 Y cooTBeTcTBYIO-
rn.HH 3JIeMeHT x (TaKoH, 'ITO b*x = a), MbI onpeJ{eJIHeM 'IaCTlI'IHYIO
orrepau.lIlO T B X : x=aTb. l1Ha'Ie MOIKHO CKa3aTb, 'ITO x=aTb B TOM H
TOJIbKO B TOM CJIY'Iae, KorJ{a b*x=a.
Onepau.lIlO T Ha3blBalOT o6panmou onepal{ueu K *. 113Y'IliM ee CBOH-
CTBa. 3aMeTHM, 'ITO TaK KaK onepau.lIH * He TOJIbKO COKpaTHMa, HO 1I
KOMMYTaTlIBHa, TO H3 b*x=a BbITeKaeT, 'ITO x*b=a. I1oKaIKeM, 'ITO J{JIH
Jl106bIX JI.BYX 3JIeMeHTOB b H C H3 X BbinOJIHHeTCH paBeHCTBO
(b*c) Tb=c, (1)
a eCJIH (a; b) E Y (1'. e. eCJIH cYIl1.eCTByeT 3JIeMeHT x=aTb), TO H paBeH-
CTBO
b* (aTb)=a,
(2)
B C3MOM .l\eJIe, ypaBHeHHe b*x=b*c HMeeT peweHHe x=c, KOTopoe e.l\HHCTBeHHo B CHJIY
COKpaTHMOCTH onepaUHH *. 3TO H 3HaQHT, QTO (b*c)Tb=c. 3THM .l\OKa3aHo COOTHO-
meHHe (I). l.JT06b1 .l\OKa3aTb COOTHoweHHe (2), 3aMeTHM, QTO H3 aTb=x CJIe.l\yeT b*x=a,
a 9TO H 3HaQHT, QTO b*(aTb)=a.
ECJIH 3aMeHlITb B paBeHCTBax (1) H (2) onepau.HlO * Ha CJlO}KeHHe, TO
T 6Y)l.eT onepau.lIeH BbI'IHTaHHH H Mbi rrOJIY'IliM 1I3BeCTHbie TO}K)l.ecTBa:
+0-b=c 
H
b+ (a-b)=a.
(4)
COOTHOIlleHlIe (4) cnpaBeJI.JIHBO JIHIllb npH YCJlOBlIH, 'ITO J{JIH napbI
(a; b) CYIl1.eCTByeT pa3HOCTb. HanpHMep, ecJIH X - MHO}KeCTBO HaTY-
paJIbHbIX 'IHCeJI, TO J{OJI}KHO BblIIOJIHHTbCH YCJlOBHe a>b.
ToqHO TaK IKe J{JIH orrepau.HH YMHO}KeHHH nOJIyqaeM COOTHOIlleHHH
(be) : b=c (5)
H
(a: b).b=a.
(6)
I1pH 3TOM (6) lIMeeT MeCTO JIHIllb npH YCJIOBIIIJ, liTO B MHO}KecTBe X
orrpe)l.eJIeHO 'IaCTHOe 'IHCeJI a H b. HanpHMep, eCJIH X - MHO}KecTBO Ha-
TypaJIbHbIX 'IHCeJI, TO J{JIH 3Toro a )l.OJIIKHO )J,e.nlITbCH Ha b.
PaCCMOTpHM Tenepb TOIK)J,eCTBa, B KOTopbIe BXOJ{HT TPH 3JIeMeHTa
a, b, c. I1pe)J,nOJIO}KlIM, 'ITO orrepau.HH 'I: He TOJIbKO KOMMYT3TlIBHa H CO-
KpaTHMa, HO, KpOMe TOro, aCCOlLHaTlIBHa. B 3TOM CJIY'Iae npli YCJIOBliH
CYIl1.eCTBOB3HHH 3J1eMeHTa aT (b*c) crrpaBeJ{JIHBO TOIKJ{eCTBO
aT (b*c)=aTcTb (7)
(B rrpaBoH 'IaCTH orrepalLlIH BbinOJIHHIOTCH no rropH)l.KY CJIeBa HanpaBO).
236

B CaMOM .l\eJIe' eCJ1H x=aT (b*c). TO x* (b*c)=a. B CHJIY aCCOll.HaTHBHOCTH Orrepall.HH *
3TO COOTHOWelIHe MOJKHO rrepenHCaTb B BH.l\e (x*b)*c=a. Ho TOr.ll.a x*b=aTc H X=
=aTcTb, qTO H TpeOOBaJIOCb .lI.OKa3aTb.
I10CKOJIbKY Onepau.HH * no YCJIOBlilO KOMMYTaTHBHa, TO HapHAY C
TOIKAeCTBOM (7) HMeeM:
aT (b*c) --:. aTbTc. (7')
J{oKaIKeM Tenepb, 'ITO eC/lU cyw,ecmeyem 8.1leMeHm bTc, mo
a* (bTc) = (a*b)Tc. (8)
LLJI5I 3Toro 3aMeTUM. qTO B CHJIY aCCOll.HaTHBHOCTH Orrepall.HH * H paBeHcTBa (2) HMeelr.
[a* (bTc)hc=a*[ (bTc)*c]=a*b.
Ho no paBeHCTIJY (2)
3HaqHT,
[ (a*b)Tchc=a*b.
[a* (bTc)hc=[ (a*b)Tc]..c,
H B CHJIY cOKpaTHMocTH Orrepall.HH * rrOJIyqaeM COOTHoweHHe (8).
AHaJIOrH'IHO AOKa3bIBaeTCH, tJTO eC/lU cyw,ecmeyem He mO/lbKO bTc,
HO U aTe, mo
a* (bTc) = (aTc)*b.
J{aJIee, eC/lU cyw,ecmeYlOm aTb u aT (bTc), mo
aT (bTc) = (a*c)Tb.
(9)
(10)
B caMOM .l\eJIe' eCJIH aT (bTc)=x. TO a=x* (bTc), U rrOTOMY a= (x*b)Tc. Ho TOr)J.a
a*c=x*b. a 3TO 03UaqaeT, qTO x= (a*c)Tb. 3HaqUT,
aT (bTc)= (a*c)Tb.
EC/lU cyw,ecmeyem U aTb, mo
aT (bTc) = (aTb)*c.
(11)
3TO YTBepIKAeHHe )l.OKa3blBaeTCH aHaJIOrH'IHO, KaK H COOTHOIlleHHe
aT (aTb) --=- b,
(12)
HMelOIl1.ee MeCTO, eCJIH cYIl1.eCTByeT aTb.
CooTHoIlleHlIH (7) - (12) MOIKHO nplIMeHHTb K caMbIM pa3Ho06pa3-
HbiM aJIre6paHtJecKliM onepau.lIHM, JIlIIllb 6bI OHli 6bIJIli KOMMYTaTlIBHbI,
aCcou.HaTlIBHbI H cOKpaTliMbI. B tJaCTHOCTlI, OHli nplIMeHlIMbI K onepa-
lLHHM CJIOIKeHHH H YMHoIKeHlIH (B cJIY'Iae YMHoIKeHlIH paccMaTpHBalOTcH
JIHIIIb tJHCJIa, OTJIHtJHbIe OT HYJIH). ECJIli * - onepau.lIH CJIOIKeHHH, TO
T - onepau.HH BbltJHTaHHH. 3aMeHHH B COOTHOIIIeHlIHX (7) - (12) 3HaK
* Ha +, a 3HaK T Ha -, nOJIY'IaeM H3BecTHble H3 IllKOJIbi paBeHCTBa:
a-(b+c)=a-b-c=a-c-b,
a+(b-c) = a+b-c=a-c+b,
a-(b-c)=a+c-b=a-b+c,
a-(a-b)=b.
(13)
(14)
(15)
(16)
237

npH 3TOM eCJIH Onepau.HH CJIO:H<eHHH H BbI'IHTaHHH paCCMaTpHBalOTCH
B MHO}KeCTBe R )l.eliCTBHTeJIbHbIX 'IHCeJI HJIH B MHO>KecTBe Q palLHOH2JIb- .
HbIX 'I HCeJI , TO paBeHcTBa (13) - (16) BepHbI 6e3 Orp3HH'IeHHH. ECJIH
}({e OHH paccMaTpHBaIOTcH B MHO:H<eCTBe N HaTypaJIbHbIX 'IHCeJI HJIH B
MHO:H<eCTBe R+ nOJIO}KHTeJIbHblX 'IHCeJI, TO 'IHCJIa a, b, c )J.OJI>KHbI ObITb
T3KHMH, 'IT06b1 CYIl1.eCTBOBaJIH cooTBeTcTBYlOIl1.He pa3HocTH.
TO'IHO TaK }Ke, 3aMeHHH * YMHO:H<eHHeM, a T )J.eJIeHHeM, nOJIY'IaeM
paBeHcTBa:
a : (bc)=a : b : c=a : c : b,
a.(b: c)=a.b: c=a: c.b,
a: (b : c)=a.c : b=a : b.c,
a: (a: b)=b.
Pa3YMeeTcSI, H <PoPMYJIbI (17) - (20) HCTHHHbI JIHIllb npH
lITO Bce 0603Ha'IeHHbie B HHX )l.eHCTBHH BblnOJIHHMbi.
(17)
(18)
(19)
(20)
YCJIOBHH,
YllpamHeHHII
f. Hati.QHTe Ha MHOJKeCTBe Q paI.{HOH3J1bHblX qHCe.JI Onepal.{RIO, 06paTH)'IO 3J1re-
6paHqeCKOii Onepal.{HH a*b=a+b+ab.
2. Cym.ecTByer JlH Onepal.{HSI, 06p3'nlaSl Onepal.{HH 06'be.QHHeHHSI MHOJKeCTB?
a qnepauHH nepeceqeHHSI MHOJKeCTB?
3. Cym.eCTByer JIH onepau:HSI, 06paTHaSl onepaUHH aDb B MHOJKeCTBe HaTyp3J1bHblX
'1HCe.JI?
4. Cym.ecTByer JIH Onepal.{HSI, 06paTHall onepau:HH YMHOJKeHHSI B MHOJKecTBe Q
paI.{HOH3J1bHblX qHCe.JI? a B MHOJKeCTBe Q+ nOJIOJKHTe.JIbHblX paI.{HOH3J1bHblX qHCe.JI?
6. HeiiTpaJIbHblii H norJloalOU(Hii aJleMeHTbI. qHCJlO 0 HrpaeT oca-
6ylO pOJIb )l.JIH Onepau.HH CJIO}KeHHSI: lIpH6aBJIeHHe HYJIH K JIlO60MY 'IHC-
JIya He MeHHeT 3Toro 'IHCJIa. HHblMH CJIOBaMH, )l.JIH Bcex 'IHCeJI a HCTHH-
HO paBeHCTBO a+O=a. TO'IHO TaK :H<e YMHO>KeHHe JIloooro tIHCJIa a Ha 1
He H3MeHHeT 3TOro 'IHCJIa: )J..JIH Bcex tlHCeJI a HCTHHHO paBeHCTBO a.l =a.
fOBOpHT, 'ITO 0 - HeHTpaJIbHbIH 3JIeMeHT OTHOCHTeJIbHO onepalLHH
CJIO>KeHHH, a 1 - OTHOCHTeJIbHO onepauHH YMH()}KeHIDI. B 06m.eM BH)l.e
nOHHTHe HeliTpaJIbHoro 3JIeMeHTa onpe.n:eJIHeTCH CJle)l.YlOIl1.l!M 06pa30M.
nYCTb B MHO:H<ecTBe X 3a)l.aHa a.nre6paH'IecKaH onepau.HH *. 3JIeMeHT e
H3 X Ha3bIBalOT HeUmpaAbHblM OTHOCHTeJIbHO 3TOH onepalLHH, eCJIH !I.JlH
Bcex 3JIeMeHTOB a H3 X BbllIOJIHHlOTCH paBeHCTBa
a''k(!=e*a=a.
B MHO}KeCTBe X He MO:m.eT 6b1Tb )l.BYX pa3JIH'IHblX HeliTpaJIbHblX Me-
MeHTOB. B caMOM ;:tene, ecJ1H e H e 1 HeHTpaJIbHbI OTHOCHTeJlbUO onepalLHH
*, TO e*tl )l.OJI:m.HO paBIDITbCH H e, H II, a 3T0 3Ha'IHT, 'ITO e=el' MWKeT
CJlY'IHTbCH, 'ITO B X HeT 8K O)l.ROro HefrrpaJIbHoro 3JIeMeHTa (HanpHMep. B
MHO}KecTBe HaTypaJIbHblX 'IHCeJI HeT 3JIeMeHTOB, HeHTpaJIbHblX OTHOCH-
TeJIbHO CJIO:H<eHHH, nOCKOJIbKY 0 He C'IHTaeTCH Ha-rypaJIbHblM 'IHCJIOM).
ECJIH. B MHO}KeCTBe X ecTb 3JIeMeHT e, HeiITpaJIbHblH OTHOCHTeJIbHO
onepalLHH *, TO B JIlOOOM BbIpaiKeHHH, CO)l.ep}Kaw;eM JIHIllb 3TY onepalLHlO,
MWKHO onycTHTb Bce 6YKBbI e BMecTe C COOTBCTBYlOIl1.HMH 3HaKaMH *.
23&

HanpHMep,
17+0+6+0+0+2+0= 17+6+2.
4HCJIO 0 HrpaeT OC06ylO pOJIb He TOJIbKO OTHOCHTeJIbHO onepau.HH
CJlo»<eHHSI, HO H <>THOCHTeJIbHO onepalL HH YMHO»<eHHH. Ho ecJIH np"
CJlO»<eHHH C HYJIeM 'IHCJla He H3MeHH.IOTCH, TO npH YMHOOKeHHH Ha 0
Bce OHH craHoBHTCH HymIMH: )l.JIH JIr060ro a HCTHHHO, 'IToa.O=O'a=O.
rOBOpHT, 'ITO 0 HBJIHeTCH nOrJIOIl1.alOw.HM 3JIeMeHTOM OTHOCHTeJIbHO
onepau.HH YMHO:H<eHHH. B 06Il1.eM BH.n:e nOHHTHe nOrJIOIl1.alOIl1.erO 3JIeMeHTa
onpe.n:eJIHeTCH CJle.n:YlOIl1.HM 06pa30M.
I1YCTb * - aJIre6paH'IecKaH onepau.HH B MHO:H<eCTBe X. 3JIeMeHT 6)
Ha3b1BaeTCH nOC/lOUJ,llIOU{UM OTHOCHTeJIbHO 3TOM: onepau.HH, eCJIH )l.JIH
JIi06oro a EX BblnOJIHHIOTCH paBeHCTBa w*a=a*w=w. TaKHM 06pa30M,
o - norJIOIl1.aIOIl1.HH 3JIeMeHT OTHOCHTeJIbHO onepau.HH YMHOJKeHHH.
KaK H )l.JIH Hf:M:TpaJIbHblX 3JIeMeHTOB, )l.OKa3bIBaeTCH, 'ITO B X He
MO}KeT 6b1Tb )l.BYX pa3JIH'IHblX nOrJIow.alOIl1.HX 3JIeMeHTOB OTHOCHTe.lbHo
onepau.HH *. B caMOM )l.eJIe, eCJIH 00 H (0)- nOrJIOmalOIl1.He 3JIeMeHTbI,
TO 00*(0) )l.OJI»<HO paBHHTbCH H 00, H Wit a nOTOMY 00=(0).
ECJIH 00 - nOrJIOIl1.alOIl1.HM: 3JIeMeHT OTHOCHTeJIbHO onepalLlIIl *,
TO JIlO60e BbIpaJKeHHe, COAep:H<aIl1.ee JIHlllb 3TY onepau.HlO, B KOTopoe
BXO)l.HT 00, MO:H<HO 3aMeHHTb Ha w. HanpHMep,
4. (-5).19.0.8=0.
YllpamBeBBH
I. Cyw.eCTByeT JIH HeihpaJIbHblii 3JIeMeHT MH onepaIlHH CJ10JKeHHH HaTypa.%HblX
qHCe.JI?
2. Haii.ll.HTe HeiiTPaJlbHblii 3JIeMeHT MH onepauHH a*b=a+b+ab B MHOJKecTBe
paIlHOHaJIbHblX qHCeJI.
3. KaKoe MHOJKecTBO HBJlHeTCH HeiiTpaJIbHblM 3JIeMeHTOM OTHOCHTe.JIbHO onepaIlHH
06J.e.ll.HHeHHH MHOJKeCTB? Cyw.eCTByeT JIB HeiiTp3JIbHblii 3JIeMeHT MH onepaIlIIH ne-
peceqeHHH MHOJKeCTB?
4. KaKoe MHOJKecTBO HBJlIIeTCH norJlom.alOllJ.HM OTHOCHTe.JIbHO onepaIlHH nepece-
qeHHH MHOJKeCTB?
5. KaKoe qHCJ10 HBJlHeTCH nOrJIow.alOUJ:HM B MHOJKecTBe .lI.eJIHTe.JIeii QHCJ1a 60
OTHOCHTe.JIbHO onepaHH aJ(b?
6. KaKoe QHCJ10 HBJlHeTCH nOrJIow.alOw.HM OTHOCHTe.JIbHO onepaIlHH a*b=a+b+ab
B MHOJKeCTBe paIlHOHaJlbHblX QHCe.JI?
7. CHMMeTpHQHble 9JJeMeHTbI. B MHo»<ecTBe Q pau.HoHaJIbHbIX 'IHCeJI
onepau.HH BbI'IHTaHHH CBO)l.HTCH K onepau.HH CJIO:H<eHHH B CHJIY paBeHcTBa
a-b=a+(-b).
TO'IHO TaK :H<e onepalI.HJo )l.eJIeHHSI B MHmKecTBe <>TJlH'IHblX OT HYJIH pa-
lI.HOHaJIbHbIX 'IHCeJI MO»<HO CBecrH K onepalLHH YMHo»<eHHSI. ECJlH b=F0,
TO
I
a:b=a'7}.
B nepBOM CJlY'Iae MbI npH6aBHJIH K a'lHCJlO - b, np<>THBOI1.OJIO:H<Hoe qHC-
I
JIY b. A BO BTOpOM CJlY'Iae YMHO>KHJIH a Ha 'IHCJIO b' 06paTHoe 'IlICJlY b.
239

CeM'IaC MbI BBe)l.eM nOHHTlie 3JIeMeHTa, CliMMeTplI'IHOrO C 3JIeMeH-
TOM a OTHOClITe.nbHO aJIre6palI'IeCKOH onepau.lIli *. 4aCTHbIMli CJIY'IaHMH
3Toro nOHHTliH HBJIHIOTCH 1I3BeCTHbIe nOHHTHH npOTliBOnOJIO}KHorO H
06paTHoro 'IlICeJI.
I1YCTb B MHO}KeCTBe X CYW.eCTBYeT 3JIeMeHT e, HeHTpaJIbHbIH OTHO-
CHTeJ1bHO aJIre6palI'IeCKOH onepau.lIli *. Ha30BeM 3JIeMeHT a CUMMempUli-
IlblM 3JIeMeHTY a, eCJIli BblIIOJIHHIOTCH paBeHCTBa
a*a=a*a=e.
(I)
ECJIli aJIre6paH'IeCKaH onepau.HH * B MHOJKeCTBe X accou.HaTHBHa,
TO Hli O)l.lIH 3JIeMeHT a H3 X He MO}KeT HMeTb )l.BYX pa3JIH'IHbIX CliMMeTplI'I-
HbIX 3JIeMeHTOB.
B caMOM )l.eJIe, eCJIli b 1I C CliMMeTplI'IHbI a, TO b*a=a*c=e. Ho B CHJIY
aCCOlLlIaTlIBHOCTH onepau.lIH * HMeeM (b*a)*c=b*(a*c). 3TO 3Ha'IHT, 'ITO
e*c=b*e, T. e. 'ITO c=b. Pa3YMeeTCH, MOJKeT CJlY'IIi'fbCH, 'ITO Y a HeT
HH O)l.HOrO CliMMeTplI'IHOrO 3JIeMeHTa (HanpliMep, B MHO}KeCTBe HaTypaJIb-
HbIX 'IlICe.n Hli O)l.HO 'IlICJIO He lIMeeT CliMMeTplI'IHOrO OTHOClITeJIbHO one-
paU.HH CJIO}KeHHH - 'IlICJIO, CliMMeTpH'IHOe HaTypaJIbHOMY, oTplIu.aTeJIb-
HO, a TaKlie 'IHCJIa He BXO)l.HT B N).
PaBeHCTBa a*a=e H a*a=e nepeXO)l.HT )l.pyr B )l.pyra npH nepeCTaHOB-
Ke 3JIeMeHTOB alia. I103TOMY, eC/lU a CUMMempUfiHO a, mo a CUMMempUfi-
110 a. I1HbIMli CJIOBaMlI, eCJIlI CYIl1.eCTByeT 3JIeMeHT a, TO )l.OJIIKHO BbmOJI-
HHTbCH paBeHCTBO
'"
a= 00
.lJ.oKaJI<eM Tenepb, 'ITO eC/lU onepal{Ufl * accol{uamue1la U y 8/leMeH11W8
a U b ecmb CUMMempU'-IHble 8/leMeHmbl a U b, mo U a*b UMeem cUMMempUfi-
IlblU 8.1leMeHm, npU'-IeM
 --
(a*b)=b*a.
(3)
B caMOM )l.eJIe, B ClIJIY aCCOlLHaTHBHOCTH onepalLHH * HMeeT MeCTO
paBeHcTBo
(a*b )*(b*a) = a*( b*b)*a= a*e*a=a*a=e.
TO'IHO TaK}Ke )l.OKa3blBaeTCH paBeHcTBo
(b*a)*(a*b)=e.
3TH paBeHCTBa nOKa3blBalOT, 'ITO 3JIeMeHTbI a*b H b*;; CHMMeTpH'IHbI.
I1plIMeHlIM )l.OKa3aHHbIe paBeHCTBa (2) H (3) K onepalLlIli CJIO}KeHlIH
QHCeJI. .lJ.JIH 3TOH onepau.lIH HeMTpaJIbHbIM 3JIeMeHTOM CJlYJKHT 'IlICJIO 0,
a nOToMY YCJIOBHe cHMMeTpHH 3anHCblBaeTCH TaK:
a+ a- O.
HHbIMH CJIOBaMlI, C 'IlICJIOM a cliMMeTplI'IHO OTHOClITeJIbHO CJIOJKeHHH
npOTliBOTIOJIOIKHOe eMY 'IlICJIO a=-a. A TOr)l.a paBeHCTBO (2) 03Ha'IaeT,
'ITO
-(-a)=a,
240

a paBeHcTBo (3) - 'ITO
- (a+b)=-a-b.
nYCTb Tenepb * - onepalLlIH YMHO}l{eHlIH qlICeJI. B 3TOM cJIY'Iae
HeMTpaJIbHbIM 3JIeMeHTOM HBJIHeTCH 'IlICJIO I, a nOToMY paBeHcTBo
a*=e npHHlIMaeT BH)I. a. a= 1. 1--1:3 Hero BlI)l.HO, 'ITO CHMMeTpH'IHbIMlI
3JIeMeHTaMH . 06JIa)l.alOT JIHIllb OTJIH'IHbIe OT HYJIH 'IHCJla. npH 3TOM eCJIlI
- 1
a*O, TO a 0603Ha'IalOT -, T. e. B 3TOM CJIY'Iae ClIMMeTplI'IHOe 'IlICJlO -
a
3T0 06paTHoe 'IlICJlO. PaBeHcTBa (2) II (3) nplIHlIMalOT BlI)I.:
I
T=a,
a
H
I I I
ab =ab'
MbI BH)l.HM, 'ITO cpopMYJIa (3) JIe>KHT KaK B OCHOBe paCKpbITlIH CK060K,
nepe)l. KOTOpbIMH CTOlIT 3HaK «MlIHYC», TaK H B OCHOBe YMHO>KeHlIH )l.PO-
6eM C 'IHCJIlITeJIeM I.
ECJIlI onepalLlIH * aCCOlLlIaTlIBHa II B Bblpa>KeHlIe, cO)l.ep>KaIl1.ee JIlIIllb
3TY onepalLlIlO, BXO)l.HT pH)l.OM )l.Ba clIMMeTplI'IHbIX )l.Pyr )l.pyry 3JIeMeHTa,
TO a*a (HJIlI a*a) MO>KHO 3aMeHlITb HeMTpaJIbHbIM 3JIeMeHTOM e, a nOTOM
onYCTHTb ero. HHbIMH CJIOBaMlI, )l.Ba CTOHIl1.lIX pH)l.OM ClIMMeTplI'IHbIX
)l.Pyr )l.Pyry 3JIeMeHTa MO>KHO onYCTlITb. ECJlH >Ke onepalLlIH * He TOJIbKO
aCCOlLHaTlIBHa, HO II KOMMYTaTHBHa, TO MO>KHO OlIYCTlITb JII06bIe )l.Ba BXO-
)J.HIl1.HX B BbIpa>KeHlIe ClIMMeTplI'IHbIX 3JIeMeHTa. HanplIMep,
4+ 7 +(-4)+3= 7 +3= 10,
2 I 3
8 '3'8' 2. 5 =5.
0603Ha'IlIM 'Iepe3 a- n 3JIeMeHT (a)n. KpoMe TOro, nOJIO>KlIM aO=e.
MO>KHO )l.OKa3aTb, 'ITO lIplI TaKOM paCIIIHpeHlIH lIOHHTlIH CTeneHlI oc-
TalOTcH B ClIJIe paBeHCTBa an*am=a n + m II (an)m=a nm . ECJllI, KpOMe Toro,
onepalLHH * KOMMYTaTHBHa, TO BepHO II paBeHCTBO (a*b)n=a"*b".
J{OKa>KeM Tenepb, 'ITO eC/lU onepal{UR * accolfuamuena U Ka3ICablU
8/leMeHm a UMeem CUMMempUftHblll eMY 8/leMenm a, mo onepatfUR * co-
KpamUMa. B caMOM )l.eJIe, H3 paBeHCTBa a*x=a*y CJle)l.YeT, 'ITO a* (a*x) =
=a* (a*y), II nOTOMY B CHJIY aCCOlLlIaTlIBHOCTlI * HMeeM,a*a)*x=(ti*a)*y.
TaK KaK a*a=e, a e*x=x II e*y=y, TO x=y. 3Ha'IHT, lI3 a*x=a*y CJIe-
)l.yeT x=y, TO'IHO TaK >Ke lI3 x*a=y*a BbIBO)l.HTCH, qTO x=y. 3HaqlIT,
OlIepalLlIH * COKpaTlIMa. ECJlH OHa KOMMYTaTHBHa, TO )l.JIH Hee cymecTByer
06paTHaH onepalLlIH, KOTopaH BbIpa>KaeTCH CJle)l.yroIl1.lIM 06pa30M:
aTb=a*b.
B caMOM )l.eJIe,
(4)
(a*b)*b=a*(b*b)=a.
241

LlJ1H OnepaIUIII C.l0)j{eHHH paBeHcTBo (4) npHHuMaeT 8H)I.:
a-b=a-H-b),
3 ,nJIH Onepau. UH YMHO>KeHlIH - BU)I.:
I
a:b= 'fi'
¥upamOeODJI
I. KaKoe '!HCJlO CHMMeTpH'!HO '!HCJ1Y 8 OTHOCHTeJIbHO Onepal\HH CJIOJKeHHJI B
MHOJKeCTBe IJ,eJJbIX '!HCeJI? a '!HCJ1Y -7?
2. KaKoe 'IHCJlO cHMMeTpH'!Ho '!HCJ1Y 10 OTHOCHTeJlbHO OrrepalJ,HH YMHo)j{eHHH B
MHOJKeCTBe paIJ,HOH3,1bHbIX '!HCeJl? a '!HCJlY -3? ECTb J1H '!HCJIO, cHMMeTpH'JHOe '!I'CJIY
o OTHOCHTe.'1bHO 3TOM OrrepalJ,HH?
3. KaKoe '!HCJlO cHMMeTpH'!Ho IJHCJIY a OTHOCHTe.JIbHO Onepal\HH a*b=a+b+ab?
 3. HEIlOTOPbIE PO.D;bI AJII'EBP
1. fpynnbI H lIOJIyrpynnbI. MbI Ha3BaJIU rpynnoH lIpe06pa30BaHllii
COBOKYlIHOCTb G npeo6pa30BaHuM MHO>KecTBa X, KOTopaH BMeCTe C JIIO-
6bIMU )l.BYMH npeo6pa30BaHuHMu <p, 'I\J CO)l.ep>KUT ux KOMlI03HU.UIO <p'I\J,
3 BMecTe C Ka>K)l.bIM lIpe06pa30BaHueM <p - 06paTHoe eMY npeo6pa30Ba-
HUe <p-l. BBe)l.eM Tenepb 06mee nOHHTue rpynnbI, 3JIeMeHTaMU KOTOpOM
MorYT 6bITb He TOJIbKO npeo6pa30BaHHH, HO U 'IUCJIa, MHOrO'IJIeHbI U T. )I..
Fpynnou Ha3bIBalOT HenYCToe MHO>KeCTBO G, B KQTOpOM onpe)l.eJIeHa
aCCOlLHaTUBHaH aJIre6paU'IeCKaH onepalLUH *, TaKaH, 'ITO B G CO)l.ep>KHT-
CH 3JIeMeHT e, HeHTp3JIbHbIH OTHOCUTeJIbHO 3TOM OlIepalLUU (T. e. TaKOM,
'ITO e*g=g*f!=g )l.JIH Bcex g E G), npU'IeM BMecTe C JI!?6bIM 3JIeMeHTOM g
rpYlIlIa G CO)l.ep)j{UT U CUMMeTpU'IHbIM eMY 3JIeMeHT g (1'. e. TaKOM, 'ITO
g*g=g*g=e).
BCHKaH rpynna npe06pa30BaHUM HBJIHeTCH rpynnoM OTHOCUTe.nbHO
onepalLHH KOMn03UlLHH lIpeo6pa30BaHUH: YCJlOBUe aCCOlLU3TUBHOCTH
BblnOJIHHeTCH, lIOCKOJIbKY KOMlI03UlLUH lIpe06pa30BaHUM aCCOlLuaTUBHa,
pOJIb HeHTpaJIbHOrO 3JIeMeHTa urpaeT TO:H<)l.ecTBeHHoe npeo6pa30BaHHe
E, a pOJIb CUMMeTpU'IHOrO C <p 3JIeMeHTa - 06paTHoe npeo6pa30BaHue
<p-l.
npHBe)l.eM npUMepbl rpynn, 3JIeMeHTaMU KOTOpbIX HBJIHIOTCH 06'beK-
TbI HHOM lIPUP0)l.bI.
II P H M e p 1. MHO:H<ecTBO Z lLeJIbIX 'IHCeJI 06pa3YeT rpYlIny OTHO-
CHTeJIbHO olIepalLJlH CJlO:H<eHHH 'IHCeJI. 3Ta onepalLHH HBJIHeTC51 aJIre6pa-
HIJecKOH 8 Z (CYMMa lLeJIbIX 'IHCeJI - lLeJIoe 'Iuc.no); OHa acCOu.uaTUBHa;
po.1Ib HeihpaJIbHoro 3JIeMeHT3 HrpaeT IJHCJlO 0, a pOJIb CHMMeTpU'IHoro C
QHCJlOM a 3JIeMeHTa - lIpOTHBOlIOJIO:H<Hoe IJHCJlO - a. OrMeTHM, 'ITO B
Z OlIepau.HH CJlOJKeHHH KOMMYTaTHBHa.
fpYlIlIbI, B KOTOpblX OlIepalLuH * KOMMYTaTHBHa, Ha3bIBalOT KOlttMyma-
mueHblMu C!pynnaMu. B KOMMYTaTHBHbIX rpynnax 06bI'IHO onepau.ulO *
0603Ha'IalOT + U Ha3bIBalOT C/lO;)fCenueM, HeHTpa,'IbHbIH 3JIeMeHT 0603Ha-
IJaIOT 0 H Ha3b1BalOT nY/leM, a CUMMeTPU'IHbIM K a 3JIeMeHT 0603Ha'IalOT
-a JI Ha3b1BalOT npomueonO/lOOlCHblM a.
242

n p H M e p 2. MHO>KeCTBO Q + nOJIO>KHTeJIbHbIX pau.HoHa.'1bHblX
'IHCeJI 06pa3YeT rpynny OTHOCHTeJIbHO Onepau.HH YMHO>KeHHH: npoH3Be-
)l.eHHe )l.BYX lIOJIO>KHTeJIbHbIX palLHOHaJIbHbIX 'IHCeJI HBJIHeTCH nOJIO>KH-
TeJIbHbIM pau.HoHaJIbHbIM 'IHCJIOM; Onepau.HH YMHO>KeHHH aCCOu.HaTHBHa;
pOJIb HeihpaJIbHOrO 3JIeMeHTa HrpaeT 'IHCJlO 1, a pOJIb 3JIeMeHTa, CHM-
I
MeTpHQHOrO C 'IHCJlOM a,- 06paTHoe eMY 'IHCJIO a = a-\
n p H M e p 3. MHO>KeCTBO MHOrO'IJIeHOB, CTelIeHb KOTOpblX He
lIpeBOCXO)l.HT n, 06pa3yeT rpYlIny OTHOCHTe.nbHo onepalLHH c.nO>KeHHH
MHorO'IJIeHOB.
n p H M e p 4. MHO>KeCTBO Ta6.nHu. C m cTpoKaMH H n CToJI6u.aMH
(HJIH, KaK HX Ha3blBalOT B MaTeMaTHKe, Mampul{) 06pa3yeT rpynny OT-
HOCHTeJIbHO olIepalLHH CJIO>KeHHH Ta6JIHlL, npH KOTOpOH CKJIa)l.blBalOTCH
cOOTBeTcTBYIOIl1.He 3JIeMeHTbI Ta6JIHu., HanpHMep:
( 2 3-8 ) + ( 3-7 1 ) = ( 5-4-7 )
1-6 2 8 5-6 9-1-4 .
n p H M e p 5. MHO)l{eCTBO HaTypaJIbHbIX 'IHCeJI N He 06pa3YeT
rpynnbI HH OTHOCUTeJIbHO onepalLHH CJIO)l{eHHH, HH OTHOCHTeJIbHO OlIe-
palLHU YMHO>KeHHH: XOTH 3TH onepau.HH H accou.HaTHBHbI B N, HO N He
CO)l.ep)l{HT HH 'IHCJla, CHMMeTpH'IHOrO a OTHOCHTeJIbHO c.nO)l{eHUH, HH
qHCJIa, CHMMeTpH'IHOrO a OTHOCHTe.'lbHO YMHO>KeHHH.
MnOJKeCTBO X. B KOTOpOM 3a.1l.aHa aCCOlJ,HaTHBHaH 6HHapHaH OnepalJ,HH *. Ha3b1-
BaeTC5! noAyepynnou. MHOJKecTBO HaTypaJIbHblX qHCeJl HBJlHeTCH no.lyrpynnoH H
OTHOCHTe.JIbHO OnepalJ,HH CJlOJKeHHH, H OTHOCHTeJJbHO OnepalJ,HH YMIIOJKeHHH.
YopaacBellRR
1. B MHOJKecTBe X= {a; b} aJIre6paHqecKaH OnepalJ,HH * 3a)l.aHa CJleltYJOllleii
Ta6JJHlJ,eii;
a b
a
b
a
b
b
a
.D:oKamHTe, qTO X HBJHleTCH KOMM}'TaTHBHOH rpynnOH. KaKoH 3JJeMeHT B X HeHTp3JJeH?
2. LLOKaJKHTe. qTO MHOJKeCTBO palJ,HOHaJIbHblX qHCeJJ 06pa3yeT rpyuny OTHOCH-
Te.JIbHO OnepalJ,HH CJlOJKeHHH.
3. 06pa3yeT JlH rpynny COBOK)'UHOCTb nO.l\MHOmecTB MHOJKeCTBa X OTHOCHTe.nbHO
OnepalJ,HH nepeceqeHHH MHOJKeCTB? .-
4. LLOK3JKHTe. qTO ecJJH cp - npe06pa30BaHHe MHomecrsa X, TO cOBOKynHOCTh
npe06pa30BaHHH BH,1J.a {cpnl nEZ} 06p33yeT rpynny.
5. 06pa3YeT JlH rpynny OTHOCHTe.JIbHO CJlOJKeHHH: a) MHOJKeCTBO qeTHbiX QeJJblX
qHCeJJ; 6) MHomeCTBO palJ,HOH3JJbHblX qHCeJJ; B) MHOJKecTBO HppauHOHaJIbHblX qHceJJ?
06pa3yeT JlH KaKoe-HH6Y.ll.b H3 3THX MHOJKeCTB rpynny OTHOCllTeJJbHO YMHOJKeHHH?
2. KOJIbu.a H nOJIH. B MHO:H<eCTBe Z lLeJIbIX qHCeJI onpe,L\eJIeHH )l.Be
aJIre6paU'IecKue onepau.uH - CJlO:H<eHHe H YMHO:H<eHHe. OrHOCHTeJIbdO
onepalLHH CJlO:H<eHHH 31'0 MHO:H<eCTBO SlBJIHeTCH KOMMYTaTHBHOH rpy'n-
243

nOM, lIpH'IeM olIepalLuH YMHO>KeHHH )l.HCTpH6YTHBHa OTHOCHTeJIbHO OlIe-
palLHH CJIO>KeHHH. TaKHe MHO>KeCTBa C )l.BYMH aJIre6paH'IeCKHMH OlIe-
palLHHMH 'IaCTO BCTpe'IalOTCH B MaTeMaTHKe. BBe)l.eM CJle)l.YlOIl1.ee olIpe-
,neJIeHHe.
MHO>KecTBO X, B KOTOpOM olIpe)l.eJIeHbI )l.Be aJIre6paH'IeCKHe ollepa-
lLHH * H 0, Ha3blBaeTCH KO/lbl{OM OTHOCHTe.nbHO 3THX olIepalLHM, eCJIH
OHO 06pa3yeT KOMMYTaTHBHYIO rpYlIlIY OTHOCHTeJIbHO OlIepalLHH *, a
OlIepalLUH ° ,nHCTpH6ynjBHa OTHOCHTeJIbHQ onepalL HH *.
HalIpHMep, MHO>KeCTBO Z - KOJIblLO OTHOCIfTeJIbHO OlIepalLHH CJIO-
}l{eHHH H YMHO>KeHHH. KOJIblLOM OTHOCHTeJIbHO onepalLHH CJIO}KeHHH H
YMHO}KeHHH HBJIHeTCH MHO>KeCTBO Z [x] MHOrOtIJIeHOB OT x C lLeJ1bIMU Ko-
3cpcpHu.HeHTaMH. 31'0 JIerKO CJIe)l.yeT H3 TOro, lITO KaK CYMMa, TaK H lIpO-
H3Be)l.eHHe MHOrOlIJIeHOB C lLe.nbIMH K03cpcpHlLHeHTaMH CHOBa HMelOT lLe-
JIble K03cpcpUlLHeHTbI.
06bI'IHO OlIepalLHH * HOB KOJIbu.e Ha3bIBalOT C/lO:JICeHueM If YMHO:JICe-
HueM H 0603Ha'IalOT + H '. 3JIeMeHT KOJIblLa X, HeHTpaJIbHbIH OTHOCH-
TeJIbHO CJIO>KeHHH (OH CYIl1.eCTBYeT, TaK KaK X - rpYlIlIa OTHOCHTeJIbHO
CJIO>KeHHH) , Ha3bIBalOT HY/leM 3TOro KOJIblLa H 0603Ha'IalOT O. 3JIeMeHT,
CHMMeTpU'IHbIH 3JIeMeHTY a OTHOCHTeJIbHO CJIO>KeHHH, Ha3blBalOT npo-
mU80nO/lO:JICHbIM 3JIeMeHTY a H 0603Ha'IalOT -a. O'IeBH)l.HO, 'ITO -(-a)=
=a. .L{JIH JIlO60ro 3JIeMeHTa a KO,1bu.a X BbinOJIHHeTCH paBeHCTBO
a+(-a)=O.
TaK KaK 0+0=0, TO -0=0.
J{JIH JII06bIX )l.BYX 3JIeMeHTOB a H b KOJIblLa X OlIpe)l.eJIeHa HX pa3-
HOCTb, T. e. TaKOM 3JIeMeHT x, 'ITO a=b+x. 31'0H pa3HOCTblO HBJIHeTCH
:weMeHT a+ (-b), 0603Ha'IaeMbIH 06bI'IHO a-b.
IIoKa>KeM, 'ITO B JII060M KOJIblLe HYJIb HBJIHeTCH lIOrJIOIl1.alOIl1.HM 3JIe-
MeHTOM, T. e. 'ITO )l.JIH JII060ro aE X HMeeM a.O=O'a=O. B caMOM )l.eJIe,
TaK KaK 0=0+0 U YMHO>KeHHe B KOJIbu.e X )l.HCTpu6YTUBHO OTHOCHTeJIb-
HO CJIO>KeHHH, TO a.O=a.O+a.O, a lIOTOMY a'O=O )l.JIH JII060ro aEX.
TO'IHO TaK >Ke )l.OKa3bIBaeTCH, 'ITO O'a=O.
B JII060M KOJIblLe ClIpaBe)l.JIHBbI npaeU/la 3HaK08:
a(b)=(-a) b=-ab, (-a) (-b)=ab.
PaBeHCTBO a(-b)=-ab c.ne)l.yeT H3 Toro, 'ITO B CHJIY )l.HCTpH6y-
THBHOCTH YMHO>KeHHH OTHOCHTeJIbHO CJlO>KeHHH )l.JIH JII06bIX a H b H3
X HMeeM
ab+a (-b) =a[b+ (-b)]=a. 0=0,
a 3T0 H 03Ha'IaeT, 'ITO 3JIeMeHTbI a (-b) H ab lIpOTHBOlIOJIO>KHbI, a (-b) =
=-ab. PaBeHCTBO (-a)b=-ab )l.OKa3bIBaeTCH TO'IHO TaK >Ke, a
(-a) (-b)=-[a(-b)I=-(-ab)='ab.
OrHOCHTeJlbHO onepalLHH YMHO>KeHHH B KOJIblLe MbI He )l.eJIaJIH HH-
KaKHX lIpe)l.nOJIO>KeHHM, KpOMe )l.HCTPH6YTHBHOCTH OTHOCHTeJIbHO CJIO-
:H<eHHH - OHa MO>KeT 6bITb HH KOMMYTaTHBHOH, HH aCCOlLHaTHBHOH.
ECJlH Bce >Ke OlIepalLHH YMHO>KeHHH B KOJIblLe X KOMMYTaTHBHa, TO X
lIa3bIBalOT KOMMymamueHblM KO/lbl{OM, a eCJIH 3Ta OlIepalLHH aCCOlLHa-
244

THBHa, TO X - acc0l{UamUBHoe IW/lbl{O. B CJIY'Iae, KOr)l.a X CO)l.ep)!{HT
3JIeMeHT e, HeHTpaJIbHbIH OTHOCHTMbHO YMHO>KeHHH, e Ha3bIBalOT eall-
HU/{eU/W.1bl{a H rOBOpHT, 'ITO X - KO/lbl{O C eaUHU/{eU. TaKHM 06pa30M,
)l.JI5I JII060ro 3JIer-leHTa a KOJIbu.a Xc e)l.HHHu.eH e BbinOJIHHIOTCH paBeHCTBa
a.e=e'a=a.
B KOMMYTaTHBHbIX H aCCOl\HaTHBHblx KOJIbl\aX C e.n:HHHl\eH BepHbI Bee <jJOPMYJIbi
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245

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247

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248

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aKCHOMax, )l.eJIaeT 3TH TeopHH Ha lIepBbIH: B3rJIH)I. CJIHIllKOM a6CTpaKT-
HbIMH H cpopMaJIbHbIMH. Be)l.b MO}KeT OKa3aTbCH, 'ITO O)l.HHM H TeM >Ke
aKCHOMaM Y)l.OBJIeTBOpHIOT pa3Hble MHO}KeCTBa 06'beKTOB H pa3HbIe COOT-
BeTCTBHH Me}K)l.Y HHMH. O)l.HaKO B 3TOH KIDKYIl1.eMCH a6CTpaKTHOCTH H
COCTOHT CHJIa aKCHOMaTH'IeCKOrO MeTO)l.a: Ka}K)l.Oe YTBep>K)l.eHHe, BbI-
Be.n.eHHoe JIOrH'IecKHM nyTeM H3 )l.aHHbIX aKCHOM, npHMeHHMO K JI 10-
6 bI M MHO>KecTBaM 06'beKTOB, JIHIllb 6bI B HHX 6bIJIH OlIpe)l.e.neHbI OTHO-
IlleHHH, Y)l.OBJIeTBOpHIOIl1.He 3THM aKCHOMaM.
MHO>KecTBO C 3a)l.aHHbIMH B HeM OTHOIlleHHHMH Ha3bIBaeTCH Mo8e/lblO
(HJIH UHmepnpemal{uetl) )l.aHHOH: cuCTeMbI aKCHOM, ecJIH B HeM BbIIIOJIHH-
IOTC Bce aKCHOMbI 3TOH: CHCTeMbJ. PaCCMOTpHM )l.Ba npHMepa.
n p H M e p 1. PaccMOTpHM aKcHOMaTH'IecKylO cHcTeMY C e)l.HHCT-
BeHHbIM OTHOIlleHHeM «a,...-b» (a 3KBHBaJICHTHO b), Y)l.OBJIeTBOpHIOIIJ.HM
TpeM aKCHOMaM (CM. c. 81):
I) 8/lfl 8ceX a 8bLnO/lHflemCfl a,...-a2;
2) 8/lfl /l106bLX a U b U3 a,...-b c/le8yem b,...-a;
3) 8M /l106blX a, b, C U3 a,...-b U b,...-c c/le8yem a,...-....
H3 3THX aKCHOM BbITeKaeT pH)I. YTBep}K)l.eHHM, HanpHMep: eCAU
a,...-b u c-b, mo a,...-c.
H3 3THX YTBep}K)l.eHHH MbI BbIBeJIH, 'ITO JII060e MHO>KeCTBO X, B KOTOpOM
3a)1.aHO OTHOIlleHHe 3KBHBaJIeHTHOCTH, paClIa)l.aeTCH Ha nOlIapHo-Hene-
peCeKalOIIJ.HecH KJIaCCbI 3KBHBaJIeHTHbIX 3JIeMeHTOB (CM. c. 81).
Tenepb MbI MO>KeM lIpHMeHHTb 3TO YTBep}K)l.eHHe K JI1060MY, MHO:meCT-
BY X, B KOTOpOM OlIpe.n.eJIeHO OTHOIlleHHe 3KBHBaJIeHTHOCTH co CBOHCTBa-
249

MH 1) -3). Bce 3TH MHOlKeCTBa 6Y)l.YT MO)l.e,'1HMH CHCTeMbI aKCHOM I) -3).
HanpliMep, B Ra'IecTBe X MOIKHO B3HTb MHOlKeCTBO Bcex OTpe3ROB, a B
Ra'IecTBe OTHOllleHHH 3RBHBaJIeHTHOCTH - OTHOllleHHe «OTpe3KH a H b
HMelOT oAHHaRoBYIO AJIHHY». JIerKo npoBepHTb, 'ITO Bce TPH aKCHOMbI B
3TOM cJIY'Iae BbInOJIHHIOTCH, a Hallle YTBeplKAeHHe 03Ha'IaeT, 'ITO MHO.
lKecTBO Bcex OTpe3KOB paCIIa)l.aeTCH Ha HenepeceKalOIII;HecH KJIaCCbI, CO-
CTOHUUle H3 arpe3ROB OAHHaKOBOM )l.JIHHbI. J{PyroH MOAeJIblO TOM lKe
CHCTeMbl aRCHOM HBJIHeTCH COBOKYIIHOCTb Bcex KOHe'IHbIX MHO:H<eCTB, B
KOTOpOM OTHOllleHHe A", B nOHHMaeTCH TaK: Me:H<)l.Y A H B CyIII;eCTBYeT
B3aHMHO-OAH03Ha'IHoe COOTBeTCTBHe. B 3TOM CJlY'Iae COBOKYIIHOCTb Bcex
KOHe'IHbIX MHOlKeCTB paCnaAaeTCH Ha KJIaCCbI 3KBIIBaJIeHTHOCTH, COCTOH-
III;He H3 paBHO'IHc.'1eHHbIX MHO:H<ecTB (MHO:H<eCTB, COAep:H<aIII;HX OAHHa-
ROBoe 'IHCJIO 3JIeMeHToB).
II p H M e p 2. PaccMOTpHM ApyrylO aKcHOMaTH'IecKYIO cHCTeMY C
O)l.HHM OTHOllleHHeM a<b (a IIpe)J.IlIeCTBYeT b) H aKCHOMaMH:
I) 8/lfl /llo6blX a u b U3 a<b c/le8yem, ttmo /103ICHO b<a,
2) 8/lfl /l106blX a, b u C U3 a<b u b<c c/le8yem a<c.
3TH aKCHOMbI 3a)1.alOT OTHOIlleHHe CTpororo nopHAKa. HHTepnpeTa-
lLHHMH 3TOM CHCTeMbI MorYT 6bITb: «'IeJIOBeK a BbIllle 'IeJIOBeKa b» B MHO-
lKeCTBe JIIOAeH, «Te.'lO a THlKe,'1ee TeJIa b» H T. A. J{06aBHB K 3TOM CHCTeMe
aKcHoM)'
3) U3 a=l=b c/le8yem, '-Imo a<b U/lU b<a,
noJiY'IHM cHcTeMY aRCHOM CTpororo JIHHeHHOro nopHAKa.
MOlKeT CJIY'IHTbCH, 'ITO )l.Be MOAeJIH )l.aHHOM CHCTeMbI aKCHOM QT-
JIH'IaIOTCH Apyr OT Apyra JIHlllb BHelllHe, a no CYTH AeJIa OHH OAHHa-
KOBb!.
HanpHMep, MHOlKeCTBa X={a; b; c} H Y={l; 2; 3} 3aAalOT MOAe-
JIH CHCTeMbI aKCHOM IIOpH)l.Ka, ecJIH nOJIO:H<HTb, 'ITO a<b, b<c H a<c, a
TaK:H<e, 'ITO 1<2, 2<3 H 1<3. J{ocTaTO'IHO nepeHMeHoBaTb a B I, b
B 2 H C B 3, 'IT06bI nepBaH MOAeJIb npeBpanIJIaCb BO BTOpYIO.
TaRHe )l.Be MOAeJIH Ha3bIBalOTCH U3QMopQ;HblMU M08e/lRMU AaHHoH
CHCTeMbI aRCHO!'.!. J{Be H30MopcpHbIe MOAeJIH AaHHoH CHCTeMbl aKCHOM
no CYTH AeJIa OAHHaKOBbI.
IT p H M e p 3. OrHollleHHe «a MeHbllle b» OnpeAeJIHeT JIHHeMHbIH
IIOpHAOR B MHOlKeCTBe N HaTypaJIbHbIX 'IHCeJI. HHbIMH CJIOBaMH, MHO-
lKecTBO N C 3THM OTHOllleHHeM HBJIHeTCH MOJIeJIblO CHCTeMbI aKCHOM
1) - 3), IIpHBeAeHHoM B npHMepe 2. J{pyroH MOAeJIblO TOH :H<e CHCTeMbI
aKCHOM HBJIHeTCH. MHO:H<ecTBO N _ OTpHlI.aTeJIbHbIX lI.eJIbIX 'IHCeJI C OT'
HOlllelmeM <<a 60JIbllle b». COOTBeTCTBHe, IIPH KQTOpOM HaTypaJIbHoMY
'IHCJIY a CTaBHT B COOTBeTCTBHe npOTHBOIIOJIO:H<Hoe eMY 'IHCJlO -a
(HanpHMep, 6- -6), 3a)l.aeT B3aHMHO-O)l.H03Ha'IHOe OT06pIDKeHHe MHO-
>KeCTBa N Ha MHO:H<eCTBO N_. IIpH 3TOM eCJIH a<b, TO, KaK H3BeCTHO,
-a>-b (HanpHMep, 4<8, HO -4>-8). II03TOMY AaHHoe OT06pa:H<e-
HHe IIepeBOAHT OTHOllleHHe JIHHeHHOro nOpH)l.Ka B MHO:H<eCTBe N B <>THO-
IlleHHe JIHHeHHoro IIOpH)l.Ka B MHo:meCTBe N_. 3HaQwr, 3TH )l.Be MO)l.eJIH
CHCTeMbI aKCHOM 1) - 3) H30MOpcpHH.
3. HenpoTHBopeIJHBOCTb, He3aBHCHMOCTb H KaTerOpHIJHOCTb CHCTeMbI
aKCROM. MbI Y:H<e roBOpHJIH, 'ITO CHCTeMa aKCHOM )l.OJI>KHa B03MO:H<HO


2JjQ




TO'lHee OTpamaTb cBoikTBa pea.llbHoro !\mpa. Ho, KpOMe Toro, OHa
)!.OJImHa Y)!.OBJIeTBOpHTb HeKOTopbIM Tpe60BaHHHM .10rJllIecKoro xapaK-
Tepa. B nepBylO Oqepe.zlb, 3Ta CHCTeMa )!.OJImHa 6blTb HenpomU80peIiU60ii.
HenpOTHBopeqHBOCTb CHCTeMbI aKCHOM 03HaLlaeT, 'lTO H3 ,!{aHHOH CHCTe-
Mbi aKCHOM HeJIb351 BbIBeCTH 3aBe)!.OMO JIOmHOe YTBepm)!.eHHe, HanpHMep,
qTO O.nHOBpeMeHHo HCTHHHbI BblCKa3bIBaHH5I A Hero OTpm.laHHe A.
TaK, He,ZI.onYCTHMa CHCTeMa aKCHOM:
1) lI./lfl AlO6oeo a cYLUfcm8yem 3AeMeHm b maKou, 1lmo a-b.
2) Hu a/lfl OaNOeO a He 6btnO/lHJ/emcJ/ a-a.
3) ECAU a-b, mo b-a.
4) ECAU a--b U b-c, mo a-c.
B caMOM )!.eJIe, B03bMeM KaKoH-HH6Y)!.b 3JIeMeHT a. no aKCHOMe l) HaH-
)!.eTC5I 3JIeMeHT b TaKoH, qTO a-b. Tor)!.a no aKCHOMe 3) b-a. Ho no
aKCHOMe 4) H3 a-b H b--a CJIe)!.YeT, LITO a-a, a 3TO nponIBopellHT aK-
CHOMe 2).
LJT06bl y6e)!.HTbc5I B HenpoTHBopeLlHBocTH CHCT€!llbI aKcnOM, )!.OCTa-
TOLIHO nocTpoHTb KaKYIO-HH6Y)!.b Mo)!.eJIb 3TOH CHCTeMbI - B€,IJ,b eC.1JH
6bI 6bIJIO npoTHBOpe'lHe B caMOH CHCTeMe, OHO npHBeJIO 6b1 K npoTJmope-
lIHIO H B 3TOH Mo)!.e.'HI.
BTOpblM Tpe60BaHHeM, KOTopoe LlaCTO npe)!.'bHB.1HlOT K aKCHOMaTHLle-
CKOH CHCTeMe, HB.1JHeTC5I He3a8tlCUMocmb. 3To 3HallHT, LITO HJI O,IJ,HY H3
aKCHOM He.1b3H BbIBeCTH H3 OCTaJIbHblX aKCHOM 3TOH CHCTeMbI. HanpH-
Mep, eCJIH K CHCTeMe aKCHOM 3KBHBaJIeHTHOCTH (npHMep I n. 2) !llbI )!.06a-
BHJIH 6b1 aKcHoMY
4) eCAU a-b U a--c, mo b-c,
TO OHa OKa3a.1JaCb 6bI JIHlllHeH - ee MomHO BbiBeCTJI H3 aKCnDr-1 I) -3).
Pa3VMeeTC5I, B CHCTeMe aKCHOM He ,!{OJImHO 6blTb H TaKJlX aKCHOM,
KOTopbIe MomHO 6blJIO Obi onpOBeprHYTb Ha OCHOBe OCTaJIbHblX aKCH-
OM - TaKa 51 CHCTeMa OblJIa 6bl npOTHBOpellJlBOH.
4r06b1 )l.OKB3BTb He3BBHCHMOCTb KBKOH-HH6Y)l.b BKCHO
bI OT OCTB.1bHblX, )1.0-
CTBTO'IHO 1I0CTpOHTb MO)l.eJIb CHCTeMbI BKCHOM, B KOTOpoii Bce BKCHOMbI, KpoMe )l.BHHOii,
OCTaJOTCII TeMH >Ke, B )l.BHHBII BKCHOMB 3BMeHlleTCII ee OTpHU3HHeM.
LLOKaJKeM, HBlIpHMep, '1TO BKCHOMbi crpororo 1I0pll)l.KB I) H 2) He3BBHCHMbl. 4T06b1
)l.OKa38Tb He3BBHCHMOCTb BKCHOMbi I), 3BMeHHM ee HB IIpOTHBOIIOJlOJKHYIO:
1') cyU¥cmeYlOm mGKUi! a H b, 'Imo a<b u b<a. AKCHOMY >Ke 2)OCT3BHM HeH3-
MeHHoii. MO)l.eJIblO HOBOii CHCTeMbi BKCHOM IIBJllleTCH MHO>KeCTBO X= {a; b; c}, B KO-
TOpOM a<a, a<b, a<c, b<a. b<b, b<c.
4r06b1 )l.OKB3BTb He3aBHCHMOCTb BKCHOMbi 2) OT BKCHOMbi 1), 3aMellHM ee HB
npOTH80IIOJlO>KHYIO:
2') cyU¥cmoylOm ma/we 91leMelimbl a, b u c, 'ItrW a<b, b<c, liO 1l0JICHO, limo a<c.
AKCHOMY >Ke 1) OCTaBHM HeH3MeHHoiL MO)l.eJIblO HOBOH CHCTeMbI BKCHOM HB.
HeTCB
MHOJKeCTBO X= {a; b; c}. B KOTOpOM a<b H b<c. HO He HMeeT MeCTB a<c.
On.teTHM eme O)!.HO CBOHCTBO, KOTOpbIM 06JIa)!.aIOT HeKOTOpble CHCTe-
Mbl aKCHoM. Ha30BeM CHCTeMY aKCHOM Kam£eopu'lecKiJii, eCJ1H BCe ee
MO)!.eJlH H30MOp
HbI ,!{pyr ,!{pyry. HH CHCTeMa 3KCHOM 3KBHBaJIeHTHOCTH,
HH CHCTeMa aKCHOM JIHHeHHoro nopH,nKa He 51BJIHIOTCSI KaTerOpHtlecKH-
MR - OHH HMeIOT MHoro HeH30Mop
HblX MO,!{eJIeH. HanpHMep, )!.JISI aK-
CHOM JlHHeHHOro nOp5l,!{Ka MO,!{eJIb MOJKeT (:()CTOSlTb H H3 Tpex, H H3 qe--
Tblpex, H .B.IDKe "3 6ecKoHeQHoro MHIDKeCTBa 3JIeMeHTOB.


251




YlIpa»HeHHH
1. ,QOKaLKHTC HerrpOTHBOpe'llJBOCTb CHCTeMb! aKCHOM CTpOrOrO JlHHeliHOrO rroplI)J.-
Ei1, rrOCTpOHB ee MO)J.eJIb H3 '1eTb!peX 3J1eMeHTOB G, b, c, d. KaTerOpH'IHa JlH na CHCTeM3
ZI(CHOM? ]Bm[eTCIi .'m OHa HenpOTHBOpe'lHBoli? ,QOKaJKHTe He3aBHCHMOCTb 3KCHOM
1) - 3) cTpororo JlHHeliHOro 1I0plI)J.!W.
2. MHOJKeCTbO X COCTOHT H3 )J.BYX !WaCCOB 06'beK'roB: «TO'IeK» H «rrpIIMblx», )J.JlII
EOTOpblX orrpe)J.e.neHo 6HHapHoe OTHOIUeHHe «TO'lKa a rrpHHa)J.JleJKHT IIp IIMoli a». 3a,lJ,a,lJ,HM
TaKYIO cHcTeMY aKCHOM:
1) J1,llfl 1l106blX iJ6YX motteK a U b cyw,ccm6yem npflJ.tafl a, IWmopou npuHaOllC:JfCam
It a U b.
2) J1,llflll106biX pa3llU4f1blX motte!( a It b cyu:,ecm6yem He 6olleeoiJHou npflMou, IWmopou
npuHaiJlleJlCam o6e 8mu mottKu.
3) Kl1:JIciJou npflMou a npuHaiJlle:JfCam no KpauHeu Mepe iJ6e Pa:3IlUttHbte rnottKu.
4) Cyw,ecm6YlOm mpu mOttKU a, b, c, He npuHaOlle:JfCaw,ue OiJHOU It mou :JfCe npflMOU.
TIoKaJKHTe, '1TO ecm! Ha3BaTb TO'lKaMH BCpIUHHbI TpeyrOJlbHOll IIHpaMH)J.bI, a
IIplIMblMH - ee pe6pa, TO aKCHOMbi 1) - 4) BblnOJlHeHbI. KaTerOpH'IHa JlH CHCTeMa
3KCHOM 1) - 4)?
3. I1cxo;.{5! 113 aKCHOM 1) - 4) npHMepa 2 ,n:OKa)J{HTe, '1TO ,n:Be rrp5!Mble HMelOT HJIH
O;.{HY 06myIO TO'lKY, HJlH HH o,n:Holi 06meli TO'IKH.
 2. AKCIIOMATUKA MHOiRECTBA HATYPAJIbHbIX qllCEJI
1. B03HHKHOBeHHe nOHSInUI HaTypaJIbHOrO lJHCJIa. KaK H Bce MaTe-
MaTHlIecKHe rrOH5ITH5I, HaTypaJIbHbIe lIHCJIa B03HHKJIli H3 rroTpe6-
HOCTeH npaKTHKH. Yme B rJIy60KoH )!.peBHOCTH Ha)!.o 6bIJIO cpaBHHBaTb
Mem)!.y C060H KOHeLlHble MHomeCTBa, LlT06bl Y3HaTb, nopOBHY JIH B HHX
3JIeMeHTOB, HarrpHMep, XBaTHT JIH opymH5I Ha Bcex OXOTHHKOB, pbl6 Ha
Bcex lIJIeHOB rrJIeMeHH H T. )!.. CaMbIM rrpOCTblM crroc060M cpaBHeHH5I )!.BYX
MHomeCTB, He Tpe60BaBlllHM,HX rrepeCLleTa, 6bIJIO YCTaHOBJIeHHe B3aHMHO-
O,lJ.H03HaLlHoro COOTBeTCTBH5I HJIH Mem)!.y caMHMH MHomeCTBaMH, HJIli
Me)i{Jl.y O)!.HI1M H3 MHomeCTB H rrO)!.MHomeCTBOM )!.pyroro MHomeCTBa.
TaK, HarrpHMep, repO)J.OT, rpeLleCKHH HCTOpHK, mHBIllHH B V B. )!.O
H. 3., paCCKa3b1BaJI, lITO rrepClI)!.CKHH Ll,apb .LJ.apHH, OCTaBHB BO BpeMfl
noxo)!.a rpeKOB )!.JI5I oxpaHbI MOCTa, rrOCTpoeHHoro HM lIepe3 .LJ.YHaH, CKa-
3aJI: {(B03bMHTe 310T peMeHb H, HaLlHHa51 C Toro )!.H5I, KaK 51 rrOH)!.y Ha
CKH<l>OB, pa3B5I3blBaHTe Ha HeM Kam)!.blH )!.eHb rro O)!.HOMY Y3JIY. Kor)!.a
MliHyeT lIHC,TJO )!.HeH, 03HaLleHHoe Y3JIaMH, H 51 He BepHYCb, rrJIblBHTe
06paTHO Ha pO)!.HHY».
ECJIH y)!.aBaJIOCb Herrocpe)J.cTBeHHo YCTaHoBHTb B3aHMHO-0)!.H03Hall-
Hoe COOTBeTCTBHe Mem)!.y )!.BYM5I MHomeCTBaMH, TO 3TH MHomeCTBa ClIH-
TaJIHCb COCT05lIIJ,HMH H3 O)!.HHaKOBoro KOJIHLleCTBa 3JIeMeHTOB. ECJIH me
COOTBeTCTBHe YCTaHaBJIHBaJIOCb Mem)!.y OLlHHM MHomeCTBOM H c06CTBeH-
HbIM nO)!.MHomeCTBOM BToporo MHomeCTBa, TO ClIHTaJIH, LITO B nepBOM
MHomeCTBe MeHbllle 3.neMeHTOB, lIeM BO BTOpOM.
O)!.HaKO TaKOH MeTO)!. 6blJI rrpHMeHHM He Bcer)!.a. HarrpHMep, C ero
nOMOIIJ,blO HeJIb351 6blJIO Y3HaTb, B KaKOM H3 )!.ByX y)!.aJIeHHbiX )!.pyr OT
,npyra CTa)!. 60JIbllle OBell., HJIH cpaBHHTb lIHCJIeHHOCTb CTa)!.a ceHlIac C
ero lIHc.neHHOCTblO ro)!. TOMY Ha3a)!.. TpY)!.HO 6blJIO H 3a<l>lIKCHpOBaTb pe-
3yJIbTaT cpaBHeHH5I )!.ByX MHomeCTB. n03TOMY )!.JI5I cpaBHeHHH )!.BYX
MH02KeCTB CTaJIH npHMeH5ITb MHomeCTBa-rrocpe)!.HHKH, COCT05lBlllHe H3
naJIbLl,eB, KaMelllKOB, paKOBHH H T. )!.. lIT06hI cpaBHHTb, HanpHMep, )!.Ba
252

CT3)!.3, 6p3JIH MHomeCTBO paKOBHH, HaXO)!.HIIJ.eeCH BO B33H:.mO-O)!.H03-
HaLlHOM COOTBeTCTBHH C nepBbIM CT3)!.OM. 3T0 MHomeCTBO paKOBHH MO)KHO
6b1JIO rrepeHeCTH K MeCTY, r)!.e HaXO)!.HJIOCb BTopoe CT3)!.0, H T3M Cp3BHHTb
ero C 3THM CTa)!.OM. Pa3YMeeTC5I, 3T3 orrep3L1,H5I yme TpeOOBaJIa 33L1aTKOB
aOCTpaKTHoro MblIIIJIeHH5I (HarrpHMep, BbIBo)!.a, LITO eCJIH B rrepBOM CTa)!.e
CTOJIbKO )Ke OBeLl" CKOJIbKO paKOBHH, a BO BTOpOM MeHblIIe OBell, LleM
paKOBHH, TO B rrepBOM CTa)!.e OOJIbllle OBeLl" LleM BO BTOpOM).
O)!.HO H TO me MHO)KeCTBO-rrocpe)!.HHK MomHO 6bIJIO cpaBHHBaTb H C
MHomeCTBOM CKOT3 B CTa)!.e, H C MHomeCTBOM YOHTbIX 3Bepet'J:, H C MHO-
meCTBOM MelllKOB rrllleH, HLI,bI H C MHomeCTBOM )!.Het'J:. n03TOMY Ha3BaHH5I
MHomecTB-rrocpe)!.HHKoB CTaJIH rrpHMeH5ITb )!.JI5I Bblpa)KeHH5I LlHCJIeHHO-
CTH cpaBHHBaeMblx C HHMH MHomeCTB, T. e. KaK HMeHa LlHCJIHTeJIbHble.
HarrpHMep, BMeCTO Toro LlTOObI CKa3aTb «rr5lTb 516JIOK», rOBOpHJIH «pYKa
510JIOK» - Ha PYKe rr5lTb rraJIbLl,eB, a ClIeT Beo1C5I rro rraJIbLl,aM. OTMeTHM,
LITO B pyCCKOM 513blKe ecTb )!.peBHee CJIOBO «rr5lCTb», 03HaLIalOIIJ.ee rr5lTb
KOCTOLleK, coe,1l.HH5IIOIIJ.HX 3arr5lCTbe C rraJIbllaMH. 'A LlT06bl CKa3aTb
«)!.Ba)!'LI,aTb CTpeJI», rOBOpHJIH «lIeJIOBeK CTpeJI» - ClIeT BeJ1CH H rro rraJIb-
llaM PYK, H rro rraJIbLl,aM Hor, a HX OOIIJ.ee LlHCJIO paBHO )!.Ba)!.LI,3TH.
nepBOHaLlaJIbHO rrpHMeHHJIH pa3JIHLlHble Ha3BaHH5I LlHCJIHTeJIbHbIX B
3aBHCHM'JCTH OT Toro, KaKoe MHomeCTBo-rrocpe)!.HHK rrpHMeH5I.lIOCb )!.JI5I
cpaBHeHH5I MHomeCTB, a TaKme OT Toro, KaKHe MHomeCTBa cpaBHHBa-
JIHCb )!.pyr C )!.pyroM (HarrpHMep, O)!.HH Ha3BaHHH lIHCJIHTeJIbHblX rrpH-
MeH5IJIH )!.JI5I O)!.YllleBJIeHHbIX rrpe)J.MeTOB, a )!.p yrHe - )!.JI5I HeO)!.YllleB-
JIeHHbIX). Ho rrOCTerreHHO Bce 3TH Ha3BaHH5I CTaJIH 3aMeH5ITbC5I 06m.HM
Ha3BaHHeM ,!{aHHoro MHomeCTBa-rrocpe)!.HHKa. 3TOT npOLl,eCC aOCTp arH-
pOBaHH5I rrpHBeJI K B03HHKHOBeHHlO oom.ero rrOH5ITH5I 0 LlHCJIaX «O)!.HH»,
«)!.Ba», «TPH» H T. )!.. npH 3TOM, rrOCKOJIbKY LlaIIJ.e Bcero CLleT BeJIC5I C no-
MOIIJ.blO rraJIbLl,eB, TaKHe MHO)KeCTBa-rrocpe)!.HHKH, KaK 10, 20, 100 ()!.e-
C5ITb )!.eC5ITKOB), HrpanH OCOOYlO pOJIb; )!.pYrHe MHomeCTBa-rrocpe)!.HHKH
rrOJIYLlaJIHCb H3 HHX rrYTeM rrpHOaBJIeHH5I HJIH BbIlIHTaHH5I HeCKOJIbKHX
3JIeMeHTOB (rOBOpHJIH «)!.Ba)!.LI,aTb 0e3 )!.BYX» BMeCTO «BoceMHa)!'LI,aTb» H
T. )!..; OT3BYKH 3Toro H ceii:lIac eCTb B Ha3BaHHH LlHCJIa «)!.eB5IHOCTO» -
OHO MOrJIO paHee rrpOH3HOCHTbC5I «)!.eC5IHOCTO» H 03HaLlaTb «oe3 )!.eC5ITH
CTO»). I1HbIMH CJIOBaMH, nepBOHaLlaJIbHO MHO)KecTBO HaTypaJIbHbIX LlHCeJI
B03HHKJIO B BH)!.e OT)!.eJIbHbIX «OCTpOBKOB», He CJIHBaBlllHXC5I B e)!.HHblH
«MaTepHK».
JIHlllb rrOTOM CTaJIH pacrrOJIaraTb LlHCJIa B O)!.HH P5l)!., rrpHOaBJIH5I
Kam)!.b1H pa3 rro O)!.HOMY 3JIeMeHTY, HarrpHMep, 1, 2, 3,4, 5, ...
TaK Bo3HHKJIO rrOH5ITHe pflaa HamypaAbHblX t/,uceA. TepMHH «HaTY-
paJIbHOe LlHCJIO» BrrepBble yrroTpeOHJI pHMCKHH YLleHblH E03llHH (475-
524 r. H. 3.).
B03HHKHOBeHHe rrOHHTH5I HaTypaJIbHOrO LlHCJIa 6b1JIO BamHeHlllHM
MOMeHTOM B xo)!.e pa3BHTH5I MaTeM3THKH. nOHBHJlaCb B03MomHOCTb H3Y-
Ll3Tb 3TH LlHCJIa He3aBHCHMO OT Tex KOHKpeTHblX 3a)!.aLl, B CB5I3H C KOTO-
pbIMH OHM B03HHKJIH. TaK n05lBHJIaCb TeOpeTHlIeCKaH HaYKa 0 LlHCJIaX,
KOTOPYlO Ha3BaJIH «apHcpMeTHKOH». O)!.HOH H3 BamHbIX rrpOOJIeM 3TOro
3Tana pa3BHTH5I M3TCMaTHKII 6b1JIO H3YLleHHe OLleHb 60JIb.lllHX lIUCeJI.
HarrpHMep, B ,lJ.peBHHX naM5ITHHKaX HH,!{HHCKOH nHCbMeHHOCTH ynoMH-
253

HaIOTC5I lIHCJIa, )!.JI5I 3anHCH KOTOpbIX HY>KHa ffiHHHIl.a,C 60JIee lIeM n5lThIO-
.lI.ecHTblO HYJI5IMH. rpOMa,lJ.HbIe lIlICJIa BCTpetJalOTCH II B .lI.peBHerpelle-
CKIIX TpaKTaTaX no MaTeMaTHKe. Pa3MbIllfJIHH H3JI. TaKIIMII tJIICJIaMH,
YlIeHbIe nOH5IJIH, LITO HaTypaJIbHbIH p5l.ll. lIlICM 6ecKOHelleH, LITO K Kam-
.lI.OMY HaTypaJIbHOMY IIHCJIy MO)KHO nplI6aBIITb e!tHHHIl.Y II nOJlyqHTb
60JIblllee lIHCJIO. -
2. KOJIHIfeCTBeHHaJI TeopHH HaTypaJIbHblX qHceJI. nOCJIe TOro KaK
B XIX B. r. KaHTOpoM 6b1JIa COO.ll.aHa TeOplI5I MHomeCTB, ija ee OCHOBe
nOCTpolIJIH TeopHIO HaTypa.1JbHblX tJHCeJI. B OCHOBY TeopHH 6b1JIII nOJlO-
>KeHbI nOH5ITIIH KOHellHoro MHomeCTBa II B3aHMHO-O.ll.HOOHallHoro COOTBeT-
CTBII5I. 1I.Ba KOHe<lHblX MHO>KeCTBa A II B Ha30BeM paBHO'lUC./leHHblMU,
eCJIH MC)K.lI.Y H8MH MomHO YCTaHOBHTb B3aIIMHO-O.ll.HOOHallHoe COOTBeT-
CTBHe. OrHollleHlIe «MHomecTBO A paBHOlllICJIelIHO MHomeCTBY B7I Pe4>-
J1eKCHBHO, CIIMMeTplIlIHO H TpaH3HTHBHO. CJIffioBaTeJIbHO, OTHoweHlIe
paBHOlllICJIeHHOCTH 51BJI5IeTC5I OTHomeHlIeM 3KBIIBaJIeHTIIOCTH II onpffie-
JI5IeT pa36HeHIIe cOBoKynHocTH Bcex KOHellHblX MHomeCTB Ha KJlaCCbI
3KBHBaJIeHTHOCTII.
B O.ll.HOM KJlacce cO.ll.ep>KaTCH caMble pa3JIIIIIHble MHomecTBa; 06IIl.HM
NJH acex HIIX 51BJIHeTCSI TO CBOHCTBO, liTO Bce OHH paBHOqIICJIeHHbI, T. e.
CQJteP)KaT O.ll.lIHaKoBoe KOJIHllecTBO 3JIeMeHTOB. HanplIMep, B KJIaCCe,
co.n;ep>KaIIl.eM MHomecTBo {a; b}, eO.ll.epmaTC5I TaKlIe MlIomeCTBa, KaK
MHOJKeCTBO rJIa3 Y IIMOBeKa, MHomeCTBO KpbIJIbeB Y nTIIIl.bI, MHomeCTBO
.lI.IIarOHaJIeH KBa,npaTa H T. .lI..
Ha30BeM HamypaJlbHbUt 'lUC/lOM 06rnee CBOHCTBO KJIaCCa HenyCTblx
KOHe<lHblX 3KBIIBaJIeHTHblX .lI.pyr .lI.Pyry MHomCCT8. TaR IIHMO 3 038a-
lIaeT He TPII naJIblJ,a, TPII naJIOIIKH, TPH CTOpOHbI TpeyrOJlbHIIKa, a TO
06IIl.ee CBOHCTBO, KOTOpbIM 06JIa,lJ.alOT Bee 3TH MHO>KeCTBa - IIX 06rnylO
KOJIHlIeCTBeHHYIO xapaKTepHCTIIKY.
KIDKJI,bIH KJIacC paBHOlllICJIeHHblX MHomeCTB BnOJIHe OnpffiM5IeTC5I
JII06bIM CBOHM npffiCTaBIITeJIeM, n03TOMY MomHO rOBoplITb 0 HaTypaJIb-
HOM lIHCJIe, OnpffieJIeHHOM JII06bIM MHO>KeCTBOM .lI.aHHoro KJIaCca.
qIICJIO, onpe.ll.MeHHoe MHO>KecTBOM M, 0603HallalOT 1M I HJIH n (M)
II Ha3bIBaiOT M014HocmblO MHO>KeCTBa M.
1I.06aBJI5I5I K JII060MY KOHe<lHOMY MHomecTBY O.ll.HH 3JIeMeHT, He CO-
.lI.epLKaIII.HHC5I B HeM, nOJIYlllIM HOBoe MHOmeCTBO, He 3KBIIBaJIeHTHOe
IICXO)!.HOMY. OCYIIl.ecTBJI5I5I 3TOT npOIl.eCC, Mbi nOJIYllHM oeCKOHellHYIO
noCJIffiOBaTeJIbHOCTb He 9KBHBaJIeHTHblX .lI.pyr .lI.pyry MHO>KeCTB H onpe-
.lI.eJleHHblH elO PM HaTYPaJIbHbIX lIHCM, 1I306pameHHbIH CIIMBOJlaMH
I, 2, 3, 4, ..., n, ... .
BOObMeM JI.Ba KOHeqHblX MHO>KCCT8a A H B. 0603HallHM lIepe3 a lIb
COOTBeTcTBylOIIl.lIe 8M HaTypaJIbHhIe tJHCJIa. MHomeCTBa A H B Moryt
OKa3aTbCH 3KBHBaJIeHTHblMH JI800 He3KBHBaJIeHTHblMlI.
ECJIII A"'B, TO M1I0>KeCTB8 A H B npHHaJI.JIe>KaT O.ll.HOMY II TOMY me
KJIaecy H COOTBeTCTBYIOInHe 11M 'lHCJIa paBHbI, T. e. a=b. ECJIII A H B
He3KBIIBaJIeHTHbI, TO MHO>KCCT88 A H B npHIIaJI.JIe>KaT pa3HbIM KJlaCCaM, 8
nOTOMY COOTBeTCTBYIOIIl.He IIM qHCJIa Pa3JIIIIIHbI.
nYCTb MHomeCTBO A co,!{ep>KHT a 3JIeMeHTOB, B- b 3JIeMeHTOB. ECJIR
MH01KCCT80 A paBIIOIIHCJIelIlIO C cOOcrBeHHblM_ nO)J.MHO>KCCT80M Bi MHO-
...

.Ji\€CTBa B, TO rOBOpHT, LITO lIHC,,10 a MeHbWe lIHc.'Ia b, H nHlllYT a<b. Ta-
KHM 06pa30M,
(a<b )<=>(A - 8 1 c; 8 1\ 8 1 =1=8 1\ 8 1 =1= 0) 1).
MomHo YCTaHOBHTb, LITO OTHOllleHHe a<6 06JIMaeT CBOHCTB3!\fH aCHM-
MeTpHqHOCTH H TpaH3HTHBHOCTH, a nOTOMY HB.1JHeTC5I OTHOlllemieM CTpO-
roro nopH)!.Ka. TaKHM OOp, a30fo.I MHomeCTBO N HaTypa.1bHbIX lIHce.l
ynopH)!.OlleHO.
KIDK)!.OH OnepaI.I.HH HM KOHellHblMH MHomeCTBaMH OTBeLlaeT onepa-
I.I.H5I Ha)!. COOTBCTCTBYIOII.I.HMH 3TJIM MHO)KeCTBaM lIHCJIaMU. HarrpHMep.
nVCTb MHomeCTBa A H 8 He nepeceKaIOTC5I, npHLleM n(A)=a II n(B)=b.
Tor)!.a MHomeCTBY C=A U B COOTBeTcTByeT LlUC,10 C, Ha3bIBae!\!Oe Cy.M),tOLI
lIHCeJI a H b H 0603HaqaeMOe a+b. .LJ.OK, 3)!{eM QTO 3TO onpe;I.e.1eHHe
He 3aBHCHT OT Bbl60pa HenepeCeKalOII.I.HXC5I MHomeCTB A H B, TaKHX,
LITO n(A)=a H n(B)=b. I1HblMH CJIOBaMH, )!.OKameM, LITO eCJm A-AI
H B-B1o rrpHlIeM An 8=Al n B 1 =0, TO C1=A I U BI-C=A U B.
B ca!OM !l,e.1e, nycTb <p - B3aHMHo-0,nH03Ha'lHOe oTo6paJKeHHe A liB AI' a \f' -
B33HMHO-Q;J,H03HB4Hoe Or06paJKeHHe B HB BI' KBJK)J.blH 3JIeMeHT c H3 C= A U B npH-
H:J)!JleJKHT JlH60 A, JIH60 B (OH He MOJKeT npHHa)J..1eJKaTb 060HM MHOJKeCTBBl, TaK KBK
OHH He nepeceKaIOTCIJ). ECJIH cE A, TO 1I0JlOJKHM f (C)=<p (c), a ec.1H cE B, TO rrOJIOJKHM
f (c)=' (c). 3THM onpe)J.eJIHeTCIJ oT06paJKeHHe f MHOJKeCTBa C B MHO)/(CCTBO C 1 . nOKB-
)/(eM, '1TO KaJK)J.bIH 3.1eMeHT c' EC 1 IJBJIIJeTCIJ 06pa30M O)J.HOro H TOJIbKO O.D,ROro 3JIC-
MeHTa C H3 C. B caMOM )J.e.1e, eCJIH c' EAlo TO B Hero oro6paJKaeTCIJ e;J,tiHCTBeHHblfl
3.1eMeHT H3 A (HanoMHHM. '1TO <p - B3aHMHO-Q)J.H03Ha4Hoe COOTBeTCTBJle Me)/(,lIy A
H AI)' a eCJlH c' E B 1 . TO c' IJBJllJeTCIJ 06pa30M O)J.Horo H TOJIbKO o,nHoro 3.eMeHTa 113
MHOJKeCTBa B. IlpH 3TOM c' He MOJKeT 6blTb KaK 06pa30M 3JIeMeHTa H3 A, TaK H 06pa301
3J1eMeHTa H3 B, TaK KaK TOr,na OH npHHaMeJKaJI 6b1 H A 1 H Bl' a 3TH MHO;KeCTBa He
nepeceKalOTCfl,
I1TaK, MLI' YCTaHOBllJIH B3aHMHO-0)J.H03Ha'lHOe COOTBeTCTBHe MeJK)J.y C H C 1 , a 110'
TOMy 3TH MHO)/(ecTBa paBH04HCJIeHHbl.
113 KOMMYTaTJIBHOCTH H aCCOI.I.HaTHBHOCTH OrrepaI.I.HH 06'be)!.HHeHlfH
MHO)KecTB BbiTeKalOT CBOHCTBa KOMMYTaTHBHOCTH H aCCOI.I.HaTHBHOCTH
CJIOmCHHH HaTypaJIbHblX lIHCeJI. .LJ.aJIee, eCJIH A H 8 - HerrYCThie He-
nepeceKalOllJ,HeC5I MHO)KeCTBa. TO A HBJI5IeTC5I c06CTBeHHbiM rrOLlMHome-
CTBOM B A U B, H rrOToMY n (A U B)=I=n (A). nO.lJ.OOHO TOMY KaK OrrepalJ,H5I
CJIOmeHH5I HaTypaJIbHblX lIHce.1 CB5I3aHa C orrepallHeH 06'beLlHHeHH5I
!\lHomecTB, orrepaI.I.lI5I BbllIHTaHH5I CBH3aHa C 06pa30BaHHeM )!.orrOJIHeHHH
K nO)!.MHomecTBY. OYCTb A - KOHeQHOe MHO)KeCTBO H 8 - ero coo-
CTBeHHoe rrO)!.MHomecTBo, rrpHlIeM n (A)=a H n (8)=b. Ha30BeM Pa3-
I:lOCmblO a-b HaTypaJIbHblX QHCeJI a H b MOIlJ,HOCTb MHomeCTBa 8 A -
LlOrrOJIHeHH5I K 8 B A. TaK KaK B A U B=A, TO (a-b)+b=a. 3rHM OO'bHC-
H5IeTCH )!.aBaeMoe B IllKOJIe orrpe)!.eJIeHHe: pa3HOCTblO HaTypaJIbHbIX LlHCeJI
a H b Ha3b1BaJQT TaKoe HaTypaJIbHoe lIHCJIO C, LITO c+b=a.
OrrepaI.I.HH YMHO)KeHHH HaTypaJIbHblX lIHCeJI CBH3aHa C nO)!.ClIeTOM
LlHCJIa 3JI€MeIiTOB B JJ.eKapTOBOM npOH3Be.l{eHHH. nYCTb n (A)=a H
n (B)=b. Ha30BeM npoU36eOeHueM HaTypaJIbHblX lIHCeJI a H b MOII.I.HOCTb
I) 3HaK = ObJJI BnepBble BBe)J.eH aHrJIHiicKHJd Y'lHTe.1eM MaTeMaTJlKH P. PeKOp-
,nOM (1510-15:58). 3HaKH C<Jt H «> Jt BnepBhle ynOTpe6HJI 3HrJlHHCKHH M3TeM8THK
T. XappHoT (1560-1621). '
255

.,-::J
MHomeCTBa A xB (CM. c. 39), T. e. MHomeCTBa Bcex nap, COCTaBJIeHHbIX
H3 3JIeMeHTOB MHomeCTB A 11 B. KaK H cYMMa, npoH3Be)!.eHHe HaTypaJIb-
HbIX lIHCeJI He 3aBHCHT OT Toro, KaKHe KOHellHbIe MHO)KeCTBa A H B Bb16-
paHbI B K3L1eCTBe npe)!.CTaBHTeJIeH lIHCeJI a H b.
XOTH )!.eKapTOBO rrpOH3Be)!.eHHe MHomecTB HeKOMMYTaTHBHO (Bo06-
llJ.e rOBopH, A X B=I=B X A), cnpaBe)!.JIHBO paBeHcTBo n(A X B)=n(B X A).
4T06bl )J.OKa3aTb ero, )!.OCTaTOllHO nOCTaBHTb B COOTBeTCTBHe Kam)J.oH
nape (a; b) H3 A xB rrapy (b; a) H3 BxA. TaK me )!.OKa3bIBaeTCH, LITO
n(A X (BxC»=n((AxB)xC). OrclO)!.a BblTeKaeT, LITO YMHOJ/CeHue
llamypa/lbHblX liuce/l KO,1l/flymamu8HO u accOl{uamu8HO.
nOKa:fKeM, LITO a.l =a )J.JIH ,'11060ro HaTypaJIbHOrO lIHCJIa a. B caMOM
,lJ.eJIe, a.l - lIHC.1JO 3.7JeMeHTOB )!.eKapTOBa rrpoH3Be)!.eHHH A X B, r)!.e
n(A)=a, a B COCTOHT 113 O)!.Horo 3JIeMeHTa, HarrpHMep B={}. Ho
IIHCJIO nap BH)!.a (a; ), r)J.e a E A, paBHO lIHCJIY 3JIeMeHTOB B MHomeCTBe
A. 3HallHT, n (A X {} )=n (A), T. e. a.l =a. .LJ.oKameM Terrepb, LITO )!.JIH
JII06blX HaTypaJIbHblX lIliCeJI a H b BbIl10JIHHeTC5I paBeHCTBO a (b+ I) =
=ab+a. B caMOM )!.eJIe, nYCTb n (A)=a, n (B)=b H B 1 nOJIYllaeTCH H3 B
.l{06aB.1elmeM O)!.Horo 3JIeMeHTa (HanpHMep, 3JIeMeHTa 1'). Tor)!.a B A X B i
BOH)!.YT, Bo-rrepBblX, Bce napbl, BXO)!.51llJ.He B A X B, a BO-BTOPbIX, napbl
BH)!.a (a; 1'), r)!.e a EA. 4UCJIO HOBblX rrap paBHO lIHCJIY 3JIeMeHTOB MHO-
meCTBa A. 310 H 3HaLlIiT, LITO n(AxB 1 )=n(AxB)+n(A), T. e. LITO
a (b+ 1)=ab+a.
nOCTpoeHHe T eopHH HaTypaJIbHblX lIHCeJI Ha TeopeTHKO-MHomeCT-
BeHHOH CCHOBe BeCbMa HarJI5I)!.HO H, rro CYTH )!.eJIa, COBna)!.aeT C )!.aBae-
MhIM- B lllKOJIe. O)!.HaKO OHO 06JIa)!.a€T O)!.HHM CYllJ.eCTBeHHbIM He)!.OCTaT'
KOM: CCHOBHoe rrOHHTHe - K 0 Hell Hoe MHO m e C T B 0 - OCTaeT-
CH rrpH 3TOM He orrpe)!.e.1JeHHbIM. MomHo CKa3aTb, LITO MHomeCTBO KOHell-
HO, eCJIH OHO He paBHOLlHCJIeHHO HHKaKOMY CBoeMY c06CTBeHHOMY rro)!.-
MHomeCTBY, HO H 3TO orrpe)!.eJIeHHe HMeeT He)!.OCTaTKH (HarrpHMep, HCXO)!.51
H3 Hero C.7JOmHO )!.OKa33Tb, LITO MHomeCTBO Bcex rrO)!.MHomeCTB KOHellHO-
fO MHomecTBa KOHellHO). n03TOMY Mbl )!.a)!.HM ceHlIac HHoe rrOCTpoeHHe
TeopHH HaTypa,1bHblX lIHCeJI, OCHoBaHHoe Ha CHCTeMe aKClfoM )!.JI5I CJIO-
meHH5I.
3. AKCHOMbi CJlo)KeHHH. nOCTpOHTb cHcTeMY aKCHOM )J.JI5I MHomeCTBa
N HaTypaJIbHblX lIHCeJI MomHO MHOrHMH pa3JIHlIHbIMH crroc06aMH.
3a OCHOBHble rrOHHTHH MomHO rrpHHHTb CYMMY LlHCeJI, HJIH OTHOllleHHe
nOp5l)!.Ka, HJIH OTHOllleHHe Herrocpe)!.cTBeHHoro CJIe)!.OBaHH5I O)!.Horo LlHCJIa
3a )!.PyrHM H T. )!.. B KaJK'J.OM cJIYlIae HymHo 3a)!.aTb aKcHoMbl, orrHCblBalO-
m.He CBoHcTBa OCHOBHblX rroHHTHH. .LJ.a)!.HM cHcTeMY aKCHOM, rrpHH5IB 3a
OCHOBHoe rrOHHTHe OrrepalJ,HIO CJIOmeHH5I.
HerrycToe MHomeCTBO N Ha30BeM MHOJ/CecmeOAt Hamypa/lbHblX liuceA,
eCJIH B HeM orrpe)!.eJIeHa 6HHapHa51 aJIre6paHLleCKa51 onepall.H5I (a; b
-+a+b (LlHCJIO a+b Mbl Ha30BeM CYMMOii LlHCeJI a H b), Ha3b1BaeMa51
C/lOJ/CeHueM 1), 06JI a)!.alOllJ.a51 CJIe)!.YIOllJ.HMH CBoHcTBaMH:
1) C/loJ/CeHue KOMMymamu8Ho: eCJIH a E N H bEN, TO a+b=b+a.
1) H3rrOMHHM, '1TO 3TO 03H3'13eT cJle,lJ.YIOIlI,ee: K3JK,lJ.bIM ,lJ.BYM H3Typ3J1bHbIM '1HCJI3M
IJ H b CT3BHTC$l B COOTBeTCTBlle H3ryp3J1bHOe 'II\CJlO a+b.
256

2) CAMlCeHUe-accOll,uamuoHo: ecJlH a EN, bE N,c EN, TO a+ (b+c)=
=(a+b)+c.
3) JI.AJl AI06blX a8YX HamypaAbHblx liuceA au b ux CYMMa a+b OmAU'l-
Ha om a, a+lr-Fa.
4) B AI060M HenycmoM noiJMHo3ICecmee A MH03lCeCmBa N eCmb maKOe
IiUCAO a, limo 8ce omAUIiHbie om a IiUCAa x E A M03ICHO npeaCtna8Umb 8 8Uae
x=a+b, zae bE N.
AKCHOM 1) - 4) )!.OCTaTOLlHO )!.Jl5I TOro, LlT06b1 nocTpoHTb BCIO apHcp..
MeTHKY HaTypaJlbHblX LlHCeJl. npH TaKOM nOCTpoeHHH yme HeJIb3fl onH-
paTbC5I Ha CBoHcTBa KOHeLlHblX MHomeCTB, He Hallle)!.IIIHe OTpIDKeHH5I 8
9THX aKCHOMax, CKOJIb 6bl HH 6bIJiH OLleBH)!.HbI 9TH CBOHCTBa. HanpHMep,
paHee MomHO 6blJlO CKa3aTb, LITO eCJlH A H B HenYCTbl, TO B OTJlHLlHO OT
AU B, H n03TOMY )!.OJlmHO BblnOJlHHTbC5I COOTHOlIIeHHe lr-Fa+b. Ho
Tenepb MbI He MomeM Y)!.OBJleTBOpHTbC5I 3THM paccym)!.eHHeM, a )!.OJlmHbI
BbiBeCTH COOTHOlIIeHHe b=f=a+b H3 aKCHOM 1) - 4). Cpe)!.H 3TH X aKCHOM
eCTb aKCHOMa 3), noxoma51 Ha YTBepm)!.eHHe, nO)!.Jlemal.IJ.ee )!.OKa3aTeJIb-
CTBY. Ho B 3TOH aKCBOMe YTBepm)!.aeTC5I, LITO CYMMa OTJlHLlHa OT nepBoro
CJIaraeMOro, a He OT BToporo. n03TOMY )!.Jl5I )!.OKa3aTeJIbCTBa HepaBeH-
. CTBa b=Fa+b Ha)!.o nOCTaBHTb b Ha nepBoe MecTO. 310 MomHO C)!.eJIaTb
C nOMOl.IJ.blO aKCHOMbl I): a+b=b+a. Tenepb yme MomHO npHMeHHTb
8KCHOMY 3) H nOJlYLlHTb, LITO b=f=b+a, a Tor)!.a, nOCKOJlbKY b+a=
=a+b, HMeeM: b=f=a+b.
Pa3YMeeTCfl, )!.OKa3aTeJlbCTBa, HcnOJlb3YlOI.IJ.He JlHWb aKCHOMbI 1) - 4)
H He npHOeralOl.IJ.He K HarJl5l)!.HOCTH, CTaHOBHTC5I OOblLlHO )!.OBOJlbHO
.l{JlHHHblMH. Ho 3aTO nOJlYLleHHble pe3YJlbTaTbI MomHO npHMeHHTb He
TOJlbKO K 06blLlHOMY MHomeCTBY HaTypaJlbHblX LlHCeJl, HO H K JlI060ii
Mo)!.eJlH CHCTeMbI aKCHOM 1) - 4).
O)!.HY H3 TaKHX Mo)!.eJleH MbI xopowo 3HaeM - 3TO MHomeCTBO LlHCeJI
{I; 2; 3; 4; . . .} C 06bILlHOH onepallHeH CJlOmeHH5I. Ho Hap5l)!.y C 3TOii
Mo)!.eJlblO CYl.IJ.ecTBYIOT H HHble, HanpHMep MHomeCTBO LlHCeJl {-I; -2;
-3; . . .}, B KOTOpOM OnepaIl.Hfl CJlOmeHH5I Tome onpe)!.eJleHa 06bILlHbIM
06pa30M.
B HeKOTopblx MO,neJI5IX CHCTeMbl aKCHOM CJIOmeHH5I OnepalJ,H5I CJlO-
meHH5I Henoxoma Ha 06bILlHYIO. HanpHMep, ecJlH B3HTb MHomeCTBO
{3; 4 5; ...} C OOblLlHOH OnepalJ,HeH CJlOmeHH5I, TO OKameTC5I Hapy-
llleHHOl! aKCHOMa 4) - LlHCJIa 4 H 5 HeJlb351 npe)!.cTaBHTb B BH)!.e CYMMbI
lIHCJla 3 H KaKorO-TO LlHCJla 3Toro MHomecTBa. Ho eC.1IH nOJlOmHTb a'i(.b=-=
=a+b-2, TO )!.Jl5I TaKoro «CJIOmeHH5I» Bce aKCHOMbl I) - 4) oKamYTcH
BblnOJlHeHHbIMH. HanpHMep, 4=3*3=3+3-2, 5=3*4=3+4-2. Bbl-
nOJlHeHbI OHH H )!.Jl5I MlwmecTBa {2; 22; 2 3 ; . . .}, eCJlH 3a «CJlom(,I1I1e»
npHH5ITb YMHomeHHe LlHCeJl.
H XOTH Bce 3TH MHOmeCTBa pa3J1HLlHbI H OnepalJ,HH B HHX He noxomH
Ha 06b1L1Hoe CJlOmeHH, Bce, LITO MbI )!.OKameM )!.J1H CJlOmeHHH HaTypaJlb-
HbiX qHCeJI, OnHpa5lCb JlHllib Ha aKCHOMbl I) - 4), 6Y)!.eT BepHo ,nJl51 JllOooii
MO)!.eJIH.
MomHo )!.OKa3aTb, LITO cucmeMa aKCUOM 1) - 4) Kame20pUliHa - Bce
ee MoOeAu U30MOpcpHbl. Hanpm.lep, YCTaHoBHM H30Mop<tm3M Me)K,lJ.y 06bILl-
HoA HHTepnperaIJ.HeH 3TOH CHCTeMbl aKcHoM, KaK MHomecTBa {I; 2; 3; . . .
P 16 tn2
257

C 06bI'IHOH orrepaUHefi: CJIOmeHHH, Ii HHTepnpeTaUHefi B MHOMeCTBe
{2; 2 2 ; 2 3 ;. . .} C onepaUHeii YMHomeHHH. .LJ.JIH 3Toro rrOCTaBHM B COOT-
BeTCTBHe Ka)K,lJ.OMY HaTypaJIbHOMY LlHCJIY n IIHCJI02 n . Tor)!.a CYMMe m+n
COOTBeTCTBYeT IIHCJIO 2m+n. TaK KaK 2 m + n =2 m 2 n , TO rrpH OTo6paMeHHH
112n OnepalJ,HH CJIOmeHHH rrepeXO)!.HT B OrrepalJ,HIO YMHOMeHHH.
YlIpamueUUJI
1. )J,oKIIJKHTe HII OCHOBe IIKCHOM I} - 4) pIlBeHcTBII:
8) a+ (b+c)= (c+b)+a;
6) a+ (b+c)= (a+c)+b;
B) (a+b)+ (c+d)= (a+d)+ (b+c).
2. OjJOPMYJIHpyihe, '1TO 03HII'IIIIOT IIKCHOMbI I) - 4) npH HHTeprrpeTIIl\HH :nOH
CHCTeMbi IIKCHOM B MHOJKeCTBe: II) l\eJIbIX OTPHl\8TeJIbHbIX '1HCeJI; 6} lJHCeJI BHAII 6n
C 06bI'IHOH OlIeplll\HeH CJIOJKeHH5I; B} '1HCeJI BHAII 3 n C Oneplll\HeH YMHomeHH5I.
IlpoBepbTe BbinOJIHeHHe IIKCHOM 1) - 4} ){JI5I 9THX MOAeJIeii:.
3. 51BJI5IeTC5I JlH MOAeJiblO CHCTeMbi IIKCHOM I} - 4) MHOJKecTBO l\eJlbIX '1HCeJI C
06bI'IHOH Oneplll\HeH CJIOJKeHH5I? KIIKII5I IIKCHOMII OKII3aJIIICb HllpyweHlwH?
4. 51BJI5IeTc5I JIH MOAeJIblO CHCTeMbI IIKCHOM I} - 4} MHOJKeCTBO, COCT05Imee H3
'1HceJl {4; 5;' ...} C 06bI'IHOH oneplll\Heii CJIO>KeHH5I? KIIKII5I H3 IIKCHOM OKI!311JIIICb
HllpYlUeHHoii:?
5. Y CTIIHOBHTe H30MOptpH3M MeJK,lJ.y 06bI'IHOH MOAeJIblO {I; 2; 3; .. .} CHCTeMbi IIK-
CHOM CJIOJKeHH5I H MOAeJI5IMH: II) {-I; -2; -3; -4; ...} C 06bI'IHbIM CJlOmeHHeM;
6) {3; 4; 5; .. .}, C OnepIIl\Heii'«CJIOJKeHH5I» a*b=a+b-2.
4. OTHoweHHe nopH)J.Ka B MHO)KeCTBe HaTypaJIbHblX IIHCeJI Hero
CBOHerBa. BBe)!.eM Terrepb B MHomecl'Be N HaTypaJIbHblX IIHCeJI OTHO-
llleHHe rropH)!.Ka. npH 3TOM Mbl " MomeM CCblJIaTbCH Ha KaKHe-JIH60 H2-
rJIH,!{Hble rrpe.n.CTaBJIeHHH 0 KOHellHbIX MH02KecTBax, IIX paCrrOJIOMeHHH.X
H OT06p3)KeHHHX, a )!.OJImHbI orrHpaTbCH JIHWb Ha rrOHHTlie CYMMbl 3JIe-
MeHTOB H aKCHOMbi I) -4). Pa3YMeeTCH, paccMOTpeHHH, CBH3aHHbIe C
KOHellllblMH MHomecTBaMH, rroMorYT B Bbl60pe onpeHHH, lITO 3Ha-
IIaT CJIOBa «HaTypaJIbHoe IIHCJIO a MeHbllIe HaTypaJIbHOrO IIHCJIa b».
Mbi 3HaeM, lITO rrpH 06'be.n.HHeHHH KOHelIHoro MH02KeCTBa A C Heny-
CTblM KOHelIHblM MH02KeCTBOM C, TaKHM., LITO A n C= 0, rrOJIYlIaeTCH
MHomeCTBO B=A U C, co)!.epmaIIJ.ee 60JIbWe 3JIeMeHTOB, LleM A. 310
,lJ.eJIaeT eCTeCTBeHHblM CJIe,!{YIOm.ee onp e,!{eJIeH He.
HaTypaJIbHoe IIHCJIO a MeHbUle HaTypaJIbHOrO b, eCJIH cyw.eCTBYeT
TaKoe HaTypaJIbHOe LlHCJIO c, lITO a+c=b. B 3TOM cJIYlIae rrHlllY. a<b.
H anpHMep, 4<9, TaK KaK cym.ecTByeT TaKoe HaTypaJIbHOe LlHCJIO 5,
lITO 4+5=9.
HCnOJIb3YH OTHOIlleHHe a<b, MOMHO COPMYJIHpOBaTb aKcHoMY
4) CJIe)!.YIOIIJ.HM 06pa30M:
4') B Alo60M HenycmoM noOMHO{)ICeCtn8e A MHO{)ICeCmea N HamypaAb-
HblX 'lUCeA ecmb HaUMeHbUlee 1) IIHCJIO (T. e. TaKoe IIHCJIO a, lITO a<x
)l.JIH Bcex OTJIHIIHblX OT a IIHCeJI x H3 A).
B caMOM ,!{eJIe, CJIOBa «,!{JIH Bcex x E A, OT JIlilIHblX OT a, Haii)l.eTCH
bE NTaKoe, IIToa+b=x» 03HalIalOT TO 2Ke CaMoe, lITO Ii «lIJIH Bcex x E A,
OTJIHIIHblX OT a, BbInOJIHHeTCH HepaBeHCTBO a<x».
1) He CJIeAYeT AYMIITb, '1TO B KIIJK,IJ.OH HHTeplIpeTIIUHH CJlOBO ftaUMeHbLUuu HMeeT
OObI'IHbIH CMbICJI. HlllIpHMep, IIpH HHTepnpeT3UHH CHCTeMbi 3KCHOM I) - 4) B MHOJKeCT-
lie N_={-l; -2; . . ., -n; . . .}cH3HMeHbIUHM» OK3JKeT l1aH60JlbIUee '1HCJlO.


.lI.oKa)KeM Tenepb, qTO OmHOw.eHue < 8 MHO{}fCeCm8e N Jl8AJlemtJf om-
HouteHUeM cmpoeoeo nOpflOKa, m. e. LITO OHO TpaH3HTHBHO H aCHMMeTpHq-
HO. B caMOM .l{eJIe, nycTb a<b H b<c. nO onpe)!.eJIeHHIO 3TO 03HallaeT,
qTO HaH)!.YTcfl qHCJla k H 1 TaKHe, qTO b=a+k H c=b+l. Ho Tor)!.a c=
=(a+k)+l H no aKCHOMe 2) c=a+ (k+l). nOCKOJlbKY k+l- HaTy-
paJlbHOe lIHCJlO, 3T0 paBeHcTBo 03HaqaeT, LITO a<c. 3HaqHT, H3 a<b H
kc BbITeKaeT a<c, T. e. OTHOllleHHe < TpaH3HTHBHO.
Tenepb )!.OKa)KeM, LITO 3TO OTHOllleHHe aCHMMeTpHqHO. B caMOM )!.eJIe,
nYCTb a<b H b<a. Tor)!.a B CHJlY )!.OKa3aHHOH Bblllle TpaH3HTHBHOCTH
a<a. Ho 3TO 3HaqHT, qTO a=a+k, r)!.e kEN. nOJIyqeHHOe paBeHCTBO
npoTHBOpeqHT aKCHOMe 3). 3HaqHT, c)!.eJlaHHOe npe)!.nOJlOmeHHe HeBepHo,
H HaTypaJlbHbIx qHCeJl a, b, )!.Jlfl KOTOpblX a<b H b<a, He Cym.ecTBYeT.
ACHMMeTpHqHOCTb OTHOllleHHfl < n03BOJIfleT YTOqHHTb' aKcHoMY
4). no 3TOH aKCHOMe B HenycToM MHomeCTBe A HaTypaJlbHbIX qHCeJI
eCTb XOTb O)!.HH HaHMeHbIIIHH 3J1eMeHT a. Tenepb MbI )!.OKameM, qTO 3TOT
3J1eMeHT 0)!.H03HaHHO onpe)!.eJIeH, T. e. qTO HHoro HaHMeHblllero 3J1eMeHTa
B A He CYIll.ecTBYeT. B C8MOM )!.eJIe, ecJlH b )!.pyroH HaHMeHblllHH 3J1eMeHT B
A, TO BbInOJlHfllOTCfl 06a HepaBeHCTBa, a<b H b<a, a 3TO, KaK MbI )!.O-
Ka3aJlH, HeB03MO)KHO.
HTaK, < - »mHOtueHue cmpoeoeo nOfJ'flOKa 8 N. .lI.oKIDKeM, LITO amoln
nopflOOK AUHeeH, T. e. LITO Mfl JlI06bIX )!.ByX pa3J1HqHbIX HaTypaJIbHbIX
qHCeJI a H b BblnOJIHfleTCH XOTfl 6b1 O)!.HO H3 oTHOllleHHH a<b H b<a.
B caMOM )!.eJIe, B03bMeM MHO)KecTBO H9 )!.ByX 3J1eMeHTOB A ={a; b}. no
aKCHOMe 4') B 3TOM MHOmeCTBe){OJImeH 6mb HaHMeHblllHH 3J1eMeHT. ECJIH
3THM 3J1eMeHTOM HBJIHeTCfl a, TO a<b. ECJIH me HaHMeHbIIIHM 3J1eMeHTOM
flBJlfleTCfl b, TO b<a.
.LJ.oKameM Tenepb, lITO CA(X}IUmae HamypaAbHblX 'lUCeA 06Aaoaem c80ii-
C11l8OM MOHomoHHocmu: ecJlH a<b, TO )!.Jlfl moOoro c E N HMeeM a+c<
<b+c (HepaBeHCTBO COXpaHfleT CMbICJI, ecJlH npH6aBHTb K 06eHM qa-
CTflM O)!.HO H TO JKe qHCJlO). B caMOM )!.eJIe, a<b 03HaqaeT, LITO Cyrn.ecTBY-
eT HaTypaJlbHOe qHCJlO k, Mfl KOToporo b=a+k. Ho Tor)!.a b+c=
=(a+k)+c. no aKCHOMaM I) II 2) nOJlyqaeM:
b+c=a+ (k+c) =a+ (c+k) = (a+c)+k.
n03TOMY b+c=(a+c)+k. A 310 H 03HaqaeT, qTO a+c<b+c.
.LJ.oKIDKeM Tenepb, LITO C/lO{}fCeHUe HamypaAbHbLX 'lUCeA cOKpamU.M.O:
eCJlH a+c=b+c, TO a=b. B CaMOM )!.eJIe, B03MomHbI TPH cJlyqafl: a<b,
b<a, a=b. Ho ecJlH a<b, TO a+c<b+c, a Mbl npe)!.nOJlO)KHJlII, LITO
a+c=b+c. 3HaQHT, CJIYQaH a<b HeB03MomeH. no TOH me npHQHHe He-
B03MO)KeH cJlYQaH b<a. OcTaeTCfl e)!.HHCTBeHHafl B03MO)KHOCTb: a=b.
YupamueUBH
1. ]:(OKaJKHTe, '1TO eCJIH a<b H c<d, TO a+c<b+d.
2. ]:(oKa>DTe, lITO ecJIH a+c<b+d H b<a, TO c<d.
3. 06'bBCHBTe CMhICJI OTHOIUeHHB a<b, ecJIH B Ka'leCTBe MOAeJIH MHOJKeCTBa Ha-
TypaJlbHbIx '1HCeJI B3BTO MHOJKecTBO:
a) OTpHll.aTeJIbHbIX UeJIbiX '1HCeJI C onepaUHeH CJlOJKeHHII;
6) {3nl n EN} C 'olIepaUHeH CJlOJKeHHSI;
9.
2'59

B) t21 E!,,} C, OlIepall.Heit Y.!"HO>KHS!;
r) {3, 4, 5, '" C Onepall.HeH a*b-a+b-2.
4. ]:(OK3>KHTe, '1TO eCJIH a<.b H c<.d, TO a+c<.b+d.
5. ]:(OK3JKHTe, '1TO eCJIH a<.b H b<.a, TO a=b.
6. ]:(oK3JKHTe, '1TO ecJIH a<.b H c<d, TO a+c<b+d.
7. ]:(oK3JKHTe, '1TO eCJIH a+b=c+d H b<d, TO c<a.
5. HeorpaHHLleHHocTb H ,ll.HCKpeTHocTb MHO)l{eCTBa HaTYPaJIbHblX
LlHCeJI. v!3 aKcHoMbl4') n. 2 BblTeKaeT, LITO B MHomecTBe N HaTypaJIbHblX
LlHCeJI eCTb HaHMeHblllee LlHCJIO. 31'0 LlHCJIO 0603HaLlalOT I H Ha3b1BalOT
eaUHuu,et1. TaK KaK I - HaHMeHblllee LlHCJIO B N, TO )!.JIfl JII06oro a E N,
TaKoro, LITO a:pl, BblnOJIHeHO HepaBeHCTBO l<a. 31'0 3HaLlHT, LITO
a=l+b, r)!.e bEN.
HaH60JIblllerO LlHCJla B MHomeCTBe N HaTypaJIbHbIX lIHCeJl He cYl.IJ.e-
CTBYeT. B caMOM JJ.eJIe, a<a+ I JJ.JIH JlI060ro a E N, H nOTOMY a He MOmeT
6b1Tb Hav.60JlblllHM LlHCJIOM B N. fOBOpHT, LITO MHWICecm80 N Hamypa.1b-
llblX 'lUC('/l OepaHU'leHO CHU3Y, HO He OepaHU'leliO c8epxy.
Cpe)!.H Bcex LlHCeJI, CJIe)!.YIOI.IJ.HX 3a LlHCJIOM a, eCTb HaHMeHblllee, a
HMeHHO LlHCJIO a+ I. B caMOM )!.eJIe, ecJIH b C.TIe)!.YeT 3a a, TO HaH)!.eTCfl
TaKoe HaTypaJIbHOe LlHCJIO C, LITO b=a+c. Ho lc, H nOToMY a+ la+c,
T. e. a+ lb. MbI )!.OKa3aJIH, 'ITO a+ I - HaHMeHblllee H3 LlHCeJl, CJIe-
,!{YIOIll.HX 3a a.
B )!.aJlbHeHllleM HaHMeHblllee H3 LlHCeJI, c.1e)!.YIOI.IJ.HX 3a a, Mbi 6y)!.eM
Ha3bIBaTb Henocpeacm8eHHo c/leaYW14UM 3a a. HTaK, B N )!.JIH K3)K)!.Oro
3JIeMeHTa eCTb Herrocpe)!.cTBeHHo CJIe)!.YIOIll.HH 3a HHM 3J1eMeHT. 31'0 CBOH-
CTBO Ha3b1BaIOT aUCKpemHocmblO MHomecTBa HaTypaJIbHblX LlHCeJI.
06paTHbIM K oTHOllleHHIO «b Herrocpe)!.CTBeHHO C.1e)!.YeT 3a a» HBJIHeTCfl
OTHOllleHHe «a Herrocpe)!.CTBeHHO npe)!.lllecTBYeT b». HHblMH CJlOBaMH, a
Henocpe,!{CTBeHHO npe)!.lllecTBYeT b B TOM H TOJlbKO B TOM CJIYLlae, Kor)!.a
b=a+ I. 4HCJIO I He HMeeT Henocpe)!.CTBeHHbIX npeJl.lllecTBeHHHKOB,
TaK KaK B CHJIY aKCHOM I) H 3) HH )!.JIH O)!.Horo HaTypaJIbHOrO LlHCJla a
He MOmeT BbmOJIHHTbCH paBeHcTBo 1 =a+ I. nOKameM, LITO 8ce Hamy-
paAbHble 'lUC/la, Om/lU'lHble om I, UMelOm OaHO U mOAbKO oiJHO HenocpeiJ-
cm8eHHO npeaUlecm8YlOUIf!e UM 'lUC/lO.
B C3MOM AeJIe, eCJIH b=l= I, TO I <b (BeAb I - H3HteHbIUee H3TypaJIbilOe '1HCJlO).
A 9TO 03Ha'laeT, '1TO cyw:ecTByeT T3Koe '1HCJlO aEN, '1TO b=l+a=a+l, T. e. TaKoe,
'1TO b HenocpeAcTBeHHo CJleAyeT 33 a. Ho TOfA3 a HenocpeACTBeHHo npeAIUecTByeT b.
HaM OCT3J10Cb nOKa3aTb, '1TO Y b HeT HHblX HenocpeAcTBeHHblx npeAIUecTBeHHHoB.
npeAnOJlO>KHM, '1TO c=/=a, HO c HenocpeAcTBeHHo npeAIUecTByeT b. TOfA3 He TOJibKO
b=a+l, HO H b=c+l, T. e. a+l=c+l. B CHJlY cBoiicTBa COKpaTHMOCTH AJlS! C.10-
>KeHHS! OTCIOA3 CJleAyeT, '1TO a=c BonpeKH npe)l;nOJlO>KeHHIO. 3Ha'lHT, Y b eCTb JlHU'b
OAlio HenocpeAcTBeHHO npeAIUeCTBYIOw:ee eMY '1HCJlO.
YnpamueuulI
I. liTO 03H3'131OT CJlOBa «b HenocpeACTBellHO cJleAyeT sa a», eCJlH B Ka'lecTBe
MOAeJIH CHCTeMbl 3KCHOM I) - 4) Bbl6p3HO MHOJKeCTBO oTpHIJ.areJlbHblX lI.eJlblx 'lIiCeJI
C Onep3IJ.Heii 06bl'lHoro CJlO>KeHH5I?
2. PeIUHTe TOT JKe Bonpoc, eCJlH IJ Ka'leCTlJe MO;J,e.1H CHCTeMbi 3KCHOM Bbl6paHO
MHOJKeCTBO '1HCeJI BHAa {2; 22; ...} C Onep3IJ.Heii YMHOJKeHHS!?
260

6. npHHLl.Hn MaTeMaTHlJecKoii HHJI.YKLl.HH. lI.oKameM em.e O)!.HO CBOM.
CTBO MHomeCTBa HaTypaJlbHblX 4HCeJI - npllHl{un ,AlameMamU'leCKOU
UHOYK!{UU.
EC/lu nooMHMlCeCm80 A MHMICecm8a N Hamypa/lbHbtx 'luce/l coiJepCHCum
1 u 8Mecme C 1WJICObiM 'lUC/lOM a cooepCHCum HenOCpeOCmBeHHO C/leoYLOUJ,ee
sa HUM 'lUC/lO a+ 1, mo A c08naOaem C N.
Har JIH)!.HbIM CMblCJI 3Toro YTBepm)!.eHHH COCTOHT B TOM, LITO HallaB C
lIHCJla I H npHOaBJl5I5I K HeMY O)!.HY e)!.HHHU;Y 3a )J.PyroH, Mbl paHO HJlH
n03)!.HO nOJIYlIHM JII060e HaTypaJlbHOe lIHCJlO. .lI.oKa3aTeJIbCTBO npHH-
ulma MaTeMaTHlIeCKOM HH)!.yKUHH npOBe)!.eM OT npOTHBHoro. npe)!.nOJIO-
mHM, LITO cym.eCTByeT MHomecTBo A C yKa3aHHbIMH CBoHcTBaMH (1 E A
H H3 a E A c.TJe)!.yeT a+ 1 E A), He COBna)!.alOm.ee C N. Tor)!.a ero JI.onOJI-
HeHHe A'=N",A HenycTo, A'=F0. 3HallHT, no aKCHOMe 4') B MHomeCT-
Be A' eCTb HaHMeHbIlIee lIHCJIO b. 3r0 lIHCJlO OTJlHLlHO OT 1, TaK KaK
1 E A, H nOTOMY 1 He MOmeT npHHaJI.JIemaTb )!.onOJIHeHHIO K A. 3HallHT,
lIHCJIO b Henocpe)J.CTBeHHO CJIe)!.yeT 3a HeKoTopblM lIHCJlOM a, b=a+ l.
TaK KaK a<b, a b - HaHMeHbIlIee lIHCJlO B A', TO a He npHHa)!.J1emHT A'.
T. e. HBJIHeTCH lIHCJlOM H3 MHomeCTBa A. Ho Tor)!.a H lIHCJIO b=a+ 1
)!.OJlmHO 6bIJIO 6bl npHHa)!.J1emaTb MHO)!{ecTBY A. nOJIYllaeTcH, 4TO b npll-
Ha)!.JIemliT Ii MHomecTBY A, Hero )!.onOJlHeHHIO A', a 3Toro He MomeT
6blTb. nO.TJYlleHHoe npOTHBopellHe JI.OKa3blBaeT, LITO A' = 0, T. e. A=N.
npHHUHn MaTeMaTHlIeCKOM HH)!.YKUHH LlaCTO HcnOJlb3YlOT )J.JIH )!.OKa-
3aTeJlbCTBa MaTeMaTHlIeCKHX TeopeM. .lI.oKameM, HanpHMep, LITO )J.J1H Bcex
HaTypa.1bHblX lIHCeJI n cnpaBe)J.J1HBO paBeHCTBO
1+3+5+...+(2n-l)=n 2 . (I)
flpOCTa5J npoBepKa nOKa3blBaeT, LITO 310 paBeHCTBO BepHO npH n= l.
B caMOM )!.e.TJe, )!.J1H J1eBOH lIaCTH paBeHCTBa lIHCJlO n nOKa3blBaeT, CKOJIb-
KO lIJIeHOB Ha)!.o B3HTb B CYMMe. n03TOMY eCJlII n= I, TO HMO B3HTb JIHWb
nepBoe CJIaraeMOe I. A npaBaH lIaCTb npH 11= I npHHHMaeT 3HalleHHe
}2=l. TaK KaK 1=1, TO npH n=1 paBeHCTBO (I) cnpaBe)!.JIliBO. BepHO
OHO H npH n=2. B 3TOM cJlYllae B JIeBOH 4aCTH Ha)!.o B3HTb )!.Ba cJlarae-
MblX, a B npaBOH BMeCTO n nO)J.CTaBHTb 3Ha4eHlle 2. nOJIYllaeM Bepnoe
paBeHcTBo 1+3=2 2 - 06e lIaCTH 3Toro paBeHcTBa npHHHMalOT 3HaLleHHe
4. TOllHO TaK me npH 11=3 nOJIYLlaeM 1+3+5=3 2 (06e lIaCTH paBHbl
9). Ho, CKOJlbKO 6bl 3HaLleHHH n Mbl Hli B3H.TJH, 113 cnpaBe)!.JIliBOCTIi pa-
BeHCTBa (I) )!.J151 3THX 3HalleHHH He C,'IeJI.yeT, LITO OHO cnpaBe)!.JIHBO H )!.J1H
OCTa,1bHbIX 3HalleHHH n-npOBepHTb Bce 3HaLleHliH n Mbl He B caCTOH-
IIHH.
n03TOMY npH6eralOT K HHOMY paccym)!.eHHIO. CliallaJia )!.OKa3blBaIOT,
LITO paBeHCTBO (I) cnpaBe)!.J1HBO npH n= I (3TO Mbl yme c)!.eJlaJIH). A no-
TOM )!.OKa3blBalOT, LITO eCJlH OHO cnpaBe)!.,1liBO )!..1H KaKoro-To 3HalleHli51
n, TO OHO BepHO H )J.JIH Henocpe)!.CTBeHHO cJle)!.YIOw.ero 3a HHM 3HalleHHH
11+ I. Y!HbIMH CJIOBaMH, )J.OKa3blBalOT TaKOe YTBepm)!.eHIIe: eCJIH BepHO,
LITO
1 +3+.. .+(2n-l) =n 2 ,
TO BepHO H paBeHCTBO
1+3+...+(2n-l)+[2(n+ 1)-1 )=(n+ 1)2
(2)
261

(B JIeBOH: qaCTH )J.06aBHJJH CJIe)J.YlOm.ee 3a 2n-1 HelleTHOe lIHCJIO
2(11+1)-1, a B npaBOH: 3aMeHHJJHnHan+l; MO)!{HO CKa3aTb H TaK,
LITO B 06eHx lIaCTHX paBeHcTBa (1) 3aMeHHJlH n Ha n+I).
no npe)J.nOJJO)!{eHHIO, cYMMa nepBblx n lIJJeHOB B paBeHCTBe (2) paB-
Ha n 2 . 3aMeHHB HX 3THM 3HaLleHHeM, nOJJYllaeM, LITO Ha)!.o )J.OKa3aTb
paBeHCTBO n 2 +[2(n+I)-1J=(n+I)2. A )J.J151 3Toro )J.OCTaTOllHO pac-
KpblTb CR06KH: n2+2n+2-I=n2+2n+I=(n+l)2.
I1TaK, )J.OKa3aHO H TO, LITO paBeHCTBO (1) cnpaBe)!.JIHBO npH 11= 1, H TO,
LITO H3 ero cnpaBe)!.JIHBOCTH npH HeKOTopOM n CJle)!.yeT, LITO OHO BepHO H
npH n+ I. A Tor)J.a MO)!{HO paccy)!{)!.aTb cJle)!.YlOm.HM 06pa30M. TaK KaK
paBeHCTBO BepHO npH n=l, TO OHO BepHO H npH n=I+I=2. Ho Tor)!.a
OHO BepHO If npH n=2+ 1 =3. np0)!.OJJ)!{aH 3TO paccy)!{)!.eHHe, Mbl paHO
HJIH n03)!.HO )!.OH:)!.eM )!.O JII060ro HaTypaJIbHOrO 3HalleHHH n, T. e. paBeH-
CTBO (I) BepHO )!.JIH Bcex HaTypaJIbHblX 3HaLleHHH: n.
npOBe)!.eHHoe paccy)!{)!.eHHe He COBceM noJJHO C TOllKH 3peHHH JIOrH-
Klf, TaK KaK CKa3aHHaH B KOHue <ppa3a «MbI paHO HJlH n03)J.HO )!.OH:)J.eM
)J.O JJI060ro 3HaLleHHH n» He nOJlYllHJla 060CHOBaHHH C nOMom.blO aKCHOM.
4T06bl oMCHOBaTb ee, nocTynHM CJIe)!.YlOm.HM 06pa30M. 0603HallHM
'lepe3 A MHO)!{ecTBO Tex 3HalleHHH: n, )!.JIH KOTOpblX paBeHCTBO (I) BepHO.
MbI 3HaeM, LITO MHomecTBY A npHHa)!.J1emHT lIHCJIO I (TaK KaK 3TO pa-
BeHCTBO BepHO npH n= I), a H3 TOro, LITO n E A, CJle)!.yeT, LITO n+ 1 E A
(eCJIH paBeHCTBO BepHo )!.JJH HeKoToporo 3HalleHHH n, TO OHO BepHO H
npH n+ 1). Ho Tor)!.a no npHHuHny MaTeMaTHLlecKOH: HH)!.YKUHH MHO)!{e-
CTBO A COBna)!.aeT co BceM MHO)!{eCTBOM N HaTypaJIbHbIX lIHCeJI, a 3TO
03HallaeT, LITO paBeHCTBO BepHO )!.JIH Bcex HaTypaJlbHbIX 3HalleHHH n.
TOllHO TaK me )!.OKa3blBaeTCH, LITO eC/lU HeKomopblU npeOUKam T (n)
CmaflO8umcfl ucmUfIHblM npu 3aMeHe 8 HeM n Ha 1 U eC/lU U3 ucmUHHocmu
am020 npeaUKama npu HeKomopoM 3Ha'leHUU n C/leayem, 'lmo OH tlCmUHeH
U npu n+I, mo MHO:JICeCm80 ucmUHHocmu amoco npeouKama c08naoaem
co 8ceM MHO:JICeCm80M N Hamypa/lbHblX 'luce/l, m. e. amom npeouKam
ucmUHeH a/lfl Bcex Hamypa/lbHblX 3Ha'leHUU n.
Y npamHeHBII
I. ):{OKmKHTe C nOMOlll,blO MaTeMaTH'leCKOii HH,aYKll,HH CJle,aYlOllI,He paBeHCTBa:
s) 12+22+... +n2 n (n+ l (2n+ I );
2 ( I I ) 2
6) 13+23+... +n 3 n n 4 , ;
B) i2_22+32_42+...+(_l)n-ln2=(_l)n+1 n(n;-I) ;
1 I 1 I n
r) D+ 4.7 + 7.1O +"'+(3n-2)(3n+l) 3n+l'
2. ):{oKmKHTe C nOMOlll,blO MSTeMaTH'IeCKOii HH,aYKll,HH, '1TO eCJlH a<b, TO a n <b".
7. AKcHoMaTHKa neaHo. nOHHTHe CJlO)KeHHH HarypaJlbHbIX '1HCeJI He HBJlHeTCH
e,aHIICTBeHHoii 6a30ii, Ha KOTOpoiitMOJKHO nOCTpOHTb aKCHOMaTHKY MHO)KeCTBa HaTY-
paJlbHblX '1HCeJI. BOJlee TOro, 3TO nOHHTHe He HBJlHeTCjj npocTeiiIUHM - KaK H3BeCTHo,
npH6aB.eHHe K HaTypaJlbHOMY 'IHCJlY n HaTypaJIbHOrO '1HCJla m MOJKHO OCYllI,ecTBHTb
«no war aM», ,a06aBJI5!H Ha Ka)K,aOM wary no o,aHoii e,aHHHll,e, HanpHMep: 5+3= «(5+
+ 1)+ 1)+ I). n03TOMY Onepall,HIO CJlO)KeHHH MOJKHO CBecTH K Donee npocToii - one.
262

paHH rrpH6aB.eHHII'IHc.a 1. nOCKOJlbKY
n+ I IIBJllleTCII HenOCpeAcTBeHHO CJle.JJ.YIO-
ll(UM 3a n '1HCJlOM, MO>KHO rOBOpHTb 0
rrepexo,ae K HenOCpe,aCTBeHHOCJIe,ltYlOlUe-
MY 'iHCJlY.
TaKHM 06pa30M, B Ka'leCTBe OCIIOB-
1I0ro OTHOIUeHHII B MHOJKecTBe HaTypaJlb-
IIbIX '1HCe.l MOJKHO BbI6paTb «4HCJlO p
Herrocpe;J:CTBeHHO CJle,ayeT 3a '1HCJlOM n».
npu 3TOM npHHHMalOT CJle,aYIOIUHe aK-
CHOMbl:
1) lI./lfl KaJIC0030 HamypaJlbHOZO 'IUC/la n cyw,ecm8yem Henocpeocm8eHHo c/leoYIOw,ee
sa HUM 'lUC/lO.
2) EC/lu 'lUC/la p U q HenocpeocmtJeHHo clleoYlOm sa 'lUC/lOM n, mo OHU paeHbt opye
OPY3Y.
3TH aKCHOMbl 03Ha'laIOT, '1TO 3a KaJKAbIM HaTypaJlbHblM '1HCJlOM n HenocpeACTBeHIIO
CJleAyeT OAHO U TOJlbKO OAHO HaTypaJlbHOe '1HCJlO, 0603Ha'lHM ero n'.
3) H U 00.'10 'lUC/lO He MOJICem H.enocpeoCm8eHlI.O CIle008amb sa 08YMfI pa3/lU'lHblMU
Hamypa/lbHblMU 'lUC/laMU: eC/lU m'=n', mo m=n.
AKCHOMbi 1) - 3) rrOKa3b1BaIOT, '1TO HaTypaJIbHble '1HCJla H,aYT O,aHO 3a ,apyrHM.
Ho UMble '1HCJla II1J.YT O,aHO 3a ,ltpyrHM, o,aHaKO cpe,aH HHX HeT caMoro nepBOro, a 8
MHOJKecTBe HaTypaJIbHblX '1UCeJI ecTb caMoe nepBoe '1UCJlO, a HMeHHO 1. OjJoPMYJlH-
pyeM 3TO YTBepJKJ\eHHe B BH,ae CJle,aYlOlUeH aKCHOMbI:
4) B },;HOJICeCm8e HamypaJlbHblX 'luce/l ecmb 'lUC/lO I, KOmopoe He C/leoyem HU sa
oiJHUM HamypG./lbHblM 'lUC/lOM.
MOJKeT CJlY'lHTbCH, '1TO OTHOIUeHHe Henocpe,acTBeHHoro CJle,aoBBHlliI 8 HeKOTOpoM
MHO>KeCTBe A y,aoBJleTBoplleT BceM aKCHOMaM 1) - 4), HO 3TO MHOJKeCTBO co,aep>KHT
«JlHIllHue» 3J1eMeHTbl. TaKoe MHO>KeCTBO H306paJKeHO Ha pHcYHKe 176. 4T06b1 HCKJlIO-
'1HTb B03MOJKHOCTb nOIlBJleHHII JlUIllHHX 3J1eMeHT08, BBe,aeM elUe O,aHY aKcHoMY (aK-
CHOMY MaTeMaTH'IeCKOH UH,aYKUHU).
5) EC/lu noiJMHOJICeCm80 A MHOJICeCmtJa N HamypaJlbHbtX 'luce/l coiJepJICum I It
8Mecme C KaJICOblM HamypaJlbHblM 'lUC/lOM n coiJepJICum HenocpeocmtJeHHo clleoYIOw,ee sa
HUM 'lUC/lO n', mo A c08naiJaem co flCeM MHOJICeCmtJOM N.
AKCHOMbi 1) - 5) 6b1J1H BnepBble $pMYJlHpOBaHbl HTaJIbllHCKHM y<IeHblM 11. nea-
110 B KOHue XIX 8. l1x Ha3b1BaJOT l1KCUOMaMU DeaHO.
l1TaK, Mbl ,aaJIH ,aBe aKCHOMaTHKH .!\JIll MHO>KeCTBa H8rypaJIbHblX '1HCeJI. B03-
HHKaeT Bonpoc: paBHOCHJlbHbI JlH 3TH aKCHOMaTHKH, onpe,aeJIlllOT JlU OHH O,aHO H TO
JKe MHOJKeCTBO? TO'lHee, 3TOT BOnpoc 03Ha'laeT CJle,aYlOlUee. llBe aKCHOMaTHKH pa3-
JlH'IalOTCII KaK OCIIOBHblMH nOHIITHIIMH, TBK H npHHHMaeMbIMH 3a OCHOBY YTBep>KAe-
HUIIMU. n03TOMY TO, '1TO B O,aHOii: aKCHOMaTHKe IIBJllleTCII aKCUOMoii, BO BTOpoii CTa-
1I0BHTCII TeopeMoii, Tpe6YlOlUeii ,aOKa3aTeJIbCTBa. Ha30BeM ,aBe aKCHOMaTHKH iJK8U8a-
/leHmHblMU, ecJlH OT Ka>K1J.oii H3 HHX MO>KHO nepeHTU K ,aPyroH, BblnOJlHHB CJle,aYIOIU.He
,aeiiCTBHH;
a) orrpe,aeJIUB OCHOBHble nOHIITHII ,apyroii: aKCHOMaTHKH '1epe3 OCH08Hble IIOHIITHII
AaHHoH,
6) ,aOKa3aB aKCHOMbi ,apyroii aKCHOMaTHKH Ha OCHOBe aKCHOM ,aaHHoH.
B rr. 4 MbI, HCX01J.II H3 aKCHOM CJlOJKeHHII, ,aOKa3aJlH cBOiicTBa OTHOIUeHHII «Herro-
cpeAcTBeHHo c.le,'tOBaTb 3a». Cpe,aH 3THX cBoiicTB OKa3aJlHCb Bce YTBepJKJ\eH, HII BXO-
,allIU.He B cHcTeMY aKCUOM neaHo. 3Ha'lHT, aKCHOMaTHKa neaHo CJle,ayer H3 aKCHOM
CJlO>KeHHII.
MO>KHO ,aOKa3aTb, '1TO H, 06p, aTHO H3 CHCTeMbl aKCHOM neaHo BblT1fJBIOT aKCHOMbi
CJlO>KeHHII. MbI He 6y,aeM ,lteTaJIbHO npOBo,aHTb 3TH ,aOKa3aTeJIbCTBa, a ftOKa>KeM JlHlllb,
KaK C nOMOIU.blO OTHOllleHHII CHenocpe,ltCTBeHHO CJle,aOBaTb 3a» MO>KHO onpe,aeJlHTb
orrepaIJ;HJO CJlO>KeHHII HaTypaJIbHblX '1HCeJI.
C/lOJICeHUeM B MHOJKeCT8e N HaTypaJIbHblX '1HCeJI Ha3bIBaeTCII 6HH8pHall aJIre-
6paH'IeCKall orrepaUHII, CTaBlllUall B COOTBeTCTBHe nape HaTypaJlbHblX HCeJl m H n
'1Hc.o m+n, rrpH'leM
1) n+l=n',
2) n+m'= (n+m)'.
113 ,aaHHoro 3,aeCb onpe,aMeHHII OTHIO,ab He BblTeK8eT, '1TO onepaUHII C Tpe6yeMblMH
.--+-.----.....---.. .
1 : J
i,
a\ l b
c
PHC. 176
263

CBOlkTBaMH cymecmyeT. C 1I0MOmblO AOBOJlbHO CJlOJKHblX paCCYJKAeHHI1, KOTOl>blf
MbI OnYCKaeM, MOJKHO .llOKa3aTb, '1TO B MHOJKeCTBe N cymecTByeT 6HHapHa!l ame6paH-
4eCKa!l OnepaLlH!I, YAOBJleTBOp!llOma!l Tpe6oBaHH!lM I) H 2), npH'IeM 3Ta onepaLlH!I
011H03Ha'lHO onpeAeJleHa H YAoBJleTBop!leT aKCHOMaM 1) - 4) n. 2.
I1HblMH CJlOBaMH, MO>KHO 1l0Ka3aTb, '1TO BBelleHHa!l onepaLlH!I y;lOEJlC'TBOp!leT
aKCHOMaM CJIOJKeHH!I.
YnpamHeHBII
I. .IJ.oKaJKHTe Ha OCHOBaHHH aK.CHOM neaHo, '!TO HH OAHO HaTypa.lbHoe '!HCJIO
He COBna,lJ.aeT C HenOCpellCTBeHHO CJIeAYlOmHM 3a HHM.
2. .IJ.OKIDKHTe Ha OCHOBaHHH aKCHOM neaHo, '1TO AJI!I JlI060fO HaTypa.lbHofO '1HCJla
n, KpoMe 1, cymeCTByer HenocpeAcTBeHHo npeAIUecTBYlOmee '1HCJIO.
3. .IJ.oKa>KHTe Ha OCHOBaHHH aKCHOM neaHo, '1TO B KaJKAOM HenycToM MHOJKeCTBe
HaTypaJlbHblx '1HCeJI ecTb HaHMeHbllJHH 3J1eMeHT.
 3. APJI<I>METJIKA IIATYPAJIbHbIX qllCEJI
1. YMH01KeHHe HaTypaJIbHblX lJHcen. Onpe)!.eJllfM Tenepb B MHO-
JKeCTBe N HaTypaJIbHblX lIlfCeJI HOB)'IO oHHapHYIO aJireOpalfllecKYIO
onepaII.HIO - YMHomeHHe. YMHoCHCeHueM HaTypaJIbHblX lIlfCeJl Ha3blBa-
etCH 6HHapHaH aJlre6paHlIecKaH OnepaII.HH B N, CTaBHm.afl B COOTBeT-
CTBHe nape (a; b), a E N, bEN, lIHCJIO ab lf3 N, KOTopoe Ha3bIBaeTCH
npoU38eOeHueM 'luce/l a u b, nplflleM:
1) P./lfl /l106OZ0 Hamypa/lbHoeo. 'lUC/la a 8btnO/lHfiemcfl paeeHCm80
a.I=a.
2) p'/lJl /l106blX HamypaJlbflblX 'luce/l a u b 8blnO/lHflemCfl paeeHCm80
a (b+ I)=ab+a.
lI.oKameM, LITO eC/lU cyw,ecm8yem 6UHapHafl onepal{Ufl 8 MHOCHCecm8e
N, 06/laOOJOU¥lJ/. c80iicm8aMU 1) U 2), mo OHa oOH03Ha'lHO onpeOe.'leHa.
4lfCJla a If b Ha3bIBalOTCH MHOmIITeJlflMlf.
npellnOJlO>KHM, '1TO onepaLlH!I * TaKJKe 06J1a,lJ.aeT cBoikTBaMH I) H 2), T. e. a*l=a
H a* (b+ l)=a*b+a. MbI XOTHM AOKa3aTb, '!TO TOrAa AJI!I mo6b1X a H b BbInOJlH!leTC!I
paBeHCTBO a*b=ab. 3a<jJHKcHpyeM 3Ha'!eHHe a H 0603Ha'lHM '1epe3 Ba MHOJKeCTBO
lIaTypaJIbHblX '1HCeJl b, AJiH KOTOpblX a*b=ab. B 3m MHOJKeCTBO BXOAHT I, nOCKOJlbKY
a*l=a H a.l=a, H nOTOMY a*l=a'l. .IJ.aJlee, nYCTb bEBa. TOfAa a*b=a.b. Ho no
cBoiicTBy2) HMeeM a*(b+ l)=a*b+a=ab+a=a(b+ I). 3T0 03Ha'laeT, '1TO b+lEBa.
I1TaK, lEBa H H3 bEBa CJIeAyeT, '1TO b+l EBa. n03TOMY B CHJlY nplfHLlHna MaTe-
MaTH'!eCKOH HHAYKLlHH Ba= N, a 3TO 03Ha'laeT, '1TO paBeHCTBO a*b= ab "PH 3aAaHHOM
a BepHO AJiH Bcex bE N. B CHJlY npOH3BOJlbHOCTH a OHO BepHO AJI!I Bcex a H b H3 N.
Cyw,eCm808aHUe 6UHapHoii onepal{uu, o6.i/QoalOw,eu C80UCm8aMU
1) H 2), ,lJ.OKa3blBaeTCH C nOMom.blO MaTeMaTHlIecKoH HH,lJ.YKII.HH no a.
0603Ha'lHM '!epe3 A MHOJKeCTBO lIaTypaJlbHblX '!HCeJl a, .l\JIH KOTOpblX TaKaH
onepaLlHH onpe)\eJIeHa (T. e. AJI!I KOTOpblX cymecTByeT Or06paJKeHHe b-+a. b, npH
I\OTOpOM '!HCJlY I COOTBeTCTByeT a, a '1HCJlY b+ 1 - '1HCJlO ab+a). 4HCJlO I npHHaA-
JIeJKHT MHOJKecTBY A, TaK KaK AOCTaTO'lHO nOJlOJKHTb I. b= b AJI!I Bcex bE N (MbI
HMeeM TOfAa 1.1=1 H I (b+l)=I.b+l=b-j-l). npeAnOJlOJKHM, '1TO aEA. nOJlOJKHM
TOrAa (a+l)b=ab+b. I1MeeM (a+1).I=a.l+l.l=a-j-l H (a-j-l)(b-j-l)=a(b-j-l)+
+ (b+l)=ab-j-a-j-b-j-l=ab-j-b-j-a-j-l= (a-j-l)b-j- (a+l). 3m 3Ha'!HT, '1TO H a-j-1EA.
Ho TOfAa B CHJlY npHHLlHna MaTeMaTH'IeCKOH HHAYKLlHH A= N. 3m H 03Ha'!aeT, '!TO
onepaLlHH YMHOJKeHH!I onpeAeJIeHa AJI!I JlI06bIX HaTypaJlbHblX '1HCeJI.
264

l10K3meM Tenepb, tJTO YMflOJICeflUt! Homypa/lbHblX flUee/l iJuempu6y.
mllBflO C/le8a omHOCUme/lbHO C,1OJlCeflUfl, T. e. tJTO )J.J151 J1106b1X H3TypaJlb-
BblX tJHCeJI a, b, e BblnOJlH5IeTC5I paBeHCTBO a (b+c)=ab+ac.
.l1oKa3aTeJlbCTBO npOBO)!.HTCSI C nOMOIL\blO MareMaTH'IeCKoii HH)!.YKL\HH no c npH
(jJHKCHpOBaHHblX a H b. IlpH c= 1 )!.OKa3b1BaeMOe paBeHcTBO npHHHMaer BH)!. a (b+ 1)=
=ab+a.l=ab+a. OHO CnpaBe,ll,JlHBO no onpe)!.eJIeHHIO }'MHOJKeHHSI. Ilpe)!.noJlOJKHM
Tenepb, '1TO ,lI,JISi HeKOTOporo c YJKe )!.OKa3aHO paBeHCTBO a (b+c)=OO+ac, H nOKIDKeM,
'1TO TOr,na a[b+ (c+l)]=OO+a(c+l). B caMOM ,neJIe, B CHJlY 8CCOL\HaTHRHOCTH CJlO-
)KeHHII
a [b+ (c+ 1)]=a [ (b+c)+ I]=a (b+ c)+ a
(no CBOHCTBY 2) YMHOJKeHHII). no npe)!.nOJlOJKeHHIO HH)!.YKL\HH a (b+c)= OO+ac, H
nOTOMY
a[b+ (c+ 1»=OO+ac+a=ab+a(c+ 1).
I1TaK, ,nOK83b1B8eMOe p8BeHcTBO IIPH 3a,n,8HHblX a H b HCTHHHO IIPH c= 1, 8 H3 ero
HCTHHHOCII'H ,lI,JIII HeKoroporo c CJIe,nyeT, '1TO OHO HCTHHHO H .l\JIII C+ 1. B CHJlY IIpHH-
llHlIa MareM8TH'IecKOH HH)!.YKllHH OHO HCTHHHO .l\JIII BCex c. nOCKOJlbKY a H b - m06b1e
H8TypaJIbHble '1HCJla, paBeHCTBO D.OKa3aHO npH Bcex HaTypaJIbHblX 3Ha'leHHSlX a, b H C.
TOllHO TaK me ,lJ.OKa3bIBaeTC5I, LITO Y.MHOJICeflUt! auempu6ymU8flO cnpa-
ea omHOeUme/lbflO C/lOJlCeHUfl, T. e. tJTO paBeHCTBO
(a+b)e=ac+be (1)
HCTHHHO ,lJ.JI5I JII06bIX HaTypaJIbHbIX a, b, c (noCKOJIbKY Mbi em.e He ,lJ.OKa-
3aJlH KOMMYTaTHBHOCTH YMHomeHHH, npHXO,!{HTC5I OT,lJ.eJlbHO ,lJ.OKa-
3bIBaTb ,lJ.HCTPH6YTHBHOCTb CJIeBa H cnpaBa).
.l1oK83aTeJlbCTBO CHOBa Be)!.ercII C nOMOW;blO MareMaTH'IeCKOii HH)!.YKllHH no c:
IIPH c= 1 HMeeM (a+b).I=a+b, a' l+b. l=a+b, a IIOTOM)' (a+b). l=a. l+b. 1.
.llaJlee, eCJIH .l\JIII HeKoroporo c HMeeM (a+b)c=ac+bc, TO (a+b)(c+l)= (a+b)c+
+ (a+b)=ac+bc+a+b=ac+a+bc+b=a(C+l)+b(c+l). 3H8'1HT, B CHJlY IIpHH-
llHna MareMaTH'IecKoii HH,nYKllHH paBeHcTBo HCTHHHO )!.JlSI Bcex H8TypaJlbHblX a, b H C.
.LJ.oKameM Tenepb, tJTO Y.MHOJICeHUe Hamypa/lbHblX 'luce/l accol{uamU8-
HO, T. e. )J.J1H JIIOOblX HaTypaJlbHbIX lIHCeJI a, b, C
(ab)e=a (be). (2)
CHOB8 3a<PHKcHpyeM a H b H IIpOBe)!.eM MaTeMaTH'IeCKYIO HH)!.YKllHIO no c. IlpH c= 1
D.OKa3b1BaeMOe p8BeHCTBO npHHHM8eT BH)!. (00)' l=a(b. 1). Ho (ab).I=ab H a(b.l)=ab,
a nOTOMY Hallie paBeHCTBO BepHO. IlYCTb YJKe ,nOKa38HO, '1TO (ab)c=a(bc). Tor,na IIO
CBOHCTBY 2) YMHOJKeHHII H ,nHCTpH6yrHBHOCTH CJIeBa HMeeM
(00) (c+ 1)= (ab)c+ 00= a (bc)+ 00= a (bc+b)=a [b (c+ I».
3Ha'lHT, ,nOKa3b1BaeMOe p8BeHcTBo HCTHHHO H )!.JlII c+ 1. B CHJlY npHHllHna MaTeM8-
TH'IeCKOH HH,nYKllHH OHO HCTHHHO D.JlII Bcex 3Ha'leHHH c. IloCKOJlbKY a H b npoHS-
BO,1bHbI, Mbi ,nOKa3aJIH, '1TO p8BeHcTBo (2) HCTHHHO npH npOH3BOJlbHblX HaTypaJlbHblx
3H8'1eHHIIX a, b H c.
.LJ.aJIee ,lJ.OKameM KO.M.Mymamu8Hoemb YMHOJICeHUfl, T. e. )!.OKa)!{eM, LITO
a/lfl /l106blX HamypaJlbHblX 'luce/l a U b 8btnO/lHflemCfl pa8eHCm80 abk.ba.
310 )!.OKa3aTeJIbCTBO HeOOXO,lJ.HMO npOBeCTH, 1I0CKOJlbKY B orrpe)!.eJleHHH
YMHO)!{eHHH MHOmHTeJlH a H b He paBHonpaBHbJ.
.l1oKaJKeM CHa'laJI8 C 1I0MOW;blO MareMaTHqeCKOH HH)!.YKllHH no a, '1TO paBeHCTBO
OO=ba HCTHHHO npH b=l, T. e. '1TO )!.JlSI Bcex aEN HMeeM a.l=l.a. IlpH a=1 3TO
265

pBBeHCTBO npHHHMBeT BH.ll I. i= 1.1 H nOToMY HCTHHHO. npeAnOJlOJKHM, '1TO Mil
KaKorO-TO a Y)Ke .llOKa3aHo, '1TO a. 1= I. a. Tor,J,a HMeeM B CHJlY .llHCTPH6YTHBHOCTH
(a+I).I=a.l+l.l=l.a+l.I=1 (a+I).
H nOToMY ,lJ.OK<13bIBaeMoe paBeHcTBo HCTHHHO H Mil a+ 1. 1-10 TOrAB B CHJlY npHli.
lJ,Hna MaTeMaTH'IeCKOii HH].\YKlJ,HH OHO HCTHHHO ].\Jl5I Bcex HaTypaJIbHblX 3Ha'leHHH a.
Tenepb .llOKaJKeM HCTIIHHOCTb paBeHCTBa ab=ba B 06rn;eM cJlY'Iae. 3a<f>HKCHPYCM
3Ha'leHHe a H 0603Ha'lHM '1epe3 Ba MHOJKCCTBO 3HB'IeHHH b, ].\Jl5I KOTOpblX 3TO paBeH-
CTBO BblnOJlH5IeTC5I. I(BK Mbl ToJlbKO '1TO ].\OKa3aJlH. OHO BblnOJlH5IeTCII npH b= I, H
nOTOMY 1 EBa. npe].\nOJlOJKIIM, 'ITO bEB a , T. e. '1TO ab=ba. TorAa HMeeM B CHJlY AH-
CTPH6YTlIBHOCTH
a (b+I)=ab+a=ba+a= (b+l) a,
II nOTOMY b+ lEBa.
3Ha'lHT, lEBa H H3 bEBa CJIe].\yeT b+1EBa. Ho TorAa B CHJlY npHHlJ,Hna ala.
TeMaTH'IeCKOH HHAYKlJ,HH, Ba=N, T. e. paBeHCTBO ab=ba AOKa3BHO M5I Bcex HaTY-
pBJlbHblX '1l1ceJl a H b.
Tenepb )!.OKameM MOHomOHHocmb npOU38eaeHUfl, T. e. ,lJ.OKameM, LITO
ecJIH a, b H C - HaTypaJIbHble lIHCJJa, npHlIeM a<b, TO ac<bc. B caMOM
.l{eJIe, HepaBeHcTBo a<b 03HallaeT, LITO b=a+k, kEN. Ho Tor)!.a bc=
= (a+k)c=ac+kc, H nOTOMY ac<bc.
J1 MHOCHCeHue HamYPaJlbHblX IiUCeA COKpamUMO: eCJlH a, b, c - HaTY-
paJlbHble lIHCJla H ab=ac, TO b=c. B caMOM )!.eJIe, eCJlH 6bl Mbl HMeJIH
b<c, TO B CHJlY MoHOTOHHOCTH YMHO)!{eHHH BbInOJlH5IJlOCb 6bI HepaBeH-
CTBO ab<ac, LITO npOTHBopellliT YCJlOBIUO. TaK 2Ke )!.OKa3blBaeTCH, LITO
HeB03MO:IKeH cJJYlIaii c<b. 3HallHT, b=c.
YnpamHeHHII
1. Onp<'.lleJlHTe YMHOJKeHlIe HarypaJIbHblX '1HCeJl B CJIe.llYJOrn;JfX MOAeJlIIX:
B) {-I; -2; ...; -n; ...} C 06b1'1HblM CJIOJKeHHeM;
6) 3; 4; 5; .. .}co «CJlOJKeHHeM» no <jJopMYJle a*b=a+b-2;
B) (:2; 22; ...; 2"; ...} co «CJlOJKeHHeM» no <popMyJle a*b=ab.
2. I(aKoH BHA npHHIIMalOT CBoHcTBa YMHOJKeHH5I B 3THX MOAeJIIIX?
3. ):{oKaJKHTe, '1TO eCJlH a, b, C - HaTypaJlbHble 'I HCJlB , npH'IeM a<.b, TO ac<.bc.
2. BbllJHTaHHe HaTypa,,1bHblX lJUCeJI. TaK KaK CJlO)!{eHHe HaTypaJlb-
HblX lIHCeJI KOMMYTaTHBHo H COKpaTHMO, TO )!.JIH HerO cym.eCTByeT Ok
H03HallHo onpe)!.eJleHHaH o6paTHaH lIaCTHlIHaH onepaUHH, n03BOJIHIOw.aH
no cYMMe H O)!.HOMY H3 CJIaraeMblX Haxo)!.HTb BTopoe CJlaraeMoe.
I1HblMH CJlOBaMH, BbILlHTaHHe HaTypa.1JbHbIX lIHCe.1J onpe)!.eJIHeTCH
cJIe)!.YIOIll.UM 06pa30M. nYCTb )!.aHbI HaTypaJlbHble lIHCJla a H b. Pa3HO-
cmblO 3TUX lIHCeJI (eC.'IH OHa cym.eCTByeT) Ha3bIBaeTCH TaKoe HaTypaJlb-
Hoe lIHCJIO c, LITO a=b+c. 3'ro lIHCJlO 0603HallalOT a-b. 4HCJlO a Ha3bI-
BalOT YMeHbUlaeMbIM, a LlUC,IIO b - 8blliumaeMblM.
BbIHCHHM Tenepb, )!.,IIH KaKHX nap (a; b) cym.ecTByeT pa3HoCTb a-b.
Mbl 3HaeM, LITO cym.eCTBOBaHHe HaTypaJlbHoro IIHCJla C, )!.J151 KOTOpOro
a=b+c. paBHOCH.lbHO BblTlOJlHeHHIO uepaBeHCTBa b<a. n03TOMY
pa3HOCmb a-b cyWf!cm8yem 8 mOM u mO/lbKO 8 mOM C/lYliae, KOcaa b<a.
npH 3TO1 pa3Hocmb onpeOe!leHa oOH03Ha1iH0, nOCKOJlbKY ecJlH a=b+c
H a=b+d, TO b+c=b+d, a Tor,lJ.a B CHJlY cOKpaTuMocTH CJIomeHHH
c=d.
266

Yl-raK, MbI onpe)!.eJIHJJH lIaCTHlIHYIO onepaUIfIO BbjqHTaHHH B MHome-
CTEe N HaTypaJ1bHblX lIHCeJJ, 06paTHYIO onepaUHH CJIomeHHH. Ee CBOW.
CTBa, BbiTeKalOw.He OTCIO)!.a, YKa3aHbi B n. 5  2 rJ1. VIII.
YupaJlIHeHH1I
I. ,l],oKa}KHTe, 'ITO YMHO}KeHHe HaTypaJlbHblX 4HCeJI )\IJCTPH6YTHBHO OTiIOCHTeJlbHO
Bbl'lHTaHH!I. T. e. a (b-c)=ab-ac.
2. ,l],oKa}KHT'. 4TO eC,lH b<a H a<c. TO c-a<c-b.
3. ,l],oKa}KHTe. 4TO eCJlH b<a H c<b. TO b-c<a-c.
4. ,l],aiiTe onpe.l1eJleHHe onepaUHH BbI'IHTaHH!I B cJ1e.l1YIOIUHX MO.l1eJl!lX HaTypaJIb-
HOro p!l.l1a 4HCeJl:
a) {-I; -2; .... -n; ...} C 06b14HOII onepaUHefI C,10}KeHH!I;
6) {3; 4; ...} C onepalI,Heii «CJlO,KeHH!I» no cpopMYJ1e a*b=a+b-2;
B) {2; 22; ...; 2"; ...} C onepaI\Heii «C,10}KeHH!I» no <jJopMYJle a*b=ab.
5. ,l],oKa}KHTe cJle.l1YlOlUHe cBoiicTBa Bbl'IHTaHH!I HaTypaJIbHblX 4HCeJl:
a) e<',.'IH b+c<a, TO a- (b-j-c)=a-b-c;
6) eCJlH b<a, c<b, TO a-(b-c)=a-b+c.
6. 06J1a.l1aeT JlH onepau.H!I Bbl4HTaHH!I HaTypaJIbHb!X 'IHCeJl cBoiicTBOM KOMMY-
TaTHBHocTH? a cBoiicTBOM accoUHaTHBHocTH?
3. eJIeHHe HaTypaJlbHbIX lJUCeJI. eJleHHe C OCTaTKOM. TaK KaK YM-
HomeHHe HaTypaJ1bHblX LlHCeJI KOMMYTaTHBIiO H COI<paTHMO, TO )J.JIH Hero
cyw.eCTByeT lIaCTH'IHaH 06paTHaH onepauHH, n03B0J1HIOw.aH no npOH3-
Be)!.eHHIO )!.BYX lIHCeJJ H O)!.HOMY H3 MHomHTe.1el'I 02l.H03HaLlHO onpe)!.eJIHTb
BTOpOH MHOmHTeJIb. 9Ty onepaUHIO Ha3b1BaIOT oeAeHueM HaTypaJIbHblX
LlHCeJI.
I1HblMH CJIOBaMH, )!.eJIeHHe HaTypaJIbHblX LJlIceJI onpe)!.e.15IeTCH CJIe-
JJ.YIOW.HM 06pa30M. nYCTb )!.aHbl )!.Ba HaTypaJ1bHblX LlHCJIa a H b. lJacmHblM
LlHCeJJ a H b (eCJIH OHO CYIll.eCTByeT) Ha3bIBalOT TaKoe HaTypaJlbHOe lIHC-
JJo C, LITO a=bc. 9T0 lIHCJIO C 0603HallalOT a : b. 4HC,T10 a Ha3blBalOT oe-
/lUMbIM, a lIHCJIO b - oe/lUme/leM. TaKHM 06pa30M, liacmHoe 1i1iCe/l a U b
cyWf!cmeyem 8 mOM U mO/lbKO 8 mOM C/lYliae, KOcoa a KpamHO b, m. e.
Kocoa cyWf!cm8yem maKoe IiUC/lO C, 'Imo a=bc. B 3TOM CJIYLlae rOBopHT
TaKme, LITO a Oe/lUmCfl Ha b.
EC/lU a Oe/lUmCfl Ha b, mo liacmHoe IiUce/l a U b OOH03Ha'1.HO onpeoe/le-
liO. B caMOM )!.eJIe, eCJIH a=bc H a=bd, TO bc= bd, a Tor)!.a B CHJlY COKpa-
THMOCTH YMHQ)KeHH5I HMeeM c=d.
I1TaK, Mbl onpe)!.eJIHJIH LlaCTHlIHYIO onepaul1Io Jl,eJIeHH5I B MHomeCTBe
HaTypaJIbHblX lIHCeJI, 06paTHYIO onepalJ,HH YMHomeHHH. CBOHCTBa 3Toft
onepaUHH, BblTeKalOIll.He H3 Toro, LITO OHa 06paTHa YMHomeHHIO, 6blJIH
nepellHCJIeHbl B n. 5  2 rJI. VlIl.
EC/lu Hamypa/lbHOe IiUC/lO a Oe/lUmCfl Ha Harnypa/lbHOe IiUC/lO b, mo
ba. B caMOM )!.eJIe, nycTb a=bc. TaK KaK lc, TO B CHJIY MOHOTOHHO-
CTH YMHomeHHH b. lbc, T. e. ba.
06paTHoe YTBepm)!.eHHe HeBepHO: H3 Toro, LITO ba, He c.rIe)!.yeT
)!.eJIHMOCTb a Ha b. HanpHMep, 5 17, HO 17 He )!.e,1HTCH Ha 5. LloKameM
cJIe)!.YIOIll.ee YTBepm)!.eHHe:
EC/lu ba U a He Oe/lumcfl Ha b, mo CYWf!cm8YlOm HamypaJlbHble 'lUCAa
q U r maKue, limo a=bq+r, npUIieM r<b. napa IiUGe//. (q; r) o.oH03Ha'lHO
onpeOeJlJlemCfl napoii (a; b).
7

Mbi 6Y)!.E'M Ha3bIBaTb '1HCJIO q HenO.11ibIM 'lGCnlHbiM OT )l.eJIeHHH a Ha b,
a 'IHCJlO r - ocmamKOM OT )J.eJleHH5I.
l1TaK, nYCTb ba, npH'IeM a He )J.eJIHTC5I Ha b. lI.oKmKeM CHaLlaJla,
LITO HaM)!.eTC5I TaKoe q, '1TO bq<a<b (q+ I). B caMOM )!.eJle, TaK Kal<
Ib, TO a=a.Iab=ba. OTCIO)!.a CJIeJI.yeT, LITO MHOJl<eCTBO S HaTY-
paJlbHblX LlHCeJl S, ,l{J151 KOTOpblX a<bs, He nYCTO (eMY npHHa)!.JIe)!{HT, Ha-
npHMep, LlHCJJO a). Ho Tor)!.a cpe)!.H LlHce.rI MHO)!{eCTBa S eCTb HaHMeHb-
lllee, 0603HaLlHM ero m. TaK KaK nO VC.TJOBHIO ba, TO m=l= 1. B caMOM )l.e-
J1e, eCJlH m=I, TO a<b.l=b. Ho 31-0 npOTHBopeLlHT HepaBeHCTBY ba.
nOCKOJlbKY m=l=l, CYIll.ecTByeT LlHCJlO q, Henocpe)!.CTBeHHO npe)!.IllecTBYIO-
lll.ee LlHCJlY m, T. e. TaKoe, LITO m=q+ 1. TaK KaK m-HaHMeHbllIee LlHC.TJO,
,l{J1H KOToporo a<b.m, a q<m, TO bq<a (paBeHCTBO bq=a HeB03MO)!{HO,
TaK KaK a He )!.eJIHTC5I Ha b). I1TaK, bq<a<bm, T. e. bq<a<b(q+l).
0603HaLlHM Llepe3 r pa3HocTb a - bq, r=a-bq. Tor)!.a a=bq+r.
npH 3TOM r=a-bq<b (q+ 1)-bq=b. 3HaLlHT, '1HCJIO a MO)!{HO npc)!.-
CTaBHTb B BH)!.e a=bq+r, r)!.e r<b.
lI.oKa)!{eM Tenepb, LITO TaKoe npe)!.CTaBJIeHHe 0)!.H03I1aLlHO onpe)!.eJle-
HO, T. e. LITO eCJlH a=bql+rl H a=bq2+r2, r)!.e rl<b, r 2 <b, TO ql=q2 If
rl=r 2 . B caMOM )!.eJIe, H3 paBeHCTB a=bql+rl H a=bq2+r2 CJle)!.yeT, LITO
bql+ r l=bq2+ r 2' (I)
npe,J,nOJlO)!{HM, LITO rl<r 2 . Tor)!.a H3 paBeHcTBa (I) CJle)!.yeT, LITO
bq2<bqlt npHLleM bql-bq2=r2-r1, T. e. b (ql-q)=r2-rl' Ho r 2 <b,
a Tor)!.a TeM 60Jlee r 2 -rl<b. C )!.PyroH CTOpOHbI, TaK KaK r 2 -rl =b (ql-
-q2), TO br2-rl' HepaBeHCTBa r 2 -rl<b H br2-rl npOTHBopeLlaT
,lJ.pyr )!.pyry. 3'niM )!.OKa3aHO, LITO HepaBeHCTBO rl<r 2 HeB03MO)!{HO. TOLIHO
TaK )!{e )!.OKa3b1BaeTC5I, LITO HeB03MomHO HepaBeHCTBO rl>r 2 . 3HaLlHT,
rl=r 2 , a Tor)!.a B CHJlY COKpaTHMOCTH CJIO)!{eHH5I H3 paBeHCTBa bql+rl=
=bq2+r nOJlYLlaeM bql =bq2' OTKy)!.a B CHJlY COKpaTHMOCTH YMHome-
HH5I HaXO)!.HM, LITO ql =q2' I1TaK, napa (q; r) 0)J.H03HaLlHO onpe)!.eJl5leTC5I
napOH (a; b).
YnpamueUBH
1. YKaJKHTe HenOJIHOe lJaCTHoe Ii OCTaTOK npli .!{eJlCHIiIi:
a) 25 Ha 38; 6) 165245 Ha 876.
2. HaH.!{IiTe IJIiCJlO, npli .!{eJleHIiIi KOToporo Ha 15421 nOJlY'IaeTCH IIenomlOC
lJaCTHoe 246 Ii OCTaTOK 6723.
3. no .!{eJlHMoMY a Ii oCTaTKY r HaH.llliTe HenOJIHoe lJaCTHoe q Ii .lleJlIiTeJlb b, eC.11i:
a) a= 100, r=6; 6) a= 148, r=37; B) a=497, r= 16.
4. no .!{eJIIiMOMY a Ii Heno.1HOMY 'IacTHoMy q HaH.!{I1Te .!{eJl!iTCJlb b Ii OCTaTOK r,
CCJlIi: a) a=371, q=14; 6) a=3l29, q=83; B) a=13 127, q=12l; r) a=42 157,
q=231.
4. nOpH,l{KOBble If KO.'IH4eCTBeHHble HaTypaJIbHbIe 4MCJIa. Hap5l)!.y
C pa3BHToM Bblwe aKCHOMaTliLleCKOH TeopHeM HaTypaJlbHblx LlHCeJI cy-
llJ.eCTByeT )!.pyra51 TeopHH, OCHoBbIBalOw.aHC5I Ha TeopHH KOHeLlHblX MHO-
meCTB, onHCaHHa51 B n. 2  2. B 3TOM TeopHH HaTypa.TJbHOe LlHC.10 onpe)!.e-
JIHeTCH KaK 06w.ee CBOHCTBO K.1acca 3!\BlfBaJleHTHbIX KOHeLlHbIX MHO-
meCTB, a )!.ei1cTBH5I Ha)J. 3THMH LlHCJIaMH BBO)!.HTC5I Ha OCHOBe )!.elICTBlfI'f
Ha)J. MHO)!{eC1BaMH.
268

KaK yme OTMella.1JOCb BbIllle, STOT no)!.xo)!. OCHOBaH, RpOMe 06Ill.HX
nOHHTHH TeopHH MHomeCTB, Ha nOHHTHH KOHellHoro MHO)heCTBa, KOTO-
POMY He 6bIJIO )J.aHO onpe.AeJIeHHH. OKa3aJIOCb, O)J.HaKO, LITO C<PoPMYJIH-
pOBaTb TaKoe onpe.AeJleHHe, T. e. YKa3aTb pa3JIHlIHe Mem)!.y KOHellHblMH
H 6ecKoHellHbIMH MHomecTBaMH, 3aTpY)J.HHTeJIbHO.
n03TOMY B H3CTOHlIJ.ee BpeMH 06bIlIHO He TeopHIO HaTypaJlbHbIX
lIHCeJI CTpOHT Ha OCHOBe nOHHTHH KOHellHoro MHomeCTBa, a, Hao60poT,
noHHTHe KOHeLlHoro MHomeCTBa onpe)!.eJIHIOT C nOMOIll.blO nOHHTHH HaTY-
paJlbHOrO lIHCJIa.
nYCTb m - HaTypaJlbHOe lIHCJlO. H330BeM ompe3KoM N m HaTypaJIb-
Horo pMa lIHCeJI cOBoKynHocTb Bcex HaTypaJlbHbIx qHCeJl k, )!.J1H KOTO-
pbIX km. MHomecTBo A Ha3bIBaeTCH KOHe'lHblM, eCJIH CYIll.eCTBYeT B3a-
HMHO-O)!.H03HaLlHoe COOTBeTCTBHe Mem)!.y 3THM MHomeCTBOM H HeKOTO-
pblM OTpe3KOM N m HaTypaJlbHoro pH)!.a LlHCeJI. 4HCJlO m Ha3bIBaeTCH
npH 3TOM 'lUC/lO.M. 8/leMeHm08 B MHomeCTBe A H 0603HallaeTCH IA I HJlH
n (A). CaMa onepalJ,HH YCTaHOBJleHHH B3aHMHO-0)J.H03HallHoro COOTBeT-
CTBHH Mem)!.y A H N m Ha3bIB3eTCH nepeCliemOM 3J1eMeHTOB MHomeCTBa A.
.lI.J1H Toro lIT06bl )J.3HHoe ceHlIac onpe)!.eJIeHHe lIHCJla 3J1eMeHTOB B MHO-
meCTBe A 6blJlO onpaB)!.aHHbIM, HMO )!.OKa3aTb, LITO oOHO U mo CHCe MHO-
CHCecmeo A He MOCHCem HaXoOumbCfl 80 83aUMHO-oOH03Ha'lHOM coomeemcm-
8UU C a8YMfl pCl3/lU'lHblMU Ompe3KaMU N m U N q HamypaJlbH020 pf1ija 'lU.
cell. Ho eCJIH 6bl TaKHe COOTBeTCTBHH MomHO 6bIJlO YCTaHOBHTb, TO CY-
IIJ.eCTBOBaJIO 6bI B3aHMHO-0)!.H03HallHoe COOTBeTCTBHe Mem)J.y N m H
N q . C nOMOIll.blO MaTeMaTHlIeCKOH HH)J.YKUHH MomHO ,!{OKa3aTb, LITO npH
m=l=q TaKoro COOTBeTCTBHH He CYIll.ecTBYeT, a nOTOMY A MO)!{eT Haxo-
,!{HTbCH BO B3aHMHO-0)!.H03HallHOM COOTBeTCTBHH J1HWb C O)!.HHM H3 OT-
pe3KOB N m . I1HbIMH CJIOBaMH, 'lUC/lO 3/leMeHm08 KOHeliH020 MHOCHCecmea
He 3a8UCUm om cnoc06a nepeCliema am020 MHOCHCecmea.
OrMeTHM TaKme 6e3 )J.OKa3aTeJIbCTBa CJle)J.YlOlIJ.He YTBepm)J.eHHH:
1) JI1060e Henycmoe noOMHoCHCecmeo Ompe3Ka N m HamypaJlbH020 pf1ija
'luce/l KOHe'lHO (H3 SToro YTBepm)!.eHHH BbiTeKaeT, LITO Henycmoe 1100-
MHOCHCecmeo KOHe'lH020 MHOCHCecmea KOHeliHO).
2) EC/lU {al; ...; an} - MHOCHCecmeo, cocmOflw,ee u3 Hamypa/lbHblX
'tuce/l, npu'te.M. Hau60/lbw.ee U3 HUX pa8HO I, mo n1 (paBeHcTBo )J.OCTH-
raeTCH, eCJlH {al;"'; a n }= {I; ...; n}).
npH nepeclIeTe 3JIeMeHTOB KOHellHoro MHomeCTBa Mbi He TOJIbKO Ha-
XO)J.HM, CKOJlbKO B HeM 3JIeMeHTOB, HO H paCCTaBJIHeM 3TH 3J1eMeHTbI B
onpe)!.eJIeHHOM nopH)!.Ke, ynopH)!.OllHBaeM HX. n03TOMY HaTypaJlbHbIe
lIHCJla CJlymaT He TOJlbKO )!.J1H OTBeTa Ha Bonpoc «CKOJlbKO?», HO H )!.J1H
OTBeTa H3 Bonpoc «K3KOM: no ClIeTY?». I1HblMH CJlOBaMH, OHH HBJlHIOTCH
He TOJIbKO KO/lU'tecmeeHHbLMU, HO H nOpflaK08blMU lIlicJlaMH. B Hall3J1b-
HOM: lllKOJIe Ha)J.O nOKa3bIBaTb 06e pOJIH HaTypaJlbHbIX lIHCeJl. CBH3b
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Hap5lJ],y C 3TOH cHcTeMoH 6bIJIa )!.pyra5l, B KOTOpOH JIerHOH 0603HaLlaJI
TbMy TeM, JIeo)!.p - JIerHOH JIerHOHOB H T. )!.. JIeo)!.p .1eo)!.poB Ha3bI-
r3JIC5I eopoHO/I, a 10 BOpOHOB - KOAoiJOU.
4T06bl 0603HaLlHTb TbMbI, 6YKBbI, COOTBeTCTBylOIll.He lIHCJIaM OT I J],O
9, 06BO)!.HJIHCb CrrJIOWHbIM KpymKOM, )!.JI5I 0603HaLleHH5I JIerHOHOB 3TH
i!e 6YKBbi 06BO)!.H.1HCb KpymKoM H3 TOLleK, a )!.JI5I o603HaLleHH5I .'1eo)!.-
pOB - KpymKoM H3 JIYlleu. B 3THX 0603HaLleHH5IX BH)!.Hbl 3aLlaTKH rro-
3HlI.HOHHOH CHCTeMbI CLlHCJIeHH5I.
Pa3YMeeTC5I, TaKHe CHCTeMbI 3arrHCH LlHCeJI, KaK pHMCKa51 HJIH rpe-
l;eCKa5l, 6bIJ1H ropa3)!.O y)!.05Hee, LleM 3apy6KH Ha 6HpK8X, TaK KaK rr03-
r,OJI5IJIH KOpOTKO 3arrHCbIBaTb 60JIbWHe LlHCJIa. O)!.HaKO BbJnOJIH5ITb )!.eH-
CTBII5I Ha)!. LlHCJIaMH B 3THX CHCTeMax 3arrHCH 6blJIO He CJIHWKOM y)!.06HO-
!: orr p06yfITe YMHomHTb MMDCCLXII Ha MCDXLVIII HJIH IjJM Ha
CIt,\,. 110151 06JIerLleHl!51 BbILlHCJIeHHH HCrrOJIb30BaJIHCb CLleTHbIe )!.OCKH -
a6aKH, Ha KOTOpbIX LlHCJIa Kam)!.oro pa3p5lJ],a H306pamaJIHCb KaMelllKa-
II1 (pHC. 177).
YnpamHeUDII
I. 3anHWHTe pHMCKHMH lI.Hq,paMH 1JHCJla 9471, 6844.
2. 3anHllJHTe apa6cKIIMII lI.Hq,paMH 'IHCJla MMDCXL VIII. MMMDCCCLX XI.
11.11
3. np01JTHTe 3anHCH <Jf1a, £A., M<JLY.
2. n03HlJ,HOHHbIe CHCTeMbl ClJHCJIeHHH. bOJIbWHM )!.OCTHmeHHeM
[} pa3BHTHH MaTeMaTHKH OKa3aJIOCb c03J],aHHe n03ul{uoHHblX cucmeM
C'/UCAeHUfl- CHCTeM, B KOTOpblX O)!.HH H TOT me 3HaK (O)!.Ha H T8 me lI.H<ppa)
1\!OmeT 0603HaLlaTb pa3.'IHLlHbIe LlHCJIa B 3aBHCHMOCTH OT nOJIOmeHH5I 3TO-
ro 3HaKa B 3arrHCH LlHCJIa (ero rr03HlI.HH). HaH60JIbWee pacrrpOCTpaHe-
Iwe rrO,1YLlHJIa )!.eC5ITHLlHa51 CHCTeMa CLlHCJIeHH5I, CB5I3aHHa51 CO CLleTOM
lIa rraJ1blI.ax. O)!.HaKO, KaK yme rOBOpHJIOCb BbIwe, BCTpeLlaJIHCb H HHbie
CHCTeMbI ClIHCJIeHHH - )!.BaJ],lI.aTepHLlHa5l, OCHOBaHHa51 Ha CLleTe )!.Ba)!.-
!IaTKa1H, )!.BeHa)!.lI.aTepIlLlHa5l, OCHOBaHHa51 Ha ClIeTe )!.1O:>KHHaMH. J1 ceH-
Llac IIpHMeH5IIOT CLleT )!.lOmHHaMH )!.JI5I HeKOTopblX TOBapOB (JIOmeK, BH-
.'IOK, HomeH, rrJIaTKOB H T. )!..). a BbJ)!.alOIll.eMC5I LleJIOBeKe rOBOp5lT «He-
)!.101I<lIHHbIH LleJ10BeK».
nepBOH rr03HlI.HOHHOH CHCTeMOH CLlHCJIeHH5I OKa3aJIaCb weCTHJ],eC5I-
TepllLlHa51 CIlCTeMa CLlHCJIeHH5I. Ee npHMeHHJIH B )!.peBHeM BaBHJIOHe.
274

BaBHJIOHHHe IlI1CaJIH Ha rJlHHHHbIX Tat'5J1HlIKaX, BbI)!.aBJlHBaH Ha rJIlme
I{JIHHbH, a nOTOM Ta6J1HlIKIi 06mHraJiH B rOHlIapHOH: nellH. BaBHJlOH-
CKaH HYMepaU HH 6uJIa OCHOBaHa Ha HCnOJlb30BaHHH J1HWb )!.BYX KJlHHO-
nHCHblX 3HaKOB r 11 .( . DepBblH: H3 HHX 0603HallaJI QHCJIa 1 H 50,
a BTOpOH - LlHC.13 10 H 600. KaK H MHOrHe CTOpOHbI CBOeH: KYJlbTypbl.
BaBHJIOIIHHe 3aHMCTBOBaJiH IllecTH)!.ecHTepHlIHYIO cHcTeMY CQHCJleHH 51
Y IllyMepoB. Ho wYMepbI npHMeH5IJlH )!.J151 lIHCeJI 60 H 600 HHble 3HaKH,
LleM )!.JIH I H 10, a BaBHJlOHHHe pa3BHJlH HX cHcTeMY 3anHCH H CTaJIH
0603HaLlaTb COOTBeTCTBeHHble lIHCJla O)!.HHaKOBbIMH KJlHHbHMH, a 3Halle-
HHe K,1HHbeB 3aBHce.1J0 OT HX nOJlomeHHH B 3anHCH lIHCJla. npH 3anHCH
LlHCeJI OT 1 )!.O 60 3HaKH )!.J1H e)!.HHHubi H 10 nOBTOpHJlH CTOJlbKO pa3,
CKOJlbKO B 3TOM LlHCJIe e)!.HHHU H )!.eCHTKOB, a lIHCJla, KpaTHbie 60, OT
60 )!.o 59.60 0603HaLlaJIH TeMH me 3HaKaMH, YKa3blBaBwHMH J1HWb
MHOmHTeJIb npH 60. BIlOCJle)!.cTBHH 3HaK r CTaJi 0603HallaTb He TOJlbKO
1 H 60, HO H 3600=60 2 , 60 3 H Bo06w.e J1106ylO CTelleHb lIHCJIa 60. A 3HaK
« cTaJI 0603HaLlaTb He TOJlbKO 10 H 600, HO H J11060e lIHCJlO BH)!.a 10. 60 n .
CJle)!.bI wecTH)!.eC5ITepHlIHOH: CHCTeMbl ClIHCJleHHH cOXpaHHJlHCb )!.O
CI1X nop: lIac )!.eJlHT Ha 60 MHHYT, MHHYTY - Ha 60 ceKYH)!., nOJlHblH:
yroJi - Ha 360 0 H T. )!.. B aCTpoHoMHlIecKHx BbIlIHCJleHHHX weCTH)!.ecH-
TepHlIHble )!.P06H npHMeHHJlHCb )!.O XVI B. H. 3.
HaH60JlbWee pacnpOCTpaHeHHe nOJiYlIHJIa )!.eCHTHlIHaH CHCTeMa
3anHCH lIHCeJI. Ew.e B III B.)!.O H. 3. 3HaMeHHTblH: )!.peBHerpelleCKHH: YlIe-
HblH: ApXI1Me)!. pa3pa60TaJi B COllHHeHHH «ncaMMHT» (<<J1ClIHCJleHHe nec-
lIHHOK») CHCTeMY ClIHCJleHHH, OCHOBaHHYIO Ha lIHCJle 10, KOTopaH n03BO-
J1HJIa Ha3bIBaTb OlleHb 60JlbIllHe LlHCJIa - BO BCHKOM CJlYlIae 6OJlbWHe,
LleM lIl1CJlO necllHHoK B wape «PMHYCOM, paBHblM paCCTOHHHIO OT 3eMJlH
)!.O C<PePbl HenO)!.BHmHblx 3 Be3)!.» , KaK IlHCaJi ApXHMe)!.. 4HCJla, KOTopble
OH YMeJI Ha3bIBaTb, 6b1J1If )!.eH:CTBHTeJIbHO rpOMa)!.HbIMH - caMoe 6oJIb-
woe lIHCJlO Y Hero paBHHJIOCb e)!.HHHue C 8.1016 HYJlHMH (CTOJlbKO HY.1eif
He nOMeCTHTCH H Ha J1eHTe, )!.J1HHa KOTOpOH: paBHa paCCTOHHHIO OT 3eMJIH
)!.O COJlHLla).
B COBpeMeHHOM BH)!.e )!.eC5ITHlIHa51 CHCTeMa ClIHCJleHHH CJlOmHJIaCh
IlpHMepHO B VI B. H. 3. B J1H)!.HH. BamHblM )!.OCTHmeHHeM HH)!.HH:CKOH Ha-
)'KH 6blJlO BBe)!.eHHe oco60ro 3HaKa )!../IH HYJlH - )!.O 3Toro He 6bl.0 OCO-
60ro 0603HalleHHH )!.JIH nponYCKa pa3pH)!.oB, a nOTOMY, HanpHMep, Y
BaBHJIOHHH lIHCJla 7.60 3 +15 H 7.60+15 0603HallaJIHCb O)!.HHaKOBO.
TOJlbKO nocJle BBe)!.eHIfH HYJlH )!.eCHTHLlHaH CHCTeMa 3anHCH LlHCeJI OKa-
3aJIaCb nOJIHOCTblO ocpopMJIeHHOH:. B03MomHO, LITO nepBOHallaJIbHO HY,1Jb
nOHBHJlCH, lIT06bl 0603HallHTb OTCYTCTBHe KaMeWKOB B COOTBeTCTBYIO-
w.eM pa3pH,ne a6aKa.
3. 3anHcb HaTypaJlbHblX lIHCeJI B .lI.eCHTHlJHOH CHCTeMe ClJHCJIeHHSI.
Ha30BeM oecJlmU'LHoii 3anUCblO Hamypa/lbHOZO IiUC/la n npe)!.cTaB.1JeHHe B
BIf)!.e CYMMbl cJle)!.ylOw.ero BH)!.a:
n=nk.lOk+nk_l.lOk-l+...+no.
275

3)!.ecb 11", I1Lit ..., 110 - ue.'1bIC neOTpHIJ.aTe.lbHble lIHCJIa, npllHIIMalO-
w.He 3HaLleHH5I 0, 1,2 ,3,4,5,6, '1,8,9. npH 3TOM 11,,=#=0.11.'151 KpaTKo-
CTH nmUYT TaK: n=n"I1"_l" .110 (LlepTa cBepxy 1l03BOJI5IeT OTJIHLlHTb ':ny
3anHCb OT npoH3Be)!.eHH5I lIHCeJI I1h, 11/1-10 ..., no; eCJIH BMeCTO 6YI<B nH-
WYTC5I IJ.H<PPbJ, TO lIepTa He HymHa). HanpHMep,
4705=4.10 3 +7.10 2 +0.10+5.
JIIO)!.H HaCTOJIbKO npHBbJKJIH K )!.eC5ITHlIHOH 3anHCH HaTypaJIbHbJX
lIHCeJI, LITO OTom)!.eCTBJI5I1OT lIHCJIa H HX 3anHCH. O)!.HaKO He CJIe)!.yeT
nYTaTb 3TH )!.Ba nOH5ITH5I - H cyw.eCTBOBaHHe, H e)!.HHCTBeHHOCTb )!.ec5I-
THlIHOH 3anHCH HaTypaJIbHblX LlHCeJI Hym)!.aIOTC5I B ,lI.OKa3aTeJIbCTBe.
nepBblM warOM B 3TOM )!.OKa3aTeJlbCTBe 51BJI5IeTC5I CJIe)!.ylOw.ee YTBep)l{-
l\eHHe:
JI./lfl AI060eo HamypaAbHDeO 'lUCAa n 8bl11oAHflemcfl Hepa8eHcm80 n< I on.
B caMOM )!.eJIe, ecJIH k<l, TO IOk<IOI, H nOTOMY LlHCJIa 10, 10 2 ,...
..., IOn nOnapHO pa3JIHlIHbl. n03TOMY HaH60JIbWee H3 HHX 60JIbIlle, LleM n,
a 3TO H 3HallHT, LITO 11< IOn. 0603HallHM Llepe3 An MHO)l{ecTBO HaTypaJIb-
HblX lIHCeJISTaKHX, LITO IOs<n. TaK KaKn<lO n , TO )!.JI5I BCeX S H3 An
HMeeM 11, H nOTOMY B MHomeCTBe An eCTb HaH60.1JbWee. 0603HallHM
ero k. Tor)!.a IOkI1<IOk+l.
110KameM cyw.ecTBoBaHHe ,lI.eC5ITHlIHOH 3anHCU JII060ro HaTypaJIb-
Horo LlHCJIa 11 C nOMow.blO MaTeMaTHlIecKoH HH)!.YKIJ.HH no k, r)!.e IOkn<
<lO k +1. npH k=O BbJnOJIH5IeTC5I HepaBeHCTBo In<lO. B 3TOM CJIY-
Llae n - 0)!.H03HallHoe lIHCJIO H 3anHCbIBaeTC5I C nOM0w.blO JIHillb o)!.Hoii
IJ,H<PPbJ. npe)!.nOJIOmHM Tenepb, LITO )!.JI5I Bcex HaTypaJIbHblX lIHCeJI,
MeHbWHX lIeM IO k , )!.OKa3aHO cyw.ecTBoBaHHe )!.eC5ITHlIHOH 3anHCH.
B03bMeM Tor)!.a JII060e lIHCJIO 11, 3aKJIIOlIeHHOe Mem)!.y IOk H I Ok+ 1: IOk
11<IOk+l. ECJIH n )!.eJIHTCH Ha IOk, TO n=l1" .IOk, r)!.e In,,<IO,
H Mbl nOJIYllHJIH )!.ecHTHlIHYIO 3anHCb 11. ECJIH me npH ,lJ.eJIeHHH n Ha IO k
nOJIYllaeTc5I OCTaTOK r(n=n" .IOk +r), TO r<IOk, H no npe)!.nOJIOmeHHIO
HH)!.YKIJ,HH r HMeeT )!.ec5ITHlIHYIO 3anHCb r=11 8 .lO s + ...+no. Ho Tor)!.a
n=n" .IOk+r=I1"IOk+ n8 .IOs+ ...+no.
3Ta 3anHCb eW.e He 51BJI5IeTC5I )!.ec5lTHlIHOH, TaK KaK npH s<k-I B
Hee He BXO)!.51T HeKOTopbIe CTeneHH 10. B 3TOM CJIYLlae )!.OCTaTOllHO )!.O-
6aBHTb HecKOJIbKO CJIaraeMblX C HYJIeBblMH K03qxpHIJ.HeHTaMH H nOJIY-
lIHTb Tpe6yeMYlO )!.ecHTHLlHYIO 3anHCb:
n=n" .IOk+ l1k _i'IOk-l+...+ no . (1)
Mbl )!.OKa3aJIH, LITO Bce lIHCJIa, MeHbWHe 10, HMelOT )!.eC5ITHlIHYIO 3anHCb,
a H3 cyw.ecTBoBaHH5I 3anHCH y lIHCeJI, MeHbWHX IOk, BblBeJIH CYW.eCTBO-
BaHHeTaKOH 3anHCH y Bcex lIHCeJI n, TaKHX, LITO IOkn<IOk+l. OTCIO)!.a,
B CHJIY npHHIJ.Hna MaTeMaTHlIecKoH HH)!.YKIJ.HH BblTeKaeT, LITO 8ee Hamy-
paAbHbte 'lUCAa UMelOm OeCflmU'lHYIO 3anUCb.
110KameM Tenepb, LITO KQ:J/Cooe 'lUCAO n UMeem mO/lbKO oiJHY OeCflmU'l-
HYIO 3anUCb. 4HCJIO k B paBeHcTBe (I) 0)!.H03HallHO onpe)!.eJI5IeTC5I YCJIO-
BHeM 1 Okn< IOk+1. nocJIe Taro KaK k onpe)!.eJIeHO, K03qxpHIJ.HeHT
276

II" on pe)!.eJI5IeTC5I Tpe60BaHlfeM nIt lOkn«nk + 1) 1 Ok. .LJ.aJIee onpe.ne-
.TI5IeM aH3JIOrH t IHbIM 06pa30M IJ,H<PPbI nIt-I, ..., no.
.LJ.eC5ITHlIJla51 3anHCb LlHCeJI 06JIerLlaeT HX H3YlIeHHe. B lIaCTHOCTIf,
COBce:\1 npOCTO pernae1C5I Honpoc 0 cpaBHeHHH )J.BYX HaTypaJIbHblX lIHCeJI
no llX )!.eC5ITHlIHOH 3anHCii.
n yCTb
n=n" .lO"+n,,_II0 k - l + ...+no,
m=ml.l01+ml_llOl-I+...+mo,
EC/lU 6bliW/lHeHO OOHO U3 C/leOYlOllU.\ ':c1OGuii:
a) k<l;
6) k=l, 1-/0 nk<m,,;
B) k=l, n"=m,,, ..., ns=ms, HO nS-I<ms-i,
mo 'lUC/lO n MeHbUle 'lUC/la m.
B CaMOI\I )!.eJIe, nYCTb k<l. Tor)!.a BbInOJIH5IeTC5I HepaBeHCTBO lOk+l
101. A TaK l<aK n<10k+1 H 10 1 m, TO n<IOk+IlOIm, T. e. n<m.
AHaJIOmLlHO, eCJIH k=l, HO n"<m,,, TO n,,+ lm", If nOTOMY
(fl,,+1).lOkm,,-10k. A TaK KaK n«n,,+l).IOk H m,,'lOkm,
TO n< (n,,+ I) .10km" .10km, T. e. n<m. TOllHO TaK me pa361f-
paeTC5I CJI'IlIaH B).
J1erKo BH)!.eTb, LITO eCJ1l1 n H m - pa3JIHlIHbie HaTypaJIbHbIe qlfCJIa,
TO BbIDO.1HHeTC5I JIH60 O)!.HO H3 YCJIOBHH a)-B), JIHOO O)!.HO H3 YCJIOBHH,
nOJIYlIaeMblX H3 HIfX H3l\leHeHHeM pOJIeH n H m. n03TOI\IY eCJIIf n=t=m,
TO YKa3aHHbie YCJIOBH5I n03BOJI5IIOT YCTaHOBHTb, LITO n<m HJIIf m<n.
4. 3amtCb lJHCeJI B .lI.pyrux n03HIJ.HOHHblX cHcTeMax ClJHCJIeHIUI.
Mbl yme rOBOpUJIH, LITO Hap5l)!.y C )!.eC5ITHtIHOH CHCTeMOH 3anHCH lIHCeJI
npHMeH5IJIHCb HHbIe CHCTeMbl: )!.BeHa)!.IJ.aTepHlIHa5l, )!.Ba)!.lJ,aTepHlIH, a51
llleCTH)!.eC5ITepHlIHa5l. CeHLlac B 6bJCTpo)!.eHCTBYIOW.HX BbILlHCJIHTeJIbHblX
MarnHHax npHMeH5IIOT )!.BOHlIHYIO If BOCbMepHlIHYIO CHCTeMbl 3anHCH lIH-
ceJI. Bce 3TH CHCTeMbl 3anHCH LlHCeJI OCHOBaHbl Ha O)!.HOM npHHIJ.Hne.
BbJ6HpaeTC5I HaTypa.1bHoe lIHCJIO p, 60JIbWee e.lJ.HHHIJ.bl - OCHOBaHue
cucmeMbl C'lUC/leHUfl. TOllHO TaK me, KaK H B CJIYlIae )!.eC5ITHLlHOH CHCTe!llbl
CLlHCJIeHH5I, )!.OKa3blBaeTC5I, LITO JII060e HaTypaJIbHOe LlHC.T1O MomHO e)!.HH-
CTBeHHb:M 06pa30M 3anHcaTb B BH)!.e CYMMbI:
n=n"pk+n,,_lpk-1 +... +nIP+nO,
r)!.e Bce ns MeHbwe p, npHlIeM n,,=t=O. 3Ty CYMMY KpaTKO 3anHCbIBaIOT
TaK: n=n . ..n op (3HaLlOK p BHH3Y YKa3bIBaeT Ha OCHOBaHHe CHCTeMbl
ClIHCJIeHH5I). HanpHMep, 3anHCb n=475 s 03HallaeT, LITO n=4.8 L t-
+ 7 -8+5=256+56+5=3171). CaJ\1O LlHCJIO p MomHO 3anHcaTb B BH)!.e
p= I 'p+O, H nOTOMY B p-HlIHOH CHCTeMe ClIHCJIeHH5I OlIO nHWeTC5I
TaK: lOp, TOllHO TaK me pk=l.pk+O'pk-l+...+O, H nOTOMY 3anHCbI-
BaeTC5I KaK 1O...Op (3)!.ecb k HYJIeH).
I1HbIMH CJIOBaMH, 8 p-U'lHoii cucmeMe C'lUC/leHUfl 3ClnUCb lOp 03Haliaem
OCH08aHUt! cucmeMbl, m. e. 'lUC/lO p, a aanUCb 1O...Op (k HY/leii) - 'lUC-
AO pk.
n,,=t=O,
ml=t=O.
I) ECJlH OCHOB8HHe CIICTCMbI C'IHCJleHHH He YK838HO, 6YAeM C'IHT8Tb, 'ITO 'IHCJlO
38nHC8HO B .l\eCHTH'IHOH CHCTeMe C'IHCJleHHH.
277

KOJIHtJeCTBO IJ.H<PP, He06xo)!.HMblx )!.JI5I 3anHCH LlHCeJI B P-HLlHOfi CHc-
TeMe CLlHCJIeHH5I, paBHO p: 0, I, . . ., p - I. 3THM 06'b51 CH 51 eTC 51 , n6-
MY B BbIIIHCJIHTeJIbHblX MaWHHax npHMeH5IIOT )!.BOHtJHYIO CHCTeMY CLlJlC-
JIeHH5I: )!.JI5I 3anHCH LlHCeJI )!.OCTaTOLlHO Bcero .lJ.Byx IJ.H<I>P, ° H I. 4ToPbI
H306pa3HTb 3TH LlHCJIa, B MaWHHe npHMeH5IIOT oc06ble YCTpOHCTBa, KO-
TopbIe MorYT HaXO)!.HTbC5I B )!.BYX H TO.TJbKO )!.ByX COCT05lHHHX (npOBO-
.n.HTb HJIH He npOBO,lI.HTb 3JIeKTpHLleCKHH TOR, HMeTb HaMarHHLleHHOCTb
onpe)!.eJIeHHoro 3HaKa H T. )!..).
CpaBHeHHe LlHCeJI B JII060H CHCTeMe CLlHCJIeHH5I npOH3BO)!.HTC5I TaK
me, RaK H B )!.eC5ITHLlHOH CHCTeMe:
ECAU
n=nkpk+...+no,
m=mlpl+...+mo,
mo n<m 8 C/leoYI014UX mpex CAYliaflX:
a) k<l;
6) k=l, HO nk<mk;
B) k=l, nk=mk, ..., ns=ms, HO nS-1<mS-l'
O)!.HO H TO me HaTypaJIbHOe LlHCJIO MomeT 6b1Tb 3anHcaHO B JIIOOOH
113 CHCTeM CLlHCJIeHH5I. 4T06bl H3 O)!.HOH 3anHCH nOJIYLlHTb )!.pyrylO, )!.O-
CTaTOLlHO HaYLlHTbC5I nepeBO)!.HTb JII06ylO H3 3anHceH B )!.eC5ITHLlHYIO, a
.n.eC5ITHLlHYIO - B JII06ylO )!.pyrylO (nO)!.OOHO TOMY KaK )!.JI5I nepeBo)!.a C
O)!.Horo 513blKa Ha )!.pyroH )!.OCTaTOLlHO HMeTb CJIOBapH )!.JI5I nepeBo)!.a C
3THX 513bIKOB Ha PYCCKHH H C pYCCKoro Ha 3TH 513bIKH).
I1TaK, pa36epeM ABe 3 a)!.aLlH.
3 a)!. a LI a I. fl y cmb oaHa p-uliHafl 3anUCb nk...nO IiUC/la n. Haoo
. p
Haiimu oeCflmUIiHYIO 3anUCb moeo OIC e ttUC/la .
D.JI5I 3Toro )!.ocTaTotJHo BMecTO nk...nOp HanHcaTb nkpk+...+no, a
IIOTOM 3aMeHHTb nk, ..., no H p HX )!.eC5ITHIIHbIMH 3anHC5IMH H BbmOJIHHTb
no npaBHJIaM, npHH5ITbIM B )!.eC5ITHLlHOH cHcTeMe, oo03HaLleHHble )!.eH-
CTBH5I. D.eC5ITHLlHaH 3anHCb pe3YJIbTaTa H OY)!.eT HCKOMblM OTBeTOM.
n p H M e p 1. nOJIYLlHM )!.eC5ITHLlHYIO 3anHCb )!.JIfi LlHCJIa 3627'
Mbl HMeeM: 3627=3.72+6.7+2= 191. 3HaLlHT, 362 7 = 191.
n p H M e p 2. nOJIYtJHM )!.eC5ITHQHYIO 3anHCb )!.JI5I LlHCJIa ex412'
r)!.e ex - HOBa51 IJ.H<ppa, 0603HaLlalOlIJ.a51 B )!.BeHa.ll.IJ.aTepHLlHoH CHCTeMe
CLlHCJIeHH5I tJHCJIO 10, a  - IJ.H<ppa )!.JI5I tJHCJIa 11. I1MeeM: ex412 =
= 10 .12 2 +4 .12+ II = 1499.
3 a )!.. a LI a 2. HaiiBeM P-UIiHYIO 3anUCb O/lfl IiUCAa, 3aoaHHoeo oecfl-
mUIiHou 3anUCblO.
DYCTb n=nkpk+nk_ipk-l+...+nlP+nO' 8T0 lIHCJIO MomHO 3anH-
caTb B BH)!.e:
n= p (nkpk-l+.. .+nl)+nO'
TaK KaR no<p, TO H3 3TOH 3anHCH BH)!.HO, tJTO no - OCTaTOR OT )!.eJIe-
I!H5I LlHCJIa n Ha p, npHLleM HenoJIHblM LlaCTHbiM npH 3TOM )!.eJIeHHH oy-
)!.eT nkpk-l+ ...+ni' TOIIHO TaK me HaXO)!.HM, liTO ni - OCTaTOR OT )!.eJIe-
HH5I 3Toro HenOJIHorO LlaCTHoro Ha p H T. .no. .
278

TaKHM 06pa30M, p-HtJHaH 3anHCb tJHCJIa n HaXO)!.HTCH CJIe.nYIOIIJ,HM
o6pa30M. 4HCnO n eJIHM (B )!.ec5lTHLlHOH CHCTeMe ClIHCJIeHHH) Ha p.
OCTaTOK OT )!.eJIeHHH )!.aCT nOCJIe)!.HIOIO IJ.H<PPY no B P-HtJHOH 3anHCIf n.
HenOJIHOe tJaCTHoe CHOBa )!.eJlHM H p. HOBbIH OCTaTOK )!.aCT npe)!.noC-
.TJeHIOIO IJ.H<PPY P-HlIHOH 3anHCH n. npO)!.OJImaH npoIJ.ecc )!.eJIeHHH, HaH-
eM Bce IJ.H<PPbI P-HtJHOH 3anHCH lIHCJIa n.
n p H M e p 3. HaH)!.eM )!.BOHlIHYIO 3anHCb tJHCJla 46.
PaCnOJIOmHM )!.eJIeHHe CJIe)!.YIOW:HM o6pa30M:
461 2
01231 2
--.--rTi I 2
--.-rs \2
lf212
OTTI2
1fO
3anHcbmaH OCTaTKH, HatJHHa51 C nOCJIe)!.Hero, nOJIYlIaeM:
46=I01110 2 .
npoBepHM OTBeT, nepeBe)!.H 101 110 2 06paTHo B )!.ecHTHlIHYIO cHcTeMY
ctJHCJIeHHH:
I011102=I.25+0.24+1.23+1.22+1.21+0=32+8+4+2=46.
n p H M e p 4. HaH)!.eM BocbMepHlIHYIO 3anHCb LlHCJIa 691.
nOCJIe)!.oBaTeJlbHO )!.eJIH tJHCJIO 691 H nOJIyqalOW:HecH lIaCTHbie Ha 8,
nOJIYtJaeM:
691 1 8
3 18618
6TiO 18
2TI18
110
3HallHT,691=1263 8 .
n pH M e p 5. HaH)!.eM )!.BeHa)!.IJ.aTepHlIHYIO 3anHCb llHCJIa 19510.
19510 I 12
10 1 1625/ 12
5 1 135 112
3111112
t1!O
.LJ.J151 OCTaTKOB 10 H II BBO)!.HM HOBbie IJ.H<PPbl a H p COOTBeTCTBeHHO. Tor-
)!.a 3anHCb lIHCJIa 19510 HMeeT BH)!.:
1950=35au.
279

n p H M e p 6. DepeBe.n.eM LlUCJ10 32 014 5 B BOCbMepuLlHYIO CUCTeMY
CLlUCJ1eHH5I. CHaLlaJ1a nepeBe.n.eM 32 014 5 B )!.eC5ITIf'IHYIO cHcTeMY CqlfC'
JIeHH5I:
32 014 5 =3.5'+2.5 3 +0.5 2 + 1.5+4=2134.
A Tenepb 2134 nepeBe)!.eM B BOCbMepH'IHYIO cHcTeMY C'IHCJIeHU5I:
2134 I 8
6 1 26618
2 13318
1T418
4jO
3Ha'IHT, 32014 5 =4126 8 ,
MbI BH)J,HM, LITO npH YBeJIH'IeHUH OCHOBaHU5I CHCTeMbl KOJIHlIeCTBO
lJ.H(J>p B 3anHCH .n.aHHOrO 'IUCJIa YMeHbillaeTC5I, HO npHXO)!.UTC5I uCnOJIb-
30BaTb 60JIbillee 'IHCJIO pa3JIH'IHbiX IJ,H<pP.
Pa3YMeeTC5I, B03MOmeH HenOCpe)!.CTBeHHblli nepeBO)!. u3 p-UlIHOH CH-
CTeMbI C'IHCJIeHH5I B q-H'IHYIO. Ho 3TO Tpe6yeT HaBblKa BbInOJIHeHH5I
apH<pMeTlf'IecKHX OnepalJ,HH B 3TUX CHCTeMaX CLlUCJIeHH5I, H Mbl He 6y-
.l{eM nOJIb30BaTbC5I 3THM cnoc060M. O)!.HaKO eCJ1H O)!.HO H3 'IUCeJI p, q
51BJ151eTC5I CTeneHblO )!.pyroro, HanpHMep q =pk, TO nepeBO)!. H3 p-H'IHOH
CHCTeMbl C'IHCJIeHH5I B q-H'IHYIO H 06paTHo He Bbl3blBaeT 3aTpY)!.HeHHH.
B caMOM )!.eJIe, nYCTb
N =anqn+... +alq+a O '
(1)
3)!.ecb an, ..., a lo a o - HaTypaJIbHble 'IHCJIa, MeHblllue q=p". I1x MomHO
npe)!.CTaBHTb B BH)!.e:
an =a.k_lP,,-l+...+Ct o ,
aO='Ak_1pk-l+.. .+'Ao.
nO)!.CTaBJI5I5I 3TH 3anUCH B (1) H 3aMeH5I5I q Ha p", nOJIY'IHM 3anUCb BH-
)!.a:
N = (Ctk_Jpk-l+.. .+a.o)p"+...+('Ak_1pk-l+. ..+'Ao).
A Tenepb )!.OCTaTO'IHO paCKpblTb cK06KH, lIT06bl nOJIYlIHTb p-U'IHYIO 3a-
nHCb 'IHCJIa N.
4T06bl C)!.eJIaTb 06paTHblH nepexo)!., Ha)!.o pa36uTb CJIaraeMble B 3a-
nUCH
N=bspS+...+b o
cnpaBa HaJ1eBO Ha rpynnbl no k CJIaraeMblX (B caMOH JIeBOH rpynne MO-
)I{(:T 6bITb MeHbllle CJIaraeMblx) H BblHecTH 3a cK06KH pk, p2k U T.)!.. 3a-
MeH5I5I p" Ha q, p2k Ha q2 H T. )!.., nOJIY'IHM q-ulIHYIO 3anHCb LlHCJIa N (BbI-
pameHU5I B cK06Kax 51BJI5IIOTC5I npu 3TOM q-HlIHbIMH IJ,H<ppar.m).
B npaKTUKe pa60TbI 6blcTpo.n.eHcTBYIOIIJ.UX BblLlHCJIHTeJ1bHbiX Ma-
lllHH oc060 BamHYIO pOJIb IfrpaeT nepeBO)!. U3 )!.BOlfllHOH CHCTeMbi ClIH-
CJIeHU5I B BocbMepulIHYIO H 06paTHo. COCTaBuM Ta6JIHIJ,Y, nOKa3blBalO-
280

m.YIO Bblpa)KeHHe Kam,noro H3 tJHCeJI 0, 1, 2, 3, 4, 5, 6, 7 B )!.BOHLlHOil CHC-
TeMe CLlHCJleHH5I:
B BcbMe- 1 0 I
pH'IHOH
B .D.BOH'IHOii I 000 I 001
I 2 I 3
1 010 1011
4
5
(j
7
100
101
110
111
4T06bl nepeBecTH LlHCJlO 25 420 8 B ,nBOHLlHYIO cHcTeMY CLlHCJleHH5I,
Jl.OCTaToLlHo 3aMeHHTb Ka)K,nylO UHq>py )!.aHHoro LlHCJIa TpeM5I COOTBeTCT-
BYIOW.HMH UHq>paMH )!.BOHLIHOH CHCTeMbi (mpuaaoii, COCT05Jw.eH H3 HYJI5I
H e)!.HHHU). nOJlYLlaeM: 25 420 8 = 10 101 100010 000 2 (nepBYIO UHq>PY 2
Mbl 3aMeHHJlH Ha 10, a He Ha 010).
nepeBe)!.eM LlHCJlO 11 0 10 001 101 000 111 2 B BocbMepHLlHYIO cHcTeMY
CLlHCJIeHH5J. 3anHcb 3Toro LlHCJla yme pa36HTa Ha TpHa)!.bI. 3aMeHHH
Ka)K,nylO TpHa)!.y cooTBeTcTBYIOW.eH UHpOH no Ta6J1Hue, nOJIYLlaeM, LITO
3TO LlHCJlO paBHo 321 507 8 ,
Bblwe yme rOBOpHJlOCb, LITO no)!.aBJl5I1Ow.ee 60JIbWHHCTBO 3.neKTpOH-
HblX BbILlHCJlHTe.1JbHbIX MaWHH pa60TaeT B )!.BOHLIHOH CHCTeMe CLlHCJleHH5I
(HeKOTopble - B TpOHLlHOH). nporpaMMY peWeHH5I TOH H.1H HHOM 3a)!.a-
LlH Ha)!.o npe,nJIaraTb MaWHHe B )!.BOHLIHOH CHCTeMe, LITO Hey,a06HO .l{JI5I
nporpaMMHCTa. Ho TaK KaK BOCbMepHLlHa51 CHCTeMa CLlHC.1JCHH5I )!.OBOJIb-
HO «6J1H3Ka» K )!.ec5lTHLlHOH Ii nepexo)!. 8-+ 2 J1erKO ocyw.eCTB.TJ5JeTCH, TO
nporpaMMbI )!.J1H MalllHH nHWYT B BOCbMepHLlHOH CHCTeMe CLlHc.1eHH5I, a
3aTeM MexaHHLleCKH nepeBO)!.5JT B )!.BOHLlHYIO.
nepeBO)!. )!.eC5JTHLlHoil 3anHCH LlHCJla B ,nBOHLlHYIO BbmO""H5IeTC5I npoc-
TO, eCJIH CHaLlaJia nepeBeCTH ee B BOCbMepHLlHYIO, a 3aTeM YKa3aHHbIM
cnoc06oM B )!.BOHLlHYIO. 3THM npHeMoM nporpaMMHcTbI nOJIb3YIOTCSJ npH
nepexo)!.e 10-.2.
n p H M e p 7. Bblpa3HM 2567 B )!.BOHLIHOH CHCTeMe.
2567=5007 8 ,
25671 8
7/ 320 18
0/4018
015/8
5TO
3HaLlHT, 5007 8 = 101 000000 111 2 .
YnpaJKHCHHlI
1. CKO.lbKO UI1<jJp HY/KHO .D.JlH 3anHCH 'IHCeJI B JJ.eBHTepH'IHOii CHCTeMe C'IHCJleHHSI?
a B .D.BeHa.D.uarepH'lHoii?
2. npH 3anHCH 'IHCJla 3 701 2a4, r.D.e a 03Ha'laeT 10, a  - 14, HCnOJIb30BaHa
HaH60JIblllaSl H3 UH<jJp CHcreMbl C'IHCJleHHH. no KaKoii CHCTeMe C'lHc.leHHH 3anHcall0
9TO 'IHCJlO?
3. BepHo Jill 3anllcaHbi '1HCJla B CeMepH'IHOii CHCTeMe C'IHc.eHHSI: 2360 7 , 35 7217.
6085l2 7 ? ..
281

4. 3anHlUHTe nepBble COpOK '1HCeJI B neBIiTepH'IHOA, Jt8eHaJtiUlTepH'IHOH H nIlTllaiJ;-
U8TepH'IHOH CHCTeMax C'IHCeHHII.
5. npH KaKOM OCHOBaHHH CHCTeMbl C'IHCJleIlHH BbmO.HHeTCIi paBeHcTBo 26= 101/
6. 3anHlUHTe 'IHc.a 70 532 8 H 160 345 8 B JJ.BOH'IHOii CHCTeMe C'IHc.eHHII.
7. 3anHwHTe 'IHca 100 III 010001 2 , I JOOOlr JOllJ0 2 B BOCbMepH'IHoi{ CH-
creMe C 'lHceHHH.
8. 3anHWHTe 'IHCJ!O 359 211 B BOCbMepH'IHOH CHCTeMe C'IHCJleHHII.
9. 3anHwHTe B JJ.eCJlTH'IHOH CHCTeMe C'IHceHHH 'IHCa 2324 6 , a48l12 (a= 10,
fI= II).
10. 4HC0 6451 372 8 aanHWHTe B JJ.BellaJJ.l\3TepH'IHOH CHCTeMC C'IHCJleHHII.
5. CnoIKeHHe lJHceJI 8 )l.eCSlTH'IHOH H JJ.pyrHx n03HIJ,HOHHbiX CHCTeMaX
C'tHCJIeHHSI. CnomHM LlHCJIa 341 H 7 238. LJ,JIH 3Toro 3amlweM HX CJIe-
.l{YIOllJ,HM 06pa30M:
341 =0 .10 3 +3 .10 2 +4.10+ I,
7 238=7.10 3 +2.10 2 +3.10+8.
3HaLlHT,
341+ 7238=(0.10 3 +3.10 2 +4.10+ 1)+(7 .10 3 +2 .10 2 +3 .10+8).
nOJIb3YHCb aCCOIJ,HaTHBHOCTblO H KOMMYTaTHBHOCTblO c.'IO)l{eHH5I, a
TaK)]{e )!.HCTPH6YTHBHOCTblO YMHO)l{eHHH OTHOCHTeJIbHO CJIOmeHH5I, nepe-
nHweM 3TY CYMMY TaK:
341 + 7238=(0+ 7) .10 3 +(3+2) .10 2 +(4+3) .10+(1+8)=7.10 3 +
+5.10 2 +7.10+9= 7579. '
T OLIHO TaK )l{e BblnOJIHHeTC5I CJIO)l{eHHe LlHCeJI:
n=ll" .10k+...+no H m=m" .lO k +...+mo, ecJIH BO Bcex pa3p5l)l.aX
HMeeM ns+ms<10 (B c,;:YLlae Kor)!.a LlHCJIO pa3p5l)!.oB B LlHCJIaX n H m
pa3JIHLlHO, Mbl MomeM ypaBHHTb LlHCJIO pa3p5l)!.oB, npHnHcaB K O)!.HOMY
1f3 Lluce., Bnepe)!.H HecKOJIbKO HYJleH). I1MeHHo, B 3TOM cJIYLlae
(nk .10 k + ...+no)+(m" .10k+...+mo)=(nk+m,,) .IOk+...+(no+mo).
CJIomlJee 06CTOlIT )!.eJIO, eCJIH XOT5I 6bI O.'1,Ha H3 CYMM ns+ms 60JIbille
HJIH paBHa )!.eCHTH, ns+ms;;;:' 10. HaH)!.eM, HanpHMep, CYMMY 748+436.
I1Meel\l:
748+436=(7.10 2 +4.10+8)+(4.10 2 +3.10+6)=(7+4) .10 2 +
+(4+3)'10+(8+6).
nOCKOJ1bKY 8+6= 10+4, TO
748+436=(7+4) .10 2 +(4+3). 10+ 10+4=(7+4)102+(4+3+ l). 10+
+4=(7+4).10 2 +8.10+4.
.LJ.aJIee, 7+ 4 = 10+ 1, H, CJIe)!.OBaTeJIbHO,
(7+4) .10 2 =(10+ I) .10 2 = 103+ 10 2 ,
a nOTOMY
I1TaK,
748+436= 103+ 10 2 +8 .10+4= 1184.
748+436= 1184.
282

B '06w.eM BH)!.e aJirOpHTM CJlOmeHH5I HaTypaJlbHbIX LlHCeJl, 3a.n.aHHblX
B )!.ecHTHLlHoa CHCTeMe CLlHCJ!eHlf5l, TaKOB.
a) 3anUUleM 8mopoe C/laeaeMoe noiJ nep8blM maK, 'lm06bl coomeemcm-
8YIOU{ue pa3p5l8bl 6btAU 3anUCGHbI8pY2 noiJ 0PY20M. B C/lYliae He06xoOUMO-
emu npunUUle.M nepeo MeHbUlUM us C/la2aeMblX HeCKO/lbKO HY/leU, Iim06bl
ypa8HJlmb IiUC/lO paSpJl008 8 060ux C/laeG£MbLX.
6) C/lO{)fCUM t{ucfJPbt pCl3pJlOa eouHut{ (3iJecb U HU{)fCe Mbl O/l5l KpamKoc-
mu 2080pUM «t{ucfJpa» 8Mecmo «ooH03HaIiHoe Ii UC/lO , U306pa{)fCaeMOe t{ucfJ-
pou»). EC/lu cYMMa MeHbUle oecfl111u, 3anUCb18aeM ee 8 pCl3p5l0 e8UHUt{ om-
8ema U nepexoiJuM K C/leOYIOU{eMY pCl3pJlOY (oeC5lmK08).
B) EC/lU cyMMa t{ucfJp eouHut{ 60/lbUle U/lU pa8Ha aeCJlmU, mo npea-
Ctna8/lfleM ee 8 8uoe no+mo= 10+10, r)!.e 10 - ooH03HaliHoe IiUC/lO, 00-
nUCb18aeM 10 8 paspJl8 e8UHUt{ om8ema U npU6a8/l5leM 1 K t{ucfJpe OeCJlmK08
nep8020 C/laeaeM020, nOC/le 'ie20 nepexoouM K (X1spfliJy OeC5lmK08.
r) jJa/lee n08mOp5le/!t me ;}Ice onepat{uu C OeC5lmKaMU, nomOM - C com-
HJlMU U m. o. n pOt{eCC SaKaHIiU8aemcJl, M20a OKa3bIBGlOmCJl C/lMlCeHHbIMU
t{UcfJPbl cmapUlUX pCl3p5l008 (npu 8mOM eC/lU ux CYMMa OKa{)fCemC5l 60/lbUle
U/lU pa8Ha oecJlmu, mo npunUCbl8aeM 8nepeOu 060ux C/la2aeMblX HY/lU,
Y8e/lUIiU8aeM HY/lb nepeo nep8blM C/laeaeMblM Ha 1, U 8e/laeM C/lO;}lCeHue:
1+0=1).
TOLIHO TaK )]{e CKJla)!.blBaIOT LlHCJla B )!.pyrHx CHCTeMax CLlHCJleHH5I.
Ha)!.o JIHWb 3HaTb Ta6J1HlJ.Y CJlOmeHHSJ 0)!.H03HaLlHbIx LlIICeJI B 3TOH CHC-
TeMe CLlHCJleHHSJ. HanpHMep, B )!.BOHLIHOH CHCTeMe Ta6JIHIJ,a CJIOmeHHH
HMeeT OLleHb npocToH BH)!.:
'>(0
o
1
o
1
1
10
B TpOHLlHOH CHCTeMe OHa HMeeT H)!.:
o
2
o
I
2
o
1
2
I
2
10
2
10
11
a B BOCbMepHtlHOH - BH,l(:
283

 I 0 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 10
2 2 3 4 5 6 7 10 II
3 3 4 5 6 7 10 11 12
4 4 5 6 7 10 11 12 13
5 5 6 7 10 II 12 13 14
 6 7 10 II 12 13 14 15
7 10 11 12 13 14 15 16
npHBe)!.eM npHMepbI CJIOMeHH5I B 3THX CHCTeMax CLlHCJIeHH5I.
+101111011 2 +121011 3 + 103571 8
11011101 2 22120 3 235 746 8
1001011000, 2202013 341 S37 8
YnpamaeHHII
I. CocnlBbTe T86J1HI.lbl CJlOJKeHHH ,lJ,JlH '1eTBepH t lHoii H ,lJ,BeH8.L\I.l8TepH'IHOii CHCTeM
CtlHCJleHHH (0603H8'1bTe 6YKBOii a I.lHcjJpy 10, 8 6YKBoii  I.lHcjJpy II B ,lJ,BeIl8,lJ,U:8TepH'I.
Hoii CHCTeMe C'IIICJleHHH).
2. H8ii.L\HTe CYMMbI:
a) 110011101 2 +11 010 111 2 ; 6) 210210 3 +120120 3 ;
B) 1312031-1+2323204; r) al12+7fu12;
.L\) 4765 8 +6437 8 ; e) 316 8 + 101 101 100 2 ,
ITpoBepbTe nOJIYlJeHHble OTBeTbI, nepeBe.L\H CJlaraeMble B ,lJ,eCHTH'IHYIO CHCTeMY
C'IHc.rIeHHH, BbIIIOJIHUB CJlOiKeHHe B 3ToiI CHCTeMe C'IHCJleHHH H nepeBe,lJ,H CYMMY CHOBa
B J!alt8HHbIe CIiCTeMbI C'IHCJleUH!I.
6. BbllJHTaHne 8 )!.eCHTHlJHOH H )!.pyrHx cHcTeMax ClJHCJIeHHH. nYCTb
H3 LlHCJIa 7845 Ha)!.o BblLleCTb 342. YpaBHHeM KOJIHLlecTBO IJ,Hcpp B YMeHb-
IllaeMOM H BbILlHTaeMOM, 3anHcaB BbILlHTaeMoe B BH)!.e 0342. 3anHweM
3TH LlIfCJIa 13 13H)!.e:
7 845= 7.10 3 +8.10 2 +4.10+5
0342=0.10 3 +3.10 2 +4.10+2.
A Tenepb pa3HOCTb 7845-0342 MomHO 3anHcaTb B BH)!.e:
7845-0342=(7.10 3 +8.10 2 +4.10+5)-(0.10 3 +3.10 2 +4.10+2)=
= 7 .103+8 .10 2 +4 .10+5-0 .10 3 -3 .10 2 -4 .10-2.
I1CnOJIb3YH CBoHCTBa CYMMbl H pa3HOCTH, 3TO BblpameHHe MomHO
nepenHcaTb CJIe)!.YIOllJ,HM o6pa30M:
7845-0342=(7-0) .10 3 +(8-3) .10 2 +(4-4) .10+(5-2)=7.10 3 +
+5.10 2 +0.10+3=7503.
TaKHM )Ke o6pa30M MomHO HaHTH pa3HOCTb n-m, r,ne
fl=fl" .lO k +...+flo, m=m" .lO k +...+mo,
284

ecJIH )!.JISJ Bcex s (sk) BblnOJlHSJeTCSJ HepaBeHCTBO nsms' B 3TOM
CJIYLlae
n-m=(nk-mk) . IOk+...+(no-mo)
(KaK H B CJIYLlae CYMMbl, Mbl ypaBHHBaeM KOJIHLleCTBO IJ,H<PP B BbILlHTae-
!110M H YMeHbwaeMOM, )!.onHCblBaSJ HYJIH Bnepe)!.H BblLlHTaeMoro).
CJIOmHee HaHTH pa3HocTb 7215-342 - ee HeJIb351 npe)!.cTaBHTb
B BH)!.e (7-0) .10 3 +(2-3) .10 2 +(1-4) .10+(5-2), nOCKOJIbKY B He-
KOTOpblX H3 CK060K BblLlHTaeMoe OKa3aJIOCb 60JIbille YMeHbiliaeMoro.
4T06bl H36efHYTb 3Toro, 3anHilleM YMeHbwaeMoe B BH)!.e 7.10 3 +2.10 2 +
+ 1 .10+5, a nOTOM nepenHweM 7 B BH)!.e 6+ 1. Mbi nOJIYLlHM, LITO 7215=
=6.10 3 + 103+2 .10 2 + 1.10+5. Ho 10 3 =900+ 100=9 .10 2 + 10 .10, a
nOToMY 7215=6.10 3 +(9+2) .10 2 +(10+ 1) .10+5=6 .10 3 + 11 .10 2 +
+ 11.10+5. 3HaLlH'f, 7215-342=(6-0) .10 3 +(11-3) .10 2 +
+(11-4).10+ (5-2)=6.10 3 +8.10 2 +7.10+3=6873.
B 06w.eM BH)!.e H:lBeCTHblH H3 IlIKOJIbHOrO Kypca aJIrOpHTM BbILlHTa-
H H5I Bblr JI SJ)!.HT CJIe)!.YIOw.HM o6p a30M.
a) Ypa8Hfle.M 'mCAO pCl3pfla08 Y.M.eHbUlae.Moeo U 8bltlUmae.Atoeo U 00-
nUUle.M 8bltlUmae.MOe noiJ Y.MeHbUlae.MblM. maK, ttm06bl oiJuHaK06bte PCl3pfli)bl
HaxoauAucb apye noa apyeo.M.
6) ECAU t{ucfJpa 8 pCl3pfli)e eaUHUt{ 8blttumae.Moeo He npe80cxoaum co-
omeemcmeylOUJ,eu t{ut/JPbl Y.MeHbmaeMoeo, 8blliUmae.M ee U3 t{UcfJPbl yMeHb-
Ulae.Moeo, nOCAe tteeo nepexoau.M K cAeaylOUJ,e.MY pa3pflay.
B) ECAU 3ICe t{ucfJpa eaUHUt{ 8blttumae.MOeo 60AbUle t{UcfJPbl eaUHUt{
Y.M.eHb , Ulae.Moeo mo>no, a t{ucfJpa aeCflmK08 Y.MeHbUlae.M020 OmAUltfta om
HYAfl, mo Y.M.eHbUlae.M t{ucfJPY aecflmK08 Y.M.eHbmae.Moeo Ha 1, oiJH08pe-
.M.eHHO Y8e/lUttU8 t{ucfJPY eauHut{ YMeHbUlae.Moeo Ha 10, nOC/le tteeo 8blttUma-
e.M U3 IiUCAa 10+ no ttUCAO m o , oonUCbl8ae.M pCl3Hocmb 8 pa3pfli)e eaUHlll{
UCKOMoeo ItUCJ1.a U nepexoiJu.M K cAeaylOUJ,e.MY PCl3pfli)y.
r) ECAU t{ucfJpa eaUHUt{ 8blttUmae.M020 60AbUle l{UcfJPbl eaUHUt{ Y.M.eHb-
Ulae.M.Oeo U t{UcfJPbl, cmOflU{Ut! 8 pCl3pflae fjecflmK08, comeH U m. a. Y.M.eHb-
UlaeMoeo, pa8Hbl HY/lIO, mo 6epe.M nep8YIO Om/lUttHYIO om HY/lfl t{UcfJPY 8
YMeHbUlaeMoM (nOC/le pCl3pflaa eaUHUt{) , YMeHbUlae.M ee Ha I, Y8e/lUliueae.1t
Bce t{UcfJPbl 8 .MJ1.aaUlUX pCl3pflaax ao pCl3pflaa aeCflmK08 8K/llOttUmeAbHo WI
9, a t{ucfJPY 8 pCl3pflae eaUHUt{ Ha 10, 8blttUmae.M mo U3 IO+no, 3anucbl-
8ae.M pa3Hocmb 8 pCl3pflae eaUHUt{ UCKO.Moeo ttUC/la U nepexoaUM K CAeaylO-
UJ,eMY pa3pflay.
)!.) B C/leaYIOUI,e.M pCl3pflae n08mopJle.M onucaHHblU npot{ecc.
e) JJpot{eCC 8blliumaHUfl OKaH'lU8aemCfl, Koeaa I1poU380aumCfl 8blliuma-
Hue U3 cmapUleeo pCl3pfli)a YMeHbUlaeMoeo.
Il,H<PPbl, nOJIYLlaeMble B pa3pSJ)!.ax HCKOMorO LlHCJIa, H )!.a)!.YT )!.eCSJTHLI-
HylO 3anHCb pa3HOCTH n-m.
TOLIHO TaK )Ke BbmOJIHSJeTCSJ BblLlHTaHHe B p-HLIHOH CHCTeMe CLlHC-
JIeHHSJ. Pa3JIHLlHe COCTOHT JIHWb B TOM, LITO npH BblllHTaHHH o,nH03HaLl-
HbiX LlHCeJI Ha)!.o nOJIb30BaTbC5I Ta6JIHIJ,eH CJIO)KeHHSJ ,nJISJ 3TOH CHCTeMbi
ClIHCJIeHHSJ, npH BbmOJIHeHHH nYHKTa a) YBeJIHlIHBaTb IJ,H<pPY Ha p
(3TO LlHCJIO HrpaeT pOJIb )!.ec5lTH), a npH BbinOJIHeHHH nYHKTa r) YBeJIHLlH-
28&

BaTb npOMemYTOLlHbIe pa3pH,nbl Ha p-I, a pa3pH,n e,nHHIHJ, Ha p. HanpH-
Mep,
37i2 8
- 645 8
3045 8 ,
B caMOM ,neJIe, no Ta6JIHue CJIO)l{eHHH 5 8 +5 8 = 12 8 , a nOTOMY 12 8 -5 8 =
=5 8 , lI.aJlee, 10-48=48 H T. )!..
YnpamoeOHH
1. Bbmo.HHTe BbI'IHT3HHH:
3) 4725 8 -647 8 ; 6) 1100011101 2 -1001100111 2 ;
B) 210211 3 -12021 3 ; r) 9l7a12-a683512'
2. B K3KOii CHCTeMe C'IHCJJeIlHH BbllIOJlHeHO cJJeJ\YIOw.ee BbI'IHT3HHe:
1340
- 265
1064 ?
7. YMHO>KeHHe "IHCeJI B )!.eCJlTHIfHOH H )!.pyrHx n03HHOHHblX CHCTeMax
ClJHCJIeHHJI. 4T06bl YMHomHTb LlHCJlO 247 Ha 1000, )!.OCTaTOLlHO npHnH-
caTb K HeMY cnpaBa TPH HYJIH: 247.1000=247000. B caMOM )!.eJle, 247=
=2.10 2 +4.10+ 7, a nOToMY B CHJlY )!.HCTpH6yTHBHOCTH YMHO)l{eHHH OT-
HOCHTe,'1bHO CJlOmeHHH H paBeHcTBa IOk.1OS=IOk+s HMeeM
247.1000=(2.10 2 +4.10+ 7) .10 3 =2 .lm+4. 104+ 7.10 3 .
.3T0 BbipameHHe MO)l{HO 3anHcaTb TaK:
247 .1000=2 .1O+4 .10 4 +7.10 3 +0.10 2 +0.10+0=247 000.
Boo6ll1.e, LlT06b1 YMHO)l{HTb IfHCJlO fl Ha lOs, HMO K )!.ecHTHLlHoH 3a-
nHCH LlHCJla fl npHnHcaTb cnpaBa S HYJleH; 3TO H )!.aCT )!.eCHTHLlHYIO 3a-
nHCb npOH3Be)!.eHHH. B caMO M )!.eJle, ecJIH
fl=flk...flO' TO n=n1l. .IOk+...+flo,
a nOToMY
fl.lO s =(nk .lO k +. ..+flo) .lO s =flk .lO k + s + ...+no .lO s +0 .IOS-l+
+...+0.
3HaLlHT,
fl .lO s =flk' ..floO.. .0.
--...-
S H)'ileA
Tenepb BbIHCHHM, KaK YMHO)l{HTb LlHCJIO fl flk...flo Ha 0)!.H03HatIHOe
IIHCJlO m. I1MeeM:
flm=(flk .IOk+...+no)m=flkm .IOk+...+nom.
4HcJla fls H m 0,nH03HaLlHbl H HX npoH3Be,lJ.eHHH )!.aHbI B Ta6J1Hue YMHO)l{e-
HHH. 3TH npOH3Be,neHHH HJlH 0,!{H03HaLlHbl, HJlH )!.BY3HaLlHbI. MO)l{HO 3a-
nHcaTb HX B BH)!.e fls .m=a s .1O+b s . )!.onycKaH CJIYLlaH, Kor)!.a as=O.
286

Tor.n.a Mbl nOJlYLlaeM, LITO
11 .m  (a" .10+b It) olO k +.. .+(ao .1O+b o )=(a" .10k+1+... +ao .10)+
+(b" .10 k +...+b o ).
BbmOJIHHH nocJle 3Toro CJlOmeHHe, nOJlYLlHM OTBeT. Hanpm!ep, 38.6=
=(3.10+8) .6=3.6.10+8.6=(10+8) .10+(4 '10+8)= 10 2 +(8+4) X
X 10+8= 10 2 +(10+2) .10+8= 10 2 + 10 2 +2 .10+8=2 010 2 +2 .10+8=
= 228. nHWYT TaK:
X 38
6
228
HaKOHelJ" pa4!CMOTpHM YMHomeHHe MHor03HaLlHblX LlHCeJl .n.pyr Ha
.n.pyra. B CHJIY CBOHCTB YMHO)l{eHHH HMeeM
n (m1IOl+...+mo)=(nm 1 ) .IOl+...+nmo.
nOCJle)!.OBaTeJlbHO YMHO)l{aH LlHCJIO n wi 0)!.H03HaLlHble LlHCJla m 1 , ... , mo.
a nOTOM nOJIYLleHHble npoH3Be.n.eHHH Ha 10 1 , ..., 1, HaH)!.eM cJlarae-
M'ble, CYMMa KOTOpblX H paBHa HCKOMOMY npOH3Be)!.eHHIO nm, r)!.e
m=m 1 0 10 1 +. . .+mo.
OnHcaHHoe npaBHJlO )!.aeT OOOCHOBaHHe 06bILlHO npHMeHHeMOMY cno-
cooy YMHO)l{eHHH «cTOJI6HKOM»:
X 295
47
2065
1180
13865
TOLIHO TaK )l{e npOH3Bo,nHTCH YMHO)l{eHHe B )!.pyrHx n03HIJ,HOHHbIX
CHCTeMax CLlHCJleHHH. npH .!')TOM Ha)!.o nOJIb30BaTbCH Ta6JIlflJ,eH YMHO)l{e-
HHH 0)!.H03HaLlHbIX QHCeJI B E.lTOH CHCTeMe. npHBe)!.eM TaKHe Ta6.1HlJ,bl ,lJ.JISJ
p=2,3,8:
 0  0 2
p=2 p=3
0 0 0 0 0 0 0
1 0 1 1 0 I 2
2 0 2 11
287

o
2
3
4
5
6 7
0 0 0 0 0 0 0 0 0
p=B 1 0 1 2 3 4 5 6 7
2 0 2 4 6 10 12 14 16
3 0 3 6 11 14 17 22 25
4 0 4 10 14 20 24 30 34
5 0 5 12 17 24 31 36 43
6 0 6 14 22 30 36 44 52
7 0 7 16 25 34 43 52 61
HanpHMep, x 35 8
£
313
164
-
2153 8
D.J1H npoBepKH 3aMeTHM, LITO
35 8 =3.8+5=29, 47 8 =4.8+7=39, 2153 8 =2.8 3 +1.8 3 +5.8+3=
=1l3I, a 29.39=1l31.
Ynpamoeooll
I. BbInOJIHHTe YMHOJKeHHe H C,lJ,eJIaiiTe npOBepKY, nepeBe,lJ,51 MHO>KHTeJIH H npo-
H3Be,lJ,eHHe B ,lJ,eC5ITH'IHYIO CHCTeMY C'IHCJleHH5I:
a) 1011111 2 .101 2 ; 6) 1202 3 .120 3 ;
B) 3 470 8 , 761 8 ; r) a8 12 . 712'
r,lJ,e a= 10, = II.
2. .l1oKaiKHTe. 'ITO B JIl060ii CHCTeMe C'IHCJleHH5I C OCHOBanHCM p I\MI YMHoiKeHHJI
,lJ,BY3Ha'lHoro 'IHCJla na II p Ha,D,O CJIOJKHTb llH<ppbl ,lJ,eC5ITKOB H e,lJ,HIlHU H, eCJIH nOJly-
'IHJIC5I 0J\1I03Ha'lHbIii OTBeT, 3anHcaTb ero Me>K,lJ,Y 3THMH UH<ppaMH. a eCJIH OTBeT ,lJ,BY-
3Ha'leH. TO UH<PPY e,lJ,HHHU CYMMbl 3anHcaTb MeiK,lJ,Y UH<ppaMH ,lJ,ec5lTKOB H e,lJ,HHHu,
YBeJIH'IHB UH<PPY ,lJ,eC5ITKOB Ha I. HanpHMep, 23 8 , 118=253 8 . 37 8 , 11 8 =427 8 ,
3. BbInOJIHHTe J\eiicTBH5I B cooTBeTcTBYIO!l.tHX CHCTeMax C'IHCJleHH5I, nOCJIe 'Iero
nepeBe,lJ,HTe ,lJ,aHHble H OTBeT B ,lJ,eC5ITH'IHYIO CHCTeMY C'IHCJleHH5I, BbInOJIHHTe ,lJ,eikTBH5I
H CpaBHHTe pe3YJIbTaTbl:
a) (1001101 2 +1110001 2 ), (1100001 2 -1001101 2 );
6) 767 8 .34 8 -2055 8 ; B) 212012 3 .201 3 +22120 3 .111 3 ,
8. )J.eJIeHMe B ,!!,eCHTMlJHOH H ,!!,pyrHx n03HHOHHblX cHcTeMax ClJHCJIe-
HHH. nYCTb Ha)!.o pa3)!.eJIHTb 256 Ha 6 C OCTaTKOM. TaK KaK 60<256<
<600, TO HenOJIHOe qaCTHOe 3aKJlIOqeHO Me)K)!.Y 10 H 100. DOJlee TOqHYIO
OueHKY )!.aeT HCnOJlb30BaHHe Ta6J1f1UbI YMHO)KeHMH: TaK KaK 6.40=240,
a 6.50=300 H 240<256<300, TO HenOJJHoe qaCTHOe 3aKJlIOqeHO Me)K)!.Y
40 H 50. 31'003HaqaeT, LITO UH<ppa )!.ec5JTKOB HenOJIHOrO qaCTHOrO paBHa
4, T. e. OHO HMeeT BH)!. q=4.IO+qo. Ho Tor)!.a )!.OJl)KHbI BblnOJlHHTbC5J
paBeHcTBa (40+qo) .6256«40+qo+ 1) .6, T. e. 240+6qo 256<
<240+6(qo+l). 113 HHX HaXO)!.HM, qTO 6qoI6<6 (qo+I). BTOPHqHO
288

npHMeH5I5I Ta6JIHl.I.Y YMHO)KeHlHI, nOJIY'I8eM, '1TO qo=2. 3H8'1IIT, HenOJl-
HOO qaCTHOe paBHo 42, a Tor.ll.a OCTaTOK (IT .lI.eJIeHHH paBeH 256-42.6=4.
I1TaK, npH )!.eJIeHHH 256 Ha 6 nOJIYLlaeTCH HenOJIHOe LlacTHoe 42 H OCTa-
TOK 4: 256=42.6+4. 31'0 .lI.eJIeHHe 3anHCbIBalOT TaK;
256 / 6
- 24 42
16
12
4
TOLlHo TaK }!{e BbinOJIHHeTCH ')!.eJIeHHe MHoro3HallHbiX LlHCeJI. HanpH'
Mep, ecJIH Ha)!.o pa3)!.eJlHTb 7824 Ha 84, TO CHaLlaJIa 3aMeLlaeM, lITO 840
7824<8400, a nOTOMY HenOJIHOe LlaCTHoe )!.BY3HaLlHo. IJ.H<pPY )!.ecHT-
KOB MO}!{HO 6bIJIO 6bl HaihH nepe6opoM, nOCJIe)!.OBaTeJIbHO YMHO}!{aH 84
Ha 10,20,30 H T. )!.., nOKa He nOJIYLlHJIOCb 6bI LlHCJIO, 60JIbillee LleM 7824.
Tor,!{a H3 HepaBeHCTBa 84.10 'ql7824<84 .10(ql + I) nOJIYLlHJIH 6bl,
liTO lI.H<ppa )!.ecHTKOB HenOJIHoro LlaCTHoro paBHa qt. 31'or nepe60p
MO}!{HO COKpaTHTb, eCJIH 3aMeTHTb, '1TO 84<90, a 80.90=7200<7824.
3HaLlHT, TeM 60JIee 80.84<7824 H paCCMOTpeTb Ha)!.o JIHWb .lI.Be B03MO}!{-
HOCTH: ql=8 HJlH ql=9. npoBepKa nOKa3bIBaeT, LITO ql=9. A Tenepb
YMHO}!{aeM 84 Ha 90, BblLlHTaeM nOJIYLleHHoe npoH3Be)!.eHHe 7560 113
7824 H K nOJIYLleHHoH pa3HocTH 264 npHMeHHeM TOT}!{e npolJ,ecc. OH no-
Ka3bIBaeT, LITO lI.H<ppa e)!.HHHIJ, HenOJIHoro LlaCTHoro paBHa 3, a nOTm.';
OHO paRHO 93. OCTaToK paBeH 264-84 .3= 12.
3TH BbILlHCJleHHH 06bILlHO 3anHCblBalOT B CJIe)!.YlOllI.eM BI1)!.e:
7824 1 84
- 756 93
264
- 252
12
TOIIHo TaK }!{e BbmOJlHHeTCH ,!{eJIeHHe B .nPyrHx n03HIJ,HOHHbIX CHC-
TeM8X CLlHCJIeHHH. npolll.e Bcero ero ,!{eJIaTb B ,!{BOHLIHOH CHCTeMe - HMO
,lJ.eJIaTb Bbl60p JIHilib MeR<JlY HYJleM H e)!.HHHlI.eft:. HanpHMep,
_ 1 101 101 011. 1 1 II 0 010.
lliOOIO Ill.
110 100 01
- II 100 10
1011Ill1
1110010
1001101
3H8I1HT, HenOJIHoe lIaCTHoe paDHO 111.. 8 OCT8TOK paBeH 1 001 101!.
YnpamueUBa
BbinOJIH HTe ,l(eJieHHe C OCTaTKOM B CJle,l(YIOUJ.HX rrpHMepax: a) 111 100 011. I 10 111 a:
6) 120 211  : 2210,: B) 7453 8 : 36 8 : r) a711 : 4a12'
10 Nt 70'1
289

fJlaBa X
)J.EJUIMOCTb IJ,EJIbIX HEOTPIU(ATEJIbHbIX '-IIICEJI
 t. OTHOmEHHE ,lJ,EJIHMOCTH II ErO CBoRCTBA
1. CBOUCTBa OTHOllleHHH JI.eJIHMOCTH. B 3TOl1 r J1aBe Mbl H3YLlHM OT-
HOllleHHe .n.eJlHMOCTH B MHomeCTBe Zo HeOTpHlJ,aTeJIbHblX lJ,eJIblX 'lHCeJI.
)lJlH KpaTKOCTH BMeCTO CJlOB «ueJIoe HeOTpHlJ,aTeJlbHoe LlHCJIO» 6YJI.eM
DHcaTb «LlHCJlO».
rOBOpHT, LITO IiUC/lO a Oe/lUmCfl Ha IiUC/lO b, eCJlH CYllI.ecrByeT TaKoe
'IHCJIO C, LITO a=bc. B 310M CJIYLlae nHWYT a : b. HanpHMep, 8 : 2,
TaK KaK 8=2.4. B TO me BpeMH 8 He )!.eJIHTCH Ha 3, TaK KaK HeT ueJIoro
Ht:'oTpHlJ,aTeJlbHOrO LlHCJla C, TaKoro, LITO 8=3c.
)loKa)l{eM HeKOTOpbl€: CBOHCTBa OTHOllleHHH )!.e.'IHMOCTH.
1) lJUC/lO 0 Oe/lUmCfl Ha AJ060e IiUC/lO,
(Vb E Zo)O: b.
B caMOM )!.eJle, )!.J1H J11060ro bE Zo HMeeM: O=b .0. TaK KaK 0 E Zo,
to no onpe)!.eJIeHHIO )!.eJIHMOCTH 0 : b.
2) Hu oOHO Om/lUIiHoe om HY/lfl 'lUC/lO He ae/lUmCfl Ha 0:
(Va E Zo)a  O.
B caMOM )!.eJle, nYCTb £l:7b0. TaK KaK O.b=O )!.J1H Bcex bE ZO. TO
paBeHCTBO a=O.b He MOmeT BblnOJlHHTbCH HH )!.J1H KaKoro 3Ha'leHHH b.
3HallHT, a He )!.eJIHTCH Ha O.
3) JIto6oe IiUCAO aeAUmCfl Ha I, (Va E Z 0) a: l.
B caMOM )!.eJIe, )!.J1H J11060roaEZo HMeeM a=I.a, 3TO H 03HaLlaeT,
'ITO a )!.eJIHTCH Ha I.
4) OmHoUleHue ae/lU.MOCmU peljJ/leK(;U6HO, m. e. /lL060e 'lUCAO oe.1Umcfl
Hll COOfl, (Va E Zo) a : a.
B caMOM )!.eJIe, )!.J1H J11060ro a E Zo HMeeM a=a'1. TaK KaK IE Zo,
TO 3TO 03HallaeT, LITO a : a.
5) ECAU a : b U a>O, mo ab.
B caMOM )!.eJIe, TaK KaK a : b, TO a=bc, r)!.e cE Zoo n03TOMY a-b=
=bc-b=b(c-l). Ho no npe.u.nOJlO)l{eHHIO a>O, H nOToMY c>O. TaK
KaK J11000e nOJlO)l{HTeJIbHoe lIHC.nO U3 ZO He MeHbllle, treM I, TO C I H,
SHaLlHT, b(c-l)O. n03TOMY a-bO, T. e. ab.
6) OmHoUleHue Oe/lll.MOCmU aHmucuMM£mpuIiHo:
(Va, bE Zo) (a : b /\ b : a}=?(a=b).
:?!JO

3)!.eCb HaO paCCMOTpeTb Ma CJlYlIa5l: a) 06a lIlfCJla a If b OTJIHLlHbI
OT HYJlH, 6) XOTH 6bl O)!.HO H3 HIfX paBHO Hymo. ECJlIf a>O H b>O, '10
no CBOHCTBY 5) 1f3 a : b H b : a BbiTeKacr, LITO ab H ba. A 3TIf Hepa-
BeHCTBa MorYT HMeTb MeCTO JIHWb nplf YCJlOBHIf, LITO a=b. B 3TOM CJlY-
Llae Hawe YTBepLK,lJ.eHHe )!.OKa3aHO. npe)!.nOJlOmHM Tenepb, LITO a=O.
B 3TOM cJlYlIae no CBOHCTBY 2) H b=O, TaK KaK If Halle b He MOrJlO 6bI )!.e-
J1HTbCH Ha a. 3HallHT, If B 310M CJIYlIae a=b.
7) OmHOUleHUi! Oe/lUMOCmu mpaH3umU8HO, m. e. U3 a : b U b : c
CAeayem, limo a : c:
(Va, b, cE Zo) (a: b A b : c)=>(a : c).
B caMOM )!.eJIe, TaK KaK a : b, TO HaH)!.eTCH TaKoe lIHCJIO k, lITO a=bk,
a TaK KaK b : c, TO HaH)!.eTCH TaKoe lIlfCJlO I, LITO b=cl. Ho Tor,lJ.a IfMeeM:
a=bk=(cl)k=c(lk). 4HCJlO lk HBJlHeTCH npOH3Be)!.eHHt'M )!.BYX IJ,eJIbiX
HeOTpHIJ,aTeJlbHbIX lIHCeJI 1 H k H nOToMy caMo HBJlHeTCH IJ,fJIbiM If HeOTplf-
IJ,aTeJIbHblM. 3HallHT, a ,lJ.eJIBTCH Ha c.
8) EC/lu 'lUCAa a U b Oe/lJlmcfl Ha C, mo U ux CYMMa OeAUtnl.Jl I a c:
f'la, b, cE Zo)(a: cAb: c)=>( (a+b): c).
B caMOM )!.eJle, no YCJIOBHIO HaH,lJ.YTcH TaKHe lIHCJIa k If I, LITO a=ck II
b=cl. Ho Tor)!.a a + b=ck+cl=c(k+/). TaK KaK k+1 - IJ,eJIoe HeOT-
plfIJ,aTeJIbHOe lIHCJ10, T-0 H3 3Toro paBeHCTBa CJIe.l{yer, Iff 0 (a+b): c.
.LJ.oKa3aHHoe YTBep>K,lJ.eHHe CnpaBe.l{J1HBO H B CJIyqae, Kor,lJ.a lIHCJIO
CJlaraeMblX 60Jlbllle )!.BYX.
ECAU Ka3ICOoe U3 'luce/l ah ..., a,. oeAumcJl Ha C, mo U UX CYMMa at +
+...+a n oeAumcJl Ha c.
.LJ.oKa3aTeJlbCTBO npOBO,lJ.HTCH TOllHO TaK me, KaK H MH MYX CJIara-
eMbiX. AHaJlOrHlIHO )!.OKa3b1BaeTCH CJle.l{YIOInee YTBepm,lJ.eHHe:
EC/lU 'lUCIla a U b oe/lflmcJl Ha C, npU'leM b, mo u UX pCl3HOCmb a-b
OeAUmL'fl Ha C.
9) ECAU 'lUC/lO a aeAUmcJl HaC,mo U 8ce 'lUCAa 8uoa ax (x E Zo) Oe/lJlmcJl
Ha c.
B caMOM ,lJ.eJle,no YCJIOBHlOa=ck, r)!.ek E Z.. Ho TOf)!.a ax= (ck)x=c X
X (kx). TaK KaK kxE ZO, TO 310 }>aBeHCTBo nOKa3b1BaeT, LITO ax: c.
113 cBoHcTB 8) H 9) BbiTeKaeT nOJle3Hoe CJIe,lJ.CTBHe:
ECAU IiUC/la at, ..., a,. oe/lJlmcJl Ha c, mo, KaK08bl 6bl HU 6bl/lu IiUCAa
Xh ..., Xn, 'lUC/lO atXt + ... +a,.Xn Oe/lumcJl Ha c.
B caMOM )!.eJle, no CBOHCTBY 9) K8)K.l{oe H3 lIHCeJI atXl! ..., a,.X n ,lJ.eJIHT-
CH Ha c, a Tor)!.a no CBOHCTBY 8) H HX CYMMa atXt +... +a,.xn ,lJ.eJlIfTCH Ha c.
10) ECAU 'lUC/lO ac oe/lumcJl Ha bc, npU'leM c=;c:O, mo a oe/lumcJl Ha b.
B caMOM )!.eJIe, no yCJIOBHIO HaH.AeTCH TaKoe k, LITO ac= (bc)k, If no-
TOMY ac=(bk)c. nOCKOJIbKy c=;c:O, H3 3Toro paBeHCTBa CJIe)!.YeT, LITO a=bk,
T. e. a: b.
YopamueUBII
I. .LI.JlII K8KHX rrap 'IHCeJI BepHY 06a OTHonreUHSI: a i b H b: a}
2. .LI.OK3JKHTe, 'ITO n (n+ 1) ./{eJIHTCII Ha 2.
3. I1cTHHHo JlH BblCKa3b1B3HHe: «ECJlH CyMM8 ABYX '1HCeJI ):\eJIHTCR H8 C, TO K8JK.IlOl!
CJlaraeMOe ./{eJIHTCII H3 C»?
1 0*
291

4. ]l.oKaJKIiTe, 'ITO eCJJIi O<b<a, rrpw.leM 'IliCJIa a H b lI,eJlhle, TO il-b3 .n.CJIHTCSI
Ha a-b.
5. ]l.oK8JKHTe, 'ITO TpeX3Ha'lHOe 'IIiCJJO, 3arrHcaHHoe TpeMSI O,1J.IiHaKOBbIMH lI,Hcp-
paMH, )l.eJIHTCSI Ha 37.
6. l(aKHe H3 CJJe,1J.YlOmHx YTBepJK.D.eHHii: HCTHHHbI ,1J.JISI Bcex HaTypaJlbHblx 'IliCeJI ai
a) ecJIH 'IHCJJO a ,1J.eJIHTCSI Ha 2, TO OHO Jl.eJIHTCSI H Ha 4;
6) ecJIH 'IHCJJO a ,1J.eJIHTCSI Ha 4, TO OHO ,1J.eJIHTCSI H Ha 2;
B) eCJJH 'IHCJJO a He )l.eJIHTCSI Ha 4, TO OHO He .D;eJIHTCSI H Ha 2;
r) ecJIH 'IHCJJO a He ,1J.eJIHTCSI Ha 2, TO OHO He ,1J.eJIHTC.SI H H 1]
7. ]l.oKaJKHTe, 'ITO cYMMa )l.BY3Ha'lHoro 'IHCJJa H 'IHCJla, 3anHcaHHoro TeMH JKe
lI,HcppaMH B 06paTHoM rropSl)l.Ke, )l.eJIHTCSI Ha 11.
8. ]l.oK8JKHTe, 'ITO pa3HoCTb KBa,npaTOB .D;BYX nOCJJe,1J.OBaTeJIbHblX He'leTHbiX 'IHCeJt
AeJIHTCSI Ha 8.
9. ]l.oKaJKHTe, 'ITO ,1J.JlSI m06b1x .D;BYX lI,eJIbiX HeoTplill,a1'eJlbHblX 'IHCeJI a H b 'IHCJlO
ab (a2-1J2) ,1J.eJIHTCSI lia 3.
10. ]l.OK8JKHTe, 'ITO cYMMa Tpex rrOCJIe,1J.OBaTeJIbHblX HaTypaJlbHblx CTerreHeii: 'IHCJJ8
3 .D;eJIHTCSI Ha 13.
11. npH KaKHX 3Ha'leHHSlX n 'IHCJJO n 2 -l Jl.eJIHTCSI Ha 3?
2. npH3HaKH )J.eJIHMOCTH. n pU3HaKOM oeAuMocmu Ha lIHCJIO aHa-
3bIBalOT npaBHJIO, n03BOJIHIOInee no 3anHCH lIHCJIa x B ,lJ.eCHTHlIHOM (HJIB
HHOM n03HIJ.HOHHOH) CHCTeMe ClIHCJIeHHH Y3HaBaTb, ,lJ.eJIHTCH JIH OHO H3
a, He BbInOJIHHH Henocpe)J.CTBeHHO ,lJ.eJIeHHH x Ha a.
PaCCMOTpHM HeKOTopbIe npH3HaKH )J.eJIHMOCTH B ,lJ.eCHTHlIHOH CHC-
TeMe ClIHCJIeHHH.
1) CHallaJIa BbIBe,lJ.eM n p H 3 H a K ,lJ. e JI HMO C T H H a 2.
3anHllleM lIHCJIO x B ,lJ.ecHTHlIHoH CHCTeMe ClIHCJIeHHH:
X=X n .lO n +...+xl'lO+xo.
4HCJIO 10 ,lJ.eJIHTCH Ha 2 H nOTOMY KIDK,lJ.Oe H3 LlHCeJI 10, 10 2 , ..., IOn
Jl:eJIHTCH Ha 2. n03TOMY B CHJlY CJIe)J.CTBH5I H3 CBOHCTB 8) H 9) n. 1
lIHCJIO Y=X n .Ion+...+Xi .10 TO>Ke ,lJ.eJIHTCH Ha 2. TaKHM 06pa30M, lIHC-
JIO X HBJIHeTCH CYMMOH lIHCJIa y, )J.eJIHInerOCH Ha 2, H lInCJIa Xo. 3HallHT.
OHO ,lJ.eJIHTCH Ha 2 B TOM H TOJIbKO B TOM CJIYlIae, KOr,lJ.a Xo ,lJ.eJIHTCH Ha 2.
310 6Y,lJ.eT B CJIYlIae, KOr,lJ.a Xo paBHO O,lJ.HOMY H3 lIHCeJI 0, 2,4, 6, 8.
Mbi )J.OKa3aJIH CJIe,lJ.YIOInHH npH3HaK )J.eJIHMOCTH Ha 2.
lJUC/lO x Oe/lumcR Ha 28 mOM u mo/lbKO 8 mOM CAY'lae, K020a eeo oeCR-
mUIlHaJI. sanUCb KOH'laemcR oiJHoii U3 lfucPP 0, 2, 4, 6, 8.
.lI.JIH KpaTKocTH pellH Mbi Ha3bIBaeM 0)J.H03HallHbIe lIHCJIa 0,1,2,3,4,
5,6, 7, 8, 9 lfuljJpaMU (Ha caMOM ,lJ.eJIe IJ.H<l>PbI - 3TO JIHlllb 3anHCb TaKHX
qHCeJI). I1.H<l>pbI 0,2, 4, 6, 8 Ha30BeM liemHblMU, a OCTaJIbHbIe - Heliem-
HblMU. Tor)J.a npH3HaK ,lJ.eJlHMOCTH Ha 2 MO>KHO c<t>oPMYJIHpOBaTb TaK:
lJUCAO x ae/lUmcR Ha 2 8 mOM U mO/lbKO 8 mOM CAYtJ£le, K02aa nocAeiJ-
HJlJl, lfucPpa e20 aecflmUI/.HOU. sanucu IlemHa.
2) TOllHO TaK >Ke )J.OKa3b1BaeTCH n p H 3 H a K ,lJ. e JI HMO C T H
Ha 5. TaK KaK 10 ,lJ.eJIHTCH Ha 5, TO H Bce lIHCJIa 10, ..., IOn ,lJ.eJIHTCft
Ha 5, a nOTOMY Y=X n IOn + ... + Xl .10 ,lJ.eJIHTCH Ha 5. n03TOMY lIHCJIO x:::
=Xn .lO n +,,,+xl"lO+xo ,lJ.eJIHTCH Ha 5 B TOM H TOJIbKO B TOM CJIYlIae,
KOr,lJ.a Xo AeJIHTCH Ha 5. A 310 B03MO>KHO, JIHlllb CCJIH Xo=O HJIH xo=5.
HTaK, MbI ,lJ.OKa3aJIH CJIe.l{YIOInHH npH3HaK ,lJ.eJIHMOCTH Ha 5.
lJUC/lO aeAUmcR Ha 5 8 moM U mO/lbKO 8 moM CAY'lae, KOeaa eeo aeCRmUIl-
HaJI. sanUCb KOH'laemcR lfUljJpaMU 0 U/lU 5.
292

3) LIT06bI BbiBeCTH n p H 3 H a K )!. e .1 HMO C T H H a 4, 3aMe-
THM, qTO 100=4.25, H nOToMY 100 )!.eJIHTCH Ha 4. Ho Tor)!.a )!.eJI5ITCH
Ha 4 H LlHCJIa 1000, 10 000 H T. )!.., T. e. LlHCJIa BH)!.a lOn, n2. nYCTb
)!.ec5lTHLlHa51 3anHCb LlHCJIa X HMeeT BH)!.:
X=X n Ion+ ",+x2I02+x l lO+xo.
Tor)!.3 no )!.oKa3aHHoMY CJIaraeMOe Z=X n .lO n +... +X2 .10 2 )!.eJIHTC5I H8
4. 3HaLlHT, x )!.eJIHTCH Ha 4 B TOM H TOJIbKO TOM CJIYLlae, Kor)!.a Xl .10+
+Xo )!.eJIHTC5I Ha 4.
P,/lfl moeo 'lm06bl 'lUC/lO X aeAU/lOCb Ha 4, He06xoiJuMO U aOCmamOtlHO,
ttm06bl Ha 4 oeAUAOCb 'lUCAO, 06pCl308aHHoe nocAeaHUMU a8YMfl l{uljJpaMu
oeCflmuliHot'i 3anucu amoeo IiUCAa.
HanpHMep, 76 832 )!.eJIHTCH Ha 4, nOTOMY qTO 32 )!.eJIHTC5I Ha 4.
A 89 062 He )!.eJIHTC5I Ha 4, nOTOMY LITO 62 He )!.eJIHTCH Ha 4.
4) TOLIHO TaK )Ke, KaK npH3HaK )!.eJIHMOCTH Ha 4, BbIBO)!.HTC5I n P H-
3 H a K )!. e JI HMO C T H H a 25. TaK KaK 100 : 25, TO H 1000, 10 000
H T. )!.. )!.eJI5ITCH Ha 25. 3HaLlHT, LlHCJIO X )!.eJIHTCH Ha 25 B TOM H TO.bKO
TOM CJIYLlae, Kor,lJ.a LlHCJIO Xl .1O+xo ;l\eJIHTC5I Ha 25. A 3TO MOmeT 6bITb,
JIlIlllb eCJIH xI'IO+xo paBHo 0, 25, 50 HJIH 75.
I1TaK, npH3HaK ,lJ.eJIHMOCTH Ha 25 LlHTaeTC5I cJIe)!.YIOw.HM 06pa30M.
lJUC/lO X Oe/lUmCfl Ha 258 mOM U mO,1bKO 8 mOM C/lYliae, Koeoa eeO i}('Cfl-
mUttHafl 3anUCb 3aKaHliu8aemcfl Au60 Ha 00, Au60 Ha 25, Au60 Ha 50,
/lu60 Ha 75.
Hanpmvlep, 74 825 )!.eJIHTC5I Ha 25, a 92 654 He )!.eJIHTCH Ha 25.
5) BbIBe)!.eM Tenepb n p H 3 H a K H )!. e JI HMO C T H H a 3 H H a
9. .LJ.JI5I 3Toro )!.OKa)KeM CHaLlaJIa, LITO 8ce IiUCAa 8uoa IO n -I ae/lflmCfl Ha 9.
B caMOM )!.eJIe, )!.eC5ITHLlHaH 3anHCb TaKHX LlHCeJI HMeeT BH)!.:
IO n -I=9 .lO n - I +...+9 .1O+9=9(lon-I+...+ 10+ I)
(HanpHMep, 10 4 -I=9.1111). OTclO)!.a CJIe)!.YeT, LITO IO n -I )!.eJIIfTCH
Ha 9.
nOCKOJIbKY 9: 3, TO no CBOHCTBY 7) OTHOllleHHH )!.eJIHMOCTH (I on-I): 3.
LIHCJIO
X=X n .lO n +...+x..10 2 +xI'IO+xo
MomHO 3anHcaTb B BH)!.e:
x=[x n (lon-l)+. ..+ x2(I02-I)+xI(lO-I))+(x n + .,,+X2+XI+ Xo).
TaK KaK Ka)K)J.oe H3 qHCeJI lOn-I, "', 10-1 )!.eJIHTC5I Ha 3, TO CYM-
Ma x n (lO n -I)+",+xI(10-1) )!.eJIHTCH Ha 3. n03TOMY LlHCJIO X )!.eJIHTCH
Ha 3 B TOM H TOJlbKO TOM cJlyqae, KOr)!.a CYMMa Xn+",+x2+xd-x,
)!.eJIHTC5I Ha 3. 3Ty CYMMY Ha3bIBalOT CYMMOii l{Ul/Jp aaHHoeO IiUC/la (KaK
Mbi yme rOBOpHJlH, 3TO Ha3BaHHe He COBCeM TOLlHO, TaK KaK IJ.1f<PPbI -
3TO He LlHCJla, a 3anHCb LlHCeJI).
I1TaK, npH3HaK )!.eJIHMOCTH Ha 3 LlHTaeTCH TaK:
lJUCAO X ae/lUmcR Ha 3 8 mOM U moAbKO 8 mOM CAYliae, K02aa cyMMa
l{uljJp eeo oecfimuttHoii 3anucu Oe/lumcR Ha 3.
HanpHMep, LlHCJlO 65124 ,lJ.eJIHTCH Ha 3, TaK KaK 6+5+1+2+4=18.
a 18 .n.e.1HTCH Ha 3.
293

n p H 3 H a K )!. e JI HMO C T II H a 9 BbIBO)!.HTC5I TOqHO TaK )Ke, ec-
Jill Y'1eCTb, qTO LlHCJIa IOn-I, ..., 10-1 )!.eJl5lTC5I He TOJIbKO Ha 3, HO H
na 9.
lfuc/lo x Oe/lUmCfl Ha 9 8 mOM U mO/lbKO 8 mOM CAY'lae, Kozoa CYMMa
4UcjJP eeo oecflmuliHoii 3anucu Oe/lumcfl Ha 9.
YupaiKueUBII
1. ,[{OK8JKHTe, 'ITO l(HCJlO ,l(eJIHTCJl H8 37 B TOM H TOJlbKO TOM CJlYl(8e, KOr,l(8 cyMM8
'IHCeJl, nOJlYl(8eMbiX p836HeHHeM ,l(eCJlTHl(HOH 38nHCH .9TOrO l(HCJl8 crrp8B8 HaJIeBO
no TPH, ,l(eJIHTCII H8 37 (nOCJIe,l(HJlJl rpynn8 MOJKeT COCTOIITb H3 ,l(ByX 1I.m!>p HJlH o,l(Hoii
UH<PPbl).
2. BblBe,l(HTe npH3H8K ,l(eJlHMOCTH H8 II (HcnOJlb3yHTe, l(TO 99=9. II).
3. I1cxo,l(1I H3 p8BeHcTB8 999=27.37, BblBe,l(HTe npH3H8KH ,l(eJIHMOCTH H8 27 H 37.
4. BblBe,l(HTe npH3H8KH ,l(eJlHMOCTH H8 8 H H8 125.
5. BMecTO 3Be3,l(Ol(eK nOCT8BbTe 1I.H<jJpbl T8K, l(To6b1 nOJlYl(HJlOCb l(HCJlO, ,l(eJIJll1l,eecII:
8) H8 5: 483*, 34* 0, *31; 6) H8 9: 179*, 54*0, 5*31; B) H8 3: 24*7, 1'1<6, *22;
r) H8 8: 257*4, 3*22, 4355*; ,l() H8 II: 2033*, 471*85.
3. npH3HaKH eJIHMOCTH B pyrHx n03HHOHHWX cHcTeMax CLlHCJIe-
UHH.
npHBe)!.eM HeKOTopble npH3HaKH )!.eJlHMOCTH B )!.PYfHX n03HIJ.HOHHbIX
CHCTeMax CLlHCJleHH5I. nYCTb OCHOBaHHe CHCTeMbl CLlHCJleHH5I paBHO p.
ECJIH P : a, TO BCe LlllCJla BH)!.a p2, ..., pn )!.eJl5lTC5I Ha a, a Tor)!.a H CYMMa
Xnpn+...+XIP )!.eJlllTC5I Ha a. n03TOMY eCAU p Oe/lUmCfl Ha au IiUC/lO x
UMeem 8 CUCmeMeC'iUC!leHUfl C OCH08aHueM p 3anUCb x=xnpn+",+XIP+
+Xo, mo x Oe/lUmCfl Ha a 8 mOM U mO/lbKO 8 mOM C/lY'lae, KOeoa Xo oe-
/lUmCfl Ha a.
HanpHMep, B )!.BeHa.u.lJ,aTepHLlHOH CHCTeMe C1JHCJleHH5I qHCJlO )!.eJIHTC5I
Ha 3 B TOM H TOJlbKO B TOM CJIyqae, Kor)!.a ero nOCJIe.u.H5I5I IJ.ll<ppa paBHa
0, 3, 6 HJlH 9. 31"0 BbiTeKaeT H3 Toro, LITO 12 .u.eJIHTC5I Ha 3.
.LJ.a)!.HM eIll.e npll3HaK )!.eJlHMOCTH LlHCJIa Ha p-I, r.u.e p - OCHOBaHHe
CHCTeMbi CqHCJieHH5I. 113 a.n:re6paHqecKoii cpopMYJlbI
pn-I = (p_I)(pn-l+pn-2+...+ 1)
BbiTeKaeT, qTO )!.JI5I J11060ro n qHCJIO pn-I )!.eJlllTC5I Ha p-l. LIHCJlO X=
=xnpn+",+XIP+XO MomHO 3anHcaTb cJle)!.YIOIlJ.HM 06pa30M:
X=[X n (pn_l)+",+Xl (p-l)]+ (xn+...+xd-x o ).
nepBa51 CYMMa Bcer)!.a .D.eJIHTC5I Ha p-I. n03TOMY IiUCAO x Oe/lUmCR Ha
p - 1 8 mOM U mOAbKO 8 mOM C/lYliae, Kocoa cyMMa l{Ucpp ezo p-uliHoii
sanucu iJe/lum.cR Ha p-I.
HanpHMep, IIHCJIO 7 214 8 .D.eJlHTC5I Ha 7. B caMOM )!.eJle, cYMMa IJ.H<pP
9Toro qHCJia paBHa 7+2+1+4=168=14, a 14 )!.eJlHTC5I Ha 7.
YnpaiKueUBII
1. BblBe,l(HTe npH3H8K ,ZJ.eJIHMOCTH H8 7 B l4-pHl(Hoii CHCTeMe Cl(HCJleHHII.
2. BbIBe,l(HTe npH3H8K ,l(eJIHMOCTH H8 36 B 12-pHl(HOH CHCTeMe Cl(HCJleHHII.
3. BblBe,l(HTe npH3H8K ,l(eJlHMOCTH H8 5 B l6-pHl(Hoii CHCTeMe C'IHCJleHHJl.
r;l

4. I(puHble H )J.eJlHTeJlH. ECJIH qHCJlO a )!.eJIHTCH Ha qHCJIo b, TO ro-
BOp5lT, LITO a KpamHO b. TaK KaK qHCJIO 0 )!.eJIHTCH Ha Bce qHCJIa TO OHO
KpaTHO J11060MY qHCJIY; B )!.aJlbHeHllleM Y)!.06HO, rOBopH 0 KpaTHX qHC-
Jla b, HMeTb B BH)!.Y JlHlllb HaTypaJlbHbIe KpaTHbIe 3TOro qHCJIa, T. e.
qHCJIa b, 2b, ..., nb, HalfMeHbllllfM H3 HHX HBJlHeTC5I qHCJIO b. Bce CBOM-
CTBa OTHOllleHHH )!.eJIHMOCTH, )!.OKa3aHHble B n. 1, MomHO C<tx>PMYJlHPO-
BaTb KaK cBoHCTBa KpaTHblx. HanpHMep, eC/lU IiUCAO a KpamHO IiUCAY b
U 'lUCAO b KpamHO IiUCAY c, mo a KpamHO C H T. )!..
B03bMeM )!.Ba HaTypaJlbHblx LlHCJIa a H b. LIHCJIO m Ha3bIBaeTCH 06-
U{UM KpatnHblM 3THX qHCeJI, eCJIH OHO KpaTHO H a, H b. O)!.HHM H3 06-
W.HX KpaTHbiX qHCeJI a H b 51BJl5IeTC5I HX npOH3Be)!.eHHe ab - OHO )!.eJIHT-
C5I H Ha a, H Ha b. MHomecTBo 06Ill.HX KpaTHbiX qHCeJI a H b HBJl5IeTCH
nepeceqeHHeM MHomecTBa KpaTHbiX qHCJIa a C MHOmeCTBOI\f .qHCeJI KpaT-
HblX b.
n p H M e p 1. MHomeCTBo A KpaTHbiX qHCJIa 4 COCTOHT H3 qHCeJI
4,8,12,16,20,24,28,32,36, ..., a MHO/KeCTBO B KpaTHbix LlHCJIa 6
COCTOHT H3 qHCeJl 6, 12, 18, 24, 30, 36, ... nepeceLleHHeM 3THX MHO-
mecrB HBJI5IeTC5I MHomecTBo
A nB={12; 24; 36; ...}.
Bce qHCJIa 3Toro MHomecTBa H TOJlbKO OHH KpaTHbi H 4, H6.
B pa306QaHHoM CeHqaC npHMepe Bce qHCJIa H3 A n B OKa3aJIHCb
KpaTHbi qHCJIY 12 - HaHMeHbweMY H3 qHCeJI 3TOro MHomecTBa. Mbi
)!.OKameM CeHqaC, qTO TaK me OOcTOHT )!.eJIO )!.J151 JII06bIX )!.BYX qHCeJI
a H b.
HaHMeHblllee H3 06UJ,HX KpaTHbiX LlHCeJI a 11 b Ha3bIBaeTCH HaU,lteHb-
UlUM 06U{UM KpamHblM 3THX qHCeJI. Ero 0603HaqalOT HOK (a, b) HJIH,
KopoLle, K (a, b) (B rJl. VIII BMecTO K (a, b) nHCaJlOCb aKb). HanpH-
Mep, K(4, 6) =12.
!I.oKameM CJIe)!.YIOUJ,ee ytBep)K,lJ.eHHe:
Jl106oe 06U{ee KpamHoe a8YX liuce/l a U b ae/lUmCfl, Ha ux HaUMeHbUlee
06U{ee KpamHoe.
B caMOM )!.eJIe, nycTb m KpaTHo H a, H b H nYCTb K (a, b)=k. Pa3)!.e-
JIHM qHCJIO m Ha k C OCTaTKOM: m=kq+r. HaM Ha)!.o )!.oKa3aTb, qTO OCTa-
TOK r paBeH HYJIIO. .LJ.J1H 3Toro 3aMeTHM, qTO H m, H k )!.eJIHTC5I Ha a, a
nOTOMY r=m-kq Tome )!.eJIHTCH Ha a. TOqHO TaK me m Ii k ,lJ.eJlHTCH Ha b,
a nOTOMY H r )!.eJIHTCH Ha b. 3HaqHT, r )!.eJIHTCH H Ha a, H Ha b. ECJIH 6b1 r
6blJIO OTJlHqHO OT HYJlH, OHO 6bIJlO 6bI 06mHM KpaTHblM a H b H nOTOMY
6b1J1o 6bI He MeHbllle k (k - HaHMeHblllee 06mee KpaTHoe a H b), T. e.
rk. Ho 3Toro He MO/KeT 6bITb, TaK KaK OCTaToK MeHbwe ,lJ.eJlHTeJlH.
3HaqHT, r He MO/KeT 6mb OTJlHqHbIM OT HYJIH. I1TaK, r=O. 3HaqHT,
m=kq, T. e. m ,lJ.eJlHTCH Ha k.
HaM nOHa)!.06HTCH B ,lJ.aJIbHeHllleM em.e CJIe,lJ.}'IOUJ,ee CBOHCTBO:
ECAU K (a, b)=k, mo OM /l106020 HamYPaJlbH020 IiUCAa C 8epHO pa-
8eHcmeo K (ac, bc)=kc.
B caMOM .l{eJle, TaK KaK k .l{eJIHTCH Ha a, TO kc )!.eJlHTCH Ha ac. TOqHO
TaK me )!.OKa3bIBaeM, qTO kc ,lJ.eJIHTCH Ha be. 3HaqHT, kc - 06w,ee KpaT-
Hoe ac H bc. ,lJ.oKa/KeM, LITO OHO HBJIHeTCH HaHMeHbWHM 06IllHM KpaT-
295

HblM 3THX qHCeJ1. npe)!.n0J10)KHM, qTO [<kc H [ )!.eJ1HTC5I Ha ac H Ha bc.
Tor)!.a [ : c<kc : c=k, npHqeM [ : c )!.eJIHTC5I H Ha a, H Ha b (CM. CBOH-
CIBO to n. I). 3T0 npoTHBOpeqHT TOMY, qTO k - HaHMeHbwee 06IIJ.ee
KpaTHoe qHCeJ1 a H b. 3HaqHT, kc=K (ac, bc).
06paTHbiM K OTHoweHHIO «LIHCJ10 a KpaTHo qHCJlY b» 51BJ151eTC5I OT-
HoweHHe «LIHCJlO b - )!.eJIHTeJIb qHCJla a». I1HblMH CJIOBaMH, b 51BJ151eTCSI
)J,eJIHTeJleM a B TOM H TOJlbKO B TOM CJIyqae, Kor)!.a a KpaTHO b, T. e. a )!.e-
JlUTCSI Ha b. JII060e qHCJIO 51BJ151eTC5I )!.eJIHTeJIeM qHCJIa O. n03TOMY MbI
6Y)J,eM paCCMaTpHBaTb B 3TOM nyHKTe J1HWb )!.eJIHTeJIH HaTypaJ1bHblX
LlUCeJI. ECJIH b 51BJI5IeTCSI )!.eJlHTeJ1eM a, TO nHwYT b la. HanpHMep, 3124,
TaK KaK 24 : 3.
Ka)K,lJ.oMY CBOHCTBY OTHOWeHH5I )!.eJlHMOCTH cooTBeTcTByeT CBOHCTBO
)J,eJlIfTeJleH. HanpHMep, TpaH3HTHBHOCTb OTHOWeHH5I )!.eJlHMOCTH 03Ha-
qaeT, qTO eC/lU a - oe/lUme/lb b, a b - Oe/lUme/lb c, mo a - oe/lUme/lb c.
.LJ.aJlee, Ka:JICOoe 'lUC/lO Jl8/lflemCfl oe/lUme/leM CaM020 ce6Jl, a 'lUCAO I -
ile;iUme/lb /l106020 'lUC/la.
ECJlH qHCJla a H b )!.eJlSlTC5I Ha qHCJIO C, TO C Ha3b1BaIOT 06U{UM oe/lU-
meA£M 3THX qHCeJI. LIT06b1 HaHTH 06Ill.He )!.eJlHTeJIH qHCeJl a H b, HMO
HaHTH nepeCeqeHHe MHomeCTBa )!.eJIHTeJIeH qHCJ1a a H MHomeCTBa )!.eJIIf-
TeJleH qHCJ1a b.
n p H M e p 2. HaH)!.eM 06Ill.He )!.eJ1HTeJIH qHCeJI 24 H 60. MHomeCTBa
A H B )!.eJIHTeJIeH qHCJIa 24 H qHCJla 60 TaKOBbl:
A={I; 2; 3; 4; 6; 8; 12; 24},
B={I; 2; 2; 4; 5; 6; 10; 12; 15; 20; 30; 60}.
nepeCeqeHHe 3THX MHomecTB COCTOHT H3 qHCeJl 1, 2, 3, 4, 6 H 12. 3T0
" eCTb 06Ill.He )!.eJIHTeJIH qHCeJI 24 H 60.
nYCTb HaTypaJlbHoe qHCJlO a )!.eJ1HTC5I Ha b. Tor)!.a Bce )!.eJIHTeJIU HaTY-
paJlbHOrO qHCJIa a He npeBOCXO)!.51T a, H nOTOMY MHomeCTBO )!.eJ1HTeJ1eH a
COCTOHT H3 KOHeqHOrO qHCJIa 3J1eMeHTOB. KOHeqHblM 51BJ151eTC5I H MHO-
meCTBO 06Ill.HX )!.eJIHTeJleH HaTypaJIbHblX qHCeJI a H b. nOTOMY B HeM
HMeeTCSI HaH60Jlbwee qHCJIO. 3T0 qHCJIO Ha3bIBalOT Hau60/lbUlUM 06U{uM
oe/lume/leM qHCeJI a H b H 0603HaqalOT HOp' (a, b) HJlH D (a, b). I1TaK,
HaH60JlbwHM 06Ill.HM .l{eJlHTeJIeM HaTypaJlbHblx qlfCeJI a H b Ha3bIBalOT
naH60Jlbwee qHCJIO, Ha KOTopoe )!.eJlHTCSI H a, H b. HanpHMep, Hau60Jlh-
lllUM H3 qHCeJI 1, 2, 3, 4, 6 H 12 51BJI5IeTCSI 12. 3HaqHT, D (24, 60)= 12.
LIHCJIO 1 51BJ1S1eTCSI 06Ill.HM )!.eJIHTeJIeM J1106b1X ,lJ.BYX HaTypaJlbHblx
1JHCeJI a H b. ECJIH Y 3THX qHCeJI HeT HHblX 06Ill.HX ,lJ.eJIHTeJIeH, TO
D (a, b)= 1. TaKHe ,naa qHCJIa Ha3b1BalOT 83aUMHO npOCmblMU. I1TaK,
qHCJIa a H b Ha3b1BaIOT B3aHMHO npOCTblMH, ecJIH HX HaH60JlbwHH 06Ill.HH
)J.eJlHTeJIh paaeH 1. HanpHMep, qHCJIa 12 H 35 He HMelOT 06Ill.IfX )!.eJIHTe-
J1eH, OTJlHqHblX OT e,lJ.HHHlI.bI. 3TH qHCJIa B3aHMHO npOCTbI.
YnpamueUBII
I. TpaH3HTHBHO JlH OTHOIUeHHe «LIHCJlO a - lleJIHTeJIb '1HCJla b»? CHMMeTpH'IHO
.1H 3m oTHoIUeHHe? Pe<j>JIeKcHBHO JlH OHO? .
2. OK8JKHre. 'ITO D (2n, 2n+2)=2 rrpH mOOoM HaTypaJIbHOM 3HatJeHHH n.
96

3. .uoKa)\<HTe, '1TO '1aCTlible OT .neJIeliliSi .nBYX '1IiCeJI lia HX liali60Jlblllliii o61ll,liii
)1<'.1HTeJlb B3aliMHO npocTbI.
. a a+b
4. .uoKa)\{IiTe, '1TO eCJlIi .np06b b' r.ne a < b, COKpaTHMa. TO Ii .npo6b b- a
lOKpaTliMa.
5. CBOHCTBa HaHMeHbwero OOIll.ero KpaTHoro H HaH6OJJbwero 06-
uJ,ero )l.eJIHTeJIH. B 3TOM nYHKTe Mbi ,lJ.OKameM HeKoTopble CBOHcTBa HaH-
MeHblllero 06w.ero KpaTHoro H HaH60JIblllero 06Ill.ero ,lJ.eJIHTe.rI5I.
1) ECAU IiUCAO C Jl6/lJlemCfl 06W,UM aeAumeAeM HamypaAbHblx 'lUceA

au b (T. e. eCJIH a=a1c, b=b1c), mo IiUCAO 1= c- 06w,ee KpamHoe IiU-
ceA a U b.
HaM Ha,lJ.O ,lJ.OKa3aTb, LITO 1,lJ.eJlHTCH Ha a H Ha b. 3aMeTHM, LITO 1=
= a1cb 1 c = a1b1c=b 1 (a1c)=bta, H nOTOMY 1,lJ.eJIHTCH Ha a. C ,npyroH CTO-
c
pOHbI, 1=al (b1c)=a1b, H llOTOMY 1 ,lJ.eJIHTCH Ha b. 3HallHT, 1 - 06w.ee
KpaTHoe lIHCeJI a H b.
HanpHMep, 4 - 06W.HH ,lJ.eJIHTeJlb lIHCeJI 24 H 60, 24=4.6, 60=
24.60
=4.15. n03TOMY lIHCJIO 1=-r=4.6.15=360 KpaTHo H 24, H 60.
ab
2) lJUCAO d=T' eae k=K (a, b) Jl6/lJlemCR Hau60AbUlUM 06w,UM aeAU-
meAeM liuceA a u b.
B caMOM ,lJ.eJIe, lIHCJIO k ,lJ.eJIHTCH Ha b. Ho TOr,lJ.a ak ,lJ.eJIHTC5I Ha ab.
A TaK KaK d= a; , TO ab=dk. 3HallHT, ak ,lJ.eJIHTCH Ha dk, a Tor)!.a a ,lJ.e-
JIHTCH Ha d. TOllHO TaK me ,lJ.OKa3bIBaeTCH, LITO b ,lJ.eJIHTC5I Ha d. 3HallHT,
d - 06Ill.HH ,lJ.eJIHTeJIb lIHCeJI a H b.
Tenepb ,lJ.OKameM, LITO d - HaH60JIblllHH 06W.HH ,lJ.eJIHTeJIb lIHCeJI
a H b. npe,lJ.nOJIOmHM, LITO cyw.ecrBYeT 06W.HH ,lJ.eJIHTeJIb C lIHCeJI a H b,
KOTOpblH 60JIbllle d. Tor,lJ.a no YTBep)K,lJ.eHHIO 1) 1=P; HBJI5IeTCH 06W.HM
ab 
KpaTHbiM lIHCeJI a H b. Ho c>d, H llOTOMY 1=7 <([=k. TaKHM 06pa-
30M, 06w.ee KpaTHoe 1 lIHCeJI a H b OKa3aJIOCb MeHbllle HX HaHMeHblllero
06w.ero KpaTHoro k, LITO HeB03MomHO. 3HallHT, C,lJ.eJlaHHoe npe,lJ.nOJIome-
IIHe HeBepHO H 06Ill.HX .D.eJlHTeJleH, 6oJIblllHX, lIeM d, Y lIHCeJI a H b Her,
d=D (a, b).
113 ,lJ.OKa3aHHoro BbiTeKalOT CJIe,lJ.YIOw.He YTBep)I{)J.eHHH:
2') flpoU38eOeHUt! Hau6oAbUleZO 06w,ezo aeAumeMl U HaUMeHbUle20
06w,ezo KpamHQ20 a8YX HamypaJlbHblX liuceA pa8HO npOU38eaeHUlO [Jmux
liuceA.
ab
D(a, b).K(a, b)=-rk=ab.
B lIaCTHOCTH, ecJIH D (a, b)= 1, TO K (a, b)=ab.
2") HaUMeHbUlee 06w,ee KpamH.oe 08Yx 83aUMHO npocmblx HamypaAb-
HblX liuceA pa8H.O npOU38eaeH.UIO amux liuceA.
3) Hau60AbUlUii 06w,uii aeAumeAb d HamypaAbHblx liuce,1 a U b ae-
AUmcR H.a AI060ii 06w,uii aeAumeAb c amux liuceA.
297

B CaMOM .u.eJle, nYCTb a: c, b: c. Tor.u.a no YTBepm)!.eHHIO 1) a: HB-
b 
JIHeTCH OOIll.HM KpaTHblM qHCeJl a H H nOTOMY )!.eJlHTCH Ha k =([.
ab ab
I1TaK, c={f m. Ho Tor)!.a dab = abcm, oTKy.u.a CJle.u.yeT, LITO d=cm.
3Hall1fT, d )!.eJIHTCH Ha c.
4) EC/lu npoU38eoeHue ab Hamypa/lbHblX liuce/l a U b Oe/lumcfl Ha Ha-
mypaJlbHoe IiUC/lO m U a 83aUMHO npocmo C m, mo b Oe/lumcfl Ha m.
B caMOM .u.eJIe, lIHCJIO ab .u.eJIHTCH Ha a H Ha m H n03ToMY HBJlHeTCH
06Ill.HM KpaTHblM a H m. 3HaQHT, OHO .u.eJIHTC5I Ha HaHMeHbillee 06Ill.ee
KpaTHoe k=K (a, m) 3THX lIHCeJI. nOCKOJlbKY a If m B3aHMHO npOCTbI, TO
k=am. TaKHM 06pa30M, ab )!.eJIHTCH Ha am. Ho Tor)!.a b )!.eJIHTCH Ha m
(CBOHCTBO 10, n. 1).
5) EC/lu Hamypa/lbHOe IiUC/lO a ae/lumcfl Ha Ka3ICaoe U3 83aUMHO npo-
cmblX 'luce/l b u c, mo OHO ae/lUmCfl U IW ux npOU38eaeHUe bc.
B caMOM .u.eJle, H3 Toro, LITO a )!.eJIIfTCH Ha b If a )!.eJIIfTCH Ha C, CJle.u.y-
eT, LITO a HBJIHeTCH 06Ill.HM KpaTHblM b If C. n03TOMY a )!.eJIHTCH Ha K (b,c).
Ho lIHCJla b H C B3aHMHO npOCTbI, n03TOMY K (b, c)=bc (CM. ytBep)K,lte-
HHe 2"). I1TaK, a .u.eJlHTCH Ha bc.
113 cBoHCTBa 5) BbiTeKalOT npli3HaKH )!.eJIHMOCTH Ha lIJiCJla, HBJlHIO-
UJ.HeCH npoH3Be.u.eHHHMH .u.BYX B3aHMHO npoCTblx lIHCeJI. HanpHMep,
a/lfl moeo 'lm06bl Hamypa/lbHoe IiUC/lO x ae/lU/lOCb Ha 6, Heo6xoiJUMo U 00-
cmamOIiHO, Iim06bl OHO oe/lU/lOCb Ha 2 u Ha 3.
B caMOM .u.eJle, eCJIH x .u.eJIHTCH H8 6, TO OHO .u.eJlHTCH Ha 2 H Ha 3, TaK
KaK 2 H 3-.u.eJlHTeJIH lIHCJIa 6. 06{>aTHo, nYCTb x .u.eJIHTCH H Ha 2, H Ha
3. TaK KaK lIHCJla 2 H 3 B3aHMHO npoCTbI, no YTBep}K)l.eHHIO 5) lIliCJIO x
.l{eJIHTCH Ha HX npoH3Be)!.eHHe, T. e. Ha 6.
TOllHO TaK /Ke .u.OKa3bIBaIOTCH YTBep)!{,lJ.eHHH:
1) J1./lfl mozo 'lm06bl HamypaJlbHoe IiUC/lO x ae/lU/lOCb Ha 12, He06xoiJUAto
u aOCmamO'lHO, lim06bl OHO ae/lUAOCb Ha 3 u Ha 4.
2) J1./lfl moeo Iim06bl HamYPaJlbHoe IiUC/lO x ae/lU/lOCb Ha 15, He06xoiJuMo
u aOcmamoliHo, Iim06bl OHO ae/lU/lOCb Ha 3 U Ha 5.
HanpHMep, QHCJlO 975 .u.eJIHTCH Ha 15. B caMOM .u.eJIe, nOCJIe.u.HHH
UHa ero )!.ecHTHlIHOH 3anHCIi paBHa 5, H nOTOM)' OHO )!.eJIHTCH Ha 5.
C )(PyroH: CTOpOHbl, 9+7+5=21 .u.eJIHTCH Ha 3. 3HallHT, 975 )!.eJlHTCH
Ii Ha 5, H Ha 3, a nOTOMY )!.eJIHTCH Ha 15.
YnpamBeUBII
1. QtOPMYJlHpyihe rrpH3HaKH .l\eJIHMOCTH: a) Ha 18; 6) Ha 45; B) Ha 75; r) Ha 36.
2. LI.oKaJKHTe, '1TO ecJIH '1HCJla a H b B3aHMHO npoCTbl, TO '1Hc.I!a as H tJS TOJKe B3a-
HMHO rrpocTbI.
3. LI.oKaJKHTe, qTO eCJIH a H b - B3aHMHO npoCTblt '1HCJla, TO
D (ac, b)=D (c, b).
4. .lloKaJKHTe, '1TO ecJIH D(a, b)=I, TO D(, a+b)=l.
5. LI.oKaJKHTe, qTO ecJlH D (a, b)= 1 H a>b, TO D (, a-b)= I.
6. .lloKaJKHTe, '1TO ecJIH n B3aHIIHO rrpocro C 6, TO nB_I AMRTCJI Ha 24.
2t8'

 2. llPOCTbIE qllCJIA II IIX cBoRcTBA
1. npOCTble H COCTaBHble lJHCJIa. Kam)!.oe HaTypaJIbHOe LlHCJlO
a HMeeT no KpaHHeH Mepe )!.Ba )!.eJIHTeJI5I: LlHCJIO I H caMO LlHCJIO a. LIHC-
JIa, 60JIbillHe e)!.HHHlJ,bl H HMelOw.He pOBHO )!.Ba )!.e.'1HTeJI5I, Ha3blBalOTC5I
npOCmblMU 'lUC;WMU. HanpHMep, 5 - npocToe LlHCJIO, TaK KaK ero )!.e-
JlHTeJI5IMH 51BJI5IIOTC5I JIHlllb I H 5.
LIHCJlO 12 He HBJI5IeTC5I npOCTbIM, TaK KaK, KpOMe 1 H 12, OHO HMeeT
ew.e )!.eJIHTeJIH 2,3,4,6. TaKHe LlHCJIa Ha3blBalOTC5I COCTaBHbIMH.
HaTypaJIbHoe LlHCJlO a Ha3bIBaeTC5I COCmaBHbIM, eCJIH OHO HJ\leeT 60-
JIee )!.BYX pa3JIHLlHbiX )!.eJIHTeJIeH. LIHCJIO I HMeeT JIHWb O)!.HH )!.eJIHTeJIb-
caMO 3TO 1JHCJIO. Ero He OTHOC5IT HH K npOCTbIM, HH K COCTaBHblM LlHCJlaM.
LIHCJIO 0 HMeeT oecKoHeLlHo MHOro )!.eJIHTeJIen. Ero TaKme He OTHOCHT
HH K npOCTbIM, HH K COCTaBHblM LlHCJlaM.
TaKHM oopa30M, MHomecTBo Zo lJ,eJIblX HeOTpHlJ.aTeJIbHbIX LlHCeJI
pa30HBaeTC5I Ha LleTbipe KJIaCca:
1) KJlaCC, co)!.epmaw.HH JIHWb LlHCJIO 0 (HMeeT oeCKOHeLlHO MHoro )!.e-
JI HTeJI eH);
2) KJIaCC, co)!.epmaw.HH JIHWb LlHCJlO 1 (HMeeT JIHWb O)!.HH )!.eJIHTeJlb);
3) KJIaCC npocTblX LlHCeJI (HMelOw.Hx pOBHO )!.Ba )!.eJIHTeJIH);
4) KJIaCC COCTaBHblX LlHCeJI (HMelOw.Hx oOJIee )!.BYX )!.emlTe.,eH, HO
OTJlHLlHblX OT 0).
.LJ.OKIDKeM HeKoTopbIe CBoHcTBa npOCTblX LlHCeJI:
I) EC/lu npocmoe IiUC/lO p Oe/lltnlCfl Ha HeKomopoe Hamypa;lbHoe IiUC/lC
n, OmAUIiHoe om I, mo OHO coenaoaem C n.
B caMOM )!.eJIe, eCJIH Obi p He paBHHJIOCb n', TO OHO HMeJIO ObI TPH )!.e-
JlHTeJI5I: I, n H p, a Tor)!.a OHO He OblJIO Obi npocTblM.
2) EC/lu p U q - pCl3/lUIiHble npocmble IiUC/la, mo p He Oe/lUmCfl Ha q.
TaK KaK p - npOCToe LlHCJIO, TO OHO MomeT )!.eJIHTbC5I JIHWb Ha I H
Ha p. Ho q no yCJIOBHIO OTJIHLlHO OT p, a TaK KaK q npOCToe, TO OHO OT-
JIHLlHO H OT l. 3HaLlHT, q He 51BJI5IeTC5I )!.eJIHTeJIeM LlHCJIa p. HanpHMep,
LlHCJla 17 H 3 npocTble, H nOToMY 17 He )!.eJIHTCH Ha 3.
3) Ec.'lu Hamypa/lbHoe IiUC/lO a He oeAumCfl Ha npocmoe IiUC/lO p,
mo a U p e3aUMHO npocmbi.
B caMOM )!.eJIe, B03bMeM HaH60JIbIllHH OOW.HH )!.eJIHTeJIb d LlHCeJI a H p.
Tor)!.a p )!.eJIHTCH Ha d. Ho npOCToe 1JHCJIO p HMeeT JIHlllb )!.Ba )!.eJIHTeJI5I:
I H p. n03TOMY JIH60 d=p, JIH60 d= 1. ECJIH Obi d paBH5IJIOCb p, TO a )!.e.1JH-
JIOCb Obi Ha p BonpeKH YCJIOBHIO. 3Ha1JHT, OCTaeTC5I JlHlllb cJIYLlaii d=l,
a B 3TOM CJIYLlae a H p B3aHMHO npoCTbI.
4) ECAU npoU38eaeHUt! aeyx HamypaJlbHblX liuce;l a U b oe,Ulmcfl Ha
npocmoe IiUC/lO p, mo XOtnfl 6bl oiJHO U3 HUX ae/lUmCfl Ha p.
B caMOM )!.eJIe, npe.u.nOJlOmHM, LITO a He )!.eJIHTC5I Ha p. Tor)!.a no YT-
Bep)K,lJ.eHHIO 3) 1JHCJla a H p B3aHMHO npoCTbI. Ho no YTBepm)!.eHHIO 4)
n. 5 eCJIH ab )!.eJIHTC5I Ha p H a H p B3aHMHO npOCTbI, TO b )!.eJIHTCH
Hap.
n p H M e p l. I1MeeM 25.69=1725. LIHCJIO 1725 .u.eJIHTC5I Ha 3,
TaK KaK 1 + 7 + 2+ 5= 15 )!.eJIHTCH Ha 3. Ho 25 He ,neJIHTCH Ha 3. 3Ha-
LlIIT, 69 )!.eJIHTC5I Ha 3 (3TO JlerKO npOBepHTb Henocpe.n.c'tBeHHO).
299

38MeTIfM, LITO 8H8JIOrlfLlHOe YTBepm)!.eHlfe )!.JIH COCT8BHbiX LlIfCeJI
yme HeBepHO: npOlf3Be)!.eHlfe )!.BYX LlIfCeJl MOmeT .l{eJIIfTbCH H8 COCT8BHOe
'lllCJIO, XOTH Hlf O)!.IfH 1f3 ero MHOmlfTeJleH H8 3TO LlIfCJIO He )!.eJIIfTCH.
HanplfMep, 8.9=72 )!.eJIIfTCH H8 12, XOTH Hlf 8, Hlf 9 Ha 12 He )!.eJIIfTCH.
5) ECAU HamypaJlbH
 'lUCAO 60AbUle 1, mo OHO UMeem XOtnfl 6bl oault
npocmoii aeAumeAb.
npoBe)!.eM )!.OK838TeJIbCTBO OT npoTIfBHoro. npe)!.nOJIOmIfM, LITO YT-
Bepm)!.eHlfe HeBepHo, T. e. CYIll.eCTBYIOT H8TypaJlbHbie LlIfCJl8, KOTopbIe
60Jlbllle I, HO He IfMelOT Hlf O)!.Horo npOCToro )!.eJlIfTeJIH. Tor)!.8 MHome-
CTao A T8KlfX LlIfCeJI co)!.epm8J10 6b1 H8lfMeHbWee LlIfCJlO. 0603HaLlIfM
STO LlIfCJlO Llepe3 a. nOCKOJlbKY Bce LllfCJl8 B MHomeCTBe A 60Jlbllle 1, TO
If a 60JlbWe I, 8 Tor)!.8 OHO MOmeT 6blTb JlIfOO npOCTbIM, JlIf60 COCT8BHblM.
Ho npOCTblM LlIfCJIO a He MomeT 6blTb, T8K K8K OHO nplfH8)!.JlemlfT MHO-
iKecTBY A, 8 Hlf O)!.HO 1f3 LlIfCeJl ,noro MHomecTB8 He If MeeT npOCTblX )!.eJIlf-
TeJleH If nOTOMY He npocTo. He MomeT OblTb OHO If COCT8BHblM. B caMOM
)!.eJIe, eCJllI ObI a 6blJlO COCT8BHbiM LlIfCJlOM, a IfMeJIO Obi H8TYP8J1bHbIH JI.e-
JIIfTeJlb b, OTJlIfLlHblH OT 1 If OT a. Tor)!.a 3TOT )!.eJlIfTeJIb ObIJI ObI MeHbwe a
If nOTOMY He nplfH8)!.JlemaJi Obi MHomeCTBY A (Be)!.b a-H8lfMeHbWee LlIfC-
110 B A). 3H8L1lfT, Y b eCTb npocToH )!.eJlliTeJlb p, 8 Tor)!.8 If a )!.eJlIfJlOCb Obi
H8 npOCToe LlIfCJIO p BonpeKIf npe)!.nOJlOmeHIfIO. Mbl nplfWJlIf K npOTIfBO-
peLlIfIO. 3H8L1IfT, Hawe npe)!.nOJlOmeHlfe HeBepHO If LlIfCeJI, OOJlbWIfX I,
HO He IfMelOIll.IfX npOCTbIX )!.eJlIfTeJIeH, He CYIIJ.eCTByeT.
6) H aUMeHbUluii npocmoii aeAumeAb COCtna8H020 IiUCAa a He npe80C-
xooum Va.
TaK K8K a - COCTaBHoe LlIfCJlO, 8 P - ero H8lfMeHbWlfH npocToii
)!.eJlIfTeJlb, TO a=pb. nplf 3TOM p
b, T8K K8K IfH8L1e npOCTOH JI.eJlIfTeJlb
LllfCJI8 b OblJI ObI MeHbwe, LleM p, 8 Tor)!.8 a IfMeJIO ObI npOCTbIe )!.eJIlfTeJlIf,
MeHbllllfe LleM p. YMHomlfM JleBYIO If np8BYIO Ll8CTIf Hep8BeHCTB8 p
b
H8 p. Mbi nOJIYLlIfM, LITO p2
pb=a. n03TOMY p2
a, T. e. p
Va.
113 YTBepm)!.eHIfH 6) CJIe)!.yeT, LITO ecJlIf LlIfCJIO a He )!.eJIIfTCH Hlf Ha
O)!.HO npOCToe LlIfCJ10, He npeBOCXO)!.HIll.ee Va, TO y Hero COBceM HeT
npOCTblX MHOmIfTeJIeH, MeHbWIfX 3Toro LlIfCJI8, T. e. 3TO LlIfCJIO npOCToe.
n p If M e p 2. .LJ.oK8meM, LITO LlIfCJIO 137 npoCToe. .LJ.JlH 3Toro 1f3BJle-
r-
LleM 1f3 Hero KB8JI.p8THblH KopeHb: II<V 137<12. n03TOMY eCJIlf 137
He )!.eJlIfTCH H8 npOCTbIe LlIfCJl8, MeHbWlfe 12, TO OHO HBJlHeTCH npOCTblM.
Ho MHomeCTBO npOCTbIX LlIfCeJI, MeHbWIfX 12, COCTOIfT 1f3 LlIfCeJl 2, 3,
5, 7 H 11. Hlf H8 O)!.HO 1f3 HHX 137 He )!.eJlHTCH. 3H8L1HT, 137 - npocTOe
LlHCJIO.


YnpamueUBII
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HHe HeBepHO, T. e. qTO COCTaBHbIX qHCeJI, KOTopbIe He pa3JIaralOTCH 0
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npe.n.nOJIO>KeHHIO a - HaHMeHblllee H3 IIHCeJI, HMelOmHX pa3JIHIIHbIe
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paJIbHbIX IIHCeJI, HMelOllI,HX pa3JIHIIHble pa3JIO>KeHHH Ha MHO>KHTeJIH, He
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Pa3JIO>KeHHe BH.n.a a=pt.. .pn, r.n.e Pl<P2<" .<Pn, Ha3bIBalOT
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60JIbllIee 1, HMeeT e.n.HHCTBeHHoe KaHOHHIIecKoe pa3JIO>KeHHe Ha npo-
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TO .lI:Jl5I JlI06oro k (lkn) .lI:OJI}/{Hbi BbInOJIH5ITbC5I HepaBeHCTB3 Yhar.
H ,\,,,It.
Tenepb JIerKO HaihH HaH60JIbIIIHH 06IUHH .lI:eJIHTeJlb D (a, b) '1HCeJI
a H b. Mb1 3HaeM, qTO D (a, b) 51BJI5IeTC5I H3H6oJIbIIIIIM H3 'iHCeJI, H3 KOTO-
pble .!I:eJlHTC5I H a, H b. 3Ha'iHT, .!I:JI5I Ka}/{.!I:OrO H3 npocTblX 'iUCeJI Pit no-
K33aTeJIb B pa3JIO}/{eHHH D (a, b) .lI:OJI}/{eH 6blTb HaH60JIblllHM H3 B03MO}/{-
Hb1X (T. e. Y.lI:OBJIeTBOP5l1OlUHX HepaBeHcTBaM ,\,,,a,, H ,\,,,,,). .:nUM
nOKa3aTeJleM 51BJI5IeTC5I MeHbIIIee H3 'iHCeJI a" H h' HanpHMep, ecJlU
a,,=4, ,,=3, TO caMblM 60JIbIIIHM qHCJlOM, Y.lI:OBJIeTBOp5l1Oll.I,HM HepaBeH-
CTBaM '\' 1t4 H '\' ,,3, 51 BJI 51 eTC 51 3, T. e. MeHbIIIee H3 'iHCeJI 4 H 3. MeHb-
IIIee H3 'iHCeJI a H P 0603Haq31OT min (a, ) (B cJIyqae KOr.ll:a a=p, nOJIa-
raroT min (a, p)=a). I1TaK, Mbl.ll:OKa33JIH, qTO eCAU
_ a, an b - 13, f3n
a-pI...Pn' -PI...Pn,
mo
D (a, b)=p¥' .. .pX n ,
eoe OAR 6GeX k, lkn, UMeeM '\'h=min (ah' Ph).
TO'iHO TaK }/{e .lI:OK33bIBaeTC5I, qTO HaHMeHbIIIee o6IUee KpaTHoe
K (a, b) 'iuceJI a H b BbIpa)KaeTC5I cpopMYJIOH K (a, b)=p1 . . .pn, r.lI.e
.l(JISI Bcex k, lkn, HMeeM: 6"=max(a,,, Ph).
Ynp8HCoeOOH
I. .1J.allbl qHCJIa a=2 5 . 3'. &\.11, b=22.3 6 .5 3 .P. Hai\JJ:HTe D(a, b) H K(a, b).
2. CKOJIbKO pa3JIH'IHblX 1I.eJIHTeJIeH HMeeT 'IHCJlo a=pa (BKJlIO'IaSi 1I.eJIHreJiH 1 H
pa)?
3. CKOJIbKO pa3J1HqllbiX .1\eJIHTeJIeH HMeeT qHCJIO a=p'. p,?
4. CKOJIbKO pa3JIHqllbiX .1\eJIHTeJJei! HMeeT 'IHCJIo: a) 216, 6) 10 125?
5. HaH1I.HTe a H b, eCJIH H3BecTHO, 'ITO:
a) a: b=11 : 13, D(a, b)=5;
305

6) D(a, b)=5, K (a, b)= 165;
B) D (a, b)=7, ab=294;
r) K (a, b)=75, 00=375;
.l\) a: b=7; 8, K(a, b)=224.
6. B TPH IllKOJIbHblX KHOCKa oTnpaBHJIH no O.l\HHaKoBoMY qHCJIy rerpa.l\eii. .1J.JIlI
O.l\HOH IllKOJIbi OrnpaBHJIH TerpMH naqKaMH, no 150 IllTYK B Ka)l{J!oH naqKe, .l\JIH Bro-
poi! - no 100 IllTYK, a .l\JIH TperbeH - no 200 IllTYK B Ka)l{.l\oi! naqKe. CKOJIbKO TeT-
pMeH OrnpaBHJIH Ka)l{.1\OH IllKOJIe, eCJIH qHCJIO Terpa.1\eH, OrnpaBJIeHHblX Ka)l{.l\OH IllKOJIe,
MCHbllle 20oo?
5. AnropHTM EBKJIH)I,a. OTbicKaHHe HaH6oJIblIlerO o6ll.1,ero ,.QeJIHTeJISI
)l,BYX HaTypaJIbHbIX 'lHCeJI no HX KaHOHH'lecKOMY pa3JIOmeHHIO Tpe6YeT
npe,.QBapHTeJIbHOrO pa3JIOmeHHSI 3THX 'lHCeJI Ha npocTbIe MHOmHTeJIH.
3TO HeCJIOmHO C,.QeJIaTb, ecJIH 'lHCJIa HeBeJIHKH, HO ,.QJISI MHOr03Ha'lHblX
'lHCeJI HaihH HX KaHOHH'lecKHe pa3JIOmeHHSI 6bIBaeT TPY.lI.HO. CYll.I,eCTBY-
eT cnoco6 OTbICKaHH5I HaH6oJIblIlerO 06ll.1,ero .lI.eJIHTeJIH, Tpe6YIOll.I,HH JIHlIlb
BbInOJIHeHHH onepaQHH ,.QeJIeHHH C OCTaTKOM. 9TOT cnoco6 npe,.QJIO)KHJI EB-
K.1H,.Q, Hero Ha3bIBalOT aJleopum.MO.M E8KJlUaa. AJIrOpHTM EBK<lH.aa oc-
HOBaH Ha CJIe,.QYlOll.I,HX Tpex YTBepm.ll.eHHSlx:
1) ECJlu a aeJlUmCfl H.a b, mo D(a, b)=b.
B caMOM ,.QeJIe, TaK KaK a : b H b : b, TO b - 06ll.1,HH ,.QeJIHTe.ilb 'lHCeJI
a H b. Ho JII060H .lI.eJIHTeJIb 'lHCJIa b He npeBOCXO.ll.HT 3Toro 'lHCJIa. n03TO-
MY Bce 06ll.1,He ,.QeJIHTeJIH a H b He npeBOCXO,.QHT b. 3Ha'lHT, b - HaH6oJIb-
WHH 06ll.1,HH ,.QeJIHTeJIb 'lHCeJI a H b.
2) ECJlu a=bq+r, eae tlUCJla a, bur OmJlUtlH.bl om H.YJlfl, mo .MH.O:JICe-
C11l80 o6U{ux aeJlumeJleu a u b C08naaQRm C .MH.O:JICeCm80.M o6U{ux aeJlumeJleu
bur.
B caMOM .lI.eJIe, nycTb d - 06ll.1,HH ,.QeJIHTeJIb b H r. TaK KaK b H r
)l,eJISlTCH Ha d, TO H a=bq+r ,.QeJIHTCH Ha d (CM. CJIe,.QCTBHe K 9) n. 1).
3Ha'lHT, JII060H 06ll.1,HH ,.QeJIHTeJIb 'lHCeJI b H r SlBJISleTCSI 06ll.1,HM ,.QeJIHTe-
JIeM 'lHCeJI b H a. 06paTHo, eCJIH d - 06ll.1,HH .lI.eJIHTeJIb a H b, TO d SIB-
JISleTCH H .lI.eJIHTeJIeM r, TaK KaK r=a-bq. 3Ha'lHT, JII060H 06ll.1,HH ,.QeJIH-
TeJIb a H b HBJIHeTCSI 06ll.1,HM ,.QeJIHTeJIeM b H r. COBna.ll.eHHe MHomeCTB
06ll.1,HX ,.QeJIHTeJIeH ,.QJIH a H b, C O,.QHOH CTOpOHbI, H ,.QJIH b H r, C ,.QpyroH
CTOpOHbI, .lI.OKa3aHO.
3) ECJlu a=bq+r, eae a, b, r - OmJlUtlH.bl om H.YJlfl, mo D(a, b)=
=D (b, r).
B caMOM ,.QeJIe, no YTBepm,.QeHHIO 2) MHomeCTBO 06ll.1,HX .lI.eJIHTeJIeH
qHCeJI a H b COBna,.QaeT C MHomeCTBOM 06ll.1,HX .lI.eJIHTeJIeH 'lHCeJI b H r.
A TOr)l,a 06a MHomeCTBa HMelOT O,.QHH H TOT me HaH60JIblIlHH 3JIeMeHT.
3HatJHT, D(a, b)=D (b, r).
Tenepb MbI yme MomeM onHcaTb aJIrOpUTM EBKJIH.lI.a .lI.JISI HaXOm.ll.e-
HHSI HaH6oJIblIlerO o6ll.1,ero .lI.eJIHTeJISI 'lHCeJI a H b. TIYCTb ab. ECJIH a
AeJIUTCSI Ha b 6e3 OCTaTKa, TO no YTBepLK,ll.eHHIO 1) HMeeM D(a, b)=b.
DYCTb npH )l,eJIeHHH a Ha b nOJIYtJaeTcSI OCTaTOK r, a=bq+r. B 3TOM
CJlYtJae no YTBepLK,ll.eHHIO 3) HMeeM D(a, b)=D(b, r) H 3a.ll.a'la CBeJIaCb K
OTbiCKaHHIO HaH60J1blIlerO 06ll.1,ero .lI.eJIHTeJISI tJuceJI b H r. ECJIH b )I,lHT-
CSI Ha r, TO, KaK Mbi yme 3HaeM, D(b, r)=r, a TOr)l,a H D(a, b)=r. ECJIH
}Ke npH .lI.eJIeHHH b Ha r nOJIytJHTCSI OCTaTOK r10 TO b=rql+ri, H nOTOMY
D (a, b)=D (b, r)=D (r, rl)'
006

npO.n.OJI)f{aH OnHCaHHblH npoQecc, Mb1 6y.n.eM nOJIY'IaTb BCe MeHbllIHC
H MeHbllIHe OCTaTKH: '10 '2, ..., 'm. B KOHQe KOHQOB, Mb1 .lI.OMeM llo
OCTaTKa, Ha KaI'OpbIH npe.n.bl.n.Yll.I,HH OCTaTOK 6y.n.eT .n.eJIHTbCH. 3TOT HaH-
MeHbllIHH OTJIHtJHblH OT HYJIH OCTaTOK H 6y.n.eT HaH60JIbllIHM 06ll.1,HM .n.e-
JIHTe,rleM 'IHCM a H b.
n p H Me p. HaH.n.eM D (2585, 7975).
Mbi HMeeM 7975=3.2585+220. Tenepb pa3.n.eJIHM 2585 Ha 220:
2585=220.11 + 165.
lI.aJIee .n.eJIHM 220 Ha 165: 220= 165+55.
HaKoHeQ, .n.MHM 165 Ha 55: 165=55.3.
lI.eJIeHHe BbInOJIHeHO 6e3 OCTaTKa. 3HatJHT, HaHMeHbIlIH" OTJIHtJHbl"
OT HYJIH OCTaTOK paBeH 55, T. e. D (7975, 2585)=55.
nOJIb3YHCb aJIrop HTMOM EBKJIH.n.a, MO)f{HO .n.OKa3aTb cJIe.n.YlOll.I,ee
YTBeplK.ll.eHHe: OJlf/. Jl106blX 08YX HamypaJlbHblX If.uceJl a u b Haaoymcf/. ma-
/Cue HamYPaJlbHble If.UCJla x U y, If.mo D (a, b)=ax-by.
YnpamneUHII
I. HaHJ.(HTe c nOMolllblO aJIropHTMa EBKJIH!la JlaH60JIblllHH 06IllHH !lCJIHTeJIb
'IHCM: a) 846 H 246; 6) 1960 H 588; B) 15 283 H 10 013.
21 120
2. CoKpaTHTe !lp06b 30720 '
III 1234
3. npHBellHTe J.(P06H 21 120 H 30720 K OlIJlOMY 3HaMeHaTeJIIO.
4. HaHllHTe JlIIHMeHblllee 06111ee KpaTHoe qHCeJI: a) 846 H 246; 6) 1960 H 588.

r JI a B a XI
PAllOHAJIbHhIE II )];EMCTBllTEJlbHbIE QllCJ1A
 t. MHOiKECTBO nOJIOiKlITEJIbHbIX
PAD:lIOHAJIbHbIX 'IlICEJI
1. H3MepeHHe OTpe3KOB. bOJIblIlHHCTBO npHMeHeHHH MaTeMaTHKH
CBOJl,HTC5I K Jl,BYM OCHOBHbIM 3aJl,aqaM: noiJCf/.emy t/,uCJla 9Jle.MeH.m08 KOH.ett-
HOZO MHO{)ICeC11UJa U U3.MepeHUlO 8eJlut/,UH. DpH nepeqHCJIeHHH 3JIeMeHTOB
KOHeqHbIX MHO>KeCTB OTBeT BbIpa>KaeTC5I HaTypaJIbHbIM qHCJIOM: qeTbipe
ap6Y3a, BoceMb rpY30BHKOB, TPH KycKa MaTepHH. DpH 3TOM He 06pa-
m.aIOT BHHMaHHSI Ha TO, qTO MaCCbI ap6Y30B MorYT OKa3aTbCSI pa3JIHqHbI-
MH, qTO KYCKH MaTepHH MorYT HMeTb pa3HYlO Jl,JIHHY, a rpY30BHKH -
pa3HYlO rpY30nOJl,'beMHocTb. OJl,HaKO qT06bI Y3HaTb, XBaTHT JIH MaTe-
pHH B 3THX KYCKax Jl,JISI H3rOTOBJIeHHSI KOCTIOMOB Ha qeTbipeX qeJIOBeK,
HaJl,O U3.MepUf11b Jl,JIHHY Ka>K,ll.oro KYCKa. Bo06me, U3.MepeHUe 8eJlUt/,UH,
T. e. cpaBHeHHe 3THX BeJIHqHH C HeKOTopOH eiJuHUt{I!u U3.MepeHUJ/. (MeT-
pOM, KHJIOrpaMMOM H T. .ll.) H BbIpa>KeHHe pe3YJIbTaTa cpaBHeHHSI qHC-
JIOM, BCTpeqaeTCH nO'ITH B JII060M BHJI,e qeJIOBeqecKoH Jl,eSlTeJIbHOCTH.
ECJIH H3MepSleMYlO BeJIHqHHY MO>KHO pa3J1,eJIHTb Ha HecKOJIbKO qac-
TeA, «paBHbIx» (B TOM HJIH HHOM CMbICJIe) eJl,HHHUe H3MepeHHSI, TO pe3YJIb-
TaT H3MepeHHSI (HJIH, KaK rOBopSlT, .Mepa 8eJlUt/,UHbl) BbIpa>KaeTCSI Hary-
paJIbHbIM qHCJIOM. qame, O.llHaKO, eJl,HHHUa H3MepeHHSI He YKJIa.lU>IBaeT-
CSlueJIoe qHCJIO pa3 B H3MepSleMOH BeJIHqHHe. D03TOMY .llJISI BbIpIDKeHHSI
MepbI BeJIHqHHbI npHXO.llHTCSI paClIlHpHTb 3anac qHCeJI, BBeJl,SI qHCJIa,
OTJIHqHbIe OT HaTypaJIbHbIX.
B 3TOH r JIaBe MbI paCCMOTpHM pa3JIHqHbIe paClIlHpeHHSI MHO>KecTBa
HaTypaJIbHbIX qHCeJI, nOCTpOHB CHaqaJIa MHO>KeCTBO nOJIO>KHTeJIbHbIX
paUHOHaJIbHbIX qHCeJI Q+, nOTOM - MHO>KeCTBO nOJIO>KHTeJIbHbIX Jl,eH-
CTBHTeJIbHbIX qHCeJI R+ H, HaKOHeu, MHO>KecTBO Jl,eHCTBHTeJIbHbIX qHCM
R. DpH .9TOM MSI KIDK,!I,orO BHJI,a qHCM 6YJl,YT onpe.lleJIeHbI OIlepaUHH
CJIO>KeHHSI H YMHO>KeHHSI, OTpa>KalOmHe onpe.lleJIeHHbIe onepaUHH HM
H3MepSleMbIMH BMHqHHaMH H eJJ.HHHuaMH H3MepeHHSI. qT06bI BbISlC-
HHTb, KaKOBa .9Ta CBSl3b, paCCMOTpHM H3MepeHHe Jl,JIHH OTpe3KOB.
CKa>KeM, 'lTO ompeaoK a paa6um Ha Ompe3KU ai, a 2 , ..., an, (HJIH CO-
cmoum H3 .9THX OTpe3KOB), eCJTH OH SlBJISleTCSI HX 06'beJl,HHeHHeM, npHqeM
a HHKaKHe Jl,Ba 83 HHX He HMelOT 06-
I'  I 'I w.eH BHYTpeHHeH TOqKH (He HaJIe-
at 6, I oJ a't 05 raIOT JJ.pyr Ha Jl,pyra), XOTSI H MD-
PRC. 178 rYT HMeTb 06w.He KOHUbi (pHC. 178).
308

ECJIH OTpe30K a pa36HT Ha OTpe3KH all a2' .. ., an, TO ero Ha3bIBalOT
CY.M.MOU 3THX OTpe3KOB H nHlUYT:
n
a=a 1 +a 2 +.. .+a n HJIH a= alt.
k=1
BbI6epeM HeKOTopbIH OTpe30K e H Ha30BeM ero eOUliUIlHblM ompe3KOM
HJIH eOUliUltireU U3.MepeHUfl OAUHbt. ECJIH OTpOK a MOiKHO pa36HTb Ha n
OTpe3KOB, Ka)f{)];bIH H3 KOTOpbIX KOHrpY3HTeH e.D,HHHqHOMY OTpe3KY e,
TO CKa>KeM, qTO OTpe30K a KpameH OTpe3KY e, H Ha30BeM qHCJIO n .Mepou
HJIH 3HatteHUe.M O.IlUHbt OTpe3Ka a npu eOUHUIlHOM Ompe3Ke e. Mepy OTpe3-
Ka a npH e.ll.HHHqHOM OTpe3Ke e 0603HaqHM me(a). ECJIH e.D,HHHqHbIH OT-
pe30K e <pHKCHpOBaH, TO BMecTO me(a) 6Y.lI.eM nHcaTb m (a) H Ha3bIBaTb
3TO qHCJIO npOCTO OAUHOU (a He 3HaqeHHeM .lI.JIHHbI) Ompe3K.Cl. He06xO.ll.HMO
HMeTb B BH.lI.Y, qTO npH nepeXO.ll.e K .lI.PyroH e.ll.HHHll.e H3MepeHHH qHCJIO
m (a) H3MemleTCH, XOTH caM OTpe30K OCTaeTCH HeH3MeHHbIM.
ECJIH m (a)=n, TO 6Y.lI.eM nHcaTb a  n 'e, qTO 03HaqaeT: OTpe30K a
COCTOHT H3 n OTpe3KOB, KOHrpY3H1.'HbIX OTpe3KY e. OqeBH.lI.HO, qTO npH
3aMeHe OTpe3Ka e KOHrpY3HTHbIM eMY OTpe3KOM f Mepa He H3MeHHeTCH,
me(a)=mf(a) (npH H3MepeHHH KaKorO-HH6Y.ll.b OTpe3Ka .lI.BYMH pa3JIH4-
HbIMH JIHHeHKaMH nOJIyqaTCH O.ll.HHaKOBble pe3YJIbTaTbI, ecJIH, KOHeqHO,
o6e JIHHeHKH O.ll.HHaKOBO rpa.n.YHpOBaHbI). 06paTHO, eCJIH a  n.e H
a  n.f, TO e  f. Pa3YMeeTCH, ecJIH a "" n'e H a "" m'e, TO n=m - O.ll.HH
H TOT >Ke OTpe30K He MO>KeT HMeTb pa3JIHqHbIX Mep npH 3a.ll.aHHOH e.ll.H-
HHue H3MepeHHH.
Ka>K.lI.OMY OTpe3KY e COOTBeTCTByeT MHO>KecTBO 8 1) OTpe3KOB, KpaT-
HbIX e. Ka)f{)];OMY H3 TaKHX OTpe3KoB MbI nOCTaBHJIH B COOTBeTCTBHe era
.lI.JIHHY npH e.ll.HHHqHOM OTpe3Ke e - HaTypaJIbHOe qHCJIO m (a). ECJIH
OTpE.'3KH a H b KOHrpY3HTHbI, TO m(a)=m(b). 06paTHO, ecJIH m(a)=
=m (b), TO OTpe3KH a H bKOHrpY3HTHbI. TaKHM 06pa30M, Ha MHO>KecTBe
8 OTHOIlIeHHH «OTpe3KH a H b KOHrpY3HTHbI» H «MepbI OTpe3KOB a H b
O.ll.HHaKOBbI» COBna.n.aIOT .lI.pyr C .lI.pyroM. Mepa OTpe3Ka 06JIa.n.aeT .lI.BYMSI
B3>KHbIMH CBoHcTBaMH - 8MHTHBHOCTblO H MYJIbTHnJIHKaTHBHocTblO.
PaCCMOTpHM 3TH CBOHCTBa.
nYCTb OTpe30K a MO>KHO pa36HTb Ha .lI.Ba OTpe3Ka b He, npHHa.ll.JIe>Ka-
lIJ.He 8, npH4eM m (b)=p H m (c)=q. Tor.ll.a Becb OTpe30K a pa36HB<JeTCH
Ha p+q qaCTeH, KOHrpY3HTHbIX e.D,HHHqHOMY OTpe3KY e, a nOToMy ero
Mepa paBHa p+q, T. e. m (a)=m (b)+m (c). HTaK MbI .lI.OKa3aJIH CJIe.ll.YIO-
lIJ.ee CBOHCTBO .lI.JIHH OTpe3KOB.
a) ECAU a=b+c, npUtteM OAUHbl ompe3K08 b U C 8blpa'J/CalOmcH. Hamy-
paAbHbtMU I(,UCAa.MU, nw aAUHa Ompe3K.Cl a pasHa CYM.Me aAUH ezo I(,(lcmeu:
m (a)=m (b)+m (c). (I)
TaK KaK nO-JIaTbIHH CJIO>KeHHe Ha3blBalOT additio, TO 31'0 CBOHCTBO .lI.JIHH
Ha3bIBaeTCH aa{)umU8HOCtnblO.
BTopoe CBOHC'{BO .llJIHH CBH3aHO C nepeXO.ll.OM OT O.llHOH e.D,HHHll.bl U3-
MepeHHH K .lI.pyroA. MbI 3HaeM, qTO ecJIH npH H3MepeHHH OTpe3Ka a MeT-
1) E - nponHCHaSi rpe'leCKaSi 6YKoa CKCH..
309

paMH. nOJIyqH.JIOCb qH.CJIO p, TO npH. H.3MepeHHH. TOro JKe OTpe3Ka CaHTll-
MeTpaMH. nOJIyqH.TCH qH.CJIO 100 p. 3To MOJKHO Bb1pa3H.Tb B BH..lI.e paBeH-
CTBa m 2 (a)= 100 .ml(a), r.ll.e qepe3 ml(a) 0603HaqeHO 3HaqeHHe .lI.JIH.Hb1
OTpe3Ka a npH. H3MepeHH.H. ero MeTpaMH., a qepe3 m 2 (a) - npH H.3Mepe-
HH.H. CaHTH.MeTpaMH.. npH. 3TOM qH.CJIO 100 nOKa3bIBaeT, CKOJIbKHM HOBb1M
e.ll.H.HH.aM H.3MepeHHR paBHa H.CX0.ll.HaH e.ll.H.HH.a H3MepeHH.H (CKOJIbKO
CaHTHMeTpOB B MeTpe).
PaCCMOTpH.M Tenepb B 06Il1.eM BH..lI.e nepexo.ll. OT O.ll.HOH e.ll.HHHILbI H.3-
MepeHlffi .lI.JIHH K .lI.pyrOH. nYCTb e l H. e2 - .lI.Be e.ll.H.HH.lI,bI ll3MepeHH.H .lI.JIH.-
HbI, npHqeM el B n paa 60AbUle, tte.M e20 T. e. el ....... n 'e2, r.ll.e n - HaTypaJIb-
Hoe qHCJIO. ECJIH. npH. H.3MepeHH.H. OTpe3Ka a e.ll.H.HH.eH ll3MepeHHH el
nOJIyqH.JIOCb qllCJIO p (T. e. ecJIH. a ....... p 'el), TO npH. H.3MepeHH.H. Toro JKe OT-
pe3Ka a arpe3KOM e2 nOJIyqH.TCH qllCJIO pn (T. e. a ....... (pn)e2). B caMOM
.lI.eJIe, OTpe30K a COCTOH.T H3 P OTpe3KOB, KOHrpY3HTHbIX OTpe3KY eIt a
KaJK).!,bIH H.3 HllX - H3 n OTpe3KOB, KOHrpY3HTHbIX OTpe3KY e2' 3HaqHT,
B OTpe3Ke a CO.ll.epJKH.TCH pn OTpe3KOB, KOHrpY3HTHbIX eh T. e. a ....... (pn)e2.
TaK KaK a ....... pel H el  ne2, TO Mb1 .lI.OK83aJIH. paBeHCTBO p (ne2)=(pn)e2'
BY.ll.eM 0603HaqaTb )J.JIH.HY OTpe3Ka a, ll3MepeHHoro e)l.HHH.qHb1M OT-
pe3KOM ei, qepe3 ml(a), a .lI.JIllHY Toro JKe OTpe3Ka, ll3MepeHHoro e.ll."-
HH.qHb1M OTpe3KOM e2t- qepe3 m2(a). Tor.ll.a ml(a)=p II m 2 (a)=pn.
YqH.ThIBaH, lITO )J.JIHHa OTpe3Ka el, H3MepeHHoro OTpe3KOM eh paBHa n,
r. e. m 2 (el)=n, paBeHCTBO m2(a)=pn MOJKHO 3anHcaTb TaK:
m2(a)=ml(a)m 2 (el)' (2)
I1TaK, Mb1 .lI.OKa3aJIH. CJIe.ll.YIOIlI.ee cBotkTBO .lI.JIH.Hb1 OTpe3Ka.
6) ECAU ompe30K a KpameH ompe3KY eIt a ompe30K el KpameH ompe31cy
e2, mo ompe30K a KpameH e)!o npulf£.M 8blnOAHflemcf/. pll8eHCm80 (2).
nOCKOJIbKY B npaBoH qaCTH. paBeHcTBa (2) CTOHT npOH.3Be.ll.eHHe
ml(a) II m 2 (el), CBOHCTBO 6) Ha3b1BaIOT .MyAbmUnAUKamU8HOCmblO Mepb1
(OT JIaTHHCKoro CJIOBa muItiplicatio - «YMHOJKeHlle»). OHO YCTaHaBJIH.-
BaeT CBH3b MeJK.ny onepalleH YMHOJKeHHH HaTypaJIbHbIX qllCeJI H nepe-
XO.ll.OM K HOBoH e)l.HHHe ll3MepeHHH.
YopamneUHR
I. 9KBHBaJIeHTHbi JlH Ha MHO)l{OCTBe nJlOCKHX <j>Hryp OTHOllleHHSI «<j>Hrypbl a H b
KOHrpY3HTHbI» H crrJlOrn;MH <j>Hryp a H b o.ZJ:HHaKoBbI»?
2. 9KBHBaJIeHTHbI JlH Ha MHO)l{eCTBe yrJloB OTHOIlleHHH cyrJlbl a H b KOHrpY3HTHbI»
H CBeJIHqHHbI yrJIOB a B b paBHI.l»?
3. .1J.oK3)I{B1'e COOiiCTB-a a,zJJI.HTBBHOCTH H MYJlbTHrrJlHKaTHBHoCTK .ZJ:JIH nJlOrn;a.ZJ:eH
,pHryp, Koropble MO)l{HO pa30HTb Ha e.1\HHH'IHble KBa,npaTbI.
4. .1J.OK3)I{HTe cBoiiCTBa 3.ZJ:,!(BTHBHOCTH H MYJlbTHnJlHKaTHBHocTH .ZJ:JIH BeJIH'IHH
yrJloB.
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/
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3e, HO KOp01Je, qeM 4e. n03TOMY
ero .lI.JIH.HY npH. e.ll.H.HllQHOM OTpe3Ke.
e HeJIb3H Bb1pa3HTb HaTypaJIbHb1M
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e
PHC. 179
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4HCJIOM. O)I.HaKO ecJIH pa36HTb OTpe30K e Ha nHTb KOHrpY9HTHbIX 4aC.
TeR H BbI6paTb O,n:HY H3 HHX 3a HOBYIO e,zJ,HHHU.Y ,n:JIHHbI, TO ,n:JIHHa OT-
pe3Ka a BbIpa3HTCSI H3TypaJIbHbIM qHCJIOM 18 - OH COCTOIIT H3 BOceMHa-
,n:u.aTH OTpe3KOB, Ka}K,l{bIH H3 KOTOpbIX COCTaBJIHeT mlTYIO 4aCTb e,zJ,HHH4-
HOfO OTpe3Ka.
.L(JIH TOrO 4TOObI BbIpa3HTb HaTypaJIbHbIM 4HCJIOM ,n:JIHHY KaKoro-
HH6y,n:b ,n:pyroro OTpe3Ka, npHllIJIOCb 6bI ,n:e.n:HTb HCXO,n:HbIH e.llHHH4HbIH
OTpe30K He Ha nfITb, a, CKmKeM, Ha 38 UJIH 217 qacreM. DpH 9TOM Ha
CKOJIbKO 6bI 4acreH MbI HH pa3,n:e.n:HJIH e,zJ,HHH4HbIR OTpe30K, MO)KHO HaHTH
OTpe30K, ,n:JIH H3MepeHHH KOTOporo npHlleTCH .lleJIffTh e,n:HHHqHbIH OTpe-
30K Ha ew.e 60JIbmee qHCJIO qaCTeH. n03TOMY y,n:06Hee nOCTym-ITb HHa-
4e - He CTpeMHTbCH Bcef.lla BbIp33HTh .llJIHHY H8TypaJIbHbIM 4HCJIOM, a
coxpaHHTb O,n:HH H TOT )Ke e,zJ,HHHlfHbIH OTpe30K, HO Ka)l{,n:bIii: pa3 YKa3bI-
BaTb, Ha KaKoe IfHCJIO 4acreu Mbi ero ,n:e.n:HM ff H3 CKOJIbKHX TaKHX 4ac-
TeH COCTOHT H3MepHeMbIH 0Tpe30K. B onHcaHHOM BbIIIIe CJIY4ae pe3YJIb-
TaT H3MepeHHH BbIpa3HTCH napOH: H8TypaJIbHbIX 'IHCe.n: (18; 5). 4aw.e
18
TaKyro napy 3amrCbmaIOT B BH,n:e ,n:p06H S'
B 06m.eM BH,!{e .41'06H "PH H3MepeHHHX B03HHKalOT CJle.LI.YIOw,HM 06-
pa30M. Ha30BeM n-u dO.lleu OTpe3Ka e TaKOR OTpe30K f, 4TO e -- nf, H
6y,n:eM nHcaTb m(a)=.E.., ecJIH OTpe30K a HBJIHerCH CYMMOii: p OTpe3KOB.
n
KOHrpY9HTHbIxn-ft ,n:OJIe e,zJ,HHUqOOrO OTpe3Ka e. B 9TOM <:JIY4ae nHillYT
TaK}f{e a """ .E..e II fOBOpSIT, 'ITO OTpe30K a COH3MepHM C e. SIcHO, 4TO
n
m(a)=.E.. B TOM H TWlbKO B TOM CJIY4ae, Kor,n:a na ;::;,p e.
n
P'.IlUfla OdH.020 U m020 3ICe ompe3Ka a M03lCem npu 3aOaH.HOM eiJUH.U"lHOM
ompe3Ke e 8btpa3lCafTtbCJl paaAu"tH.blMU iJp06JtMu. B caMOM ,n:e.n:e, ecJIH na 
 pe, TO npH JIlO00M HaTyp8JIbHOM m HMeeM (nm)a """ (pm)e, a nOTOMY
,n:JIHHa TOro )l{e OTpe3Ka a BbIpa)l{aeTCH He TOJIbKO ,n:P06blO : ' HO H
,n:P06blO pm . JJ.OKIDKeM CJIe,n:ylOw.ylO TeOpeMY.
mn
T e 0 p eM a. P'AJl m020 "lmo6bl iJpo6u £.. U  8btpa:JlCa.llU d.llUHY odH.020
n q
U m020 3ICe ompe3Kt% a, H.eOOXoOuMO U dOCmamOttH.O, "lmo6bt 8blnOAHJl/lOCb
pa8eH.Cm80 Hamypa/l.bH.blX "lUCeA pq=nt.
B caMOM ,n:e.n:e, eCJIH m (a)=£" H m (a)=l., TO na """ pe H qa  te. Ho
n q
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BO MO)l{er HMt'!fb MeCTO JIHWb npH YCJIOBHH, qTO pq=nt. 3Ha4HT, YCJIO-
BHe pq=nt HeOOXO,n:HMO MH TOro, QT06b1 ,n:p06H 1!... II  BhIpa}f{aJIH
n q
,n:JIHHY o,n:Horo H Toro )l{e OTpe3Ka.
06paTHo, nYCTb pq=nt n £.. - ,n:JIHHa OTpe3Ka a, a l. - MHHa OT-
n q
pe3K8 b. Tor,n:a na """ pe II qb  te. OrcIO,lI,a CJIe,lJ,yer, 'ITO (fU{)a  (pq)e
H (fU{)b=(nt)e. DOCKOJIbKY pq=nt, nOJIyqaeM: (fU{)a """ (fU{)b. 31'0 HMeer
MeCTO, JIHllIb ec.TIII OTpe3KH a II b KOHrpY3HTHh1. 3Ha'IHT, YCJIOBHe pq=nt
3U

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n q
9HTHbIX OTpe3KOB, HJIH, liTO TO Me caMoe, .u,JIHHY o.u,Horo H Toro Me OT-
pe3Ka.
B .u,aJIbHeihneM MbI 6y.u,eM Ha3bIBaTb .u,Be .u,P06H J!.... H .!......, TaKHe, qTO
n q
pq=nt, 3K6U8aJ/.eHmHbLMU. Mbi BH.u,HM, qTO oee opo6u 3K6UeaJ/.eHmHble mOM
U mO.llbKO e mOM C.IlY'1ae, Koeoa OHU eblpa;)fCQlOm O.llUHY OOHoeo U moeo
:JICe Ompe3Ka.
3. nOJI01KHTeJIbHble paQHOHaJIbHble 'IHCJIa. DOCKOJIbKY .u,JIHHa OT-
pe3Ka .u,OJIMHa BbipaMaTbcH O.u,HHM qHCJlOM, 9KBHBaJIeHTHbie .u,P06H HB-
JIHIOTCH pa3JIH'IHblMH npe.n.craBJIeHHHMH o.u,Horo H Toro Me qHCJIa. l.JHc7
JIa, KOTopble MOMHO npe.u,craBHTb (3anHcaTb) B BH)le .u,po6eH, Ha3bIBalOTCH
nO.llO;)fCUme.llbHblMU pal{UOHa.llbHblMU 'iUC./UlMU. MOMHO CKa3aTb, qTO n 0-
nOMHTeJIbHblM paUHOHaJIbHblM 'iHCJIOM Ha3
B a e T C H MHO M e C T B 0 9 K B H B a JI e H T H bI X .u, P 0 6 e H.
1 2 3
TaKHM 06pa30M, HH 2' HH 4' HH "6 He HBJIHIOTCH caMH no ce6e nOJIOMH-
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2 3 n }
4; 6; ...; 2n ; . .. HBJIHeTCH nOJIOMHTeJIbHbIM paUHOHaJIbHbIMqHCJlOM.
1 2
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Cpe.n.H Bcex 3anHceH .u,aHHoro nOJIOMHTeJIbHOrO paUHOHaJIbHOrO qHC-
JIa B BH.u,e .u,p06H MOMHO BbI,ll,eJIHTb 3anHCb, B KOTOpOH qHCJIHTeJIb H 3Ha-
MeHaTeJIb B3aHMHO npOCTbI. TaKHe .u,p06H Ha3bIBalOT HeCOKpamU.MblMU.
I1TaK, CnpaBe,lI,JIHBa CJIe.u,YIOw.aH TeopeMa:
T e 0 p eM a. D./lR .Il106oeo nO.llO;)fCUme.llbHoeo pal{uoHa.llbHoeo 'iUC./Ul a
(m. e. O/lR .IlKJ6oeo MHocmea 3K6U8aJleHmftblX opo6eu) HauoemcR OOHa U
nw.llbKO oOl-Ul npeOCI1lG8.1lR1OUJ,llR eeo op06b, tlUC.llUme.llb U 31-UlMeHame.llb KO-
mopou eaaUMHO npOCtnbl.
P
B C8MOM l1Me, cYllleCTByeT XOTb 6bI o.ZJ:Ha .ZJ:P06b - , npe.ZJ:cTaBJHllOlllaH 1JHCJIO a.
n
nYCTb d - HaHCiOJlblllHH 06111HH l1e.rIHreJib qHCeJI P H n. Tor.ZJ:a P = PId, n = nld,
fl1e Pi H ni B3aHMHO npOCTbI. TaK KaK PIn = nIP = dPInlt TO .ZJ:p06H !!.. H PI
n nl
!lKBHBaJleHTHbI, 8 nOTOMY Pi TO)\{e npe.llCTaBJIHeT qHCJIO a. 3HaqHT, PI _ HeCOKpa-
ni nl
THMaH aaUHCb 1JHCJIa a.
.LJ:oK3)I{eM, 1JTO HHOH HecGKpaTHMOH sanHCH 1JHCJlO a He HMeeT. npe.ZJ:nOJlO)\{HM,
qTO ,llp06b .!. t TO)I{e npe,llCTaBJIHeT 1JHCJIO a H OTJlH1JHa OT .ZJ:poCiH Pl. Tor.na snl=Plt.
nl
TaK KaK JleBaH qaCTb 3TOro paBeHCTBa ,lleJIHTCH Ha nl' TO ero npaBaH qaCTb .ZJ:eJIHTCH
lIa nl' npH 3TOM TaK KaK nl H PI B3aHMHO npoCTbI, TO t ,!(eJIHTCH Ha nJ' t=nJq. 4HCJIO
PI S " u
q OTJlH1JHO OT e.nHJlHUbI, TaK KaK HHa1Je .np06H - H -coBna.ZJ:anH ubi. nTaK, sn I = Plnlq,
nt t
s
H nOTOMY S=Plq. 3HaQHT. S 'l'O)\{e ,lleJIHTCH Ha q. H nO'l'OMY .np06b t MO)l{HO COKpaTIITIt
Ha q. TaKHM 06pa30M, Jlr06aR OTJlH'lHaR OT Pi saUHCb 1JHCJIa a COKparHMa.
nl .
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ECJIII t - HaTypaJIbHOe qIICJIO II a  te, TO .n.JIH JIlo60ro HaTypaJIbHOrO
qIICJIa n IIMeeM na "' (nt)e. 3'1'0 nOKa3bIBaeT, qTO .n.JIIIHY OTpe.3Ka a MOMHO
n
BbIpa3HTb He TOJIbKO HaTypaJIbHbIM qIICJIOM t, HO II .n.P06HMII BII.n.a-.
n
HHbIMH 'CJIOBaMII, H.amypaJlbH.Oe ttUCAO t c08n.ai:Jaem C nOAO;)fCUmeAbH.bIM
pal{UOH.a/lbH.blM. ttUCAOM 8uoa:
{ t . 2t . . nt. }
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YnpamOeOBJI
1. ,noKa)l{HTe, 'Ira OTHOllleHHe  - : o6J1a.ZJ:aeT cBoiicTBaMH cHMMeTpH'IHOCTH.
pe<j>JIeKcHBHoCTH H TpaH3HTHBHOCTH.
2. .1J.oKa)l{HTe, 'Iro OTHOllleHHe coH3MepHMocTH oTpe3KoB 06JIa,n,aeT cBoiicTBaMM
pe<j>J!eKcHBHoCTH, CHMMeTpH'IHOCTH H TpaH3HTHBHOCTH.
3. HaH.D.HTe .D.P06b, !lKBHBaJIeHTHYIO .D.p06H  H HMelOlllYIO 3HaMeHaTeJIb 111 11 I.
37 118 78 1415
4. CoKpaTHTe .np06H: 999 ' 413 ' 650 ' 1981 '
4. CnO)l{eHHe nOllOJKHTeJIbHblX paQHOHaJlbHblX 'IHCeJI. B STOM nYHKTe
MbI onpe.n.eJIIIM Onepall,HIO CJIOMeHIIH B MHoMecTBe Q + nOJIOMHTMbHbIX
pall,HOHaJIbHbIX qHCM. CHaqaJIa .n.OKIDKeM CJIe.n.YIOw.ee YTBep}K)J.eHIIe.
JI106b1£ 000 ttUCAa.a u b U3 Q+ MO;)fCH.O npeiJcmll8Umb 8 8uoe op06etl,
UMeIOI4UX oOUH.aK06bl£ 3H.a.MeH.ameAU.
B caMOM .n.Me, nYCTb qIICJIO a npe.n.CTaBJIeHO .n.p06bIO P-, a qIICJIO b-
n
Ap06blO ..!... Tor.n.a 9TH Me qIICJIa MOMHO npe.n.CTaBIITb ,lI,p06HMH pq H
q nq
nt
-, HMelOw.IIMH o.n.HHaKOBbIe 3HaMeHaTMH.
nq
3aMeHa .n.po6eH J!.. H.!... 3KBIIBaJIeHTHbIMII HM .n.p06HMH, IIMeIOw.IIMII
n q
o.n.HHaKOBbIe 3HaMeHaTMH, Ha3bIBaeTCH npueeOeH.ueM K oOH.OMY 3H.aMeH.a-
me/llO. HaIIMeHbllIHM 06w.HM 3HaMeHaTMeM .n.po6eH P- H .!... HBJIHeTCH
n q
HaHMeHbllIee 06w.ee KpaTHoe qHCM n H q. ECJIH k=K (n, q), TO k=nl=
=qm, H nOTOMY .n.p06b.!!.... 3KBHBaJIeHTHa .n.p06H pI I = P k l , a .n.p06b..!..-
n n q
tm tm
AproH qm = T'
4 II
n p H Me p 1. npHBe.n. eM K 06w.eMY 3HaMeHaTMIO .n.p06II 35 H 15 '
4.15 60 11.35 385
MOMHO 3aMeHHTb 3TH .n.p06H Ha 35.15 = 525 H 15.35 = 525 ; qaw.e Haxo-
AHT HaHMeHbllIee 06w.ee KpaTHoe qHCM 35 H 15: K (35, 15)= 105, nOCJIe
4.3 12 11.7 77
qero 3aMeHHIOT 9TH AP06H Ha 35.3 = 105 H 15.7 = 105 ' r.n.e 3=105: 35,
7=105 : 15.
al3

B03MO>KHOCTb npHBeJl,eHHH Jl,pO-
) 6eH K OJl,HOMY 3HaMeHaTeJIIO 03Ha-
'IaeT, 'ITO ecJIH OTpe3KH a H b co-
H3MepHMbI C eJl,HHH'IHbIM OTpe3KOM
pt.
e, a '" - e, b  - e, TO HMeeTCH
n q
OTpe30K f, KOTOPOMY KpaTHbl OTpe3KH a, b H e. B Ka'IecTBe TaKoro
OTpe3Ka MO>KHO B3HTb, HanpHMep, nq-IO Jl,0JI1O e. 3'1'OT OTpe30K f Ha3b1-
BaeTCH 06U{eu eaUH.Ul{eu U3MepeH.UR OTpe3KOB a H b. ECJIH Bbl6paTb ero
3a HOBbIH eJl,HHH'IHbIH OTpe30K, TO Jl,JIHHbl OTpe3KOB a H b Bblpa3HTCH
HaTypaJIbHbIMH 'IHCJIaMH pq H tn. nOCJIe 3Toro He COCTaBJIHeT TpYJI,a
HaHTH Jl,JIHHY CYMMbl H pa3HOCTH 3"fHX OTpe.3KOB, Y3HaTb, KaKOH H3
3"fHX OTpe.3KOB J],JIHHHee, H T. JI,.
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n n
nYCTb C - CYMMa 3THX OTpe.3KOB (pHC. 180). TorJl,a na  pe, nb  te, H
nOTOMY nc  n (a+b)na+nb  pe+te  (p+t)e.
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n
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PHC. 180
P-+i.= p+t
n n n'
n03TOMY npHHHMaeM no onpeJl,eJIeHulO:
CJmo6bl CflOJICUmb nOflOJICun'/.eflbH.ble pal{UOH.aflbH.ble 'luc,1£l U U b, H.aaO
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n n
flU; CYMMa 3mux Ilucefl npeaCl1W1JflRemcR ap06b1O pt (T. e. Jl,p06blO C TeM
>Ke 3HaMeHaTeJIeM, 'IUCJlHTeJIb KOTOpOH paBeH CYMMe 'IHCJlHTeJIeH CJIarae-
MblX Jl,p06eH).
JIerKo npoBepuTb, 'ITO npu 3aMeHe Jl,p06eii P- U ..!... 3KBHBaJIeHTHblMH
n n
HM Jl,p06HMH p+t TO>Ke 3aMeHUTCH 3KBHBaJIeHTHOH Jl,p06blO. 3Ha'IHT,
n
cYMMa 'IHCeJI H3 Q+ He 3aBHCUT OT Bbl60pa HX 3anHCH B BHJI,e Jl,po6eH.
ECJIH 'IHCJIa a H b npeJl,CTaBJIeHbI Jl,p06HMH C pa3JIH'IHblMH 3HaMeHa-
TeJIHMH, HaJl,O CHa'IaJIa npHBeCTH 3TH Jl,p06H K OJl,HOMY 3HaMeHaTeJIlO, a
nOTOM npHMeHHTb c4><>pMYJIHpOBaHHoe BblIIIe npaBHJIo.
13 22 .
n p H M e p 2. CJImKUM Jl,p06H 60 H \05 ' HaHMeHbwHu 06W.UH
3HaMeHaTeJIb paBeH K (60, 105)=420. 3Ha'IUT,
13 +=_ + 22.4 = 91+88 = 179
60 105 60.7 105.4 420 420 .
Bo06w.e .!!.. +..!... = pq+tn .
'n q nq
11 To eCTb nOTpe6oBaTb, 'lT06b1.AJIHHa CYMMbI OTpe3KOB paBHSlJlaCb CYMMe Hit WlHH.
31.

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J,tymamU8Hocmu, accOl{uamU8Hocmu U COKpamUMocmu: .n:JIH JIl06blX 'IHCeJJ
a, b, C H3 Q+ HMeeM a+b=b+a H a+(b+c)=(a+b)+c, a H3 a+c=
=b+ccJIe.n:YeT, 'IToa=b. KpoMeToro, JIerKO npoBepHTb, 'ITO JI.JIH mo6blx
'IHCeJI a H b H3 Q+ HMeeM a+b+a.
[loKa)j{eM, HanpHMep, 'ITO a+b=b+a. npe.n:CTaBHM 'IHCJIa a H b
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n n n
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n
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npaBHJIO .n:aeT TOT me pe3YJIbTaT, 'ITO H CJIOmeHHe B N. B caMOM JI.eJIe,
HaTypaJIbHOe 'IHCJIO p MomHO 3anHcaTb .n:p06blO  ' 'IHCJIO t - JI.p06blO
-f-, a CYMMa 3THX .n:po6eii paBHa pit , T. e. HaTypaJIbHOMY 'IHCJIY p+ t.
CKameM, 'ITO 'IHCJIO a H3 Q + 60JlbUJ£ 'IHCJIa b H3 Q +, eCJIH cYlIJ.ecrByeT
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ECJIH 'IHCJIa a H b BblpameHbI .n:PO(jHMH .!!... H ..!.., HMelOlIJ.IIMH o.n:HHa-
n n
KOBbie 3HaMeHaTeJIH, TO a>b B TOM H TOJIbKO B TOM CJIY'Iae, Kor.n.a p>t.
ECJIH me OHH BblpameHbl .n:P06HMH !!.... H ..!.., TO a>b B TOM H TO,'1bKO B TOM
n q
CJIY'Iae, KOrJI.a pq>nt.
OTHOllIeHHe nopH.n:Ka B MHomeCTBe Q+ o(jJIa.n:aeT .n:BYMH CBOHCTBaMH,
OTJIH'IHbIMH OT CBOHCTB TaKoro me OTHOllIeHHH B MHomeCTBe N HaTY-
paJIbHblX 'IHCeJI. HanoMHHM, 'ITO cpe.n:H HaTypaJIbHblX 'IHCe,'! eCTb HaH-
MeHbllIee''IHCJIO 1, a, KpoMe Toro, MHomecTBo HaTypaJIbHblX '1HCeJI .n:HC-
KpeTHO - MH Kam.n:oro HaTypaJIbHOrO 'IHCJIa eCTb HenOcpe..'l.CTBeHHo
cJIe.n:YIOlIJ.ee 3a HHM 'I HCJIO. I1Ha'le 06croHT lI.eJIO B MHomt'CTBe Q +:
a) 8 JIHO;)ICeCm8e Q + Hem HauMeHbweeo 'lUCJla;
6) Me;)IC0Y Jl106blMU 08YMJI pa3JlU'lHblMU 'lUCJlaMU a U b U3 Q+ aaKJlIO'le-
HQ 6eCKOHe'lHO MHoeO 'luceJl moeo ;)ICe MHO;)ICeCmea.
[loKameM CHa'laJIa OTCYTcTBHe B Q+ HaHMeHbllIero 'IHCJIa. B caMOM
.n:eJIe, nycTb a - KaKoe-TO 'IHCJIO H3 MHomecTBa Q +. Ero MomHO npeA-
CTaBHTb B BHJI.e .n:PO(jH .!!.... Ho Tor.n:a .n:p06b 2 p - 3anHCb 'IHCJIa, MeHb-
n n
llIero 'IeM a.
B03bMeM Tenepb JII06ble JI.Ba pa3mI'IHblX nOJIOmHTeJIbHblX paUHOHaJIb-
HblX 'IHCJIa a H b. Tor.n:a O.n:HO H3 HHX MeHbllIe lI.Pyroro, HanpHMep a<b.
n p eJI.CTaBHM 'IHCJIa a H b .n:p06HMH m H!!..... TaK KaK a<b, TO m<p.
n n
B03bMeM 'IHCJIO C, npe.n:craBJ'1HeMoe .n:p06blO mp . TaK KaK m<p, TO
2m m+p 2p b T.f
2m<m+p<2p, H nOToMY 2n < 2n ' T. e. a<c< . nTaK, Me-
)!{,l{Y JIIO(jblMH .n:BYMH 'IHCJIaMH H3 Q+ 3aKJIIO'IeHO XOTH 6bl O.n:HO I.JHCJIO C,
315

raK)!{e nplHla)J..'le)!{aw.ee Q+. A nOTOM MO)!{HO BbI6paTb )J.Ba (IHCJIa, 3aKJIIO-
qeHHble Me)!{)J.Y a H c H Me)!{)J.Y C H b. np0)J.OJI)!{aH 3TOT npou,ecc, Mil
HaH)J.eM Me)!{)J.Y a H b 6ecKoHeqHO MHoro pa3JIHqHbIX qHCM H3 Q+.
OrMeTI1M, <ITO cpeou 'lUCeA U3 M.HOJfCeCf'Ma Q+ Hem flllu60AbUli!eo:
ll/l51 JII06oro 'IHCJIa a, npe)J.CTaBJleHHOro )J.p06blO!!...., CYl.l.lecTBYeT qHCJIO,
n
KOTOpoe 60JIbllIe Hero, HanpHMep qHCJIO, npe)J.CTaBJIeHHOe )J.pOOblO p+1 .
n
Onpe)J.eJlHM Tenepb onepau,111O 8bl'lUmaHUJ/. 8 Q+. nYCTb a>b. Tor)J.a
no onpe)J.eJIeHHIO cyw.ecTByeT TaKoe C E Q+, <ITO a=b+c. nOKa)!{eM, qTO
ql1CJIO C O)J.H03HaqHO onpe)J.eJIeHO. B caMOM )J.eJIe, nyCTb a=b+d, r)J.e
dE Q+. Tor)J.a I1MeeM b+c=b+d. B CHJIY cOKpaTHMocTH CJIO)!{eHI1H B
Q+ OTCIO)J.a cJIe)J.YeT, <ITO c=d, a 3TO H 3HaQHT, QTO c O)J.H03HaQHO onpe-
)J.e.TJeHO.
ECJIH CYl.l.leCTByeT TaKoe c E Q+, QTO a=b+c, TO QHCJIO C Ha3bIBalOT
pa3HOCtnb1O QUCM a H b H 0603HaQalOT a-b. 5IcHO, QTO eCJIH a H b BbIpa-
p t p-t
)!{eHbI )J.P06HMH - H -, TO a-b Bblpa)!{aeTCH )J.pOOblO -. ECJIH)!{e
n n n
a l! b npe)J.CTaBJIeHbl )J.P0651MH !!.... H ..!..., TO a-b npe)J.CTaBJI51eTC51 )J.po6blO
n q
pq-nt p t
-. MO)!{HO npHBecTH )J.pOOH - H - K 3HaMeHaTeJIIO K (n, q).
nq n q
HanpHMep,
13 22 91-88 3 1
60 - 105 - -:t2O = 420 = 140 .
¥npamUeUBJI
P PI t t 1 P t Pi t 1
t. .D:oKa)KUTe, '1TO eCJIH - - - H - - -, TO - +- - -+ - (T. e.
n nl q ql n q nl ql
'1TO CYMMa '1HCeJl He saBHCHT OT Toro, KaKHMH .np06HMH npe.ZJ:CTaBJleHbI .9TH '1HCJIa).
2. )];0 Ka)l{HTe, '1TO CJIO)l{eHHe B Q+ KOMMYTaTHBHo, aCCOUHaTHBIlO H coKpaTHMo.
3. .1J.oKa)l{HTe, '1TO B Q+ BbInOJlHHIOTCH COOTHOllleHHH:
a) a-(b+c)=a-b-c (ecJIH a>b+c);
6) a+ (b-c)=a+b-c (eCJIH b>c);
B) a- (b-c)=a-b+c (eCJIH a>b>c).
4. lI.oKa)l{IITe, qTO eCJIH .!!... - PI H ..!... - .!!.. TO H3 .!!... > .!.. BbiTeKae-r, '110
n nl q ql' n q
Pi > .!1. .
nl ql
5. .1J.oKa)l{HTe, '1TO eCJIH a>b, TO a+c>b+c.
6. .1J.oKa)l{HTe, '1TO eCJIH a=t=b, TO a>b HJIH b>a.
7. BbinOJIHHre .1\eitCTBHH:
17 19 5 1
a) 10 80 + 2 48 + 1 32 + 96 ;
5 I 2 5 13
6) 44 +5i-+4rr+ 66 + 44 ;
n) (20-19 : )+(17 : -17)+(2+- : );
r) 2  - ( : + 1 + 1 2 5 ) .
316

14 105 531
8. PaCnOJIO)l{HTe B nopJl.l1Ke B03pacTaHHH CJIe.ZJ:YlOlllHe .ZJ:P06H: 27 ' 216 ' 1280 '
6. YMHO)l{eHHe H ,ll.MeHHe nOJIO)l{HTeJlbHblX paQHOHaJIbHblX 'IHCM.
nYCTb OTpe30K a cOH3MepHM C e.n.HHHqHblM OTpe3KOM elt a oTpe30K el -
C OTpe3KOM e 2 , npHqeM a "' elt el= ; e 2 , T. e. na "' pelt qel  te2'
Tor.n.a (nq)a "' (pq)elt (pq)el '" (pt)e 2 , H nOToMY (nq)a  (pt)e2' 31'0 nOKa-
3blBaeT, qTO ,ll.JIHHa OTpe3Ka a npH e.n.HHHqHOM OTpe3Ke e 2 BbJpaJKaeTCH
.n.P06blO PI , T. e. qTO qHCJlO m 2 (a) BbJpa:>KaeTCH .n.P06blO PI , m 2 (a) =
nq nq
= pl . Ho no YCJlOBHIO ml(a)=!!..., m2(el)=.l... nO.3TOMY ecml nOTpe-
nq n q
60BaTb BbinOJIHeHHH cBoikTBa MYJIbTHnJIHKaTHBHocTH m 2 (a)=
=ml(a)m2(el), TO .n.OJI:>KHO BblTIOJIHHTbCH paBeHCTBO pi =.!!.....
nq n q
TaKHM 06pa30M, npaBHJlO YMHOJfCeHUJl nOJIOJKHTeJIbHblX paUHOHaJIb-
HbIX qHCM ecTecTBeHHO C<POPMYJIHpOBaTb CJIe.n.YIOW.HM 06pa30M:
JJpoU36eoeHue 'lUC./la a, 8blpaJfCaeM020 op06blO .!!..., Ha 'lUCJ20 b, 8blpa-
n
JfCaeMoe Op06blO .!...., 6blpaJfCaemcfl Op06blO pt (B llIKOJle rOBopHT: npo-
q nq
H3Be,ll.eHHeM .lI.Byx .n.p06eH HBJIHeTCH .n.p 06b , qHCJIHTeJIb KOTOpOH paBeH
npOH3Be,ll.eHHIO qHCJIHTMeH MHO:>KHMblX, a 3HaMeHaTeJIb - npoH3Be.n.e-
HHIO HX 3HaMeHaTeJIeH). HanpHMep,
34 15 34.15 5
87 . 68 = 87.68 = 58 .
JIerKo npoBepHTb, qTO npH 3aMeHe .n.p06eH .!!... H .!...., npe.n.CTaBJIHIOw,HX
n q
qHCJIa a H b, .3KBHBaJIeHTHbIMH HM .n.p06HMH PI H t} .n.p06b pi 3aMe-
n i ql nq
HHeTCH .3KBHBaJIeHTHOH eH .n.P06blO Plt l . n03TOMY npOU38eoeHue 'lUCeA
n I ql
a u b He 3a8ucum om 8bl6opa npeocl1lG8/1JUOux ux op06eu.
J1 MHOJfCeHUe 8 Q + 06JlOi}aem C80UCmBaMU KOMMymamu8Hocmu, acco-
14uamU8Hocmu U COKpamUMocmu: ,lI,JIH JII06blX qHCeJI a, b, C H3 Q+ HMeeM
ab=ba, a (bc) = (ab) c, a H3 ac=bc BbITeKaeT a=b. 3TH YTBep)K)J.eHHH JIer-
KO .n.OKa3aTb, 3aMeHHB qHCJIa a, b, C npe.n.cTaBJIHIOw.HMH HX .n.p06HMH.
KpoMe Toro, YMHOJfCeHue 8 Q+ oucmpu6ymU8Ho omHocumeAbHO CAOJfCeHUfl
U MOHomOH1W: .n.JIH JII06bIX Tpex qHCen a, b, C H3 Q+ HMeeM a (b+c)=
=ab+ac, a H3 a>b BbITeKaeT ac>bc.
B CJIyqae Kor.n.a nepBblH MHO:>KHTMb - HaTypaJIbHOe qHCJIO m,
.n.aHHoe BbIllIe onpe.n.MeHHe YMHO:>KeHHH COBna,n.aeT C onpe.n.MeHHeM YM-
HO:>KeHHH KaK CYMMbI m CJIaraeMblX, paBHbIX BTOPOMY MHO:>KHTeJIIO, T. e.
ma=a+a+.. .+a (m pa3). B caMOM .n.eJIe, qHCJIO m MO:>KHO 3anHcaTb B
BH.n.e .n.p06H  l ' a  l ..!!... = mp = J!.. +... +.!!... (m pa3). OrclO.n.a CJIe.n.yeT,
q q q q
B qaCTHOCTH, qTO ecJIH m H n - HaTypaJIbHbIe qHCJIa, TO HX npOH3Be.n.e-
HHe B Q+ COBna.n.aeT C HX npOH3Be.n.eHHeM B N. KpoMe Toro, ,lI,JIH mooo-
317,

ro a E Q+ HMeeM 1 'a=a, T. e. '-IHCJIO 1 HeihpaJIbHO OTHOCIITeJIbHO YM-
HO>KeHH5I B Q+.
OnepaUH5I .QeJIeHII5I B Q+ onpe.QeJI5IeTc5I KaK onepaUIISI, 06paTHaSi
YMHO>KeHHlo. B OTJIII"IIIe OT onepauIIII BbI'IIITaHIISI OHa onpe.QeJIeHa .QJISI
Bcex nap (a; b) nOJJO>KIITeJIbHbIX pauHOHaJIbHbIX 'IIICeJI: .QJISI JIl06bIX 'III-
CeJI a H b H3 Q+ HaH.QeTCSI TaKoe 'IHCJIO C E Q+, 'ITO a=bc. B caMOM .QeJIe,
ecJIII lIIICJIO a npe;I.CTaBJISleTCSI .QP06blO .t., a 'IIICJIO b - .Qp06blO .!...., TO
n q
.QOCTaTO'IHO nOJIO>KIITb C=  . Tor.Qa 'IIICJIO bc npe.QCTaBJISleTCSI .Qp06blO
pq t t , 3KBHBaJIeHTHOH .Qp06II P-, a 3TO II 3Ha'IIIT, '-ITO bc=a.
n q . n
Ynpa1lCBeBBJI
t. .1J.oKa)l{UTe, '1TO: a) a: (bc)=a: b : c; 6) a (b : c)= (ab) : c; 13) a : (b : c)=
= (a : b)c.
2. .1J.oKa)l{HTe. qTO YMHO)l{eHHe B <1+ 06JIa.n.aer cBOHCTBaMH KOMMYTaTHBHoCTH.
aCCOUHaTHBHOCTH. .1\HCTPH6YTHBHO OTHOCHTeJJbHO CJIO)l{eHHH. cOKpaTHMo H MOHO-
TOHHO.
3. BbHlOJIHHTe CJIe.1\YlOlllHe .ZJ:eHcTBHH:
a) ( -+- )( _..!. + .!._ ) . 1;
2 3 4 5 6 8 10 12 91
6) ( .!. +1 1O . _2.  ) . s-!,..
5 2 17 5 3 17 3 '
B) (12++4  ):(7-).
7. AKCHOMaTH'IecKoe nOCTpoeHHe TeopHH nOJIO)l{HTeJlbHblX pal\HO-
HaJlbHblX 'tHCM. MbI Onpe;I.eJIIIJIII nOJIO>KIITeJIbHble paUIIOHaJIbHble 'IIICJIa
I{ onepauIIII Ha.Q HIIMII, IICXO.QSI II3 3a.Qa'III 06 II3MepeHIIII OTpe3KOB, T. e.
H3 reoMeTpIIlIecKIIX cOO6pa>KeHIIH. O.QHaKo nOJIO>KIITeJIbHble pauIIOHaJIb-
HbIe 'IIICJIa HY>KHbI He TOJIbKO MSI II3MepeHIISI MIIH, HO II .QJJSI II3MepeHIISI
Macc, nJIOlIJ.a.QeH, o6DeMoBII T. .Q. n09TOMY meJIaTeJIbHO nOCTpOIITb Teo-
pIIIO TaKHX 'IIICeJI, He OnIIpalOlIJ.YIOCSI Ha reOMeTpHliecKIIe nOHSITIISI. LlJISI
9Toro .QOCTaTO'IHO YKa3aTb CIICTeMY aKCIIOM, KOTOpblM .QOJI>KHbI Y.QOBJJe-
TBOp SlTb 3TII 'IIICJIa.
Onpe;I.eJIIIM MHO>KecTBO Q+ C nOMOlIJ.blO CIICTeMbI aKCIIOM, onIICbIBalO-
lIJ.eR CBOHCTBa onepauIIII CJIO>KeHHSI B Q+ H onepauIIH yMHomeHIISI Ha Ha-
TypaJIbHOe'IIICJIO, KOTOpa51 TO>Ke CBO.QIITCSI K CJI02KeHIIIO: na=a+.. .+a
(n pa3). 3Ta CHCTeMa aKCHOM TaKOBa:
1) M fLO:JICeCmBO Q + cooep:JICum MHO:JICeCmBO N Hamy paAbHblX ttUCeA.
2) B .MliO:JICeCmBe Q+ onpeoeAeHa onepal.fWl CAO:JICeHUR, Ctna8RUIpJl 6
coomeemcmBue A106blM 06YM ttUCAaM a u b U3 Q+ ttUCAO a+b tn020 :JICe
MHC»fCecrnea, H.a3b18aeMOe CYM.Mot1 a U b. Ha noOMHO:JICeCmBe N onepal.fUR
C.IlO:JICeHWl c08naiJaem C OnpeOeAeHHOU 6 N.
3) Onepal.fuR CAO:JICeliuR 6 Q+ KOMMymamU8H.a, accOl.fuamU6lia U CO-
KpamUMa.
4) p'AJl Ar06oeo a E Q+ Hat1iJymcR maK.Ue H.amgpaAbHble ttUCAa pun,
'II1W na=p.
&11

5) I1.Afl ,1/o6blX liamypaAbliblX '-IuceA pun ecmb maKoe a E Q+, limo
na=p.
6) ECAU na=nb, mo a=b.
MO)j{HO .lI.OKa3aTb, qTO 3Ta CIiCTeMa aKCIiOM HenpoTIlBOpeqHBa H O.ll.HO-
3HaqHO onpe.ll.e,'151eT MHO)j{eCTBO Q+ II onepalI.111O CJIO)j{eHI151 B HeM.
.1J.JIH 3raro, no.'1b3YHCb aKcHoMoH 4), Mbi CTaBHM B COOTBeTCTBHe K)I{.aOMY u E Q+
Bce napbI HaTypaJIbHblX qHCeJI (p; n) TaKHX. 'Ira na= p, T. e. .1\p06H J!..... H nOKa3blBaeM.
n
'lT0 reM C&lbiM Ka)l{J(OMY qHCJlY aE Q+ COOTBeTcTByeT cOBoKynHocTb 3KBHBaJIeHTHbiX
.ZJ:p06eH. nOCJIe 3raro J(OKa3bIBaeM, 'Ira onepaUHH CJIO)l{ellHH B Q+ CBOJ(HTCH K 06bIQ-
HOMY CJIO)l{CHHIO J(p06eH. 910 nOKa3blBaeT. qTO 3aJ(aHHaH CHCTeMa aKCHOM onpeJ(eJIHer
Q+ H onepaUHIO CJIO)l{ellHH B Q+ 0J(H03HaqHO. HenpOTHBOpeqHBOCTb CHCTeMbi aKCHOM
1) - 6) J(OKa3bIBaeTCH nYTeM nOCTpoellHH MO.1\eJIH. B KOTOpOH '1HCJIa pea.1H3ylOTcH
KaK cOBOKynHocTH 3KBHBaJIeHTHblX .1\p06eH.
HHOf.1\a BMeCTO 3aJ(aHHH CHCTeMbl aKCHOM cpa3Y onpeJ(eJIHIOT nOJIO)l{HTeJIbHbIe
pallHOHaJIbHble '1HCJIa KaK cOBoKynHocTH 3KBHBaJIeHTHblX J(p06eH, cooTBeTcTBYlOlllHM
06pa30M onpe.r\eJIHH .1\eHCTBHH HaJ( HHMH.
 2. AECHTHqHbIE APOBH
1. eCHTH'IHble APo6H H AeikTBHH HaA HHMH. MbI BH.lI.eJlII, qTO nmlB-
JleHl1e Ap06eA CBH3aHO C nepeXO.ll.OM K HOBM e.ll.IIHHlI.aM 113MepeHIIH,
npllqeM 3HaMeHaTeJIb .lI.P06H nOKa3Baer, Ha CKOJIbKO .lI.OJJeH .lI.eJlHTCH
HCXOAHaH e.ll.I1HRlI.a 113MepeHII5I. B HaCTOHmee BpeMH nOqTII BO Bcex CTpa-
Hax Mllpa .lI.eHcTByeT MempU'leCKafl cucmeMa eauliut.f, B KOTOpOH HOBbIe
eAIIHIIlI.bI nOJIyqalOTCH 1IJ111 YMeHbIlleHHeM IICXO.ll.HbIX B 10, 100, 1000
II T. A. pa3. I1JIII YBeJIllqeHlleM I1X B 10, 100, 1000 H T. A. pa3. HanpliMep,
I KM=IOOO M=I 000000 MM, 1 m=1000 KZ=I 000000 e II T..lI.. no-
3TOMY AJJH npaKTIIKII oco(jeHHO Ba)j{HbI AP06H, 3HaMeHaTeJlII KOTOpbIX
HBJIHIOTCH CTeneHHMIl qllCJla 10, T. e. Ap06H BIlAa 1; ' f.ll.e mil n - Ha-
TypaJIbHbIe qllCJIa. TaKlle Ap0611 Ha3BaIOT aeCflmU'llibLMU.
nYCTb .lI.eC5ITllqHa51 3anHCb LJHCJlHTeJJH HMeer BHA m=mk' . .mo, T. e.
m=mk .10 11 +.. .+mo.
TOf.ll.a no npaBHJlaM .lI.eikTBHH H8,!I, CTeneHHMH npH nk RMeeM:
.!!!... _ mil' 10 11 + .., +m n .IO"+m n _l.lon-l+... +mo_
IOn - IOn
_ 1 0 1l-n + + + mn-l + + mo
-mil' ... m n )() .., ]0" '
0603HaLJIIM HaTypa.'1bHOe LJHCJlO m k lO lI - n +.. . + Inn 6YKBOH M. npH-
m
HHTO 3anHcbIBaTb .lI.p06b IOn CJle,D.ylOmHM 06pa30M: M, fnn-i. . .mo. 1a-
. m
KHM 06pa30M, npH 3anIICH Ap06H lO" nOCJIe,D.HHe n nwpp ,lJ;ecHTHLJHOH
3anHCli qllCJla m OT.lI.eJJHIOT 3anHTOH. HanpHMep,  =5,71. ECJlH LJHC-
JIHTeJIb CO,lJ;ep)j{HT MeHee n ,lJ;eCHTH'IHbIX 3HaKOB, TO nepe,lJ; HIIM nHIIIYT
CTOJIbKO HYJleH, LJT06bI nOJlYLJHJlaCb n+l Ll,H<ppa, nocJle LJefO OT,lJ;eJJHIOT n
3t

3HaKOB C KOHI.J.a 3all51TOiL HanpUMep,
32 000 32
W = -.or = 0,0032.
KaKOBbI 6bI HH 6bIJIH HaTypaJIbHbIe "IHCJIa m, n, s, .llpOOH ln H
IO s m
10 n +s 9KBHB'aJIeHTHbI. B caMOM .lleJIe, m.lOn+s=lOn.lOSm.
IO s m
lh06bI nOJIYlIHTb 3anHCb .llpOOH IO nH ' Ha,zJ,O B lIHCJIe m/t. . .moO. . .0
(S HYJIeH) OT.n.e.JIHTb 3all5ITOH n+s l.lu<pp cnpaBa. B peaYJIbTaTe nOJIYlIHTCfl
m IO s m
.n.ecflTH4HaH .n.p06b M, m n -1. . .moO.. .0. TaK KaK .n.pOOH IOn H 10n+s
9KBHBaJIeHTHbI, TO 9KBHBaJIeHTHbI .n.P06H M, Tnn.-i. . .mo H M, m n -1' . .
.. .moO.. .0.
MbI .n.OKa3aJIH, TaKHM OOpa30M, liTO eCAU npunucamb K aeCRmUttH.OU
apo6u M, Tnn.-1' . .mo Jl1060e ttUCAO H.YAeU cnpaea, mo nOAyttumcR aeCRmUtt-
H.aR ap06b, 3K8U6a.AeH.mH.aJl aaH.H.Ou. 31'0 CBOHCTBO n03BOJIHeT BecbMa npo-
CTO npl1BOJl.HTb Jl.ecHTHlIHbIe .n.p06H K OJl.HOMY 3HaMeHaTemO. ECJIH Y nep-
BOH .llP06H nOCJIe 3anHTOH CTOHT n l.lH<PP, a Y BTOpOH P l.lH<PP, npH4eM
n<p, TO .n..'IH npHBe.n.eHHH 9THX Jl.po6eH K OJl.HOMY 3HaMeHaTe.JI1O Jl.OCTa-
TOllHO l1pHnHCaTb K nepBOH .n.pOOH p-n HYJIeii cnpaBa. Tor.n.a Y 06eHX
.llp06eH nOCJIe 3anHTOH 6y.n.eT CTOHTb nopOBHY l.lH<PP, a 9T0 H 3HalllfT,
liTO OHH HMelOT O.n.HH H TOT Me 3HaMeHaTe.JIb.
lI.JIH .n.p06eH, HMelOmHX O.llHHaKOBbIe 3HaMeHaTe.JIH, CJIOMeHHe H BbI-
lIHTaHHe CBO.n.HTCH K COOTBeTCTBYlOmHM Onepal.lHHM Ha,zJ, HX lIHCJIHTe.JIH-
MH. 31'0 n03BOJIHeT CBeCTH CJIOMeHHe H BbIlIHTaHHe .n.eCHTHlIHbIX Jl.pOOeH
K OnepaI.J.HHM Ha,zJ, HaTypaJIbHbIMH lIHCJIaMH. HanpHMep,
2,54 + 3,7126 = 2,5400 +3,7126 =
= 25 400 + 37126 = 62 526 = 6 2526
10000 10000 10000 ' .
B 06meM BH.lle' npaBHJIO CJIOMeHHH Jl.eCHTHlIHbIX Jl.pOOeH <P<>PMYJIH-
PyeTCH TaK:
D.AR moeo ttm06bl CAO;)fCUmb aee aecRmuttH.ble ap06u, H.Y;)fCH.O:
I) ypa8H.Rmb 8 3mux ap06Rx ttUCJlO iJeCRmUttH.blX 3H.aK08 nocAe 3llnJl-
moil, npUnUCb16aJl cnpaea, B CJlyttae H.e06xoaUMOcmu, K oafloil U3 3mux
apo6eil fleCKOJlbKO flYJlei1.;
2) om6pocumb 8 nOAyttU8UlUXCJl ap06RX 3llnJlmble u CJlO;)fCUmb nOJlY-
"leflH.ble npu 3mOM H.amypaJlbflble ttUCJlll;
3) 8 cyMMe omiJeAumb 3llnJlmoi1. cmOAbKO {)ICe ::maK08, CKOJlbKO omaeAeflO
8 Ka;)fCaOM U3 CJlll2aeMblX.
TOllHO TaK Me BbIBO.llHTCH npaBHJIa cpaBHeHHH H BbI4HTaHHH Jl.ecH-
THlIHbIX Jl.p06eH. HanpHMep, ttm06bl CpaeflUmb aee aecRmUttH.ble ap06u,
HOiJo ypaeflJlmb 8 flUX ttUCAO aeCRmUttflblX 3flaK08 nOCAe 3llnRmOU, om6po-
cumb 3llnJlmble u CpaeflUmb nOAyttOJ0W.ueCR flamypaAbflble ttUC/Ul:
4,62517>4,623, TaK KaK 4,623=4,62300, a 4,62517>4,62300, no-
CKOJIbKY 462 517>462 300.
320

PaeCMOTpHM Tenepb YMHOmeHl1e .lI.eeHTl1l1HbIX .lI.p06eii. fIycTb
M, mn-i.. .m O If P, Pq-I" .PO - .lI.Be .lI.eCHTl1tIHble .lI.P06H. Hx MomHO
sa Hca m p H m p - mp q 6
n Tb B BH,1{e IOn H TQ7i' 0 IO n ' lOq - lO n +q ' TO bI aanHCaTb
 6 u
lOn+q es 3HaMeHaTeJIH, HymHo B .lI.eeHTlfllHOH 3anHCIf HaTypaJIbHOrO
tIHCJIa mp OT.lI.emlTb 3anHTOH n+q nOCJIe.ll.HIfX II.1f<pP. OrclO.lI.a nOJIY'laeM
c.ne.ll.YIOIlI,ee npaBHJIO YMHO}KeHIfH .lI.ecHTH'IHbIX .lI.p06eH.
l.Jm06bl Hilumu npoU38eaeliue a6ljX aeCH.mUttlibLX apo6eti, liY:JfCliO:
1) om6pocutnb 8 sanucu amux opo6et1 3anH.mble,
2) liaumu np0U36eoeliue a6YX nOAyttU6UlUXCH. liamypaAbliblX ttUCeA,
3) 8 np0U38eae/-IUU OmaeAUtnb 3anJlmotl CmOAbKO nOCAealiUX 4U,pP,
CKOAbKO UX OmaeAeliO 6 n.ep80.M U 8mOpO.M .MliO:JfCUmeAH.X 8.MeCme (T. e.
n+q II.H<pp, eeJIIf B nepBOM MHOmlfTeJIe OT.lI.eJIeHO n II.H<PP, a BO BTOpOM q
u.1f<pp) .
BeebMa npOCTO BbInOJIHHeTCH YMHomeHHe lleCHTlf'lHbIX .lI.p06eH H3
QHCJIa Blflla lOP. no npaB:fJIaM .lI.eikTBIfH Ha.ll. CTeneHHMIf HMeeM:
m P loPm m
IOn .10 =""TI)iJ = IOn-p '
TaKIfM 06pa30M, eeJIH B 3a.ll.aHHOH .lI.p06H 6bIJIH OT.lI.eJIeHbI 3anHTOH no-
c.ne,ll,Hlfe n II.1f<pP, TO, 'IT06bI nOJIY'IHTb ln 'lOP, Ha.ll.O OT.lI.eJIIfTb 3anHTOii
nlfIllb n-p nOCne.ll.HIfX u.1f<PP, T. e. nepeHecTlf 3anHTyIO Ha p II.1f<pP Bnpa.
BO. ECJIIf 'IHCJIO II.H<PP nOCJIe 3anHTOH B 3anHCIf .lI.p061f ln MeHbIlle, 'IeM p,
TO Ha.ll.O npe.ll.BaplfTeJIbHO nplfnHCaTb CnpaBa eOOTBeTCTBYIOIlI,ee 'IHeJIO
OYJIeH.
C nOHHTlfeM .lI.eeHTlftIHOH .lI.p061f TeeHO CBH3aHO nOHHTlfe npo/{elima.
1
npOu.eHTOM Ha3bIBalOT .lI.p06b 100 ' Ee 0603Ha'laIOT 1 %, a p% 0603Ha-
QaeT .lI.p06b 160 . npOII.eHTbI If npO.MUAAU (T. e. sanlfclf Blf.ll.a p%o = 10 )
6bIJIIf BBe.ll.eHbI, KOr.ll.a He cyr.u,ecTBOBaJIO .lI.ecHTlf'lHbIX .lI.p06eH. lJT06b1
npOlf3BO.ll.HTb paC'IeTbI no 3aHMaM, Onpe,ll,eJIHJIH nplfpOCT KanlfTaJIa 1f3
paCtIeTa Ha 100 .lI.eHe>KHbIX e,lI,HHIfII.. 3rOT nplfpocT If Ha3bIBaJIH 'IHCJIOM
npOu.eHToB (pro centum - Ha CTO). B HaCTOHIlI,ee BpeMH nOHHTlfe npo-
u.eHTa HaXO.ll.lfT nplfMeHeHlfe B caMblX pa3HbIx 06nacTHx (3KoHoMIlKe,
XHMIfIf, npH COCTaBJIeHlf1f OTtIeTHOCTIf If T. ))..).
YnpalKnenBR
BblnOJH!HTe CJlell.YlOlllMe lI.eHCTBHII:
tI) (1,6: 1,28)+(1,5: 0,24)+ (1,1 : 0,08);
6) (1,14+0,76); (1,14-0,76)+0,54: 0,012;
B) 1 : 2.5+1,44 : 3,6+3,6 : 1,44. (0,1-0,02);
r) (0.45: 0,9+0,9 : 0.45+ 1,5: 3+0.242 : 0,11) : (2,3-1,26).
2. npeoopa30BaHlfe 06b1KHoBeHHblx JJ.pOOeH B JJ.ecSlTHQHble. l{p06b
8 32
25 3KBHB8JIeHTHa .lI.P06H 100 ' Ii nOToMY ee MO/KHO 3anHcaTb B BH)).e
11 N. 702 321

0,32. BbIHCHHM, npH KaKOM yCJIOBHH .n.p06b  9KBHBaJIeHTHa .n.ecHTH'IHOH
p06H.
/J/lR m020 Ilm06bl HecOKpamU.M.aR apo6b  6wla 8K1JU8a/leHmHa aecR-
mu'Utou op06u, He06xOOUMO u OocmamOllHo, Ilm06bl 8 pCl3/lOOlCeHUe ee 3fUl-
MeHame/lR n Ha npOCmble .MHOUme/lU eXoOU/lU /lUUlb npOCmbte IlUC/IfJ
2 u 5.
B caMOM .n.eJIe, nYCTb pa3JIO)f{eHHe n Ha npOCTble MHO)KHTeJIH HMeeT
BH.n. n=2' .5 8 H nYCTb, HanpHMep, rs. Tor.n.a, YMH02KaH 'IHCJIHTeJIb H
BH8MeHaTeJIb .n.p06H  Ha &,-8, nOJIYllaeM:
n
m m 5,_8 m 5,_8 m
n = 2'5 8 "" 2'6 8 5,-8 = 2'5" .
Ho 2' .5'= 10', u nOToMY
m {Y-8m
n"" JijT'
81-0 8HaIlHT, liTO .n.P06b .!!!.. 9KBHBaJIeHTHa .n.ecHTHtIHOt% .n.P06H.
n
06paTHO, nYCTb HecoKpaTHMaH .n.P06b ; 9KBHBaJIeHTHa .n.eCSITH'IHOH
a
.!tp06H 10" T. e. lO'm=an. ECJIH 6bI B pa3JIO)f{eHHe n Ha npOCTble MHO-
>KHTeJllf BXO.!tHJIO npOCToe qHCJIO p, OTJIH'IHOe OT 2 H 5, TO lO'm ,AeJIHJIOCb
()bI HS p. Ho B pa3JIO)f{eHUe lOr Ha npOCTbIe MHO)KIfTeJIH BXO.n.HT JIHillb 2
H 5, a nOTOMY lOr He.n.eJIHTCH Ha p. Tor.n.a (CM.  2 rJI. X) m .n.eJIHJIOCb
m
6bI Ha p H .n.p06b - MO)KHO 6bIJIO ObI COKpaTIITb Ha p BonpeKH npe.n.no-
n
JIO)f{eHHIO. nOJIY'IeHHoe npOTIfBOpe'lHe nOKa3bIBaeT, 1JT0 n He MO)f{eT
HMeTb npOCTblX MHO)f{HTeJIeH, OTJIH'IHblX OT 2 H 5.
191
n p H M e p 1. TaK KaK 250=2.5 3 , TO .!tpOOb 250 9KBHBaJleHTHa
Jl,ecHTHqHOH .n.p06H:
_ d91 _ 764 -0764
250 - 2.5a - I 000 -, .
9
n p H M e p 2. D.p06b 14 HecoKpaTHMa, npH'IeM 14=2.7. TaK KaK
B pa3JIO)f{eHHe 3HaMeHaTeJIH BXO.n.HT MHO)f{HTeJIb 7, OTJIH'IHbIH OT 2 H 5,
9TY .n.p06b HeJIb3H npeBpaTHTb B .n.ecHTHIIHYIO .n.P06b.
195
n p H M e p 3. B pa3JIO)KeHHe 3HaMeHaTeJI H .n.P06H 260 BXO,AHT
MHO)f{HTeJIb 13, OTJIHIIHbIH OT 2 H 5. Ho 9Ta .n.p06b cOKpaTHMa. COKpaIlI,aH
3
ee Ha 65, nOJIyqHM .n.p06b 4' KOTOpylO MO)f{HO npeBpaTHTb B ,AeCHTH'I-
3
nylO: "4=0,75.
322

Yo'paJKnenM" -
I. Onpe.1\eJlHTe, KaKHe H3 CJIe,u.YIO!I.(HX .lij>o6eii MQJKHO 3aOHCaTb B BII,D,e KOHe'lHblX
J{ecHTH'IlIblX JqJ06eii:, H 3anHIllHTe HX B 3TOM BII,D,e: .
17 42 52 385
a) 640 ; 6) 875 ; B) 75 ; r) ro8 '
2. 3anHlllHTe CJle.1\YlOlllHe J{ecHTH'IHble ./I.P06H B BH.1\e HecoKpaTHMYX OOwKHOBeH-
HbiX ltp06eii::
a) 0,125; 6) O,25; B) 0,1375; r) 0,2454.
3. 6ecKOHeIfHble nepHO,ll,KIfOCKHe. ,ll,ecHTH'IHIoIe )Q)06w. .lI.p06b 
HeB03MO)KHO npeBpaTHTb B KOHe'lHYIO .necHTH'lHYIO .l(p06b. Ho, .l(eJIH I
1 I
Ha 3, nOJIY'laeM, 'lTO 0,3<i-<0,4. D.a.n:ee HaXO)I,HM, 'lTO 0,33<3'<
1
<0,34,0,333<3<0,334 H T.lI.. BoooIlI,e, .l(JHI JIIOOoro n HMeeM:
1
9,33. . . < 3< ,33. . .4;
n  H$ P n  H<PP
BMeCTO Taro 'IToOb! nHCaTh oeCKOHe'lHOe MHO>KecTBO HepaBeHCTB, rOBO-
1
pHT, 'ITO .l(p06H 3 COOTBeTCTBYeT oeCKOHe'lHaH .n:eCHTH'IHaH J!.p06b
0,333. . .3... 31'0 03Ha'laeT, 'ITO ecJIH oropOCHTb B 6ecKOHe t lHOH: .n:p06H
1
Bce I.J.H<pPbI, Ha'lHHaH C HeKOTOpOH:, TO nOJIY'lHTCH 'lHCJIO, MeHbIllee 3' a
ecJIH B nOJIY'IeHHOM 'lHCJIe YBeJIHtlHTb nOCJIe.n:HIOIO I.J.H<PPY Ha 1, TO nOJIY-
1 '
IJHTCSI HCJIO, OOJIbillee 3' .
KOHe'lHble .n:ecHTH'IHb!e .n:pOOH MOlKHO TO)Ke 3anHcaTb KaK oecKoHe'l-
HbIe .n:p06H, npHnHcaB K HHM crrpaBa nOCJIe.n:OBaTeJIbHOCTb HYJIeH::
0,25=0,25000. . .0. .. . 3.l{ecb npH OTt5paCb!BaHHH Bcex mpp, Ha'lHHaH
C HeKOTopoH, nOJIY'lHTCH 'IHCJIO, He npeBOCXO.l(HIlI,ee 'lHCJIa 0,25 (Ha-
npHMep, ecJIH OCTaBHTb JIHillb o.n:HY I.J.mpPY nOCJIe 3anHTOH, TO nOJIY'lHTCH
'lHCJIO 0,2, MeHbillee 0,25, a eCJIH OCT8BHTb TPH I.J.H(ppb! nOCJIe 3anHTOH:,
TO nOJIY'IHTCH 'lHCJIO 0,250, paBHoe 0,25). ECJIH )Ke nOCJIe OTOpaCb!BaHHH
YBeJIH'lHTb nOCJIe.n:HlOIO OCTaBIllYIOCH I.J.mpPY Ha 1, TO nOJIY'IHTCH 'IHCJIO,
OOJIbIllee 0,25 (HanpHMep, 0,3 HJIH 0,251).
Mb! BH.n:HM, tITO Ka)K.n:oe nOJlO)KHTeJIbHoe pau.HoHaJIbHoe 'IHCJIO MO>K-
HO rrpe.ncTaBHTb B BH.n:e 6eCKOfletf.flOll aeCRmUtf.flOU apo6u. nOJIY'lalOIlI,HecH
npH 3TOM .n:ecHTH'IHble .n:p06H OKa3b!BaIOTCH nepuoiJUtf.eCKUMU. 3T0 3Ha-
'lHT, tlTO, Ha'lHHa5J C HeKOToporo MeeTa, OHH nOJIY'IaIOTCH OeeKOHetlHblM
nOBTopeHHeM Oll.HO" H TOft: )Ke rpynnbl H<!>P. HanpHMep, tlHCJIO ;1 BbI-
paJKaeTCH OecKOHe'lHOH .n:ecsIm'lHOH: .zq>OOblO 0,27272727.. .27. .., a
'lHCJIO :5 - 6ecKOHe'lHOH .n:eeSITH'IHOK .l(p06blO 0,1454545.. .45. .. .
l{JIH KpaTKOCTH nepBYIO H3 3THX .n:poOeH: nHIllYT B BH.n:e 0, (27), a BTa-
PYIO - B BH.n:e 0,1 (45). B CKOOKH 3aKJIIO'IaIOT nOBTOpHIOIlI,YIOCR rpyn-
oy I.J.H<!>P, KOTOp)'1O Ha3bIBalOT nepuOOo"" 3roH ,lij>oOH. OrMeTHM, 'lTO BMe-
11*
3tl3

CTO 0, (27) MO)!{HO 6bIJIO 6bI HanHcaTb HO,2(72), HO 9Ta 3anHCb 6b1JIa 6bI
Jl.JII{HHee.
OpHlJHtla nOHBJIeHHH nepHOJl.a COCTOHT B CJIeJl.YIOeM. 0YCTb Ha,no
o6paTHTb B 6ecKOHelJHYIO ,necHTHlJHYIO Jl.p06b lJHCJIO a, npe,ncTaBJIHeMoe
HecoKpaTHMoH ,np06blO !!!..... lI.JIH 9Toro Ha,no BbinOJIHHTb Jl.eJIeHHe HaTY-
n
paJIbHoro 1iHCJ1a m Ha HaTypaJIbHOe lJHCJIO n. OpH 9TOM 6y,nYT nOJIYlJaTb
CH OCT8TK, MeHbillHe n, T. e. lJHCJIa BH,na 0, I,. . ., n-l. ECJIH XOTH 6b1
OJl.HH H3 oer8TKoB OKameTCH paBHIM HYJIIO, TO nOCJIe Jl.eJIeHflH nOJIYlJHT-
CH KOHeQH8H Jl.ecHTHlJHaH Jl.pOt5b (HJIH, lJTO TO )!{e car-me, 6ecKoHelJHaH
lI.ecHTHlJHaH lI.P06b, 3aKaHlJHBalOaHCH nOCJIe,zI.OBaTe.'IbHOCTblO HYJIeH).
ECJIH me Bce OCTaTKH OTJIHlJHbI OT HYJIH, TO .Jl.eJIeHHe HHKorJl.8 He 3aKOH
'IHTCH, HO nOCKo.'IbKY KOJIH'lecTBO pa3JIHlJHbIX OCTaTKOB KOHelJHO, TO, Ha-
'IHHaH C HeKOToporo wara, KaKOH-TO OCTaTOK nOBTopHTCH, a TOrJl.a B
'laCTHOM HalJHYT nOBTopHTbCH I.I.H<PPbI.
MomHo Jl.OKa3aTb, lJTO eCJIH B pa3JIo)J{eHHe 3HaMeHaTeJIH HecoKpaTH-
MOH lI.P06H Ha npocTble MHO)J{HTeJIH He BXOJl.HT UH<pPbI 2 H 5, TO npH 06-
paeHHH ,noH Jl.P06H B 6ecKoHelJHYIO Jl.ecHTHlJHYIO Jl.p06b nOJIYlJHTCH
'lucmaJl nepuoiJut.tecKaJl iJP06b, T. e. ,np06b, nepHoJl. KOTOpOH HalJHHaeTCH
cpa3Y nOCJIe 3anHToH. ECJIH )!{e B pa3JIO)J{eHHe 3HaMeHaTeJIH BXOJl.HT
MHO)!{HTeJIb 2 HJIH 5, TO nepHOll.HlJecKaH Jl.p06b OKa3bIBaeTCH CMeutafl-
flOU - Me)l{)l.Y 3anHTOH H HalJaJIOM nepHOJl.a 6YJl.eT HeCKOJIbKO UH<pP (a
I1MeHHO CTOJIbKO, KaKOB 60JIbWHH H3 nOKa3aTeJIeH CTeneHeH MHomHTe-
JleH 2 H 5). HanpHMep, ecJIH n = 2 3 .5 2 .7.11, TO Me)l{)l.y 3an HTOH H HalJa-
JIOM nepHOJl.a OKamYTCH TPH UH<PPbI.
HH)!{e Mbi Jl.OKa)!{eM, lJTO Ka)!{Jl.OH 6eCKOHelJHOH Jl.ecHTHlJHOH Jl.p06H
COOTBeTCTBYeT lJHC,'10, npHlJeM Ha,lI, 6ecKoHelJHbIMH ,lI,eCHTHlJHbIMH ,nP06H-
MH MomHO BbInOJIHHTb ,lI,eHCTBHH aHaJIOrHlJHO TOMY, KaK 9TO ,lI,eJIaeTCH
}1,JIH KOHelJHbIX ,lI,eCHTHlJHbIX ,lI,p06eH. OOKIDKeM, KaK, nOJIb3YHCb 3THM,
MomHO JII06yIO nepHO,lI,HlJeCKYIO ,lI,p06b (KaK lJHCTYIO, TaK H CMewaHHYIO)
m
3anHcaTb B BH,lI,e ,lI,p06H BH,lI,a -.
n
nYCTb Jl.aHa nepHO,lI,HlJeCKaH ,lI,p06b 0, (24), T. e. 0,242424.. .24. ,. .
0603HalJHM cooTBeTcTBYIOee eH lJHCJIO lJepe3 a. ECJIH nepeHecTH 3anH-
TYIO Ha ,lI,Be UH<PPbI BnpaBo, lJHCJIO a yBeJIHlJHTCH B 100 pa3 H MbI nOJIY-
lJHM, lJTO lOOa=24,242424.. .24..., T. e. 100a=24+0,242424.. .24.,.=
24
=24+a. PewaH ypaBHeHHe 100a=24+a, nOJIYlJaeM, 'lTO a= 99 ' T. e.
8 24
a= 33 ' 3aMeTHM, '1TO 24 O,lI,HOBpeMeHHo H lJHCJIHTeJIb ,nP06H 99 ' H
nepHO,lI, Jl.P06H 0, (24).
TOlJHO TaK )J{e 06paaeTCH B 06bIKHoBeHHYIO ,lI,p06b JII06aH lJHCTaH
nepHo,lI,HlJeCKaH ,lI,eCHTHlJHaH ,lI,P06b.
flpu 06pa'flUU 8 06blK.fl08eflflYIO iJpo6b t.tucmou nepuolJuttecKOU iJecfl-
muou po6u flOAY'laemCfl ()p06b, 'lUCAUmeAb KOmOP?ll pCl8efl nepuooy, a
8fla.MeflameAb cocmoum U3 maKOZO t.tuCAa iJe8JlmOK, CKOAbKO ltuq;p 8 ne-
puoJe iJpo6u:
0,(35) = : ; 0,(489) = :: =  H T. ,n.
324

AHaJIOrHqHO BbIBO)l.HTCH npaBHJIO )l.JISI 06pam.eHHSI B 06blKHOBeHHYIO
)l.P06b CMeUlaHHon nepHO)l.HqeeKOn )).eeSlTHqHOn )l.p06H.
Be/lu t{i!.IlaJl IfO.Cmb anwu ap06u pCl8H.a flY/lIO. nw tW/lYttaemcR OP06b.
'l.UC/lUme/lb KOnwpOU pCl8efl pa3flOCmU .M£3fCijy 'l.UC/lOM. sanucaflflblM 14UCP-
paMU, emORU{UMU 00 flatta/la enwpoeo nep uoiJa , U 'l.UC/lOM, sanucaflHblM
14ucppaMU, emORU{UMU 00 H.atta/la nep8020 nepuoiJa; 3HaMeH.ame/lb eoemoum
U3 maK020 ''l.UC/la OeeJU1WK, eKO/lbKO 14wjJp e nepuoiJe. U maK020 'l.UC/la flY-
/leU, eKO/lbKO 14UCPP emoum ao H.atta/la nep80eQ. nepuoiJa. HanpHMep.
O 7(61) _ 761-7 _ 754 _ 377
, - -ggo--990-495"
YopamHeHBR
t. CJJeilYlO!l1He 06b1KHoBeHHble ,/.\p06H 06paTHTe B 6ecKoHe'lHble ilecllTH'IHble .ZJ:P06H:
3 4 17 36
a) 7; 6) 35 1 B) 24 ; r) 77'
2. CJJeilYlO!l1He nep:IO,/.\HqecKHe ,/.\ecllTHqHble ilp06H 06paTHTe B 06b1KHOBeHHble:
8) 0, (31); 6) 2,(75); B) 0,34(9); r) 0,27(15).
9 3. IIOJIOJKIITEJILHLIE ,nEftCTBIITEJILHLIE 'IIICJIA
t. HecoH3MepHMble OTpe3KH. C nOMOblO nOJIOmHTeJIbHblX paUHo-
HaJIbHblX qHCeJI MomHO C JII060n CTeneHblO TOqHOCTH Bblpa3HTb peaYJIb-
TaT H3MepeHHSI Ton HJIH HHon BeJIHqHHbI. nYCTb. HanpHMep, H3MepSleTCSI
J].JIilHa OTpeaKa OA H Tpe6yeTCSI nOJIyqHTb 3HaqeHHe 3ToA )l.JIHHbI C OUlH6-
v u I v D.
KOH, He npeBOCXO)l.SIeH IOn -H e)).HHHqHOrO OTpe3Ka. JISI 3Toro no-
CTynHMCJIeJ].YIOHM 06pa30M. Ha npSlMoA OA OT TOqKH 0 OTJIOmHM 8
HanpaBJIeHHH TOqKH A O)l.HH 3a )).PYfHM OTpeaKH, )l.JIHHbI KOTOpblX Bbl-
1
pIDKaIOTCSI )).p06blO IOn ' TOqKa A nona)).eT Ha O)).HH H3 3THX OTpe3KOB
(CM. pHC. 181, r)).e Bbl6paHo n= 1). 3TO 3HaqHT, qTO eYUl/!Cmsyem Heompu-
tIllme/lbHoe 'l.UC/lO m. 06/laoalOU¥!e eAeiJYIOUJ,uM e80UcmsoM: ompe30K, O/lUHa
KOmopoeo pCl8Ha In ' .M£flbLUe Ompe3Kll OA. a ompeaoK. uMelOUJ,Uu O/lUH!I
m+ 1 OA
I"()IJ' 60/lbLUe, If.eM ompe30K .
TaKHM 06pa30M, )l.JIHHa OTpeaKa OA )).OJImHa 6b1Tb 3RKJIIOqeHa Me}f{)l.Y
rn m+ 1
'lHCJIaMH IOn H ""T(jn. TOqHO TaK me MomHO B3BecHTb JII060e TeJfO C TO"-
1 O m m+ 1
HOCTblO )1.0 IOn e H T.)).. )).HaKO paUHOHRJIbHble qHCJIa IOn H Ion )).alOT
JIHUlb npHMIDKeHHbIe 3HaqeHHSI J].JIHHbI OTpe3Ka OA no He)l.ocraTKY H no
H36bITKY, HO He )).alOT OTBeTa Ha Bonpoc: 'leMY B TOqHOCTH paBHa J].JIHlIa
9Toro OTpe3Ka? D.eJIO B TOM. lITO OTBeTHTb Ha 9TOT Bonpoc. OrpaHHqHBa-
SlCb JIHUlb paUHOHaJIbHblMH qHCJIa- I I I I I I I I I I I I I I I I 1 I , I ..
MH. BO MHOfHX CJIyqaHX HeB03MO)K- a 2 A
HO - cyeCTBYIOT OTpeaKH, Heeo-
H3MepHMble C e)).HHHqHbIM OTpe3KOM PHC. 181
325

e, T. e. OTpe3Kl-i, .nJlHHY KOTOpbIX HeJlb351 BbIpa3HTb, nOJIb3Y5JCb .1JHllIb
paUHOHaJIbHbIMH 'lHCJIaMH. CYlll.-ecTBOBaHHe TaKHX OTpe3KOB BblTeKaeT
H3 CJIe.nYIOw.ero YTBep)K)J.eHHH: .
P'uaeoHa/lb K8Cldpama HeCOU3MepUMQ. C ezo CRWfJOflOU.
B caMOM JIe, nYCTb AJIHHa cropOHbI KBa.npaTa palma 1. npe.nnoJIo-
>KHM, 'lT0 .nH8rOH8JIb AC KBa.npaT8 ABCD cOHSMepHMa C ero CTOpOHOA
H ee .nJIl-IHa BbIpIDKaeTCH HecoKpaTHMoH .np06blO .E... no TeopeMe nH<pa-
q
fopa Tor.na HMeJIO Obi MeCTO paBeHCTBO IABI2+ IBC]2= IACI2, T. e. l2+
2
+12= P2 . 113 Hero CJle.n.YeT, Il'ro pl=2q2. 3Ha1J:HT, p2 - 'IeTHOe 'lHCJIO, a
q
Tor.na H 'lHCJIO P 'leTHO (KBa.npaT He'leTHOrO qHCJIa He Mor 6bI 6bITb 'leT-
HbIM). I1TaK, p=2pi. 3aMeHHB B paBeHCTBe pl_2q2 'IHCJIO P Ha 2Pi, no-
nY'laeM 4pf=2q2, T. e. 2p=ql. OrclO.na CJIe.n.YeT, qTO q'J qeTHO, a, cJIe.no-
BaTeJIbHO, q - 'leTHoe 'IHCJIO. I1T2K, qHCJla p H q 'IeTHbI, a nOToMY .nP06b
1- O>KHO COKpaTHTb Ha 2, qTO npOTHBOpeqHT npe.nnOJIO)f{eHHIO 06 ee
q
HecoKpaTHMOCTH. 31'0 npOTHBopeqHe nOKa.3bIBaeT, 'ITO ecJIH 3a e.n.HHHUY
)l.JIHHbI BbI6paHa cTopoHa KBa.npaTa, TO .nJIHHY .nHarOHaJIH 3Toro KBa.npaTa
HeJIb3H BbIpa3HTb paUHOHaJIbHblM qHCJIOM, T. e. 9Ta .nHarolIaJIb HeCOH3-
JdepHMa co cropoHoA KBa)l.paTa.
lI.JIH Toro qT06bI MO)f{HO 6blJIO BbIpa3HTb qHCJIOM pe3YJIbTaT H3Mepe-
IHIH JI106oro OTpe3Ka, Ha.no pac III H P H T b MHO>KecTBO Q + rrOJIO)f{H-
TeJIbHblX pau,HOHaJIbHbIX 'lHceJI, nOnOJIHHB ero HOBbIMH qHCJIaMH. nOJIY-
lJalOw.HecH npH 9TOM qHCJI3 Ha3blBaIOT nOAoumellbHbtMU det1cmeumellb-
HblMU qHCJIaMH, a MHO)f{eCTBO TaKHX 'lHCeJI 0603HaQaIOT R+. Pa3YMeeT-
C51, Ka)K)J.oe nOJIO>KHTeJIbHoe p8lI.HOHaJIbHOe qHCJIO )lpJUKHO npHHaMe-
>KaTb R+. T. e. .nOJI)f{HO BbIllOJIHHTbCH BKJIlOqeHHe Q+ c:.R+. KpOMe Toro,
Ha.no onpe.neJIHTh Onepau.HH CJIO)f{eHHH H YMHO}KfHHH B R+ TaK, 'lT06b1
M51 paUHOHaJIbHblX qHCeJI OHH COBna.naJIH c TeMH, KOTopble onpe.neJIeHbI
B Q +, H IJTOObl MepbI OTpe3KOB H nOCJIe pOCIIFHpeHHH MH<m<ecTBa qRCeJI
06JIa.naJIH cBolicTBaMH 8,JUI,HTHBHOCTH H MYJIbTHl1JIHKaTHBHOCTH (CM.
 I, n. 1, a) 6).
2. nOJlO>KHTeJIbHble )l,eHCTBHTeJIbHble 'IRCJla H 6ecKOHe'lHble )l,eCHTH'I-
tlble .npOOH. Mbl nOKa>KeM B 3TOM nYHKTe, lITO pesYJIbTaT H3MepeHHH
JII060ro OTpe3Ka MO>KeT OblTb 3anHcaH B BH.l1.e 6ecKoHeqHOH .necHTHqHOH
.nP06H (BoooW.e rOBopH, HenepHo.nH'lecKOH). B caMOM .ne.ne, nycTb Bbl-
t)paH eJJ;HHH'lRbIfi OTpe3OK e H sa.s:aH HeKOTOpbIfi OTp.e3OK a. Tor.na JIH60
a MeHbllIe, qeM e, mIDo u.aH.neTCH TaKoe illnypaJIbaoe 'lHCJIO n, qTO
n 'ea«n+ l)e. 31'0 HaTypaJIbHOe qHCJIO n (HJIH qHCJIO 0, ecJIH a
MeHb!lle e) Ha30BeM I{eAOU tto£mbJO Q/lUHbl Ompe3Kll a.
ECJIII a  n 'e, TO JI.JIHHa OTpe3Ka a BblpIDKaeTC51 HaTypaJIbHbiM qHC-
nOM n. B npOTHBHOM CJIY'lae a  ne+ai, r.ne al<e. Ho Tor.na HaH.neTCH
qHCJIO ni, npHHHMalOw.ee O)l,HO M3 3HaqeHHH 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
nl __ nl+1 ( + nl + <
TaKoe, qTO 10 'eal<' H nOTOMY n 10 1"""",n -e 81
«n+  )e. 31'0 3HaqHT, 'lT0 (n, nl)ea«n, nl+0, I)e).
J) n, nl - .IIecRTHqHaJl .IIp06b, HanpHMep 7,6.
326

ilpo.n:OJDKaH OllHCaHHbIH npoLlecc Il3Me.peHHH .n:aJIee, IIOJIyqliM 'lHCJIa
n2, na, . .., nk, .." npIlHIlMalOUl,He oAHO H3 3HaqeHI<IH 0, 1, 2,3,4, 5,
6, 7, 8, 9 II TaKHe, qTO .lJ.JIH JII06oro k OTpe30K a He MeHbllle, qeM OTpe30K
1
(n, nln 2 . . . nk)e, HO MeHblIle OTpe3Ka (n, nj. . .nk+ lOk )e.
OrtJeT 0 npou.ecce H3MepeHHSI orpe3Ka a MG1KHO By,qa3HTb 6eCKOHeq-
HOH .lI.ecSlTHqHOH .n.p06blO n, nln2' . .nlt. .. . ECJIH OT6pOCHTb Bce UIl<t>pbl
,nOM .n:po6n, HaqHHaSl C HeKOTopOH, TO nOJIyqHTCSI qHCJIO n, ni. . .n,,,
MeHbIIlee .n:JIHHbI H3MepHeMOro OTpe3Ka; ecJIH )Ke YBeJUlqHTb B nOJIyqeH-
HOM qHUle nOCJIe.n:HIOIO un<t>py Ha 1, TO nOJIyqHTCSI qHCJIO, KOTopoe 60JIb-
lIle .lI.JIHHbI .:noro OTpe3Ka. n03TOMY MbI CKaJKeM, tJTO .n:JIHHa OTpe3Ka a
BbIpa)KaeTCSI ,ltpo6blO n, nl', .nk"', T. e. qTO m(a)=n, nl', .nk'"
HanpHMep, m(a)=3,1764... .
HCHO, tJ:TO ,ltJIH JIlo6oro k BbInOJIHeHbI HepaBeHCTBa:
1
n, nl', .nltm(a)<n, nl" .nk+ 10 k '
npH H3MepeHHII OTpc3KOB HHKor.n:a He nOJIyqHTCH .n:po6b, 3aKaHIIHBalO-
Ul,aSlCSI 6ecKoHe t IHoH nOCJIe.n:oBaTe.n:bHOCTbIO ,lteBHTOK, HanpHMep .n:P06b
BH.lI.a 0,499. . .9. .. . .lI.e.n:o B TOM, qTO Hii O.n:HO qHCJIO x He MO)KeT y.n:OB-
JIeTBOpHTb BceM lIepaBeHCTBaM:
0,4x<O,5
0,49x<0,50
0,49. . .<0,50. . .0.
ECJIH BMecTO 3THX HepaBeHcTB HanOC8Tb
0,4<x0,5
0,40,50
0,49. , .9<xO,50. . .0,
TO HM 6y.n:eT y,ltOBJIeTBOpHTb qHCJIO 0,5. n03TOMY CqIlTaIOT, qTO .n:ecSl-
THqHaH .n:p06b 0,4999...9... =0,4 (9) HBJIHeTCH .n:pyroH 3anHCblO ,lI,JIH
qHCJIa 0,5. Bo06Ul,e, ecJIH B3HTb KOHeqHYIO .n:ecHTHqHYIO p06b, YMeHb-
IllHTb ee nOCJIe,zI,HIOIO UIl<t>py Ha 1 H npHnHcaTb K HeH cnpaBa 6ecKoHeIIHYlO
nOCJIe,zI,OBaTe.n:bHOCTb ,lteBHTOK, nOJIyqIlTCH 6ecKOHeqRaH ,ltecHTIlqHaSl
,ltp06b, paBHaH 3a,n.aHIlOM. HanpHMep,
0,232=0,23199. . .9. . .;
7,8=7,79.. .9... .
MbI nOCTaBHJlH B CoOTBeTCTBHe KIDK,lI.OMY OTpe3KY 6ecKoHeIIHYIO .n:e-
CHTHqHYIO .n:p06b. 06paTHo, )I.JIH KCUK.lI.oi;j 6ecKOHeqHOH P,ecHTHqHOH
.n:p06H, He 3aKaHqIilB8JOw.eHCH nOCJIe,zI,OB.aTe.n:bHOCTblO .n:eBSlTOK, HaH.n:eTCSI
OTpe30K, MHHa KOTOpOro BbIpIDKaeTCS 3ToH p,p06blO.
0603HaqHM MecTBO 6ecKoHeqHbIX p,ecRTHqHbIX .n.po6eii, OTJIHqHbIX
OT 0,00. . .0. . . H He 3aKaHqHBalOUl,HXCH nOCJIe.n:OBaTeJIbHOCTblO ,lteBHTOK,
qepe3 R+ H Ha30BeM MHO)KecTBOM nOAO:JfCUmeAbHblX aeficmeumeAbHblX
'-tuceA.
327
_.

LLJIfl K3}1{)1,OrO nOJIO)l(HTeJIbHOrO jl.eHCTBHTeJlbHOrO "IHCJI3 x MO)l(HO
y K333Tb ero npH6JIH)I(eHHble 3H3'1eHHH. [J pU6AU:JfCeHHOe 3HatteHue no
1
t/eoocmamKY C TO'lHOCTblO jl.0 IO k nOJIY'IHTCH, ecJIH OCT3BHTb ueJIylO tI3CTb
'1HCJI8 H nepBbIe k UH<I>P nOCJIe 38nHTOH, 8 Bce OCT8JIbHbie UH<I>PbI OT-
6pOCHTb. Ero 0603HaQ8IOT Xh' I1HblMH CJIOBaMH, eCJIH x=n, ni...
1
" .nh'", TO Xk=n, n1.. .nh' npH68BJIHH K 3TOMY tIHCJIY IO k ' nOJIYQHM
npU6AU:JfCeHHOe 8H.aIleHUe Xk OAR x no U36bl11lKY, Xk=n, nj.. .nh+ 1k '
ECJIH UH<I>P8 nh OTJIH'IH3 OT 9, TO jl.JISI nOJIytIeHHH Xk }J.OCT8TO'lHO YBeJIH-
'lHTb nh H3 1.
H3npHMep, ecJIH x=4,7128356..., TO xs=4,712 H x;=4,713. OtIe-
BHJJ.HO, tITO }J.JIfl JII06oro nOJIO)l(HTeJlbHOrO jl.eHCTBHTeJIbHOrO tlHCJI8 x
cnp3Be}J.JIHBbI Hep3BeHCTB3 XhX<Xk'
YnpaacoeoHH
1. HanullluTe npH6JIu)I{eHHble 3HaqeHUJI no HeI(ocTaTKY H no H36b1TKY I(JIJI qHCJla
=3,847198... C roqHOCTblO I(O: a) 0,01; 6) 0,0001; B) 0,00001.
2. npU6JIU)I{eHHOe 3HaqeHUe no neI(oCTaTKY C TO'lHOCTblO 11.0 0,001 ,!Iml 'IUCJIa x
paBHo 1,754. 4eMY paBHo npU6JIU)I{eHHOe 3HaqeHHe no H36b1TKY ,!IJIJI 3roro 'IHCJIa C
roqHOCTblO .ZJ:O O,OOI? a C TOqHOCTblO .ZJ:O 0,01? MO)l{eT JIH qHCJIO x 6b1Tb 60JIbllle, qeM
1,756?
3. OTHoweHHe nopH}J.Ka B MH01KeCTBe R+. nYCTb 38j1.3HbI }J.B8 no-
JIO)l(HTeJIbHbIX }J.eHCTBHTeJIbHbIX tIHCJI8 X H y:
x=m, mi.. .mh""
y=n, ni.. .nh' .. .
1.IHCJIO X .MeHbUle tIHCJIa y, ecJIH m<n HJIH ecJIH H8H}J.ercSi T8Koe k,
'ITO m=n, ml=nl,..., mh-i=nh-j, HO mh<nh' B 3TOM CJIY'l3e npH
s;?k npH6JIH)I(eHHOe 3H3'1eHHe X; }J.JISI X no H36bITKY OK8)1(erCSI He 60JIb-
we npH6JIH)I(eHHoro 3H3'1eHHSI Ys }J.JISI Y no He,n,OCT8TKY, xyS' n03TOMY
MO)l(HO CK333Tb, QTO x<y B TOM H TOJIbKO B TOM CJIYQ3e, KOrjl.3 H3Hjl.erCSI
raKoe S, tITO X;s'
JIerKo npOBepHTb, tITO omHoUleHue «<» 8 MHO:JfCeCtn8e R+ R8ARemCR
cmpoeuM AUHeUHblM nopRiJKoM, m. e. OHO lEUM.MempWl.HO, mpaH3umueHo
u npu X=FY Au60 x<y, Au60 y<x. B MHO)l(ecTBe R+, K3K H B MHO>KecTBe
Q " Her HH HaHMeHbwero, HH H8H60JIbwero 3JIeMeHT8. KpOMe TOro,
MC>K}J.Y JII06bIMH jl.BYMSI tIHCJI3MH H3 R+ JIe)l(HT 6eCKOHe'lHO MHoro p3UHO-
H3J1bHbIX tIHCeJI
O}J.HHM H3 BIDKHeHwHX CBOHCTB OTHOllleHHSI nopSl}J.K3 B MHO>KecTBe
R +, OTCYTCTBYIOIlI,HM B Q +, flBJISIerCSI CBOHCTBO Henpepbl8HOcmu. 1.IT06bI
('lOPMYJIHpOB3Tb 3TO CBOHCTBO, BBe,n,eM CJIe}J.YIOIlI,He nOHSITHSI.
LJUCA08blM MHO:JfCeCmeOM H83bIB8lOT JII060e nO}J.MHO)l(eCTBO MHO)l(ecTB8
R,1) (H8npHMep, MHO>KecTB3 N, Q+, MHO>KecTBO nepHMerpOB np8BHJIb-
1) n03)1{e MbI 0606!l1HM nOHJlTHe 'IUCJIOBOrO MHO)l{eCTBa, onpe,/.\MUB ero KaK no,/.\-
r, ill )KeCTBO MHO)l{eCTBa Bcex AeHCTBHTeJIbHblX qHCM.
,)":8

HbIX MHoroyrOJIbHHKOB, BnHCaHHblX B .n:aHHYIO OKPY)l{HOCTb, qHCJIOBue
mpe3KH H npOMe)KYTKH H T. .n:.).
CKa)KeM, qTO qHCJIOBOe MHO)KecTBO X pacnO/lO:JICel-lO C/le8a OT qHCJIOBO-
ro MHO)KecTBa Y, ecJIH .n:JHI JIIOOblX x EX H Y E Y BbInOJIHHeTCH HepSBeH-
CTBO xy. HanpHMep, OTpe30K [I; 4] pacnOJIO)KeH CJIeBa OT OTpe3K8
[6; 10], a MHO)KecTBO nJIOIlI,a.n:eH MHoroyrOJIbHHKOB, BnHcaHHblX B .n:au-
HbIll Kpyr, pacnOJIO)KeHO CJIeBa OT MHO)KecTBa nJIOIlI,a.n:eH MHoroyrOJIb-
HHKOB, onHcaHHblX BOKpyr TOro )Ke Kpyra.
B03bMeM OTpe.3KH [1; 4] H [6; 1OJ. l.fHCJIO 5 oOJIa.n:aeT TeM CBOHCTBOM,
4TO OTpe30K [I; 4] pacnOJIO)KeH CJIeBa OT 3Toro qHCJIa, a OTpe30K
!6; 10] - cnpaBa OT Hero. 5y.n:eM rOBopHTb, qTO 3TO qHCJIO pa3.n:eJIHeT
OTpe.3KH [1; 4] H [6; 1OJ. T04HO TaK )Ke nJIOIlI,a.n:b Kpyra pa3.n:eJIHeT MHO>Ke-
CTBO nJIOIlI,a.n:eH BnHcaHHbIX MHoroyrOJIbHHKOB H MHO)KeCTBO nJIoIlI,a.n:eii
onlIcaHHblX MHoroyrOJIbHHKOB, a qHCJIO 4 - OTpe3KH [1; 4] H [4; 7J.
Bo06IlI,e, CKa)KeM, qTO tJUCAO C pQ30eAJl.em tJUC/lOOble .MHO:JICeCtn8a X u
Y, eCJIH .n:JIH JIIOObIX x EX, y E Y HMeeM xcy (B 3TOM CJIyqae X pac-
nOJIO)KeHO CJIeBa OT Y). CooUcmeo Henpepbl8HOCmU MHO)KeCTBa R+ <pop-
MYJIHPYeTCH CJIe.n:YIOIUHM oopa30M.
EC/lU tJUC/l060e .MHO:JICeCtn80 X pacnO/lO:JICeHO C/leoa om tJUC/1080eO .MHO-
:JICeCf11sa Y, mo HauoemcR xomR 6bl OOHO 'l.UC/lO, pQ30eAflJOU¥e amu .MHO-
:JICCClI1oa.
CMbICJI 3TOro cBOHcTBa CTaHeT SlCHblM, eCJIH MbI nonp06yeM Y.ZJ:aJIHTb H3 MIIO)l{e-
CTBa R+ XOTSI 6b1 O.ZJ:HO 'lHCJlO, HanpHMep 6.0603HaquM TOr.ZJ:a 'lepe3 X MJlO)l{eCTBO
qHCeJI, MeHblllHX 6, a qepe3 Y - 'lHCM, 60JIblllHX 6. XOTSI X patnOJIO)l{eHO CJIesa
OT Y, nOCJIe Y.ZJ:aJIeHHSl 'IHCJla 6 HeT HH O.ZJ:Horo qHCJla, pa3.ZJ:eJISilOlllerO 3TH MJlOJKeCTBa.
3Ha'lUT, CMblCJI CBOHCTBa nenpepblBHOCTH COCTOHT B TOM, qTO B MHO)l{eCrae R + HeT Ue
TOJIbKO TaKHX «CKa'lKOB», KaK, HanpHMep. B MHO)l{eCTBe N HaTypaJIbHbIX qHCeJI, HO
H TaKHX «lll,eJIeH», KaK B MHO)l{eCTBe Q+ nOJIO)l{UreJIbHblX paUHOHaJIbHblX qUCM.
YnpamBeBBJI
I. PaCnOJIO)l{HTe B nopSl.ZJ:Ke B03paCTaHHSl CJle.ZJ:YlOlll,ue qHCJla: 7,3165...j 7,315989
...; 7,31667... .
2. .1J.oKa)l{HTe, qTO B MHO)l{eCTBe R+ HeT HU HaUMeHblllero, HU HaH60JIbillerO qHCJIa.
3. HaH.ZJ:HTe qHCJlO x, TaKoe, qTO
2,31599987.. . <x <2,31660000. ..
4. CJi01KeHHe H YMH01KeHHe B MH01KeCTBe R+. Onpe.n:eJIHM Tenepb
onepau.HH CJIO)KeHHH H YMHO)KeHHH B R+. DYCTb ,lI,SHbI qHCJIS:
x=m, mI' . .mh' ..
y=n, nl' . .nh" .
H3 R+. Tor.n:a npH JII060M k HMeeM HepaBeHCTBa Xh<Xk H YhY<Yk'
5ICHO qTO qHCJIO x+y .n:OJI)KHO OblTb He MeHbllIe, qeM JIio60e H3 qHCeJI
Xk+y'k, HO MeHbllIe, qeM JIlo60e H3 qHCeJI Xk+f/k' IJHblMH CJIOBaMH, x+y
.n:OJI)KHO pa3.n:eJIHTb MHO)KecTBa {Xh+Yh} H {Xk+Yk}. MO)KHO .n:O K a3aTb,
qTO 3TH MHO)KeCTBa pa3.n:eJIffIOTCff JIHllIb o.n:HHM qHCJIOM. D03TOMY BBe.n:eM
TaKoe onpe.n:eJIeHHe:
329

C YMMOU nOAO;)/(;UmeAbHblX 0et1cm8umellb1tblX IlUceA x U Y Ha3b18afOm
'lUCAO, pa3deARfOw,ee MHO;)/(;eCtn8a {Xk+Yk} U {Xi,+Yk}, r,ZJ:e Xk H Yk -
,ZJ:ecSITHIJHble npHOJTH)f{eHHSI 9THX 'IHCM no He.n.o'c'ra'fKY, a xft H Yk - no
H30bITKY.
MO)f{HO ,lI,OKa3aTb, IJTO onepalfUJl. C.IIOO/CeHUJl. 8 .MHO;)/(;ecftl6e R+ KOMMY-
mamU8Ha, accotfuanuum.a U COKpamU.Ma. DpH 9TOM eCAU x<y, mo dAR
Ilf06oo z E R+ UMee.M x+z<y+z. KpOMe Toro, HU dAR KOJCUX X U Y U3 R+
He 8blnOAHH.etnCR pa.eeHCtn80 x=x+y.
10'lHO TaR: JKe onpe,n:eJISIeTCSI YMHOJKeHHe B MHO)KecTBe R+. [JpOU38e-
oeHueM qHCeJI x H Y Ha3blBalOT e.n.HHCTBeHHoe qHCJTO, pa3.n.eJlSIIOm.ee MHo-
>KecTBa {XkYk} H {x;'Yk}.
OnepalfUR YMHo:»ceHUR 8 R+ IWMMymamuBHa, accolfuamU8Ha U COKpa-
mUMa. OHa ducmpu6ymU8Ha omHocumeAbHO CAO;)/(;eHUR. HaKOHelI" tIHCJIO
1 .R8AH.etnCJl HeUmpaAbHblM omHocumeAbHO Y.MHo:»ceHUR: eCJTH a E R+, TO
1.a=a.
MO)f{HO .n.oKa3aTb, tffO O/lUHbl ampe3K08 06AooafOm C80UCf/'lIJa.MU 00-
oumu8Hocmu U .MYAbmUnAUKamU8Hocmu: eCAU ompe30K a cocmoum U3
ompe3K08 b U c, mo eeo dAuHa palma cYMMe dAUH amux Ompe3K08, a eCAU
ml(a) H m 2 (a) - 8HalleHuR dAUHbl ompe3Ka a npu eOUHU'lHblX Ompe8KaX
el H e 2, mo 8blnOAHRemCR pa.eeHCtn80
m2(a)=ml(a)m 2 (el)'
MbI onYCKaeM ,lI,OKa3aTeJlbCTBO 3THX YTBep)f{.n.eHHH.
,lJ,JISI mOOLlX ,W!YX qHCeJI a If b H3 R+, TaKHX, tITO a>b, HaK)l.eTCSI Ta-
Koe C E R+, tJTO a=b+c. 3m 'lHCJIO H33LlBalOT pa3HOCmbfO 'lHceJT a H b
H o603HaqalOT a-b. 5ICHO. tITO onepalI,HSI BbItlllTaHHSI B R+ oopaTHa
onepau,HH CJlo*eHHSI:
H
(x+Y)-Y=X
(x-y)+y=x,
ecJIH x>y.
lI.JISI JIIOOLlX .n.BYX tIHCeJI x H Y H3 R+ HaH.n.eTcSI TaKoe tIHCJIO Z, Iff 0
x=yz. 3roIJHCJlO Ha3b1BalOT 'laCmHbl.M aT )l.eJIeHHSI x Ha Y H 0603HatIalOT
x: y. OnepalI,HSI )l.eJleHHSI B R+, TaKHM 06p2l30M, Bcer.n.a BbInOJTHHMa.
npHtIeM OHa oopaTHa onepalI,HH YMHO)f{eHHSI:
(xy) : Y=X,
(x : y). Y=X.
YopamHeHHH
113BecTHo,qTO V2=I,4142..., Y3=1,7320... .HaH.ZJ:HTe3Ha'leHHH V3+J"2
H Y3 - Y2 C TO'illOCTblO ]J,O 0,001 no He,u.ocTaTKY H no H36b1TKY.
5. AKCHOMaTHKa MHOJKecTBa rr6lI61K11TeJlbHblX )l.eiiCTBRTeJlbHblX IJHCeJI.
MbI Ha3BaJm 8b1me nOJTOJKHTeJIbHbfMlf ,ZJ:eikTBHTeJlbHblMH tIHCJIaMH TaKHe
qHCJIa, KOTopbIe MorYT 6b1Tb 3anHC3HbI B BH)l.e oecKoHe'lHblX ,ZJ:ec5lTHtIHbIX
,lI,pooeH. O.n.HaKo, CTporo rOBopfl, OecKoHetIHbIe )l.ecSITHtIHbIe .n.pOOH SIB-
nSIIOTCSI JIHlllb <PoPMOH 3anHCH )l.eHCTBHTe.rlbHbIX ljHCeJT. nOJIO)f{HTeJIbHbIe
330

JLeikTBHTeJIbHble llHCJIa MOXiHO 3anHcaTb He TOJIbKO B Blfe 6ecKoHellHblX
.lI.eCHTHllHblX .lI.po6eH, HO If B Blf.ll.e OecKoHellHblX .lI.BOlfllHblX, oeCKOHellHblX
TpOlfllHblX H T. .lI.. .lI.P06eH. Ec.lIlf npe,a,CTaBJIHTb 9TH IIHc.lIa B BH.lI.e 6eCKo-
HellHblX .!I.BOHIIHbIX .lI.P06eH, TO nOJIYllaIOTCH 3anlfCH, COCTOHI1I.He H3 HYJIeH
H e.n.OHHU, HanpHMep: 101, OllOlllO... .
lhoobl He CBH3blBaTb nOHHTlfe nOJIO>KHTeJIbHOro .lI.eHCTBHTeJIbHOrO
lllfc.IIa C roA: HJIO HHOD cpopMOD sanlfClf TaKHX qHceJI. Ha.n.O ccpopMYJIHpO-
BaTb aKCOOMhl, KOTOPblM OHlf Y.lI.OBJIeTBOpHIOT. O.ll.Ha H3 TaKHX ClfCTeM
aKCHOM onHpaercH Ha CBOHCTBa onepaUHH CJIO>KeHlfH. B HeD Heonpe.ll.e-
JIHeMhlMO nOHSlTHHMlf HBJIHIOTCH e,a,HHlfua H onepau:lfH c.lIO>KeHlfH. 8To
nOHHTlfH .lI.OJI>KHhI Y.lI.OBJIeTBOpHTb c.lIe.n.YIOIIJ.eD CHCTeMe aKCHOM:
I) Nc:.R+.
2) Onepall;UJl C./lO:JICeIlUJl C111DIJum 8 COOI118e1nCtn8ue AJ06oii, nape (a; b)
Iluce./l U3 R+ IlUC./lO a+b m020 :JICl! MIlO:JICl!Ctn8a. 3mo ItUCAO Ha3btBaemcJ1.
c!JM.Mou 'tUCeA a U b. a C(l.M.U a U b Ill13b«JQlOl1lCJl C./lCl2aeMbUtU. H a N
C./l(X}fCellue coenaiJaem co C./lO:JICellueM HafllYpaIlbllbtX Itucelt.
3) Onepall;UJl CPlXJICellUJl 8 R+ IWMMymamU8lla: O./lJl AJ06blX a u b U3
R+ UMee.M a+b=b+a.
4) Onepall;UJl C./lO:JlCl!IlUJl 8 R+ accoll;uamu8lla: O.llJl AJ06btx a, b u C U3
R+ UMee.M (a+b)+c=a+(b+c).
5) Bc./lu a U b npUHaOIle:JICam R+, mo a+b=Fa.
6) Bc./lu a U b npullai).Ile:JICam R+, nputte1Jl a;:I=b, mo U./lU llauOemcJl ma-
KCJe C E R+-Ilt1W a=b+c, U./lU flaUoel1lCR. m.atroe c E R+t ttmo b=a+c.
7) /l,.IlJl lIJ06oeo a E R+ u AJ06oeo llamypfJ./l.bflO2O 'lU£.ll(J n lla!loemcJl
eduIlCtn8elllloe bc:.R+ TaKoe, ttt1W a=b+b+.. .+b (n pa3).
AKCOOMbl 1)-7) n03BOJIHIOT BBecTH B R+ OTHomeHlfe nOpH.lI.Ka:
a<b B TOM H TOJlhKO B TOM CJIyqae, KOf.ll.a HaA:.a.eTCH TaKoe c E R+, liTO
b=a+c. LlOJI>KHa BhlnOJIHHTbCH aKCHOMa HenpepbIBHOCTH:
8) EC.Ilu 'tUC./loooe .M1lO:JICCCmBO X Ae:JICum C.IleBa om 1tUC/l()lJ(W) .MIlO:JICccm-
8Q. Y (T. e. XY.ll.J1H JIIOObIX x EX, 9 E Y), mocyw,ccmiJflem 'lUC./lO a E R+,
pa3OeltJllOUft!e X 0 Y (TaKoe, liTO x .lI.J1H JIIOObIX x e X H Y E V).
C nOMOI1l.blO 9TOH cHcreMhl aKCHOM MO)KHO .1t.OKasaTb, lITO JII060e IIHc.lIO
H3 R+ npe.n.CTaBOMO B BH.lle 6ecKoHetHIOD .lI.ecHTHIlHoH p06H, onpe.n.e-
JIOTb onepau:HIO YMHmKeHHSI B R+ 0 T. .lI.. M.bI He 6Y.lI.eM OCTaHaBJIHBaTbCH
Ha 9THX BOnpocax.
6. HSMepeHHe BeJIHqRH. MbI BBeJIH MHOJKecTBO R+ nOJIO>KHTeJIbHblX
.lI.eDl'TBHTeJIbHhlX IIHceJI, IIToOO nOJIyqHTb B03MO>KHOCTb Bhlpa>KaTb .lI.J1H'
HY JIl060ro OTpe3Ka. C nOMOI1l.bIO 9THX qHCeJI MO>KHO Bhlpa>Karb H Pe3YJIb-
Tar H3MepeHHH .lI.PYrHX BeJIHtUIH - nJIOI1l.a.n.eH, OO'beMOB H T. .lI.. Ocra.
HOBHMCH H3 OlIpe.n.eJIeHOH o6w.ero nOHHTHSI BeJlH'IHHbl. 11 npH HaXO>K.ll.e-
HHH .lI.J1HH OTpe3ROB, H IIPH BhlqOC-lIeHHH nJIoI1l.a.n.eA: <pHryp, H npH OThlC-
KaHHH 06'beMOB TeJI MhI HMeeM .lI.eJIO C HeKOTOphlM MHO>KeCTBOM o6'beK-
TOB, B KOTOpoM onpe.n.eJIeHbI .!I.Ba OTHOllIeBHH - OTHOllIeHHe 3KBHBaJIeHT-
HOCTH (HanpOMep, <t>HrYPbl FJ. H F. KOHrpY9HTHhI) H OTHOllIeHHe «oo'beKT
a COCTOOT 83 OO'beKTOB f} 0 'V. (HanpHMep, OTpesoK AB cocrOHT H3 OT-
pe3KOB AC H CB).
fIo9TOMY paCCMOTpHM HeKOTopoe MHO>KecrBO OO'beKTOB Q, B KOT0E...0M
onpe.ll.eJIeHhI OTHOllIeHHe 9KBHBaJIeHTHOCTH a",p H OTHOllIeHHi a=EH'Y,
331

IfHTat:\loe «ex COCTOHT H3  H v». CKa)!(eM, '11'0 Ha 3TOM MHO)!(eCTBe Ofl!JC-
)J.eJIeHa onepa/{Uf/. U3MepeHUf/., eCJUI KaLK,lI.OMY 3JIeMeHTY a H3 Q MO)!(HO
conOCTaBHTb nOJIO)!(HTeJIbHOe qHCJIO m (ex.) - Mepy ex., TaK, qTQ BbIllO.1-
HSiIOTCfl YCJIOBHfl:
a) H3 ex.N BblTeKaeT m (ex.)=m () (3KBHBaJIeHTHbIe 061>eKTbI HMeIOT
paBHbIe MepbI);
6) H3 a=EBv BbITeKaeT m (ex.)=m ()+m (V) (aMHTHBHOCTb MepbI).
CKa)!(eM, qTO Q flBJIfleTCfl nOAeM onpei)eAeHUf/. eeAUltuHbl, eCJIH Ha HeM
onpe,ll.eJIeHa onepaUHfl H3MepeHHH, npHqeM ,lI.Be pa3JIH'lHble onepaUHH
H3MepeHHfl m H ml MorYT OTJIHqaTbCfl ,lI.pyr OT ,lI.pyra JIHllib nOCTOflHHbIf
MHO)!(HTeJIeM, T. e. CYllI,ecTBYeT TaKoe nOJIO)!(HTeJIbHOe qHCJIO A, 'lTO
ml(a) =Am (a) ,lI.JIfl Bcex a E Q.
OpHMepoM nOJIfl onpe,ll.eJIeHHfl BeJIHqHHbI MQ)KeT CJIY)!(HTb MHO)Ke-
CTBO Q Bcex oTpe3KoB. B 9TOM MHO)!(eCTBe 3anHCb ex.N 03HaqaeT, 'l10
OTpe3KH ex. H  KOHrpY3HTHbl, a 3anHCb ex.=EBv - 'lTO cYllI,eCTByeT
roqKa, pa36HBalOllI,afl OTpe30K ex. Ha OTpe3KH  H V. OnepaUHfl H3Mepe-
HHH CTaBHT B COOTBeTCTBHe Ka)!(,lI.oMY oTpe3KY ex. ero ,lI.JIHHY m (ex.) , npn-
lfeM, O'leBH,lI.HO, BblnOJIHHIOTCfl YCJIOBHfl a) H 6), BbIpa)!(alOllI,He uHeapu-
OHmHocmb H aiJaumU8Hocmb ,lI.JIHHbl. JII06ble ,lI.Be onepaUHH H3MepeHH5I
,lI.JIHHbI ,lI.alOT pe3YJIbTaTbI, OTJIHqalOllI,HeCfl ,lI.Pyr OT ,lI.pyra JIHllib nOCTOflH-
HbIM MHO}J{HTeJIeM (B CHJIY CBOM:CTBa MYJIbTHnJIHKaTHBHocTH).
ECJIH Q - nOJIe onpe,ll.eJIeHHfl BeJIH'lHHbI, TO B HeM MO)!(HO BBeCT!!
OTHOllIeHHe paeH08eAUKocmu, 03HaqalOllI,ee, 'lTO m (ex.)=m (). 31'0 OTHO-
meHHe 06JIa,ll.aeT CBOM:CTBaMH pe<PJIeKCHBHocTH, CHMMeTpHqHOCTH H TpaH-
3HTHBHOCTH, a nOTOMY onpe.n;eJIfleT pa36HeHHe MHO)!(ecTBa Q Ha KJIaCCbI
9KBHBaJIeHTHOCTH. 31'0 pa36HeHHe H Ha3blBalOT eeAUIJ.UHOU, coomeemcm-
6YIOU¥!U nOAIO Q. B cJIY'lae KOr,ll.a Q COCTOHT H3 OTpe3KOB, OTHOllIeHHe
paBHOBeJIHKOCTH COBna,ll.aeT C OTHOllIeHHeM 9KBHBaJIeHTHOCTH - ,lI.Ba OT-
pe3Ka KOHrpY9HTHbI B TOM H TOJIbKO B TOM CJIyqae, Kor,ll.a HX ,lI.JIHHbI paB-
HbI. B CJIyqae nJIOllI,a,ll.eM: ,lI.eJIO OOCTOHT HHaqe - ,lI.Be qmrYPbI MorYT
HMeTb paBHbIe nJIOllI,MH, He 6Y,lI.yqH KOHrpY3HTHblMH (HanpHMep, npflMO-
yrOJfbHHK CO cTopOHaMH 6 CM H 24 CM H KBMpaT co CTOpOHOM: 12 CM).
7. H3MepeHHe nJloMeii. OOKa)!(eM, KaK CTpOHTCH 06llI,aH TeoplUI
H3MepeHHfl nJIOllI,MeM: qmryp. Bbl6epeM Ha nJIOCKOCTH .QBe B3aHMHO
nepneH,lI.HKYJIflpHbIe npflMble 1 H m H e,ll.HHH'lHblM: OTpe30K e. T.Jepe3 Q
0603Ha'lHM cOBoKynHocTb npflMoyrOJIbHHKOB, C1'OpOHbl KOTOpblX napaJI-
JleJIbHbI 3THM npflMblM, npH'leM 6Y.QeM CqHTaTb, qTO aN 03HaqaeT KOH-
rpY3HTHOCTb,ll.BYX TaKHX npHMoyrOJIbHHKOB,
m a a=EBI' - 'lTO npflMoyroJIbHHK a pa36H-
BaeTCfl Ha npflMoyrOJIbHHKH  H V npflMOM:,
napaJIJIeJIbHOM: O,ll.HOM: H3 npflMblx 1 HJIH m
P (pHC. 182).
nOKa}KeM, 'lTQ eCTb e.n;HHcTBeHHblH cno-
"t c06 onpe.n;eJIHTb nOHflTHe nJIOIItMH S (a) Ha
MHO}KeCTBe TaKHX npSlMoyroJIbHHKOB, 'lT06bl
o l nJIOIItMb e,ll.HHH'lHOrO KBa,ll.paTa paBHSlJIaCb 1.
.lI.JIH 9Toro HMO 33,QaTb nJIOIIt3,Qb npflMo-
Puc. 182 yrOJIbHHKa CPOPMYJIOM: S (a)=ab, r,ll.e a Ii b -
332

AJIHHbI ero CTOpOH. B caMOM ,n.e.ne, ecJIH a H m
b - HaTypaJIbHbIe 'IHCJ1a, TO npHMoyrOJIbHHK
a MO)KHO pa36HTb Ha ab e,zJ.HHH'IHbIX KBa.n.pa-
TOB, H nOTOMY ero nJIom.a.n.b ,n.OJI)KHa paB-
HHTbCH c}'MMe ab nJIom.a.n.eA TaKHX KBa,n.pa-
TOB, T. e. QHCJ1Y ab. .lJ.aJIee, ecJIH ,n.JIHHbI CTa-
pOH a BbIp8)KalOTCH ,n.ecHTH'IHbIMH ,n.p06HMH
a b
IOn H lon ' TO npHMoyrOJIbHHK a MO)KHO pa3- 0
6HTb Ha ab KBa.n.paTOB CO CTOpOHOH ln ' a e,n.H-
HHtIHbIA KBa.n.paT - Ha lOIn TaKHX KBa.n.paTOB.
 .1
lO,n.a CJ1c.n.yeT, 'ITO nJIow.a.n.b KIDK.!{Oro KBa.n.paTa CO CTOpOHOM ton pao-
1 00
HaTIjiii' a nJIom.a,n.b Bcero npHMoyrOJl'bHHKa paBHa J02n ' T. e. npOH3De-
a b
)l.eHHIO qHCeJI TQii H JDii' CJIyqaA, KOr)l.a xont 6bI o,n.Ha H3 CTOpOH npHMo-
yrOJIbHHKa HMeeT HppanHOHaJIbHYIO )l.JIHHY, CBO)l.HTCH K paCCMOTpeH-
HbIM - B STOM CJ1Y'Iae H nJIow.a)I.b npHMoyrOJIbHHKa, H 'IHCJ1O ab pa3)1.e-
JIHIOT MHO)KeCTBa qHCeJI X={anbn} H Y={ab}, r.n.e lln H b n - npH-
6JIHmeHHH qHCeJI a H b no He,n.ocTaTKy, a a H b - npH6JIIDKeHHH rex
)Ke 'IHCeJI no H36bITKY.
MbI .n.oKaS8JIH, lITO ecJIH Y npHMoyrOJIbHMKOB eCTb nJIow.a.n.b, TO OHa
,n.OJImHa BbIp8)KaTbCH qHCJIOM ab. opH H3MeHeHHH e,zJ.HHH'IHOrO OTpesKa
MeHHIOTCH 'IHCJ1a, BbIP8)KaJOW.He .n.JIHHbI CTOpOH npHMoyrOJIbHHKOB, a
TeM caMbIM H qHCJ1a, BbIp8)KalOw.He HX nJIow.a.n.H. npH STOM Bee 3TH 'IHCJla
nOJIyqalOTo.n.HH H TOT me nOCTOHHHbIH MHOmHTeJIb. MomHo .n.OKa3aTb,
qTO nJIOw.a)I.b, onpe,zJ.eJIHeMaH cpopMYJIoA S (a.)=ab, .n.el{cTBHTeJIbHO 
JIa.n.aer CBOHCTBaMH a) H 6) H3 n. 6. TeM CaMbIM 3aBepweHa TeopHH HSMe-
peHHH nJIow.a.n.eH: MH npHMoyrOJIbHHKOB, CTOpOHbI KOTOpbIX napaJIJIeJIb-
HbI npHMbIM I H m.
He COCTaBJIHeT Tpy.n.a BBeCTH nOHHTHe nJIow.a.n.H MH <pHryp (SoJIee
06w.ero BH)l.a - TaK Ha3bIBaeMhIX CTyneH'IaTbIX <pHryp. MbI 6y.n.eM Ha-
3bIBaTb <pHrypy a. cmyneH'-W,moa, ecJIH OHa HBJIHeTCH o61.e,zJ.HHeHHeM nps-
MOyroJIbHHKOB, HHKaKHe .n.Ba H3 KOTopbIXHe HMeIOT 06w.eA BHYTpeHHeif
TOqKH (pHC. 183). ECJ1H CTyneH'IaTaH <pHrypa COCTOHT H3 npHMoyrOJIb-
MHKOB lt..., n' TO Ha30BeM ee nJIOII.J.8)l.b1O CYMMY nJIOw.a.n.eH 9'IHX
npHMoyroJIbHHKOB. MOJKHO .n.OKa3aTb, lITO nJIOw.a.n.b cTyneHqaTOH <pHry-
pbI He 3aBHCHT OT TOro, KaK OHa pa36HTa Ha npHMoyrOJIbHHKH. KpoMe
TOro, nJIOw.azlbcryneHqaroA <PHrYPbI He H3MeHHeTCH npH napaJIJIe.nb-
HOM nepeHoce, a ecJIH <pHrypa a. pa30HTa Ha CTyneH'IaTbIe <PHrYPbI 
H '\', TO ee nJIOw.azlb paBHa CYMMe nJIow.a.n.eA 9THX <pHI'yp.
nOKa MbI He YMeeM. HaxO.n.HTb nJIow.a.zlH HH TpeyrOJIbHHKOB, HH Kpy-
rOB, HH .n.8)KenpfiMoyrOJIbHHKOB CO CTopOHaMH, He napaJIJIeJIbHbIMH npSl-
MbIM I H m. T.JTOI5bI paCWHpHTb oMaCTb npHMeHHMOCTH 9TOro nOHHTHSI,
BBe,zJ.eM Macc Qf KBa.n.pHpyeMl)Ix <pHryp (T. e. <pHryp, OOJIazlalOw.HX nJIO-
UJ,aJI,blO). CBHmeM C KIDKJI.oA <pHrypoA a. )l.8a 'IHCJ10BhIX MHomecTBa X Ii
Y. MHomecTBo X COCTOHT H3 -nJIow.a.n.eA CTyneHqaTbIX 4>uryp, neJIHKOM
l
Puc. 183
333

cO,!J.ep>Kalll,liXCH B qmrype a, a Y - li3 nJIOllI,a,!J.eii CTyneHlf8TbiX $H-
ryp, LI.eJlliKOM cO,!J.ep>Kalll,liX qmrypy a. 5lCHO, lfTO HJlOlll,a,!J.b BHYT-
peHHell CTy"eHlfaTOll qmrYPbI He npeBOCXO,!J.lIT 11J10lll,a.n:H BHenIHeH cry-
neHlfaTOll qmrYPbI. D03TOMY ecJlH x EX H Y E Y, TO x, T. e. MHO-
>KeCTBO X pacnOJlO>KeliO C,'1eBa OT MHO>KeCTBa Y. n03TOMY CYllI,ecTByeT
XOTH 6b1 O,!J.HO lfliCJlO, pa3,!J.eJIHlOllI,ee 3Tli MHO>KecTBa.
HaSOBeM qmrypy a K8aiJpupyeMoii, ecJlli COOTBeTCTBYIOlll.He eM lfliC-
J10Bble MHO>KecTBa X li Y pa3JJ.eJIHIOTCH J1lillib O,!J.HHM lfHCJlOM. 3TO
lfHCJlO S (a) li Ha3bIBalOT n/lOw,aablO cpUZYPbl a. TaKliM 06pa3OM, nJlo-
W.a.Il.b qmrYPbl He MeHbllIe nJlOlll,a,!J.ell Bcex co.n:ep>Kam,UXCH B HeR CTyneH-
lfaTblX qHfryp, HO He 60JlbllIe 11J10lll,a.Il.eH: Bcex co.n:ep>Kalll.HX ee cTyneHlfa-
TblX q>Hryp.
MO>KHO ,!J.OKa3aTb, lfTOHJlOrn.a.Il.b <PHrYPhl 06J1a,!J.aeT CBOHCTBmI a/I.'J,!!-
THBHOCTH: ecJlli )!,Be KBa,!J.plipyeMble qmrYPbl He liMelOT 06ll1,HX BIiYTpeH-
HHX TOlfeK, TO nJlOlll.3J(b liX 06'beJiHHeHliH paBHa CYMMe nJlOlll.a.ll.ell 3T!!':
<pliryp. .lJ.aJlee, nJlow.a,!J.b qJHrYPbl He H3MeH51eTCH npli nepeMelll,em1H
.9TOH: q>HrYPbI. Pa3YMeeTCH, lfHCJlO, Bblpa>KaIOw.ee nJlow.a.ll.b <PHrYPbl, MC-
HHeTCH npH liSMeHeHHH eJl.l1lmlfHOrO OTpeSKa. ECJliH e=kf, TO Sr(a)==
=k 2 Se(a), r,ll;e Se(a) - nJlOlll,a.n:b <PHrYPbl a, Kor.ll.a 3a e.ll.liHHLI.Y li3W-
peHHH .lI.JlHH B6paH OTpe30K e, a Sf(a) - KorJ(a 3a TY >Ke eJ(HHIfLI.Y BU.
6paH OTpe30K f.
COBepllIeHHO TaK me CTpOliTCH TeopliH H3MepeHliH 06'beMoB TeJI. Teo-
plfH li3MepelfHH .lI.J1HH KpHBbIX JlHHHll 60Jlee CJlO>KHa, H Mbl He 6Y.ll.er,1 ua
Hell OCTaHaBJlHBaTbC5I.
8. nJlom.aAb KpHBOIIHeHHOH l'paneu,HH. OYCTb <PYHKLI.liH y=f (x)
HeOTpliLI.aTeJIbHa H HenpepblBHa Ha OTpesKe [a; bL ct>nrypy, orpaHlilfeH-
HYIOCHli3Y OCblO a6CLI.HCC, cBepxy - rpa<pHKoM 3ToH <PYHKLI.liH, a C 60-
KOB - 0p.II:HHaTaMli x=a H x=b, Ha3bffiaiOT Kpu80/!uHeufwu mpane/{ueu
(pHC. 184). B 9TOM nYHKTe Mbl .lI.OK8>KeM, lfTO 11J10lll.b .9TOH TpaneLI.liH
BbIpa>KaeTC5I q,opMYJlOH:
b
S= f(x)dx,
a
(1)
b
r.ll.e, HanOMHHM, lfepe3  f (x)dx 0603HalfeHa pa3HoCTbF(b)-F (a) 3Halfe-
a
Hlill nepBo06pa3HOH: F (x) .lI.J151 <PYHKLI.HH f (x).
LJT06bl .lI.OKa3aTb 9TO YTBep>K.ll.eHHe, 6Y.ll.eM
U f ClfliTaTb npaBYiO rpaHHu,y Tpaneu.nli nepeMeH-
! Hon H 0603HalfHM lfepe3 S (x) 11J10lll.a.Il.b lfaCTli
1 TpaneLI.HH, 3alIITpHXOBaHHoA Ha pliCYHKe 184.
! .lJ.oK3>KeM, lITO.9Ta <PYHKLI.liH HBJlHeTCH nepBO-
06pa3HoH .lI.JlH f(x), T. e. S'(x)-f(x). .lJ.ml
3Toro .lI.a.Il.HM x I1plipaw,eHHe Ax. COOTBeTCT-
I BYIOID,ee npnpalll,eHHe as t1>YHKII.HH S (x) paBHO
-OJ a x X+nx b X nJlOrn:a.ll.H nOJlOCKH, H306p3>KeHHOH HapHcYHKe
184. .3Ta nOJlOCKa nOlfTH He OTJllilfaeTC5I OT
Pill:. 184 npHMoyrOJIbHliKa C OCHOBaHHeM ax H BbICOTOR
334

t (x), a rrOTOMY ee rrJIOlll,a,!J,b rrpH6JIH)I(eHHO paB-
Ha f(x)6.x. TaKHM 06pa30M, t1Sf(x)6.x, H
/!S
nOTOMY -rxf (x). l.JeM MeHbllIe llIHpHHa rro-
.1OCKH 6.x, TeM 60JIbllIe Haliia rrOJIOCKa Hano-
MHHaeT rrpHMoyrOJIbHHK, T. e. TeM T04Hee
/!S
npH6JIH)I(eHHOe paBeHCTBO  f(x). ECJIH
yCTpeMHTb 6.x K HYJIIO, TO 3TO npH6JIH)I(eH-
Hoe paBeHCTBO rrepeA:,!J,eT B T04Hoe, H Mbl rrOJIY-
1 . s 1 . s
'IHM, 4TO 1m - =f(x). Ho Im_ =S'(x),
Llx-+O /!x Llx_O x
H, 3Ha4HT, S' (x)=f (x). TeM caMblM ,!J,OKa3a-
HO, 4TO S () rrepBo06pa3HaH,!J,.1IH f(x).
Mbl XOTHM HaA:TH nJIOlll,a,!!,b KpHBOJIHHeA:-
HOH TpanelI.HH. OHa paBHa 3Ha4eHHIO <PYHK-
lI.HH S (x) rrpH x=b, T. e. S (b). C ,!J,pyroA:
CTOpOHbl, S (a)=O, TaK KaK «nJIOlll,a,!J,b» 4ac-
TH TparrelI.HH OT x=a ,!J,O x=a paBHa HYJIIO. n03TmlY MO)l(HO 3arrH-
caTb, 4TO S=S (b)-S (a). TaKHM 06pa30M, S - 3TO pa3HocTb 3Ha4e-
HHH rrepBo06pa3HoM: ,!J,JIS'I <PYHKlI.HH f (x) B T04Kax b H a, T. e. orrpe,!J,e-
JIeHHblH HHTerpaJI OT 3TOfi <PYHKlI.HH no OTpe3KY [a; bJ. PaBeHcTBo (1)
,!J,OKa3aHO. _
n p H M e p. HaH,!J,eM nJIOlll,a,!!,b, orpaHH4eHHYIO OCblO a6ClI.HCC U
KpHBOH y=9-x 2 .
1'13 pHCYHKa 185 BH,!J,HO, 4TO rpaq)HK <PYHKlI.HH y=9-x 2 pacrrOJIO)l(eH
Bblille OCH a6cu.HCC Ha OTpe3Ke [-3; 3] (B Toqkax x=-3 H x-3 <PYHKUHfl
y=9-x 2 06palll,aerCH B HYJIb). na9TOMY HCKOMa rrJIOlll,a,!!,b BblpffiKaerCH
HHTerpaJIOM:
+3. 3 
S  (9- X2)dx: 9dx x'dx.
!J
9
x
2
PHC. 185
Bbl4HCJIHH .9TOT HHTerpaJI, rrOJIY4aeM, 4TO
1 3 xl 1 3 33 [ t:p 3 ]
S=9x -- =9.3--'-- 9(-3)- = 36.
-3 3 -3 3
YnpallCUeUBJI
1. Hail,aHre nJIOIUa.ZJ:b KpHB9JIHHeilHp" TpaneUHH. OrpaHHqeHHOil OCblO a6CUHCC,
Op,!(HHaTaMH x= 1, x=4 H rpa<j>HKoM <j>YHKUHH yx3.
2. HaH.ZJ:HTe nJIOlUa.ZJ:b, OrpaHHqeHHYIO OCblO a6cUHcc H rpa<j>HKoM <j>YHKUHH Y=
= 16--x..
3. HaH.ZJ:Hre nJIOllla.ZJ:b, OrpaHHqeHH)'IO rpa<j>HKaMH <j>YHKUHiI y=r H y= 
 4. MHomECTBO ;rJ;E.0CTBIITEJIbHbIX QIlCEJI
1. nOJlO)l{UTeJlbHble M oTpuu,aTeJlbHble "MCJla. C nOMOUJ.blO nOJIO)l(H-
Tei1bHblX ,!J,eHcTBlneJIbHblX 4HCe.'I MO)l(HO Bblpa3HTb pe3YJIbTaT H3Mepe-
HHH JIlOWH CKaJIHpHOH BeJIH4HHbI: ,!J,JIHHbI, nJIow.a,!J,H, 06'beMa, MacCbl
335

Ii T. .lI.. Ho Ha npaKTliKe 4aCTO 6blBaeT HY)KHO Bblpa3liTb 4liC.lIOM He pe-
aY.lIbTaT li3MepeHliH BeJIli4liHbI, a ee li3MeHeHlie, T. e. nOKa3aTb, Ha
CKO.lIbKO li3MeHli.lIaCb 3Ta BeJIli4liHa. 113MeHeHlie BeJIli4liHbI MO)KeT li.ll.Tli
B .lI.Byx HanpaB.lIeHliHX - OHa MO)KeT KaK YBeJIli4liBaTbCH, TaK li YMeHb-
llIaTbCH, a MO)KeT li OCTaTbcH Heli3MeHHoit n03TOMY, 4T06bI BbIpa3liTb
li3MeHeHlie BeJIli4li, Hbl KpOMe nO.lIO)Kli're.lIbHblX .lI.eiicTBliTe.lIbHblX 4liCeJI,
HY)KHbI liHble 4liC.lIa, HY)KHO pacnmpliTb MHO)KecTBO R+. Mbi pacnmpliM
ero, nplicoe.ll.liHliB K HeMY 4liC.lIO 0 (HY.lIb) li OTpliI.I,aTe.lIbHble 4liC.lIa.
I1TaK, B03bMeM MHO)KecTBO R+ nO.lIO)KlITeJIbHbIX .lI.eiicTBliTeJIbHbIX
IIliCe.lI li nOCTaBliM B COOTBeTCTBlie Ka)K.lI.OMY 4liC.lIY x li3 R+ HOBoe 4liC.lIO,
KOTopoe 6Y.lI.eM 0603Ha4aTb -x (4liTaeTCH «MliHYC x»). HanpliMep, 4liC-
.lIy 5 CTaBliTCH B COOTBeTCTBlie 4liC.lIO -5, 4liC.lIY 8,14 - 4liC.lIO -8,14
li T. .lI.. 4liC.lIa Bli.ll.a -x, r.ll.e x E R+, Ha30BeM ompUUpme.llbH.blM.U 'iUC.Ila-
MU, a liX MHO)KecrBo 0603Ha4liM R-. KpoMe TOro, B03bMeM 4liC.lIO O.
06'be.ll.liHeHm MHO)KecTB R+, R- li {O} Ha3blBaeTCfl MH.O:JICeCmeOM aeucm-
8Ume.llbHblX 1iuce.ll li 0603Ha4aeTCfl R. TaKliM 06pa30M,
R=R+ u R- u {O},
npli4eM MHO)KeCTBa R+, R- li {O} nonapHO He nepeceKalOTCH (Hli O.ll.HO
llliC.lIO He MO)KeT 6bITb cpa3Y li nO.lIO)KliTe.lIbHbIM, li OTpliI.I,aTeJIbHblM li.lIli
li nO.lIO)KliTe.lIbHbIM li HY.lIeM).
EC.lIli BeJIli4liHa liMe.lIa 3Ha4eHlie x, a nOTOM npliHH.lIa 3Ha4eHlie y,
r.ll.e Xli Y npliHa.ll..lIe)KaT R+, TO npli x<y ee li3MeHeHlie BbIpa)KaerCH no-
.lIO)KHTe.lIbHbIM 4liC.lIOM y-x (HanpliMep, ec.lIli 3Ha4eHlie BeJIli4liHbl 6bI.lIO
6, a CTa.lIO 10, TO OHa li3MeHli.lIaCb Ha 10-6, T. e. Ha 4). EC.lIli )Ke x>y,
TO 6Y.lI.eM rOBopliTb, 4TO BeJIli4liHa li3MeHli.lIaCb Ha OTpliI..!,aTe.lIbHOe 4liC-
.lIO -(x-y) (HanpliMep, ec.lIli 3Ha4eHlie BeJIli4liHbI 6bl.lIO 6, a CTa.lIO 2,
TO OHa li3MeHli.lIaCb Ha -(6--2), T. e. Ha -4). TaKliM 06pa30M, CKa3aTb,
liTO BeJIli4liHa li3MeHli.lIaCb Ha -a, paBHOCli.lIbHO TOMY, 4T06bI CKa3aTb,
4TO OHa YMeHbIllli.lIaCb Ha a.
nO.ll.06HO TOMY KaK nO.lIO)KliTeJIbHble .lI.eiicTBliTeJIbHble 4liC.lIa li306pa-
)KaIOTCH T04KaMli KOOp.ll.liHaTHOro .lIY4a, npOli3BOJIbHble .lI.eiiCTBliTeJIbHble
4liC.lIa li306pa)KaIOTCH T04KaMli KOOp.ll.liHaTHOii npHMoii (CM. n. 1  1
r.lI. V). npli 3TOM nO.lIO)KliTeJIbHble " OTpliI..!,aTeJIbHble 4liC.lIa li306pa>Ka-
10TCH T04KaMli .lI.BYX npOTliBOnO.lIO)KHblX .lIY4eA, a 4liC.lIO 0 - 06ll1.liM Ha-
4a.lIOM 0 3TliX .lIY4eii.
4liC.lIa x li -x, r.ll.e x E R+, li306pa)KalOTCH T04KaMli KOOp.ll.liHaTHOii
npHMoii, CliMMeTpli4HO pacnO.lIO)KeHHblMli OTHOCliTe.lIbHO Ha4a.lIa OTC4e-
Ta O. 3Tli 4liC.lIa Ha3bIBalOT npomU80nO.llO:JICH.bl.MU .lI.Pyr .lI.pyry, nplilleM
C4liTalOT, 4TO -(-x)=x. HanpliMep, -(-6)=6. 4liC.lIO 0 CllliTalOT
npOTliBOnO.lIO)KHbIM caMoMY ce6e, -0=0.
PaccToHHlie .OT Ha4a.lIa OTC4eTa .lI.O TOIIKli KOOp.ll.liHaTHOH npHMoH,
li306pa>KaIOllI.eH 4liC.lIO x, Ha3bIBalOT MooY.lleM 3Toro 4liC.lIa li 0603Ha4alOT
Ixl. TaKliM 06pa30M,
{ X, ec.lIli x>o,
Ixl= -x, ec.lIli x < o,
0, eCJIli x-O.
HanpliMep, 112/= 12; 1-91=9; 101=0.
336

nYCTb npH H3MeHeHHH Ha a E R qHCJlO X E R+ nepellIJIO B qHCJlO
Y E R+. TorAa CKIDKeM, qTO AeficTBHTeJIbHOMY qHCJIY a COOTBeTCTBYt'T
napa nOJIO)KHTeJlbHbIX AeficTBHTeJIbHbIX qHCeJI (x; y). HanpHMep, napa
(7; 2) cOOTBeTCTByeT AeHCTBIITeJIbHOMY qHCJIy -5, TaK KaK npH H3MeHe-
HHH Ha -5 qHCJIO 7 nepexoAHT B 2, a napa (3; 8) cOOTBeTCTByeT qHCJIY 5,
TaK KaK npH H3MeHeHHH Ha 5 qHCJIO 3 nepexoAHT B qHCJIO 8.
OAHOMY H TOMY )Ke AeHCTBHTeJIbHOMY qHCJIY COOTBeTCTBYeT 6ecKo-
HeqHOe MHO)KeCTBO nap. HanpHMep, qHCJIY 4 COOTBeTCTBYIOT TaKHe na-
PbI, KaK (1; 5), (1,5; 5,5), W2; 4+Y2) H T. A., <IHCJIY -3 - napbl
(4; 1), (10; 7), W 29; Y 29- 3) H T. A. BbHICHHM, KorAa ABe napbI (Xl; YI)
(X2; Y2) COOTBeTCTBYIOT OAHOMY H TOMY )Ke AeficTBHTeJIbHOMY qHCJIY a.
ECJIH qHCJIOa nOJIO)KHTeJIbHO, TO 3TO 6YAeT, KorAa YI=XI+a H Y2=
=X2+a. Ho B 3TOM CJIyqae XI+Y2=XI+(X2+a)=(XI+a)+X2=YI+X2'
ECJIH )Ke a OTpHl.laTeJIbHO, TO Yi=XI-(-a) H Y2=X2-(-a), H nOTOMY
XI=YI+(-a), X2=Y2+(-a); B 3T0M CJIyqae TO)Ke X,:+Y2=X2+Yt.
AHaJIOrHqHO pa36HpaeTCH CJIYllaH a=O. Bo Bcex CJIyqaHX napbI (Xl; YI)
H (X2; Y2) OTBeq3IOT OAHOMY H TOMY )Ke AeficTBHTeJIbHOMY qHCJIY a B ToM
H TOJIbKO B TOM CJIyqae, KorAa XI+Y2=X2+YI'
n03TOMY nOHHTHe AeHCTBHTeJIbHOro qHCJIa MO)KHO onpeAeJIHTb H
qepe3 napbI nOJIO)KHTeJIbHbIX qHCeJI. .lI.JIH 3Toro B03bMeM AeKapTOB KBaJ.l-
paT R MHO)KeCTBa R+, T. e. MHO)KeCTBO nap BHAa (X; y), rAe X E R+,
Y E R+. CKIDIreM, liTO napa (Xl; YI) 8K8U8aJlet-tmt-ta nape (X2; Y2), ecJIH Xl+
+Y2=X 2 +YI. J1erKo npOBepHeTCH, qTO OTHOllIeHHe (Xl; Yl)--(X2; Y2)
06JIaJ.laeT cBoficTBaMH pe<PJIeKCHBHOCTH, CHMMeTpHqHOCTH H TpaH3HTHB-
HOCTH, a nOTOMY 3aAaeT pa36HeHHe MHO)KecTBa R+ Ha KJIaCCbI 3KBHBa-
JIeHTHbIX nap. KIDK)I.bIfi TaKOM: KJIaCC MbI H Ha3bIBaeM aeUctrl8UmeAbt-tblM
flUCAOM. npH 3TOM eCJIH x<y, TO qHCJIO, COoTBeTCTBYIOI.Qee nape (X; y),
paBHO Y-X H nOJIO)KHTeJIbHO, a eCJlH x>y, TO 3T0 qHCJIO paBHO-(X-Y)
H OTpHl.laTeJIbHO. HaKOHeI.l, ecJIH X=Y, TO nape (X; y) COOTBeTCTBYeT
qHCJIO O.
KIDK)I.YIO napy (X; y) MO)KHO H306pa3HTb HanpaBJIeHHbIM OTpe3KOM
Ha qHCJIOBOM JIyqe, HMelOw.eM HaqaJIo X H KOHeI.lY (MH npOCTOTbI peqH
Mbl 0603HaqaeM qepe3 X TOqKY C KoopABHaTofi X). 3KBHBaJIeHTHbIM napaM
COOTBeTCTBYIOT OTpe3KH, HMelOw.He o.zwHaKOBYIO MHHY H OAHHaKoBoe
HanpaBJIeHHe. Ha30BeM TaKHe HanpaBJIeHHble OTpe3KH 8K8U8aAet-tmt-tbl-
MU. TorAa MO)KHO CKa3aTb, qTO aeucm8umeAbt-toe flUCAO U306pa[}fCaem
K/lGCC. 8K8U8aAet-tmt-tbLX t-tanpa8Aet-t1-(,blX ompe3K08.
YnpaJKBeBBJI
I. Hait.ZJ:HTe '1HCJlO x, eCJIH (3x-2,4)- (2x+ 1,7).
2. Hait.ZJ:HTe 'IHCJIO, B KOTOpoe nepexo.ZJ:HT '1HCJIO 12 npH H3MeHeHHH Ha: a) 4,
6)-7, B) O.
2. Cn01KeHHe H BbllfHTRHHe )l,elkTBHTeJIbHblX IfHCeJI. nyCT& HeKOTO-
poe qHCJIO X E R+ CHaqaJIa H3MeHHJIH Ha a, a nOTOM Ha b, npHlleM
qHCJIO X HaCTOJIbKO BeJIHKO, qTO 06a 3TH H3MeHeHHH He BblBOAHT HS MHO--
)KecTBa R+. Ha30BeM CYMMOU flucell a U b AefiCTBHTeJIbHOe "HCJIO, Bbl-
7

pmKalOw.ee Pe3YJIbTHpYIOw.ee H3MeHeHHe. HaI1pHMep, ecJIH CHaqaJIa c)I,e-
JIaTb H3MeHeHHe H8 4, a IlOTOM Ha 7, qHCJIO 12 I1epeH)I,eT CHaqaJIa B 16,
a IlOTOM 16 nepeH)I,eT B 23. Ho qT06bI 12 I1epeIllJIO B 23, Ita.zw H3MeHHTb
ero Ha 11, 3HaqHT, 4+7= 11, KaK H )l,OJImHO 6bITb. ECJIH me CHa4aJIa
C)I,eJIaTb H3MeHeHHe H8' -4, a IlOTOM Ha -7, TO 12 I1epeH)I,eT CHaqa,la B
8, a IlOTOM B 1. Ho qT06bI H3 12 I10JIyqUTb 1, Ha)l,O H3MeHHTb 12 Ha -11.
OrclO)I,a cJIe)l,YeT, qTO (--4) + (-7) = -11.
Boo6w.e, eCJIH a Ii b - I10JIOmHTeJIbHbIe )l,eHCTBHTbHbIe qHCJIa H
x>a+b, TO "PH H3MeHeHHH Ha -a qHCJIO x I1epeXO)l,HT B Jf-a, a I10TOM
"PH H3MeHeHHH Ha -b qHCJIO x-a I1epeXO)l,HT B (x-a)-b, T. e. B x-
-(a+b). Ho qT06bI I10JIyqHTb x-(a+b), Ha)l,O H3MeHHTb x Ha -(a+b).
::no I10Ka3bIBaeT, qTO (-a)+(-b)=-(a+b).
PaCCMOTpHM TeI1epb CJIOmeHHe qHCeJI "POTHBO"0JIO)KHbIX 3HaKOB.
HaqHeM co CJIyqmr, Kor)l,a CJIaraeMbIe - "pOTHBO"0JIOmHbte qHCJIa.
()qeB, H)I,HO qTO eCJIH H3MeHHTb qHCJIO x CHaqaJIa Ha a, a I10TOM Ha -a,
TO I10JIyqHM CHoBa x. I1HbIMH CJIOBaMH, x+(a+(-a»=x. TaK KaK, C
.ll.pyroii CTOpOHbI, H x+O=x, TO Ha)l,O I10JIOmHTb a+(-a)=O. I1TaK,
CYMMa npomueonO/lO:JICl-tblX aeUCmeUme/lbl-tblX QUcell pael-ta I-tY/lIO.
TeI1epb HaH)I,eM CYMMY a+(-b) B 06w.eM CJIyqae (MbI CqHTaeM, qTO a
H b - I10JIOmHTeJIbHbIe qHCJIa, a I10TOMY - b QTpm1.aTeJlbHO). ECJIH
a>b, TO a=(a-b)+b, H IlOTOMY a+(-b)=(a-b)+b+(-b). Ho 110-
CJIe)l,OBaTeJIbHbIe H3MeHeHHH qHCJIa x Ha a-b, b H -b MOmHO 3aMeHRTb
H3MeHeHHeM Ha a-b (H3MeHeHHH Ha b H -b aa.?HMHO YHHqTOmaIOTCH)..
n03TOMY I10JIOmHM a+(-b)=a-b, eCJIH a>b. OqeBH)I,HO, qTO I1PH a>b
H (-b)+a=a-b.
nYCTb TeI1epb a<b. B 3TOM CJIyqae HMeeM -b=(-a)+[-(b-a)),
H IlOTOMY a+(-b)=a+(-a)+[-(b-a)]=-(b-a). 3HaqHT, I1PH a<b
HMO I10JIOmHTb a+(-b)=-(b-a). TOT me pe3YJIbTaT nOJIyqHTCH I1PH
CJlOmeHHH -b H a: (-b)+a=-(b-a).
nOJIyqeHHbIe I1paBHJIa CJIOmeHHH )l,eHCTBHTeJIbHbIX qHCeJI MomHO
CcpOPMYJIHpOBaTb CJIe)l,YIOw.HM 06pa30M:
flpu C/lO:JlCel-tUU aeyx aeuCmeUme/lbl-tblX Quce/l 0{}l-t020 u mozo :JICe 3l-taKlJ
M/lYQaemcf/. 'iUC/lO mozo :JICe 3l-taKa, MOOY/lb KomopOZO paeel-t cY/fIMe MO{}Y/let1
CJlQ2aeMblX. n pu C/lO[){Cel-tUU 'iUCeA pa3/lU'il-t020 3l-taKa nO/lY'iaemcf/. tIUC/lO,
3HaK Komop020 c08naiJaem co 3l-taKOM C/102aeM020, UMelOUJi!20 60/lbWUU MO-
aY/lb, a MOOY/lb paeefl pa3l-tocmu 60/lbute20 U Mel-lbUle20 MOaY/leU C/lazae-
MblX. CYMMa npomueonO/lO:JICl-tbIX 'iuce/l palma I-tY/lIO, a C/lO{)fCel-tue C I-tY/leM
I-te .M£l-tf/£m 'iUC/la.
JIerKo I1pOBepHTb, qTO C/lO:JICel-tue B R 06/laOaem ceOUCtrl8aMU KOMMY-
mamU8l-tOCmU, accou,uamU8l-tocmu U COKpamUMocmu. 113 )l,aHHOrO BbIllie
OI1pe)l,eJIeHHH BH)I,HO, qTO I-tY/lb - l-teUmpa/lbl-tbIU 3/leMel-tm oml-tOCUme/lbl-tO
C/lO:JICel-tWl. e R.
Bbl'iUmal-tUe e Ml-tO:JICeCmee R oI1pe)l,eJIHeTCH KaK OI1epaI1.HH, 06paTHaH
CJlOmeHHIO. nOCKOJIbKY Kam)l,oe qHCJIO b B R HMeeT I1pOTHB0I10JIOmHoe
eMY qRCJIO -b, TaKoe, qTO b+(-b)=O, TO BbIqHTaHHe qHCJIa b paBHO-
CHJJbHO CJIOmeHHIO C qHCJIOM -b:
a-b=a+(-b).
338

B CaMOM ,!!,eJIe, ,!!,.rHI JII06blX a H b HMeeM:
[a+( -b)]+b=a+ [( -b)+b J =a,
a 3TO H 03HaqaeT, qTQ a-b=a+(-b).
LLJIH nOJIO}KHTe,lbHbIX qHCeJI a H b, TaKHX, qTO a>b, HX pa3HOCT!J
n .. !J 6Wla H3MeHeHl-ieM, npH KOTOpOM b nepexO,!!,HT B a. no aHaJIorHH C
c...1Hl\I Ha30BeM ,!!,JIfl JllOtThrx )J.eHCTBHTeJIbHbIX qHCeJI a H b qHCJlO a-I}
U3.A!H£HUeM.., nepeBO)I.SllII.HM b Ba. OHO nepeBO,!!,HT TOqKY ° B TOqKY a-b.
KaK H ,!!,JIH nOJIO}KHTe.TlbHbIX J\eHCTBHTeJIbHbIX qHCeJI 3TO H3MeHeHHe reo-
Me-rpHt.JecKH H306pa}Kae-rcfl HarrpaBJIeHHbIM OTpe3KoM, H'!!'YllI.HM H3 TO'!-
KH b B TOtJKY a. Ero ,!!,JIHHa pasHa paCCTOHHHIO OT HaqaJIa OTCqeTa ,lI,O
TOqKH a-b, T. e. 11O,!!,YJlIO qHCJUI a-b. MbI ,!!,OKa3aJlH cJIe,!!,YIOllI.ee BaLK-
Hoe YTBep}K,!!,eHHe:
flAuHa omp-.'!3K.Cl, uaywpo U3 mOtlKU b 8 mOtlKY a, paeHa la-bl.
BBe,n.eM B MHD}KeCTBO R OTHOlilelIHe nopfl,!!,Ka. By.n:eM CtJHTaTb, qTO
a>b B TOM H TOJIbKO B TOM CJIyqae, Kor,!!,a pa3HOCTb a-b nOJIO}KHTeJIbHa.
J1erKo ,lJ,OKa3aTb., tJTO 3TO <YfHOllleHHe aCHMMeTpHqHO H TpaH3HTHBHO, T. e.
HBJlfleTCfl OTHOIIIeHHeM CTpororo nDpH,!!,Ka. npH "TOM ,!!,JHI. JIJ.06bIX a H b
H3 R cnpaBe)J.JIHBO O)I.HO H TOJIbKO O,!!,HO H3 OTHOllleHHH: a=b, a>b, b>a.
T. e. OTHOllleHHe nopfl,!!,Ka B R JIHHeiiHO. nOCKOJIbKY a-O=a, TO a>O,
eCJIH aER+, H a<O, -eCJIH aER-.
HeTpY,!!,HO ,!!,oKa3aTb, qTOeCJIH a>b, TO ,!!,JIH JII06oro c E R HMeeM
a+c>b+c.
YnpamnenHH
1. .1J.oKa)l{HTe, qTO eCJIH 1JHCJlO a 3a,l\aeTCH napOH (Xl: YI), a qUC.9 b - napoii
(X2: Y2), TO a+b 3a.1\aeTCH napoH (XI+X 2 : YI+yJ.
2. .1J.oKa)l{HTe, 11TO .eCJlH a 3a,l\aeTCH napoii: (x: y). TO 'IHCJlO -b Sa,l\aeTCH na-
POH (y; x).
3. .1J.oKa)l{HTe KOMMYTaTHBHocTb H aCCOUHaTHBHOCTb CJI0)l{eHHH B MHO)l{ecTBe
.ZJ:eHcTBHTMbHblX 1JHCe..
3. YMHO)l{eHHe H ){eJIeUHe B MUO)KeCT.Be )J,eHCTBHTeJlbHblX IJHCeJJ.
ECJIH .lI.JIHHa OTpe3Ka a paBHa X, TO MbI nHCaJIH a=x -e, r,!!,e e - e.n.H-
HHqHblii OTpe30K. LLJIfl HanpaBJIeHHbIX OTpe3KOB, JleJKaIIJ:HX Ha O.ll.Hoii H
Toii}Ke npflMoii, Mbl TaK}Ke 6y,!!,eM nHcaTb a=x .e, CqHTafl, qTO x>O, ecJIH
OTpe3KH a H e O,!!,HHaKOBO HanpaBJIeHbI, H x<o, ecJIH HX HanpaBJIeHHH
npOTHBOnOJIO}KHbl (B 060HX CJIyqaflX Ixl paBeH ,!!,JIHHe OTpe3Ka a np"
e.ll.HHHlJ,e H3MepeHHfl e). ECJIH e=y .f, TO a=x (yf). Onpe.n.eJIHM npoH3Be-
,!!,eHHe qHCeJI x H Y K8K TaKoe qHCJIO Z, qTO a=zf, T. e. nOJIO}KHM, qTO z=
=xy B TOM H TOJIbKO B TOM CJIyqae, Kor,!!,a z .f=x (yf).
4T06bl BbIflCHHTb, KaK nOJIyqHTb xy, ecJIH 3a,n,aHbI qHCJIa x H y, 3aMe-
THM, qTO B CHJIY CBOHCTBa MYJIbTHnJIHKaTHBHOCTH ,!!,JIHHa OTpe3Ka a npH
e,!!,IIHHlJ,e H3MepeHHfl f paBHa Ixl.lyl. HanpaBJIeHHH }Ke OTpe3KOB a H f
COBna,!!,aIOT, ecJIH 3HaKH qHCeJI x H y O,!!,HHaKOBbI, H npOTHBOnOJIO}KHbl.
ecJIH 3TH 3HaKH pa3JIHqHbI. HanpHMep, ecJIH x<O H y<O, TO OTpe3KH a
H e HMelOT npOTHBOnOJIO}KHbIe HanpaBJIeHHfl, paBHO KaK H OTpe3KH e II
f, a nOTOMY HanpaBJIeHHfl OTpe3KOB a=[OAJ H f=[OFJ COBna,n,aJOT
(pHC. 186). ECJIH }Ke x>O H y<O, TO HanpaBJIeHHH OTpe3KOB a=[OAI
339

.
f
.
o
.
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F
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l
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A
0)
5)
PHC. IB6
H e=lOE) COBna,!!,aIOT, a OTpe3KOB e Ii f=[OP) npOTIiBOnOJIOmHbI, a no-
TOMY HanpmmeHHH OTpe3KOB a Ii f llpOTIiBOll0JIOmHbI.
113 CKa3aHHOrO BbIllie CJIe,!!,yeT, '1TO llpOIi3Be,!!,eHIie ,!!,eikpBHTeJIbHbIX
'1HCeJI Ollpe,!J,eJIHeTCH CJIe,!!,YIOw.IiM 06pa30M.
npOU38eaeHueM liuceA x U y Ha3bIBaeTCH '1IiCJIQ Z, MO,!!,YJIb KOToporo
paBeH llpOH3Be,!!,eHIiIO MO,!!,YJIeH MHOmlITeJIeH, Izl= Ixl'lyl, a 3HaK ll0JIO-
mHTeJIeH, ecJIlj 3HaKIi MH<DKIiTe.rrdi O,!!,IiHaKOBbI, Ii OTpliIJ,aTeJIeH B npOTIiB-
HOM CJIY'Iae. .il.JIH JII060ro '1HCJIa x IiMeeM x '0=0 'x=O.
JIerKo ,!!,OKa3aTb, '1TO Ii 8 MHOJICeCI1Z8e R YMHOJICeHue 06/laaaem C80a.
Ctrl8aMU Kf)MJ.tymamu8Hocmu, acCOl{uamU8Hocmu U aucmpu6ymU8HOCmU
omHocumeAbHO CAOJICeHUJl. CBoiicTBOM COKpaTIiMOCTIi OHO YJKe He 06JIa-
){aeT, TaK KaK H3 Zx=zy HeJIb3H C,!!,eJIaTb BbIBO.ll.a, '1TO x=y; MOmeT CJIY-
qHTbCH, 'ITO z=O, HO x=/=y, TOr.ll.a Bce paBHO zx=zy=O. ECJIIi me Z:#=O,
TO Ii3 Zx=zy CJIe.ll.YeT x=y. TaKHM 06pa30M, paBeHCTBa MomHO COKpa-
maTb JIHllIb Ha OTJIIi'lHble OT HYJIH '1IiCJIa.
ECJIH x OTJIH'IHO OT HYJIH, TO .lI.JIH JII060ro y E R HaH.lI.eTCH TaKoe z,
'1TO x=yz. 31'0 'IIiCJIO Ha3bIBalOT t{(ll;mHblM OT ,!!,eJIeHHH x Ha y Ii 0603Ha.
qalOT x : y. TaKIiM 06pa30M, B R onpe.ll.eJIeHO .lI.eJIeHIie Ha JII060e '1RCJIO,
OTJIH'IHOe OT HYJIH.
YnpamueUBU
1. 1l0Ka)l{HTe, qro YMHO)l{eHHe B R KOMMyrarHBJlO u aCCOUHaTHBJlO.
2. .1J.oKa)l{HTe, qTO YMHO)l{eHHe B R AHCTpHoyrHBHO OTHOCHTeJJbHO CJlO)l{eHHH H
BbIqHTaHHH.
3. 4eMY pBBeH 3HaK npoH3Be,!\eHHH 27 nOJlO)l{HTeJIbHblX H 320TpHuaTeJIbHblX
MHO)l{HTeJJeH?
4. 4eMY paBHo npoH3Be.ZJ:eHHe UeJIbIX qHCeJl OT -7 .1\0 IO?
5. BbIqHCJlHTe 3HaqeHHH CJle.ZJ:YIOIll,HX Bblpa)l{eHHH:
a) 702,3- (59-389,56 : 6,8). (59,3-5,64 : 9.4);
0) (6.8.52,4-256,32). (77.34+61,32: 7.3)-919.6.

nPElI.METHblA YI(A3ATEJIb
A6cuHCC OCb 101
AzutHTHBHOCTb 309
AKCHOMbl l 248
- CHCTeMa 249 H CJI.
- - KaTerOpHqHOCTb 251
- - MO,!(eJIb 249
-- - He3BBHCHMOCTb 251
-- - HcnpOTHBOpeqHBOCTb 251
AJIre6pa 226
AprYMeHT 141
-- npoMe)l{YTOqHblii 147
ACHMnTOTbl 171
AccoUHaTHBHocTb 228
60P.l\IOPbl 214
BeKTOp 200
- ItJIHHa 200
- HanpaBJIeHHe 200
BeJIHqHHa 332
- H3MepeHHe 332
-- nOlle Onpel1MeHHR 332
ROpOH 274
Rblpa)l{eHHe 106 H CJI.
-- C nepeMcHHblMH III
- -- 06JIaCTb onpe,!(MeHHH (CYlllecT'
BOBaHHH) 112, 141
- -- TO)l{l1ecTBeHHo paBllble 112
-- 'IHCJIOBoe 106
-- -- 3Ha'leHHe 107
BbICKa3b1BaHHH 12 II CJI'
-- l1H3'bIOHKUHH 18
- HMnJlHKanHH 20
- KOH'bIOHKUHH 16
- - OTpHuaHHe 14
- - ABOHHoe 15
-- COCTaBHble 13
1 4epTa 0603l1aqaeT CJlOBO. HanpH.
Mep,- CHCTeMa-aKCIIOM CHCTeMa, a
--MO,!(eJIb-aKCHOM CHCTeMbi MO.l\Mb
(HJIH MOlleJIb CHCTeMbi aKCHOM).
- - paBHOCHJlbHble (3KBHBaneJITHble)
13
- 3KBHBaneHUHH 23
- 9J1eMellTapHble 13
BblqHTaeMoe 266
rOMOTeTHH 217
- Ko3<j><!>HuHeHT 217
- ueHTp 217
rpalj> - 69
- opHeHTHpoBaHHblA 69
- OTHOllleHHH 76
- COOTBeTCTBHH 69
rpalj>HK
- OTHOllleHHH 73
- COOTBeTCTBHH 69
ypaBHeHHIJ 122
- Ij>YHKUHH 142
rpynna 242
napaJIJIeJIbHblX nepeHocoe 205
- nepeMelll.eHHH 193
- nOBopOToB 207
- n01106HH 219
- npe06pa30BaHHH 189
- caMOCOBMelll.eHHA 212
- <j>e110pOBCKalJ 214
lI.e MopraHa <j>opMYJIbi 20
.1J.eJIHMoe 267
.1J.eJIHMOCTb 290
- npH3HaKH 292 H CJI.
- cBoiicTBa 290
.1J.eJIHTeJIb 267, 296
- 06111HH 296
- - HaH6011bWHA 296
.1J.H3'bIOHKUHH 18, 56
.1J.HCTPH6YTHBHOCTb 232
.1J.HI!xpepeHUHan 176 H CJI.
- aprYMeHTa 177
- <PYHKUHH 176
.1J.OIIH 311
.1J.p06b 311
- BbI'IHTaHHe 316
34(

-- eCHTHqHaH 319
- - 6eeKOHeqHaH nepHO.!lHqeeKaa 323
-- - CJlO)l{eHHe 320
- - YMHO)l{eHHe 321
-- HeeoKpaTHMaH 312
-- 06p3llleJlHe B .!IeeH1 HqHYIO 322
- CJlO)l{eHHe 314
 iYMHO)l{eHHe 317
- 8KBHBaJIeHTHOCTb 312
EBKJlH4a aJIrOpHTM 306
E4HHHua H3MepeHHH HH 96
3aMKHyrocrb (nO.l\MHO)l{ecTBa OTHOCH.
TeJJbHO aJIr. onepauHH) 225
3HaKoB KpHBaH 121
3HaKoB npaBHJla 244
1t30M'0p4>H3M aJIre6p 226
HMnJlHKaUHH 20
-- 06paTHaH 21
- npOTHBOnOJlO)l{H3H 22
HHllYKUHH MaTeMaTHqeeKaH 261
I1HTerpan 182 H CJI.
- Heonpe,aeJleHHblH 182
- OnpeJleJIeHHblA 184
KaCaTeJJbHaH 178
- ypaBHeHHe 179
KBaHTop 53 H CJI.
- 06111HOCTH 53
-- CYlllecTBoBaHHH 53
KOJI04a 274
1(000bUO 244
- aCCOuHaTHBHoe 245
- C eJlHHHUeA 245
- KOMMyraTHBHoe 244
I(.Y6HK8.TOp1r1KII 42 H CJL
l(oMMYTaTHBHOCTb 230
l(oMn03HUHH
- napaJIJleJIbHblX nepeHocoB 204
- nOBopOTOB 207
- npe06pa30B8.BHA 188
- CHMMerpHA oceBblX 199, 207
I(OHTpan03HUHH 3aKOH 22
I(OHbIOHKUHH 16, 55
l(oopHHaTbI 96 H CJI.
- eKapTOBa npHMOYroJlbHaa CBGTeM8
102
- - - -- HaqaJIO 102
- H8 npHMoA 96
- - -- HaqaJIO 96
-- npe06pa30BaBHe 98
l(oPTe>K 40
- HHa 40
- KOMnOHeHn.t 40
-- paBeHCTBo 41
1(00000000HuHeHT yrJlOBOH 129
I(PaTHoe 295
- 06111ee 295
H8HMeHbillee :l9a
4M2
JJerHoH 274
JIeop 274
JIorHKa MaTeMaTHqeCKaH 12
JIyq qHCJIOBO!i 97
-- -- OTKpbITblH 97
MHO)l{ecTBO 25
- 4eKapToBo npoH3BeAeHHe 39
- llHcKpeTHoe 88
- 06be,aHHeHHe 33
- nepeeeqeHHe 31
- paBeHCTBo 27
- pa3HOCTb 36
- CqeTHoe 95
- YHHBepC3J1bHOe 30
- ynopH.!IOqeHHOe 46, 87
- - JlHHeAHO 87
- 9KBHBaneHTHOCTb 96
MYJlbTHnJlHKaTHBHOCTb 310
HepaBeHCTBo 110
- rpalj>HK 125
- 4BoHHoe 109, 111
- MHO)l{eeTBO pellleHHii 118
- paBHOCIIJlbHOCTb 119
- CHCTeM3 119
- CJleJlCTBHe 119
- 'lHCJlOBoe 108
- - CBoHcTBa 108 H CJI.
-- qHCJIOBbIX Bblpa)l{eHlI1I 110
HYJlb 270
06pa3
- 9JIeMeHTa 69
- MHO)l{eeTBa 90
OKpeeTHocTb ToqKH 166
OKPY)I{HOCTH ypaBHeHHe 124
OnepauHH aJlre6p8HqeeKaa 223
- - 6HHapHaH 223
- - KOMnOHeHTbi 223
- - n.apHa.ll 223
- - 06paTHa.ll 235
- - pe3YJlbTaT 223
- - yaapHti 223
- - qacrHqHaH 225
- - - 06JIacrb onpeJleJIeHHH 225
- - - nycraH 225
OpllHHaT OCb 101
Op4HHaTa HaqaJIbHaH 130
OpHaMeHT 214
OcraToK (npH IlMeHHH) :l68
OrHollleHHe 72 II CJI.
- aHTHpeKcHBHOCTb 00
- aH] HCHMMerpH'lHOCTb 80
- aCHMMerpHqHOCTb 80
- 6HHapHoe 72
- rpalj> 76
- rpalj>HK 73
- MHOHteCTBO aaJiaillUI 73

- 1l0pH)Ka 84 H CJI.
- - JlHHeHHDrO 87
- - HecTpororo 86
- - CTpororo 84
- pa3J1H'IHJf 73
- pe<j>JIeKcHBHble 79
- cHMMeTpH'IHOCTb 80
- TO)I{.ZJ:ecTBa (paBeHCTBa) 73
- TpaH3HTHBHOCTb 80
- 3KBHBaJIeHTHOCTH 81
Or06pa)l{eHHe 89 H CJI.
- 6HeKTHBHoe (B3aHMHO-0.l\1I03Ha'iHoe)
92
-- HHeKTHBHoe 92
-- 06paTHoe 92
clOpeKTHBI!Oe 93
Orpe30K 308 H CJI.
-- ,!(eJIeHHe B ){aIIHOM orHollleHHH 100
-- e.nHHHqHblH 309
- KOOp.l\HHaTbI Cepe.l\HHbI 105
-- HaTypanbHoro pH){a 'IHce. 269
- HecoH3MepHMOCTb 325
- OTKpblTbIH 97
-- pa36HeHHe 308
-- qHCJlOBOH 97
napa ynopH.l\OlJellllaH 38
-- -- KoMnOIJCIITbI 38
naCKaJlH TpeyrOllbHHK 50
nepeMew.eHHe 193
nepeHoc napanJIeJIbHbIH 200
nepecTaHoBKH 6C3 nOBTopeHHH 46
nepHO.l\ 6eCKOHeqHOH .necRTH'IHOH .l\Po-
6H 323
I1.nOCKOCTb KOOp.l\HHaTHaH 102
I1.nOw.a.l\b 182
- KpHBOIIHHeHHoH 'fpaneUHH 335 .
noBOpor 207
nO,!(MHO)l{ecTBo 28
- ){onOllHeHHe 35
- HecOOcTBeHHoe 28
- co6CTBeHHoe 28
no.lt06He 219
nOlle 245
- 'IHCJIOBoe 245
nOllyrpynna 243
nOllYKOJIbUO 246
nOHHTHe 5
-- 06beM 6
-- cO.l\ep)l{aHHe 7
nOCJIe.l\OBaTeJ\bHOCTb 'IHCJIOBaH 154 H
CJI.
- - 6eCKOHeIJHO MaJIaH 159
- - 6eCKOHeqHO 60JIbillaH 159
-- -- B03BparuaH (peKyPpeHruaH) 156
-- - 06w.HH IJJIeH 155
npe.neJI 161
-- nOCJIe.l\OBaTeJIbHOCTH 161
-- <j>YIIKUHH B TOIJKe 167
-- <j>YHKUHH lIa 6ecKoHe'lHOCTH 164
npe.l\HKaT 51 H CJI.
- .l\H3IOHKUHH 56
- HMnJlHKaUHH 57
-- HCTHHHOCTH MIIO)l{eCTBO 51
-- KOH'bIOHKUHH 55
-- MHoroMecTHblH 60
-- O.l\HOMecTHblii 51
-- onpe,!(MeHHH MHO)l{ecTBO 51
-- orpHuaHHe 55
-- 3KBHBaJl'eHTlIocTb 52
-- 3KBHBaJleHUHH 58
npe06pa30BaHHe 186
- reOMeTpH'IeCKOe 191
-- rpynna 189
-- e.l\HHHIJHOe 187
-- KOMn03HUHH 188
-- 06paTHoe 187
- CTeneHb 184
- TO)I{.ZJ:eCTBeHHoe 187
npH3HaK 7 H CJI.
- .l\MHMOCTH 292 H CJI.
- .l\OCTaTOIJIIbIH 
- He06xo.l\HMblH 7
nporpeccHH 156 H CJI.
-- apH<j>MeTH'lecKaH 156
-- reOMerpHqeCKa5I 157
npOH3Be.l\eHHe 264
npOH3Bel1eHHH npaBHJIO 44
-- -- 0606w.eIIHoe 44
npOH3Bo.ZJ:HaH 178
npoMe)l{YToK qHCJIOBOH 197
npOMHJlJle 321
npo06pa3 nOllHblH 70
nponopUHOHaJlbHOCTb 143 H CJI.
-- npHMaH 144
-- -- rpa<j>HK 144
- 06paTHaH 145
-- -- rpa<j>HK 146
npoueHT 321
npHMaH 129 H CJI.
-- KOOp.l\HHaTHaH 96
-- napan,1eJIblIOCTH YCJIOBHe 132
-- nepneH.l\HKYJIHpHOCTH yCJIOBHe 132
-- nyqOK 133
-- TO'lKa nepeceqeHHH 138
-- YroJI 135
-- ypaBHeHHe 06w.ee 176
-- ypaBHeHHe C yrJloBbiM Koa4Kt>Hl.(HeH.
TOM 129
PaBHOBeJIHKOCTb 332
Pa3JIO)l{eHHe KaHOHH'IecKoe 304
Pa3Mew.eHHH
- 6e3 nOBTopeHHH 47
-- C nOBTOpeHH5IMH 45
Pa3HOCTb 266
Pa3pblBa TO'lKa 173
P03eTKH 213
843

CBoikTBa 6 H CJI.
-- He3aBHCHMble 8
-- HecoBMecTHMble 8
-- nporHBOnOJIO)l{Hble 8
-- paBHOCHJlbHble 8
-- cYllleCTBeHHble 7
-- XapaKTepHCTHqecKHe 1
CJKaTHe K npHMoH 220
CHMMeTpHH 195 H CJI.
-- oceBaH 196
-- -- OCb 196
-- nepeHocHaH 208
-- nOBoporHaH 211
--,QeHTpaJIbHaH 210
-- -- ueHTp 210
CoKpaTHMocTb 235
COOTBeTCTBHe 67 H CJI.
-- rpalj> 69
-- rpalj>HK 69
-- MHO)l{ecTBO 3Ha'leHHH 70
-- MHO)l{eCTBO onpe,/J,eJIeHHH 70
-- HecoBMecTHMble 71
-- 06J1acTb ornpaBJIeHHH 69
-- 06JIacTb npH6b1THH 69
-- 06paTHble 72
-- 06De,/J,HHeHHe 71
-- nepece'leHHe 71
-- nOJlHoe 70
-- nporHBOnOJlO)l{Hble 71
-- nyCToe 71
-- CJIe.ZJ:CTBHe 71
CoqeraHHH 6e3 nOBTopeHH!i 48
CYMMbI npaBHJlO 42
CqHCJleHHH CHCTeMa 271 H CJI.
-- -- BbI'IHTaHHe 284
-- - ,/J,eJIeHHe 288
-- - .ZJ:ecHTHqHaH 275 H CJI.
-- -- .ZJ:peBHerpe'lecKaH 273
-- -- .ZJ:peBHeerHnercKaH 272
-- -- .ZJ:peBHepyccKaH 274
-- - Hen03HUHOHHali 272
-- -- n03HUHOHHaH 274
-- -- npH3HaKH .ZJ:eJIHMOCTH 294
-- -- pHMcKaH 273
-- -- CJIO)l{eHHe 282
-- -- YMHO)!{eHHe 286
-- -- lllecTH.ZJ:ecHTepH'IHaH 275
TaBTOJIOrHII 24
TeopeMa 61 II CJI.
-- 3aKJIlOqeHHe 62
-- 06paTHaH 64
-- nporHBOnOJlO)l{HaH 65
-- -- 06paTHoH 65
-- pa3DHCHHTeJJbHaH 'IaCTb 62
-- yCJIOBHe 62
TO'lKa e.ZJ:HHH'IHaH 96
TpaneQHH KpHBOJlHHeHHali 334
TpH3,!la 281
TbMa 274
344
YMeHblllaelOe 266
YpaBHellHe 113 H CJI.
-- rpalj>nK 122
-- KopeHb 114
-- MHO)/{eCTBO pellleHHH 114
-- OKpY)I{HOCTH 124
-- npHMoH 129, 136
-- paBHOCHJlbHble 115
-- CHCTeMa 126
-- CJIe.ZJ:CTBHe 115
YCJIOBHe 63 II CJI.
-- ,/J,OCTaTOqIlOe 63
-- He06xo,/J,HMoe 63
-- He06xo,/J,uMoe H ,/J,OCTaTO'lHoe 64
4JHrypa reOMeTpH'IecKaH 29
-- -- KB3,/J,pnpyeMaH 334
-- -- KOllrpY3HTHOCTb 194
-- -- nO,/J,06He 219
-- -- cTyneH'IaTaH 333
<l>YHKUHH 140 H CJI.
-- 6ecKoHe'lHO 60J1bUJaH 168
-- 6eCKOHe'lHO MaJiaH 166
-- B03pacTalOlllaH 162
-- .ZJ:H<I><I>epeHUHan 176
-- ,/J,P06HO-JlHHeHHaH 152
-- KBa,/J,paTH'IHaH 151
-- KOMn03HUHH 147
-- JIHHeHllaH 144
-- MHO)l{eCTBO 3Ha'leHHH 141
-- HeOrpaHH'IeHHaH 165
-- HenpepblBHaH B TO'lKe 173
-- HenpepblBHaH Ha orpe3Ke 174
-- 06.nacTb onpe.ZJ:eJIeHHH 141
-- 06paTHaH 148
-- OrpaHH'IeHHaH 165
-- nepBo06pa3HaH 182
-- npHpallleHHe 175
-- npoH3Bo,/J,HaH 178
-- y6b1BalOlllaH 163
-- 'IHCJlOBaH 140
LJacTHoe 267
-- HenOJlHoe 268
4eTBepTH 102
4HCJ10 247 H CJI.
-- ,/J,eHcTBHTeJJbHoe 336
-- -- BbI'IHTaHHe 338
-- -- .ZJ:eJleHHe 340
-- - M0.ZJ:yJlb 336
-- -- orpHUaTMbHoe 336
-- -- nporHBOnOJlO)l{Hbie 336
-- -- CJlO)l{eHHe 337
-- YMHo)l{eHHe 340
HaTypaJIbHoe 247 H CJI.
-- aKCHOMaTHKa 256, 257
-- aKCHOMaTHKa neaHo 263
-- B3aHMHO npocTbIe 296
-- BbI'IHTaHHe 266
-- -- eJleHHe 267

-- -- -- C OCTaTKOM 267
- -- OTIIOllleHHe nopH.ZJ:Ka 258
-- -- npocTble 299
-- -- CJIO)l{eHHe 256
-- -- COCTaBHble 299
-- -- YMHO)l{eUHe 264
-- nOJlO)l{HTeJJbHoe lI.eiicTBIITeJIbHOe 326
-- -- -- 3KCHOMaTHKa 331
-- -- -- BbI'IHTaHHe 330
-- -- -- .ZJ:eJIeHHe 330
-- -- -- CJIO)l{eHHe 330
-- -- -- YMHO)l{eHHe 330
-- nOJlO)l{HTeJIbHoe pa!\HOHaJIbHoe 312
-- -- -- 3KCHOMaTHKa 318
-- BWqHTaHHe 315
-- .ZJ:eJIeHHe 318
-- CJIO)l{eHHe 313
-- YMHO)l{eHHe 317
lleJIOCTHOCTH 06JIacTb 245
3HJIepa.BeHHa .ZJ:HarpaMMbl 29
3KBHBaJleHUHH 23, 57
3J1eMeHT anre6pw
-- -- HeihpaJlbHWH 238
-- -- norJlolllalOlllHH 239
3JIeMeHT MHO)l{ecTBa 25
- -- cHMMeTpHtlHwii (.!(aHHoMY) 240

346
CnHCOK 0603HaQeHHH
A - OTpHuaHHe BblCKa3bIBaHHH A, 14
H - HCTHHa, 15
J1 - JlO)l{b, 15
1\ - KOH'bIOIIKUHH, 16
JJ - 3aBelIoMO JlO)l{Hoe BblCKa3blBaIlHe, 17
V -lIH3'bIOHKUHH, 18
H - 3aBe,!(OMO HCTHHiloe BblCKa3bIBafIHe, 19
 - HMnJlHKaUHH, 20
 - 9KBHBaJleHUHH 23
E - 3HaK npHHa,!(JIe)l{IIOCTH, 25
fit: - oTpHuaHHe npHHalIJIe)l{HOCTH, 25
N - MHO)l{eCTBO HarypaJlbHblX qHCM, 25
Zo - MHO)l{eCTBO ue,'1b1X HeOTpHuaTeJIbIlblX qHCeJl, 25
N _ - MHO)l{eCTBO ueJibIX orpHuaTeJIbHblX 'IHCM, 258
N m - oTpe30K {I, 2, ..., m} HarypaJibilOrO pH.na 'IHCM, 269
Q - MHO)l{ecTBO paUHOHaJlbHblX qHCM, 25
Q+ - MHO)l{eCTBO nO,'10)l{HTeJJbHblX paUHOHaJIbHbIX 'IHCM, 313
R - MHO)l{ecTB,i ,AeiiCTBHTeJJbHblX 'I HCeJl, 25
R + - MHO)l{eCTBQ OJlO)l{HTMbHblX .neiicTB. 'IHCM, 326
R- - MHO)KeCTBO OTpHuaTeJJbHblX ,!(eHcTB. 'IHCeJI, 336
{Xl;' . . ,X n } - MHO)l{eCTBO, COCTOHlllee H3 3JIeMeHTOB XI , ..., X n , 27
{xIP(x)} - MHO)l{eCTBO 3J1eMeilTOB, 06JIa.naJOlllHx CBOHCTBOM P(x), 27
c: - 3HaK BKJllOqeHHH 28
n - 3HaK nepeceqeHHH MHO)l{eCTB, 31
U - 3HaK 06'be,!(JlHeHHH MHO)l{eCTB, 33
B A - lIOnOJIHeHHe K B B A, 35
B' - ,!(onOJlHeHHe K B B YHHBepCaJIbHOM MHO)l{eCTBe, 35
" - 3HaK pa3HOCTH MHO)l{eCTB, 36
X - 3HaK .neKapToBa npoH3Be,!(eHHH. 39
(al;...;a n ) - KOpTe)l{
n(A) - qHCJIO 3JIeMeHTOB MHO)l{eCTBa A. 42
A - qHCJIO pa3MellleHHH C nOBTopeHHeM H3 m 3JIeMeHTOB no k, 45
P n - qHCJlO nepecTaHoBoK H3 n 3JieMeHToB, 46
nl - <j>aKTOpHaJl, 46
A - 'IHato pa3MellleHHH 6e3 nOBTopeHHH H3 m 3JIeMeHTOB no k, 47
C - qHCJIO CO'leTaHHH 6e3 nOBTopeHHH H3 m 3J1eMeHTOB no k, 47
V - KBaHTop 06111HOCTH, 53
3 - KBaHTOp cYlllecTBoBaHHH, 54
,..., - 3HaK 3KBHBaJleHTHOCTH (npelIHKaToB, MHO)l{eCTB). 50, 94
aRb - COOTBeTCTBHe R Me)I{.ZJ:Y 3J1eMeHTaMH a H b, 68
R(a) - 06pa3 a npH cooTBeTCTBHH R, 69
R-I(b) - nOJlHblii npo06pa3 b npH COOTBeTCTBHH R. 70
R -1 - 06paTHoe COOTBeTCTBHe, 72

(a,+oo
f
f : x -->- Y, x -->- Y - or06pa)l{eHHe f MHO)l{eCTBa X B MHO-
)l{eCrBO Y, 89
f
f : x-->-y, X -->- y, y = f (x) - or06pa)l{eHHe f EweMeHTa x B Ewe-
MeHT y, 89
1-1 - 06paTHoe Or06pa)l{eHlle, 94
M(x) - roqKa C KOOp.l1.HHarOH x, 96
[,] - 00, a] - 'IHCJIOBbie JIY'IH, 96
la, + 00 ( - OTKpblTbli\ 'IHCJIOBOH JIyq. 97
[a.b J - qHCJIOBOH Orpe30K, 97
)a,b( - qHCJIOBOH npOMe)l{YTOK, 97
M(x,y) - TO'lKa C KooPllHHaTaMH x H y, 97
lirn an - npelleJi nOCJIellOBaTeJIbHOCTH an , 161
n-.oo
lirn I (x) - npellen <j>YHKUHH f B TOqKe a, 167
;Ie....a
lirn I (x) - npellen <j>YHKUHH I Ha 6ecKOHeqHOCTH, 171
;Ie.... 00
/!"x, /!"g- npHpHHe x, y. 175
dy - llHcjxfJepeHUHaJi <j>YHKUHH. 176
dx - llHcjxfJepeHuHa.!J aprYMeHTa, 177
t'(x) - npoH3Bo,!lH811 !\JYHKnHH f. 178
II(x)dx - HeOnpelleJIeHHblA HHTerpaJi <PYHKUHH f, 1&3
b
II(x)dx - onpe.lleJIeHHblK HHTerpaJi <j>YHKUHH I no orpesxy lal bJ. 184
a
E - TO)l{lleCTBeHHoe npe06pa30BaHHe, 187
S--! - oceBaH cHMMeTpHH C OCblO I, 196
a - nap8JlJleJJbHblH nepeHoc (oeKTOp), 200
......
AB - HanpaBJIeHHbJH OTpe30K C Ha'laJIOM A H KOHUOM B, 199
R - noBOpor BOKpyr TO'lKH 0 Ha yrOJi a, 207
o. ..., T - anre6paHqecKHe onepaUHH, 226
(X, 0) - aJIre6pa X C onepauH 0, 226
e - HeHrpaJIbJlbIH 9JIeMeH'!', 238
(J) - nor JIOIIlalOIIlHii 3JJeMeHT, 239
;; - 9JleMeHT, CHMMeTpH'lHyii 3JIeMeH'ry a, 240
: - OTHOUJeHHe lleJI HMOCTII , 290
a K b. K (a, bj - HaHMeHbillee 06IIlee .lpaTHoe a H b, 224, 295
aD b, D (a, b) - HaHOOJIblllHH 06IIlHH lleJIHTeJlb a H b, 224, 296
V m (a) - Mepa OTpe3Ka, 309
e; - KOHrpY3HTHOCTb, 194
EB - COCTOHT H3, 331.

OrJIABJIEHHE
rnaoa I. BYCKa3YOaHHH H OnepaQHH H8JI. HHMH
 1. C6011C11i6a U 6blCKa3bl8aHUR
I. 06"eKTbI, KJlaCCbI 06'beKTOO H HX CBoACTBa (5). 2. J10rHqeCKHe paccY)l{.ZJ:eHHH (9).
3. HCTHHHble H JlO)l{Hble BblCKa3blBaHHH (12).
 2. Onepa/{uu HOiJ 6bICKa3bl8aHURMU ,
I. OTpHU3HHe BblcKa3blBaHHH (14). 2. KOH'bIOHKII,HH BYCKa3b1BaHHH (16). 3. .1J.H3'bIOHK-
UHH BblCKa3b1BaHHH (18). 4. HMnJlHKaIl,HH BblCKa3b1BaHHH (20). 5. 9KBHBaneHII,HH BY-
CKa3b1BaHHH. T aBTOJlOrHH (23).
rnaBa II. MHOJKeCTBa, KOpTeJKH, KOM6HHaTopHKa
 1. Onepa/{uu HuB MltO:JICeCmeaMU
I. MHO)l{ecTBa (25). 2. Cnoc06b1 3a.ZJ:aHHH MHO)l{ecTB. PaBHble MHO)l{ecroa (27). 3. nO){-
MHO)l{ecTBa. .1J.HarpaMMbl9iiJlepa-BeHHa (28). 4. nepeceqeHHe MHO)l{ecro (31). 5. 06'b-
e,!\HHeHHe MHO)l{ecTB (33). 6. llanll.l1HeHHe K nOAMHO)l{ecTBY. Pa3HOCTb MHO)l{eCTB
(35).7. .1J.HsrpaMMbI 3HJlepa-BeHHiI H cjJoPMY JlOmqecKHx paccY)I{IleHHA (37).
 2. 3AeMenmbl KOM6uIiIltnOPUKU
I. .1J.eKapToBo npoH3BeJ(eHHe IlBYX MHO)l{ecro (38). 2. KOpTe)l{H (40).3. l(oM6HHaTOpHKa.
OpaBHJlO CyMMbI (42). 4. npaBHJlO npOH3BelleHHH (43). 5. Pa3MeIIl,eHHH C nOBTOpeHHHMK
(45). 6. YnopHIlOqeHHble MHO)l{ecTBa (46). 7. CoqeraHHH ()ea nOHTOpeHHH (48). 8. CBOH-
CTBa qHce.n C(49). .
rnaaa III. npeJl.HKaTY H TeopeMY
 1, n peiJuKambl U onepa/{uu nuB HUMU
I. OIlHoMecTHbie npellHKaTbi (51). 2. I(BaHTOpbl (53), 3. OnepaIl,HH Hall npellHKaTaMR
(54). 4. HMnJlHKaII,HH H 3KBHBaJleHUHII npellHKaToB (57). 5. MHorOMeCTHye npeJl.HKa-
I'bI (59).
 2. TeopeMbI
I. CTpoeHHe reopeMbi (61). 2. TeopeMa, 06paTHaH JlaHHoH (63). 3. TeopeMa, npOTHBO-
nOJlO)l{HaH Jl.aHHOH (65).
rnaoa IV. CooTBercToHH, OTHoweHHH, oTo6paJKeHHH
 1. CoomaemcmeUR U onepalfuu HOiJ nUMU
I. 6HHapHble COOTBeTCTBHH (67). 2. HeKOTOpble THnbl COOTBeTCTBHH. OnepaII,HH H8JI.
COOTBeTCTBHIIMH (70). 3. OrHollleHHH Ha MHO>Kecroe (72). 4. rpalj> 6HHapHoro OTHOIlle-
HHH (76).
 2. OmnoweHuR !1KIJU8a1lenmI-WCmu
I. Pn6HeHHe MHO)l{ecroa Ha nonapHo HenepeceKalOIII,HeCH nO,II.MHO)l{ecTB8 (78). 2. Oc.
HOBRbIe cBoACTBa OTHollleHHA (79). 3. OrHollleHHH 3KBHBaneHTHOCTH (81).
3ot8

I 3. OmHomeHuJt nopMKa
J. OrHollleHHH cTpororo nopH.ZJ:Ka (84). 2. OmollleHHH HecTpororo nopH.ZJ:Ka (86). 3. Yno-
pH.ZJ:OqeHHble MHO)l{ecTBa (87).
 4. Om06pa:J/CeHuJ!
I. Oro6pa)l{eHHH (89). 2. nOJIHbli\ npoo(\pa3. 06paTHoe oT06pa)l{eHHe (91). 3. 9KBHBa-
JlfHTHble MHO)l{ecroa (94).
rJlaBa V. l(oopJl.HRaTY, ypaoReRHII H RepaBeRcTBa
 1. KoopBuHambl H(l npRMou
1. l(oop.ZJ:HHaTbI Ha npHMoi\ (96). 2. npe06pa30BaHHe KOOP.l\HHT Hj;\ npHMoH JIHHHH (98).
3. HeKoropble 3a.ZJ:aqH aHaJIHTHqeCKOH reOMeTpMH Ha npHMoA JlHHHH (99).
 2. KooplJwlO.mbl Ha nJtOCKocmu
1. .1J.eKapToBa npHMoyrOJIbHaH CHCTeMa Koop.ZJ:HHaT Ha nJIOCKOCTH (101). 2. npe06pa-
30BaHHH Koop.ZJ:HHaT Ha nJIOCKOCTH (103). 3. HeKoTopble 3a.ZJ:aqH aHaJIMTHqeCKOH reo-
MeTpHH Ha nJlOCKOCTH (104).
 3. lJuCJt06ble U 6yK6eHHble 8btpa:J/CeHUR
1. 4HCJIOBble Bblpa)l{eHHH; (106). 2. 4HCJIOBble HepaBeHcTBa (108). 3. PaBeHCTBO H He-
paBeHCTO 'IHCJIOBblX Bblpa)l{eHHH (110). 4. Bblpa)l{eHHH C nepeMeHHblMH (111).
 4. Ypa8HeHUR U Hepa8PHCm8a
1. YpaBHeHHH C O.ZJ:HOH nepeMeHHoH (113). 2. TeopeMbI 0 paBHOCHJIbHOCTH ypaBHeHHA
(117).3. HepaBeHcTBa C O.ZJ:HOH nepeMeHHoH (118). 4. YpaBHeHHH C .ZJ:ByMH rrepeMeHHblMH
(122). 5. YpaBHeHHe OKPY)l{HOCTH (124). 6. rpa<pHKH HcpaBeHCTB (125). 7. CHCTeMbI
ypaBHeHHH H HepaBeHcTB (126).
 5. JIuHeUHble ypa8HeHUR
1. YpaBHeHHe npHMoH JIHHHH C yrJloBblM K03<j>lj>H!\HeHTOM (129).2. YCJIOBHII napaJIJIeJJb-
HOCTH H nepneH.ZJ:HKYJIHpHOCTH npHMblx (131). 3. YpaBHeHHe nY'IKa npHMblx, npoxo-
.ZJ:HI!\HX 'Iepe3 3a.ZJ:aHHYIO roqKY; ypaBHeHHe npHMoH, npoxo.ZJ:Hllleii qepe3 llBe TO'lKH
(133). 4. YroJI Me)I{.ZJ:y .ZJ:BYMH npHMblMH (135). 5. 06111ee ypaBHeHMe npHMoi\ (136).
6. TO'lKa nepeCeqeHHH npHMblx (138).
rJlaBa V I. cl>YHKIIHII, npe.ZJ:eJI, npoM3BO.l\HaH, HHTerpaJI
 1. lJuCJt06ble cfJYHK/{UU
1. cl>YHKUHH H Bblpa)l{eHHH (140). 2. npHMaH nponopUHOHaJIbHOCTb, JIHHeHHaH 3aBHC""
MOCTb H HX rpa<j>HKH (144). 3. 06paTHasr nponOpllHOHaJIbHOCTb H ee rpa<j>HK (145),
4. l(oMno3HUHH <j>YHK!\HH (CJIO)l{HaH <j>YHKUHH) (146). 5. 06paTHaH <j>YHKUHH (147).
 2. nocmpoeHue epacfJuK06 cfJYHK/{UU
1. nocTpoeHHe rpa<j>HKa «no TOqKaM» (148). 2. nocTpoeHHe rpa<j>HKa nYTeM napaJIJlenlt-
Horo nepeHoca CHCTeMbi KoopllHHaT (149). 3. rpa<j>HK KBa.ZJ:paTH'IHOii <j>YHKUHH (151).
4. rpa<pHK .ZJ:p06Ho-JIHHeHHoi\ <j>YHKIIHH (152).
 3. nOCJteB06ameJtbHOCmU
1. 4HCJIOBble nOCJlellOBaTeJIbHOCTH (154). 2. PeKyppeHTHble nOCJIellOBaTeJIbHOCTH (156).
3. 5eCKOHe'lHO 60JIbillHe H 6ecKOHe'lHO MaJIbie nOCJIeJI,OBaTeJJbHOCTH (158). 4. npelleJJ
nOCJIe.ZJ:OBaTeJJbHOCTH (161).
 4. npeaeJt cfJYHK/{Ull
1. B03pacTaHHe H y6b1BaHHe <j>YHK!\Hii (162). 2. OrpaHHqeHHble H HeOrpaHH'IeHHue
<PYHKUHH (164). 3. 5eCKOHe'lHO MaJIble <PYHKUHH (165). 4. npe.ZJ:eJI <j>YHKUHH B TQ'Ie
(167). 5. npe.ZJ:eJI <j>YHK!\HH Ha 6eCKOHeqHOCTH (169). 6. HenpepblBHble <PYHKUHH (172).
7. CBoHCTBa <j>YHKUHH, HenpepblBHblx Ha oTpe3Ke (174).
 5. JI,ucfJcfJepeH/{UaA, npoU36oiJHaR, UHmeepaA
1. npHpal!\eHHe <PYHKIIHH (175). 2. .1J.HqxpepeHUHaJI <PYHKUHH (176). 3. npOH3BOJl.HH
(177). 4. MeXaHHqeCKHH CMblCJI npoH3Bo.ZJ:HOH (179). 5. cl>oPMYJlbI llHcjxfJepeHUHpoo8HHII
(180). 6. Heonpe.ZJ:e.'!eHHbli\ HHTerpaJI (182). 7. Onpe.ZJ:eneHHblH HHTerpaJI (184).
rJIaBa VII. reOMeTpH'IeCKMe npeo6pa30BaHHH
 1. F Pynna npe06pQ.3()6QHuu
I. npe06pa30B8HHH MHO)l{eCTB (186). 2. reOMeTpHqeCKHe npe06pa30BaHHII (190).
349

 2. Fpynna nepeMeU{eHuu n/lOCKOCmu U ee nodepynnbt
1. rpyrrna nepeMe1UeJlHIi nJlOCKOCT-H (193).2. Oceeall cHMMeTpHH (195). 3. napaJlJreJlb'
HYIi nepeHoc (199). 4. OnepauuH Ha.ll: napaJlJleJlbHblMH nepooocaMH {204). 5. nOBOpOTbi
nJlOCKOCTU (206). 6. LLeHTpaJlbHaH H rrOBopOTHaH cUMMeTpuH (210). 7. rpyrrrra caMO-
C()8MeIll,eHHi\: cjmryJ>b!. 50p;nopbl II Opll8MeHTY (212).
 3. FeOMempU'leCKUe npe06pa3()(IQHUR noiJ06uR U CJlCamUR
l. rOMOTeTHH (217). 2. rOMoTeTuH U rrpe06pa30BaHuH rro.ll:06uu (218). 3. npe06pa30'
Ballue C)I{aTHH K rrpHMoH (220).
rJIaBa VIII. AMe6pall'lecKRe onR R anre6pY
 1. A/lee6paU'IeCKUe onepa/{uu U a/lee6pbl
1. 06w.ee rrOHHTRe aJlre6paM'IecKoi\: onepauuH (222). 2. AJIreOpbl (226).
 2. C60ucmea aA2e6paU'lf!CKUX onepa/{uri.
1. AccoUHaTHBHble aJlre6paHqeCKHe orrepaUHH (228). 2. 1(0MMYTaTHBHocTb (230). 3. .1J.H-
CTPH6YTHBHOCTb (232). 4. COKpaTHMoCTb (234). 5. 06paTHble onepaUHH (235). 6. He[T'
paJIbHbIH H nOrJIOlllalOlll,UH 3,'1e!1leHTbI (238). 7. CUMMerpH'IHble 3.MeHThI (239).
 3. HeKomopble poiJbl aA2e6p
1. rpynnbl H nOJlyrpynnbl (242). 2. 1(0JlbUa H nOJlH (243).
rnaBa IX. HarypanbHble '1Hena
 1. CucmeMbl aiccuoM U ux C8ol1cmea
L AKcHOMarHqecKHA Mero.ll: B MaTeMarHKe (247). 2. MO.ll:eJJH CHCTeMbI.aKcHoM (249).
3. HenpoTHBOpeqHBOCrb, He3aBHCHMocrb H KaTerOpU'IHOCrb CHCTeMbI aKCHOilf (250).
 2. AKcuoMamuKa MHo:HCeCmea HamypallbHblX '1uceA
L B03HHKHoBeHHe nOHHrHH HarypaJlbHoro qHCJIa (252). 2. 1(0JlHqecroeHHaH TeOpHH
HarypaJlbHb!X qHCen (254). 3. AKCHOMbI CJ!O)l{eHHH (256). 4. OrHollleHHe nopH.lI:Ka B MHO-
:lKerBe HarypaJlbHblx qHCeJl Hero cl!DlicrBa (258). 5. HeOrpaHHqeHHOCrb H .lI:HcKpeT-
Hocrb MHO)l{ecroa HarypaJlbHblx qllt!l!n (260). 6. npHHuHn MaTeMarHqeCKOH HH.lI:YKUHH
(261). 7. AKcHoMarHKa neaHo (262).
 3. Apuq,MemuTw namypallbHblx !jUce/l
1. YMHO)l{eHHe HarypaJlbHbIX qHCM (264). 2. BbI'lHraHHe HarypaJlbHblX qHeeJI (266).
3. .1J.eneHHe HarypaJlbHb!X qHcen. .1J.eJJeHHe c ocr3rKOM (267). 4. flopH.lI:KOBbie H KOJlH'
'IeCTBeHHble HarypaJIbHble qHCJla (268). 5. HYJlb (270).
 4. CucmeMbl C'lUC/lenUR .'
1. Heno3HUHOHHble CHCTeMbt C'IHCJleHHH (271). 2. nOOIUI;IIDRHbIe CHCTeMbt C'IHCJleHHH
(274).3. 3anHcb HarypaJlbHblx qHCeJl B }J,ecHTH'tl10H CHCTeMe C'IHCJIetml1 (275). 4. 3anHcb
qHCeJJ B .lI:pyrux n03ltUHOHHblX CHCTeMaX CqHCJlellHH (277). 5. CJ10JKeUHe 'fRceJI B .l\e-
CHTI1'1HoM H JlPyrl1X II03HUHOHHb!X CRCTeMaX C'IHCJIeHHH (282). 6. BbI'lRTI1HHe B .lI:eCH-
rR'qHojj H .lI:pyrnx CHCTeMaX CtfMCJIeHHH (284). 7. YMHO)l{eHire 'IHCeJl B .lI:ecHTH'IHOH H
.l\pyrux n03HUHOHHblX CHCTeMaX CqHCJleHHH (286). 8. LLeneHHe B .lI:ecHTH'IHOH H APyrux
n03HUHOHl!blX CHcreMax CqHCJleHHH (288).
rnaoa X. J),eJlHMOCTL QeJIYX HeO'l'JMltUtTeJ\bHIoIX '1l1CeJI
 1. OmH.OUleH.ue ikAuMDcmu u ezo C60Ucnwa
l. CBoAcTBa OTHOJIIeHHR AeJlHMOCTH(290).. 2. npH3H8KH "eJlHMOC11I (292). 3. npH3H<lKH
J{eJJHMocrH B APymx CKCTeM8X C'lIfCJ1eHHR (294). 4. I(paTHbIe H J{eJIH1'eJIH (295).
5. CBoHcrBa HaHMeHblllero 06111ero KparHoro H HaHOOJIblllerO 06111ero .lI:eJJHTeJJH (297).
 2. npOCmble 'lUC/la U ux c601lc11l.6a
l. npOCTble H cocraBHble qHCJIa (299). 2. Peillero 9paroc<j>eHa (300). 3. OcHoBHaH reo-
peMa apH<j>MerHKH HarypaJlbHblx qHCeJl (302). 4. l(aHOHHlJeCKHe p&3JI0!KeHHH H .lI:eHCr-
BHR HaJ{ 'lHCntilH (304). 5. AnropHTM EBKJIIIJI,8 (006).
-

rJIaBa x I. PaUHOHaJlbHble H aeiiCTBH-:-eJlbHye 'IHCJIa
 1. M Ho:HCeCmeO nOAo:HCUme/lbHblX pa/{uOHaAbHblX '1uce/l
1. H3MepeHile oTpe3KoB (308). 2. 9KBHB3JIeHTJlble .np06H (310). 3. nOJIO)l{HTeJIbHble pa.
UHOH3JIbHble qHCJI3 (312). 4. CJ!O)l{eHHe nOJIO)l{HTeJIbJlblX P3UHOH3JIbJlblX 'lHceJI (313).
5. CBOHCTB3 CJIO)l{eJlHH. BblqHT3HHe (315). 6. YMHO)l{eJlHe II ,l\eJIeHHe nOJIO)l{H1'eJIbJlblX
P3UHOH3JIbHblX quceJI (317). 7. AKCHOM3TH'ICCKoe nOCTpOeJlUe TeOpHH nOJIO)l{HTeJIbJlblX
P3UHOH3JIbJlblX qHce.'1 (318).
 2. JI,ecRmU'lHble Bp06u
1. .1J.ecHTHqHble ,l\p06H H ,l\eHCTBHH H3,l\ HHMII (319). 2. npe06p330B3HHe O(SbIKHOBeJlHbIX
.npo6eH B aeCHTHqHble (321). 3. 5eCKOJle ' IHbIe nepHo,!(HQeCKHe ,l\eCHTHQJlble ,l\p06H (323).
 3. nOAOJ/CUme/lbHble BeUCmeUme/lbH.ble 'lUC/la
1. HecoH3MepHMble oTpe3KH (325). 2. nOJIO)l{HTeJIbHble ,l\eHCTBUTe,'1bJlble QHCJI3 H 6ec-
KOHeQHble .neCHTHQHble ,l\p06H (326). 3. OTllorneHHe nOpH,l\K3 B MJlO)l{eCTBe R + (328).
4. CJ!O)l{eJlHe H YMH6)\{eHHe B MHO)l{eCTBe R+(329). 5. AKCHOM3THK3 MJlO)l{CCTB3 nono-
)\{HTeJIbJlblX ,!(eHcTBHTeJIbHbIX QHCM (330). 6. H3MepeHHe BeJIHQHJI (331). 7. H3MepeJlHe
nJIOllla,!(eH (332). 8. nJI01ll3,!(b KpHBOJIHHeHJlOH TpaneUHH (334).
 4. MHO:JICeCm80 BeuCm8Ume/lbHblX 'luce/l
1. nOJIO)l{HTeJJbHble H OTpJlU3TeJJbJlble QHCJIa (335). 2. CJIO>Kenue H BblQHT3HHe ,!(eHcT-
BHTeJIbJlblX QUCM (337). 3. YMJlO)l{eJlHe H ,l\eJIeJlHe B MHO)l{CCTBe .neHcTBUTeJIbJlbIX QH-
CM (339}.,
n peiJM.emHblI1 YKa3ame/lb (341).
CnucOK 0603Ha'leHuu (346).

H5 N2 1284


BUl/.eHKUH Hay. HK06l/.e6UIt
U&tUJ,Kal/.O AHarOl/.UU MUXaUl/.OBUIt
PO:HtiJecr6eHCKast Bal/.epust BOpUI:06Ha
CrOUl/.OBa JI10606& Uerp06Ha


MATEMATHKA


Pe,l\aKTOp A. M. A6paMoB
XY,l\O)\{eCTBeHHbIH pe,l\aKTOp E. H. l(apacHK
TexHuqecKHH pe,l\aKTOp C. A. nTHUYHa
1(0ppeKTop H. H. HOBHKOBa
C,l\aHO B Ha60p 29/IV 1977 r. nO,l\nHCaHO K neqaTH
21/X 1977 r. 60X90 1 he. 5YMara THn. Ng I. neq. n. 22.
Yq.-H3.l\. n. 24,15. THpa)\{ 100 TblC. 3K3. 3aKa3 Ng 702.
Op,l\eHa TpY,l\OBOrO I(pacHoro 3HaMeHH
H3,l\aTeJlbCTBO «npOcBellleHHe» rOCY,l\apCTBeHHOrO
KOMHTeTa COBeTa MHHHCTpOB PC<I>CP
no ,l\enaM H3,l\aTeJlbCTB, nonHrpa<j>HH H KHH)\{HOH TOprOBJlII.
MocKBa, 3-iI npoe3.l\ MapbHHoH pOlllH, 41.
OrneqaTaHO C MaTpHU op,l\eHa OKTH6pbCKOH PeBomOl.(HH
H Op,l\eHa TpY,l\OBOrO I(pacHoro 3HaMeHH
nepBoA 06pa3uoBoil THnorpa<j>MH HMeHM A. A. >K.!\aHoBa
Ha l(aJlHHHHCKOM op,!(eHa TpY,l\OBOro I(pacHoro 3HaMeHH
nOJIHrpa<j>KoM6HHaTe ,!(eTcKoH JlHTepaTypbl
HM. 50-.
eTHH CCCP POCrJlaBnOJlHrpa<j>npoMa
rOCKOMH3.l\aTa COBeTa MHHHCTpOB PC<I>CP.
l(aJIlIHHH, npocneKT 50-JlenlH OKTH6pH, 46.
UeHa I PY6. 20 Kon.




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