Matematika sa zbirkom zadataka prvi razred gimnazije

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MATEMATJ1KA Yv6eH11K ca 3611pKoM 3a~aTaKa . 3a rrpB11 pa3pe~ r11MHa3MJa 11 cpe~IDJ1X CTpyqH11x IIIKOJia

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Jfa;i:aBa'IKa icyha ,,Klett" µ.o.o. CBe-I03apa liopoBJIDa 15/IV, 11000 Beorpa,;::i; TeJI.: 011/3348-384, aKc: 011/3348-3 5 [email protected], vwi! ·Jdett.rs Jf3,:i;asa11:

3a H3;J;aBa"'la: fop;:i;aHa KHe)l(eBllii Oprrllii faasHJI ype;J;HJIK: An:eKcaH,D;ap PajKoBllii Ype,ll;B1IK: ,;:i;p BpamtcrraB IImrnBJID PyKoBo~an; npojeKTa: AneKcaH,D;ap PajKOBM illT3.Mila: Pornrpa
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Pe'I ayTopa Y1,16em1K rrpe,n; BaMa je HarrwcaH Ha ocHoBy B(l)Keher rrporpaMa MaTeMarnKe 3a rrpBu pa3pe;:1. rKMHa3Hja (rrpwpo,n;Ho-MaTeMaTH'fKOr, orrmTer H ,n;pymTBeHo-je3WIIeIDI OKO 3aBO}la 3a yttanpel)HBa.Ibe o6pa30Ba.J:b3. H BaCIIHTalba (http://www.zuov.gov.rs). J.13Mel)y OCTaJior, Ba.2KHY HOBIDI}' y HameM o6pa30BHOM CJlCTeM)' rrpe,n;cTaBJbajy CTaH,n;ap,n;H rrocTHrnyha 3a Kpaj ocHOBHor o6pasoBaIDa (yCBojem1

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Y3 craJIHo TPara.Ibe 3a CBe 6oJhuM HaqJIBJrna rrpe,n;cTaBJba.Iba OHora

mTo Tpe6a HayqHTH. BepyjeMo ,n;a he Taj TpeH)I; fo1n1 rrpeno3H.aT y KH:.U3J£ npe.u Ba.Ma. CMaTpaMo ,n;a je ,n;o6po 6apeM y npBoM paspe,n;y, Ka,n;a TO rrporpamt !IOII)'DITajy, no~1 3aje;:i;1m'IAJ' KlhHry 3a CBe cpe,n;IbolllKonn;e. Ilpe cBera, TO je KopucHo 36or yCBaja1i>a 3aje;:mwnrn:x crami;ap;!,a 3a cpe;IH>e o6pa3oBalbe H onmTe TeH,n;eID+Hje ,n;a ce rrpo,n;pKJ1 nepuo,n; y KojeM ce MJia;l;JIM Jbyri;uMa npyxajy no;:rje;:maxe maHce Ka,n;a je y IIHTalhY o6pa30BaJbe. HanorneTKJ, BepyjeMo .a;a Ha raj Ha'Um MO:lKe .::i:orm .::i:o no;noan.a HHBOa MaTeMaTWJ:Ke IIHCMeHOCTH, a THMe

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CBaKJt o,n;eJbaK je o6JIHKOBaH y cKJia,n;y ca yo6wiajeHOM CTpYKTYPOM MaTeMaTJfqKOr TeKCTa. ,ll;eqnrnmi;Hje Ji TeopeMe ca ,Il;OKa3HMa HajBaJKHHjH cy ,n;eJIOBH TeKCTa H rroce6Ho cy HCTaKHJTH. 3aBpmeTaK CBaKor ,n;oKa3a 03Haqea j e KBa,n;paTMheM . IIpaTe mc ,n;eTaJbHO ypat)eHH rrpHMepH (KojJrr y KlhH3H :HMa 244). Kpaj rrp:HMepa je 03aaqea poM6oM . IlpHMepe rrpaTe 3a,n;au;H KOjH cy 6HpaHl1 TaKO ,n;a H>HXOBO pemaBalhe ,n;orrp:HHOC:H 6oJbeM pa3y-MeBaibY ca,n;p)l(aja o KOjHMa je peq. 3aTO rrpe,n;naJKeMo ,n;a pemaBaH>e OBMX 3ap;aTaKa 6y,n;e cacTaBHH ,n;eo yrro3HaBalba ca o,n;roBapajyliMM ca,n;p)l(ajHMa. Pemelba OBHX 3ap;aTaKa H:HCY ,n;aTa Ha Kpajy KlhHre jep CMaTpaMO ,n;a MX CBaKo MO)l(e pemHTH Ha ocHoBy o6jamlbelba H rrpHMepa KojH HM rrpeno,n;e.

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JIOnfKA If CKYIIOBJ.1 ........................................................................ 9 JionflIKJt Be3HJilUL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . l(J", Jfc1ey-tlKIUrje ..................................................................................... 32 L'Ie eHTit KO 61maTopnKe ...................................................................... 36 KBaHTHcpuxaTopu .............................................................................. 38 33.Aa.wt .......................................................................................41

PEAJIID1 liPO}EBJ1 ........................................................................... 45 ,Uel(J{M
IIPOIIOPIUfOHAJIHOCT .......... . ......................................................... 81 Pa3Mepe u nporropmrje ............... . ................... .......... ................... ......... 82 )J,HpeKTHa H o6peyra nponop.qi1:0Ha.JIHOCT......................................................... 86 IlpJtMeHe rrponop~nja .......................................................................... 90 3a,lla.qJt .......•...............................................................................98

YBO,IJ; Y fEO .. fETPHJY ...................................................................... .101 TaqKe, rrpase Jt pasmL 0,D;Hocn np1maAaH.a. ...................................................... 102 IJapaJie.'IHOCT ................................................................................. 110 Pacnope!l: TaqaKa. ,ITpK, nmrynpasa H rrorrypasaH .................................................. 115 B(l)f(He re0Merp1rjcxe cpmype. Yrao H MHoroyrao .................................................. 119 IIo.zzy;:xapaocT ,:ryJ!Oi n MepeH.e ,lzy)KIDie A)')KH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 KpfJKHJma n Kpyt ............................................................................. 126 OpnjeHTamrja pasm1 .......................................................................... 128 3a;iami ...........................•.......................................................... 130

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IIO,D;Y,D;APHOCT ............................................................................ 135 ITo,n;y,n;apHocT ,n;y)K11 ..............................................................••........... 136 ITojaM 1130MeTpHje. ITo,n;mapHocT qnuypa ...........................................•............ 139 ITo,n;y,n;apHoc1: yrnoBa ........................................................................•. 144 HopMaJIHOCT rrpaBMX ......................... ........................ ......................... 151 YrnoB11 Ha TpaHcBep3aJIH ......................................................................154 YrnoBH Tpoyrna ...................... ........ .. ............................. .................. 157 CTaBOBH rro,n;y,n;apHOCT11 TpoyrnoBa ...................... ........................ ................ 159 Oco6Htte je,n;HaK0KpaI01X TpoyrnoBa 11 noCJiewm;e ..... ...........•................................ 164 CrnB rro,n;y,n;apHocrn CCY .....................................••....•...•....•••............... 168 TpH Ba)KHe rroCJie,n;m.i;e CTaBOBa no,n;mapHocrn .......... ........••.............••.....•..........• 170 ITpHMeHe CTaBOBa rro,n;mapHoCTM Ha Tpoyrao ...................••.•...........•..•....•.......... 171 ITpHMeHe cTaBoBa no,n;mapHocTM Ha Kpyr ... ........... ........••...•...........••...•.........•• 176 ITp11MeHe CTaBOBa rro,n;mapHOCT11 Ha qeTBopoyrao ...... ........•.•..•............•.•...........••• 178 ITp11MeHe CTaBOBa rro,n;mapHOCTM y KOHCrpyi<:qMjaMa ...........•••.•...........•...•..........•••• 186 HopMaJIHOCT rrpaBe H paBHH ................................. .••................•......•......•• 193 J;11e,n;ap. HopMaJIHOCT paBH11 ................................•••.....•.......•.•••......•..••••• 197 3a,n;au;11 ........................................... ........•••.......•.....•••............•.•• 201

BEKTOPJ1 .................................................................................. .213 YcMepeHe ,n;~1111 BeKTop11 .....................................................................214 Orrepau;Hje ca BeKTOpHMa ......................................................................217 Jl11HeapHa 3aBI1CHOCT BeKTOpa ... .... ...................................... ..................... 224 Koop,n;HHaTe BeKTopa y paBHM ..................................................................228 3a,n;au;H .......................... ......... ..... .............................•..............••230

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PAID10HAJIHII AJITEBAPCIGI H3PA3H ..................................................... 253 IIojIDI n3pa3a. L{pso 1opa.3a .................................................................... 254 Uenn anre6apcKn 1opa3H ...................................................................... 256 IIommoMu ca ·e.lHOM rrpoAeH;'bJIBO t . • . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . • • . . . . • • • • . . . . . . . . . . . . • • • . • • 265 H3ll u H3C rro;nmoMa ........................................................................ 270 ParrnoHanHH a,Tie6apcxu H3pa3n ................................................................ 272 HeKe oCHOBHe Heje;l,HaKOCTJL ................................................................... 276 fumeapm1 H3pa3Jt n jewia
CillflIHOCT ................................................................................ 313 iepeH>e ,il)')K.U . . . . . . . . . . . . . • • • . . . . . . . • . • . . . • . • • • . . . • • . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . 314 Pa3Mepa n rrponopmmHanHocr ;:JpKH ............................................................ 316 TanecoBa reopeMa ............................................................................. 318 Xo 1orenrja .................................................................................. 326 CmfqHOCT .................................................................................... 330 CTaBOBn CJIJ!'IHocrn rpoyrnoBa ................................................................. 333 IIpnMeHe CTaBOBa mrrnocn1 Ha rrpa.Boyrmt: rpoyrao .............................................. 336 IIp11:MeHe CTa.BOBa CIDfqHOCTJ! Ha Kpyr ...........................................................338 3a.n;~t ...................................................................................... 339 TPJfl'OHO~

t:ETPUJA IIPABO}'TJIOf TPOYfJIA ............................................... 345 TpHTOHOMerpnjCKe }'HKUJrje OIIIrpmr 'TIIOBa ................................................... ~346 Bpep;eocrn rp1rroH0Merpujc1rn:x ipyma.µrja OII!Tpillr yrnoBa ......................................... 350 OceoBmt rpnroHOMerpHjcirn Jt,qeHTwreni ........................................................ 352 IIpnMeHe rpnroeoMerpHjcKmr ipym
_

JIOfMKA J1 CKYIIOBM ~on1qKJf Be3HH~H

(IO)

A IlojaM 11cKa3a; .i:1eqmH~je norWIIopMyne. TayT011oru:je (14)

A ,D;e¢11H11u;11ja HCKa3He ¢opM)'Jie; HCTHHHTOCHa Ta6JI11u;a ¢opMyne; .i:1e¢11H11u;11ja TayTOnornje (14)

E MeTo):la CBol)elba Ha ancyp.i:1 (17) TayT011oru:je H 3aKOHH 3aKJ1>yqu:Ba11>a (18)

A Jlom11K11 eKBHBaJieHTHe ¢opMYJie (18)

ApucrOTen {384. ape H. e. - 322. npe H.e..'

E HeKe Ba)l(He Ta)'TOJiornje H 3aKOHH 3aKJb)"IlIBaIDa (19) CxynoBH (22)

A Ha1111H11 3a):laBalba cKynoBa; jep;1rnKocT CKynOBa; IIOllPCJ'II CK}'Ila; CKy:UOBHe orrepau;11je: ymi:ja, npeceK, pa3mn
(26)

A ,1J;e¢HH11u;Hja ypel)eHor rrapa; .i:1e¢ifHHll;11ja )J;eKaprnBor npmt3Bona ,U:Ba cKyna (26)

rorqipn;,1 BIDIXe.J.Y Jiaj6mm (1646-1716)

E J1.i:1eHTHTeT11 ca )J;eKapTOBHM rrpoH3BO):IOM; ypel)eHe TJ'OjKe, 11ernopKe wr;n;., )J;eKapTOB rrpo113BOA TPH u: BHIIIe CK)'IIOBa (27) ·BHapHe pe11~je (28)

A ,D;e¢11H11u;11ja 611HapHe penau;zje HeKor CKyna; oco6IDie fomapmfX penau;11ja (28) a~u:je nopeTKa H eKBHBaJie~e

(30)

E Penau;11je rropeTKa (;n;e¢HHJ1ll;Hja H npHMepu) (30) E Penau;11je eKB11BaJieHll;J1je H KJiace eKBHBane1muje (31) K~Hje

(32)

A ,1J;e¢11H11u;11ja ¢YHKu;Hje H rrpKMepH ¢}'HKI{Hja (32) E KoMII03Hll;Hja ¢YHKu;Hja; oco6HHe H BpcTe ¢ya::Ku;n:j a (34) ;ieMeHTH KOM6HHaTOpHKe (36)

A Ilpe6pojaBalbe pa3MHIIIJbalbeM (36) .:HKaTOpH (38)

B Ymrnep3aJIHH 11 er3HCTelil\IfjaJIHu KBattTK¢nxarop; noni'IKJ1 3aKom1 H KBaHrnHKTOPH (38)

I

-

--

-- --

- - -----,.""'"-

A ~ ieljy AHoroc5pojmm 3a.KOIDfMa Koje rrpoyqasa MaTeMan1Ka rroce8Ho BY1DIBaH>e je y BeiIJIKOM 8pojy cnyqajeBa 3aCHOBaHO Ha TBp,D,lhaMa Koje Mory 811TJ1 Ta"tffie J ;m HeTaqJie. TaKBe TBpAJbe Ha3KBajy ce MCKa3u.

Ilp101ep I. Peqemm;a ,,8poj 6 je MaibH O)l 8poja 7" np11Mep je Ta'!Hor 11cKa3a. Peqemrua ,,8poj 5 je 38up 8pojeBa 3 u 4" rrp11Mep je 11cKa3a Koj11 H11je Ta'laH. Hcxa3e he~m 03Haqasaru MaJIUM cJIOBHMa nanrn11Qe ca

J1JlJ.1

+

8e3 HH,D,eKca:

a, b, c. .. .,p, q, r, ····P1• qi... y ;nrreparypn HCTIIHOCHeBpe:.uiocu1

I 1 0 HeTa'IHO), ~iecro ca -.- u ......

HcrumrroCHY Bpe.o;Hocr wa'iHO 03Ha'lasaMo ca T (1n1Ta..'10 ,,Te"), a 11cT11H11TocHy Bpe.:mocT Hewtl'iHO ca .l ('IJITaMO ,,He Te").

03Ha'IaB<( ca

(Ta'IBO JI

AKO je HeKJ1 nCKa.3 p Ta'la.H, rnnne m -c(p) = T ('IlITaj ,,Tay op, rre je,n,HaKo Te"), a aKo je UCKa3 p HeTa'la.H, mnne o T(p) =.l. ,UaKJie, ca T 03Ha
3 ;IaTaK O;:i;pe;:i:u JtCTJ1HJ.1TOCHe Bpe,D,HOCTH 11cKa3a p,

q 11 r, Tj. T(p ), T(q) 11 -r(r), aKo je:

p: 8poj I_ je Behli op, 8poja 0,33; 3

q: nocroju npMpOAaH 8poj x TaKaB ,n,a je x + 2 = l; r. 3a CBaKJI npupo.n;att 8poj x Ta'lHa je Heje,D,HaKOCT x + 1 > x. O::i; HeKJIX norraJIDfX HCKa3a rpap,11Mo HOBe 11cKa3e y3 rroMoh TaK03BaHJ1X JIOr:Jf'IKHX Kao IlITO onepamtjoY cafoq>a.Iba napy 8po ·eBa x HJ npu,::rpy-&yj 0 HO.BU bpoj X -'-), TaKO jIQDf'll.'.O onepaqajoM u.;ru, Ha npHMep, napy HC1Gl3a p H q IIpltJIPJ'KtieMO HO.BU HCKa3

,,,p JLTII q"_

O.IlHOCHO, JIOI'Jf'IKHX orrepaQHja. 0Be orrepaQ11je cy 3acH0BaHe Ha peqJ1Ma Be3IDm;Jllla) : mm, JI, He (mrje), a.KO ... OH,D,a, a.KOH Ca.MO a.KO.

Be3HJIKa,

3a;:xara · HeKa cy ;:i;an1 UCKa3U:

p: 8poj 3 je Behu o,n, 8poja 4, q: 8poj 2 je Maihn o;u; 8poja 5. OIIpeJJt nCTHHJrroCHe Bpe,D,Hocn1 HCKa3a p u c,1e::i:erurr peqemm;a:

q, a 3aTHM pa3MOTp11 11cTJ1Hl1TOCT

1) p u q: 8poj 3 je :sehu o.n; 8poja 4 H 8poj 2 je Malbl1 op, 8poja 5, 2) p JtmJ q: 8po; 3 j e Belm op, 8poja 4 mm 8poj 2 je Malhl1 op, 8poja 5, 3 mtje p: Hirje 8poj 3 Behu o,n; 8poja 4, 4 mtje q: mrje 8poj 2 MaibH op, 6poja 5.

0

,ll;Hcjy111aµt:ja ,rµJa HCKa3a p H

q je HCKa3 nP HJrn: q" KOjH ce 03HaqaBa cap V q.

,[I;HcjyttKIJ;Hja p V q je TaqHa aKo je 15ap jep;att og HCKa3a p, cy o l5a HCKa3a p,

q TaqaH, a HeTaqHa je aKO

q HeTatfHa.

IlpHMep2.

T(2+3 =5V2 ~ 3) =T -r(2 < 3V3 < 2) =T -r(3 > 5V2 < 5) =T -r(l < OV2 < 0) =l. KoHj}'lUG~Hja p;Ba HCKa3a p H

+

q je ltCKa3 nP Jt q" KOjH Ce 03Haqaaa cap/\ q.

KoHjyttKIJ;Hja p /\ q je Taqtta aKo

cy o8a HCKa3a p, q TatfHa, a HeTatfHa je axo je fu.p jep;aH op; HCKa3a p, q HeTaqaH. i:_{e) T

T

fipHMep 3. -r(2+3=5/\2~3)=T

-r(2<3/\3<2)=1. -r(3>5/\2<5 =l.

T 1 1<0A2<0

Heran;Hja HCKa3a p je HCKa3 ,,HHje p" Koja ce 03naqaBa ca •p. Heramtja

.l

=J_

..!..

•P je Ta'Illa

aKo je HCKa3 p HeTaqaH, a HeTatfHa je aKo je JtcKa3 p Ta'laH.

fipHMep4.

T

T(•2 < 4) =l.

-r(•3 > 5) =T

y ~taTeMannm jecre cy oPM)JlllcaHa y -JU1KY aKO p, oap;a q. v1cKa3e OBOr OOJilt:Ka 03HaqaBaMO cap ~ q. y roBopHOM je;nn..-y

-:orHqKa onepau;11ja Koja no 3Haqajy 3aY3J1Ma QeHYp
p

HH ce HCKa3yjy 11 Ha jep;a.H op; rnep;eilli:x Ha'IJma:

p

~a

q,

p IIOBJia'IJI q, q je IIOCJieWt:D;a npeTIIOCTaBKe p,

H3 p rnew1 q,

'

q

~

q j e IIOTpeOaB ycJIOB 3a p.

p je JJ;OBOJbaB }'CJIOB 3a q,

1e ;:\OBO:&a.H fc.-IOB 3a

q je DOYpeOaH ycn:oB 3 a

IlpHMepS.

-:peTITOCTaBHMO p;a cy Heq11j11 pop;JfTeJbH H3jaBHJIH:

-.KO OBe UlKOllCKe iogu11.e oygeut UMQO CBe iieiUu~e, K)'UUfle,\to WU pait)111ap. Ba 113jaBa je oOJIHKa p ~ q, rp;e je p HCKa3 iUu hew. oee wxollCKe iogu11.e U..WliUu cee -eillu~e, a q HCKa3 MU heuo iUu K)'UUiUu paity11.ap. Kaga CMarpaMo )la poJOITelbu roBope cnrny? AKo yqeHHK HMa cBe neTHD;e H pop,JtTeJbu M}'" K}'Ile paqyaap,

rj. aKo cy ol5a

_cKa3a TaqHa, p;aTa H3jaBa pop;JfTeJba je, CBaKaKO, ratfHa. -.Ko yqeHHK HMa CBe neTJ1Qe, a pop;J1TeJbJ1

MY He K)'lle pa~'Hap, ou.;ta cy, Hajl5na.JKe

: _qeHo, pop;HTeJbH naraJIJ1. ,ll;aKJie, aKo je p TatfHO, a _--...."
.

q HeTatfHo, OIL'.la je p ~ q HeTatfHo.

MY KYJie paqyHap, OH:Ia po.:urre;I>e OfITPHO q Ta'IHo, oaua p ~ q CMaTpaMo

~eheMo cMarparH JI(l)KOBHMa. ,[I;aKJie, aKo je p HeTatfHo, a

qHoM H3jaBOM. :iaj3ap;, npeTnocraBHMO p;a yqemfl< HeMa cBe nen~e H po;iirre.nH M)' mrcy K}'IDL"Ill :: -qyHap. IIomTo yqeHHK m1je HCnyttJ10 CBojy oOaBe3y, oOaBe3e 'X>)J;HTeJbH.

J1 y

cy oc:mboneml n

OBOM cnyqajy p;ary H3jaBy CMaTp~m Ta'IHOM. •

11

· - - - - - - - - - - - - -- - - -- - - - - - - - - - - - - -

J1bm:;mxaunja ;:i;sa HCKa3a p n J1MILllIKa.Q.lrja p =>

AeqmHUiilfla

q je JICKa3 "aKo p, OHAa q" Koj11 ce 03Ha'!aBa cap=> q. q je HeTa'llia aKo je HCKa3 p TaqaH H 11cKa3 q HeTa'IaH, a Ta'!Ha je y CBJfM OCTaJIHM cnyqajem1Ma.

T(p)

T(q) T(

T T

T

J_

T

J_

J_

J_ T

2 +3 =5 => 2 :$; 3) =T > 5 => 2 < 5) =T

T ,3

T(2 < 3 => 3 < 2) = .l < 0 => 2 < 0) = T

T(l

+

3a~aTaK

Koja OA pelfii uoiUpebHO

J1Jlli

9060lbHO TPe8a Aa CTOjJ1 YMeCTO

*** Aa 811 CJieAehe

peqe1rn1w fome Ta'IHe? ,la 8u 8poj forn Berm OA 100, *** je Aa bfAe Beh11 0,1.1, 1000. 2) Ja 'IeTBopoyrao 5y;:re rrpaBoyraoHMK, *** je ,1.1,a bfAe rrapaJieJiorpaM. 3) :La 8poj 8y::.i;e ;::reJI>HB ca 5, ,..,.,. je ,1.1,a &eroBa rrocne):\&a u;m}>pa bfAe 5. -., ) ,'.la rpoyrao by;:re j e;uraKoI
YM.ecro axo H: aum axo qecro ce A.-pahe rrnme ainco. AHa.~orno acpammaJ:De je yo&H'Iaj eHO y YHODIM je3m{HMa. Ha rrpm.cep, y e1m1eCKoM ce if and only if crpahyje ca iff.

J.illrDnnran;zje p => q H q => p cy jeAHa ):\pyroj o8paTHe. Kmtjymm;nja L(Be ofipaTHe ~11IJIJ1Kal.\Hje (p => q)A(q => p) Ha3Jrna ce eKBJIBaneHJ:\Jlja H o8eJie)IGIBa ceca p ~ q. McKa3H oBor obJIJiKa ce y roBOpHOM je3JnIDKaBaj y Ha jep,aH OA rne,1.1,e1il1:x Ha'IJiHa:

pa.KOH CCUIO a.KO q,

p je eKBrma.JieHTHo ca q, aKO p. O.Rp;a q H oOpaTHO, p;oBOJball ycrroB sa

p je norpeba.H n

q.

q je HCKa3 »P aKO l1 CaMO aKO q" KOjl1 ce 03Ha'IaBa cap ~ q. EKBHBan:emu1ja p ~ q je Ta'llia aKo cy J1CTe HCTHHHTOCHe Bpe,1.1,HOCTH J1CKa.3a p, q, a HeTa'JHa je aKO cy UCTI1HJ1.TOCHe BpeAffOCTii J1CKa3a p, q pa3JIJ1'IJ1Te.

fuamsan:e~a l{Ba JICKa3a p H ~e4>mmmI ·a

T(p)

T(q) TE. <'> !1.

T T

T

T

J_

l..

.t

T

l.

l.

.t

T

L-

IlpJIMep 7.

T(2+3=5 ~ 2 :$; 3) =T T(3 > 5 ~ 2 < 5) =.l

T(2 < 3 ~ 3 < 2) =.l T(l < 0 ~ 2 < 0) = T

+

3a;:i;aTaK OgpeAU HCTHHHTocHY BpeAffOCT cne,1.1,erui:x 11cKa3a:

: 2. 6 > 0,5. 12 => 23 > 32 ; 2 2 + 3,1 > 5 => 3,3 -1,4 < 2; 2 2 2 2 3) -6+4 < 6-4 ~ 3-7 < 7-3; "1 2 +}2 < 4 ~ J2+4 < 5 ; 2 5 ) 11 · 9 < HY => 101 · 99 < 100 ; 6 2 :$; 2 => 2 < 2. - (O,l > 0,01 A -0,l > -0,01) => (3 2 < 3 · 2 V2 · 3 < 23);

)

12

3 2 < -5) => •( •0,1 . 0,1=0,01). -2 > -/\-{ 3 2 5 2

crp]llJpa pe'lemme

iJOl"JI'IKa

IIpHMep8. O,n;pe,n;11Mo norwiKy CTPYKTYPY (pacnope,n; eneMeHTapma: HCKa3a H norwIKHX BeJIDIKa) peqettmi;e: Atta he unaiUuiUu paiUy 3a ayiUo unu y cyupoiUHOM ttehe Mohu ga ia 3a9J>11CU. IlpBO yoqJ1MO eJieMeHTapHe J1CKa3e (Tj. J1CKa3e y KOjHMa ce He jaBJbajy JIOrwIKH Be3ttmi;11) o,n; Kojmc ce oBa peqett11u;a cacToj11: p: Atta be ITJiaTHTH paTy 3a ayTO; q: Atta be MOilH ,n;a 3a,n;p)l(Jf ayTO. Kop11CTeh11 nor:tt'-!Ke Be3HHKe, nona3HY peqett:ttu;y MO)KeMo 3an11caT11 Ha rne,n;eh11 Ha'-!HH: Atta he rmanITH pary 3a aYTO

v •(Atta he nnaTHTH paTy 3a ayTO) /\.•(Atta be MoilH p;a 3aJW)KH ayrn), O,!l;HOCHO,

pv(--.pA•q). •

ap;araK

/

....;.aTH cy HCKa3H: p: ttacnpaMHe CTpaH11u;e qeTBopoyrna cy napanentte; q: ttacrrpaMHe CTpaH:ttu;e qeTBopoyrna cy rroJzyAapHe; r: qeTBopoyrao je napanenorpaM. ;:i;pe,n;11 nornqKy CTPYKTYPY peqemni;e: AKo je tteiUaopoyiao uapane,wipaM, ottga cy -:eioae ttacupaMHe ciUpattu4e uapanentte u uogygapHe.

IIpHMep9. cy ,n;aTH eJieMeHTapHH HCKa311: p: M1rnetta je ttacne,!l;J1Jla 300 000 eBpa; q: M11neHa be oTHilH Ha Mope; r: M11netta be KynHTl1 CTaH. pM11pajMo o,n; ,n;aTmc eneMeHTapIDIX HCKa3a peqeHlflJ;)' '-Il1ja je nora'lK.a CTPYKTYPa :a r. Tpa)Ketta peqeHlilJ;a rnarn: :> Munetta Hacnegu 300 000 eapa u He oge Ha Mope, ouga he otta KyfluiUu ciUaH.

~.eKa

+

3ap;aTaK

7H cy J1CKa3H: p: Au;a je ,n;o8110 Ha nyTpaj11; q: Au;a 11,n;e Ha o,n;Mop; r: Au;a be Kyn11T11 ayToMofam. O,n;pe,n;11 JIOrH'-IKY CTPYKTYPY peqeH:ttu;e: AKo At{a goc5uje Ha nyiUpuju. oiiiuhu he Ha :igMop unu he KyuuiUu ayiU01 r.

pv(-p/\-.q)

-:p

l1cKa3He
cpopM}lla. Cenrre ce .:i:a 5pojeBHe e:3pa3e rpa;orno o;:r npoMeHlhmrnx

(rnoBa Koja 03Ha
T T

T

IlpJ.IMepH HCKa3HJ1X cpopM)'lla cy: TA •.L, p v T) ..., (p /\ q => •r),

1.

l.

A

....I

T

.1

1.

l.

HCKa3Ha cnoBa H ;rornqxe KOHCTaHTe cy HajjeJJ.HOCTaBHHje 11CKa3He cpopM)'ne. AI B) 11 (A.;;. B) TaKol)e HCKa3He cpopM)'lle.

I1p11n11KOM 3arr11c11Ba1Da 11CKa3Hl1X cpopMyna rro.n;pa3YMeBa ce rne.n;etlli .n;oroBop o np1mp1neTy non1'IKI1X Be3Hl1Ka: ..., je Be3Hl1K HajBener rrp11op11TeTa, 3a 1D11M cne,n;e V 11 /\, Koj11 cy rro.n;je,n;HaKor rrp11op11TeTa, a 3a 1D11Ma => 11 .;;., TaKol)e je,n;HaKor rrp11op11TeTa.

A I

l1CKa3He cpopxy;ie cy nonRKH H3pa3H KojHMa ce orrn:cyjy cTpJKType ( 08mm;11) 11CKa3a. JlCKa:me cpopM)Ae IpllAJfMO o.n;: • UCKa3Hn:x CJIOBa a, b, c, ... , p, q, r, ... , p 1, q1 ••• • 1I0rJ{qJ(J£X KOHCTaHTii T H l. nonf'IIG!X orrepar:µrja V, A,•, =>, .;;. yrroipeboM 3arpa;ia Ha yo8K'lajeHJ.1 Ha'Ilffi.

l.

·P /\ q => p 11 TaKO ,n;aJbe.

Ha OCHOBY rn8mma Koje cy rrpHKa3aHe Ha CTpaHaMa 11 H 12, je,n;HOCTaBHO ce o.n;pel)yje J{CTJlllJITOCHa Bpe;J;HOCT cpopM)'Jla y KOjJafa cy Ca.MO JIOfHqKe KOHCTaHTe IIOBe3aHe ;JOflfql(J!M onepaiµrja..,1a Ha HeKJ.1 Ha'Ilffi. Ha MapfJ1ID1 cy ,n;aTe Tafomu;e nor11tIK11x orrep~a Hap; JIQf}fq}(J{M KOHCTaHTaMa. IlpHMep 1. J13pa'I}'HaBa!be HCTKHHTOCHe BpeJJ.HOCT11 cpopMyJie

TV•l. => T -,

~

=.l,•.L=.,..

n3Bororuo Ha ocHoBy rabillfi\a ,n;aTI1X Ha Mapr11HH.

(TV•.L => T) =(TVT => T) =(T => T) =T => .,...

l.

.;;.

l.

T T T

+

J.. .l

T

T T

.!.

l..

T

.l

J1fapaqyHaj JICTIIBJITOCHe BpeAffOCTH ¢opMYJia:

2 •(T => ..., .L) .;;. T /\ ••l.; 4) (T.;;...., .l).;;. (1. => T); 6) T /\ (.L V (1. => .L)).

T => .1 VT; 3 (T Al.) V "T /\ T ) => •.L /\ T; 5 T => (T => (T => .1));

~am

3a Koje x 113 {T,.L} cy cne,n;ehe ¢opMyne TaqHe:

• xi\ T => l.; 3) x/\x.;;. xVx; - (x => T) .;;. x; 7) xV .1 <=> x;

2) 4) 6) 8)

•x => x; x.;;. •x; (T => x) .;;. x; xVT.;;. J.?

14

-

~-

- -:raJ.fTOCHa Bpep;HOCT cpopM)'lle y Kojoj ce nojaBJbyjy KCKa3Ha CJIOBa 3aBJfCM o~ "'11:TOCHJ1X Bpep;HOCTJ1 Koje cy p;op;eJbeHe TJ1M CJIOBJ1Ma.

Bpe:umcr mpa.3a x~;

· z 3aBJtOt

o;:i: BpeJ.Hocrn

ep2.

KOje CJ .:!O:XelbeHe rrpo.MeILnJIBIIM

'™O J1CTJ1HJ1TOCHY Bpep;HOCT cpopM)'lle

x. J

u :::.

F: •(pV• q) => p/\.q --;fp) =T J1 -r(q) =.L "''1 aHarrornjy ca c5pojeBHJ1M H3pa3J1Ma, 3aKJb~eMo p;a je -r(F) jep;HaKo

OCHOj Bpep;HOCTJ1 cpopMyrre

•(TV •.l) => T /\. .l 11111111a i! iocSHjeHa J13 cpopM)'lle F 3aMeHOM crroBa .\({11111m11c::urraMa: p ca TM q ca .1. 11~craBHO ce p;oc5Hja p;a je

p M q op;roBapajyrutM rronffI
+

-r(F) = T.

~~TaK

'•"~ HCTJ1HJ1TOCHY Bpep;HOCT cpopMyne F M3 npeTxop;Hor npHMepa, aKo je:

=THT(q)=T; =.LH-r(q)=T; =l. J1 -r(q) =.L ce y HeKoj J1CKa3HOj cpopM)'llJ1 nojaBJbyjy caMO p;Ba J1CKa3Ha CJIOBa, OHp;a Ha - - qeTJ1PJ1 Ha•mHa MO:>KeMO p;a p;op;elll1MO J1CTJ1HJfTOCHe Bpe){HOCTI! Tifil CJIOBHMa. -: ucKa3Hoj cpopMYJIH nojaBJbyjy TPH cnoBa, oHp;a J1MaMO oca.'1 Moryhttocru, aKo .=.i:.yjy qeTJ1PJ1 CJIOBa, OHp;a J1MaMO 16 Moryh.Hocn1, K yoIIIIITe, aKO ce y KCKa3HOj rn11 m11111m11•,.,._.,1 nojaBJbyje n J1CKa3HJ1X CJIOBa, OHp;a J1MaMO 2n i\mryhHoCTK. aH>e Bpep;HOCTJ1 HeKe cpopM)'lle 3a CBaKO p;op;eJbKBalbe J1CTI1HKTOCHKX J1CKa3HJ1M CJIOBJ1Ma Hajjep;HOCTaBHJ1je je IIpKKa3arn: y Oc5J1JfKY racSene KOjy 0 HCTHHHTOCHa TaOnmi;a opMJ71e.

.:TJ1

nt

ep3. CTJ1HJ1TOCHe Tac5nmi;e cpopM)'lle

•(p => q)/\.p y CKJiap;y ca HaqJ1HOM Ha KOjK K3paqyttaBaMO BpeAHOCT OBe cpopM)7Ie 3a :Tiie Bpep;HOCTK KCKa3HKX CJIOBa p K q.

"r-'
I

{!_ T T

.l .l

: I

I

9. T

: I

{!_ => T

9.

; •({!_

~

I

.l

.l

I

T

T

I

.l

T

I

.l

I

T

.l .l

l. T .l

l.

I

+

~TaK

aj KCTJ1HKTOCHY Tac5mn:i;y cpopMyne p =>

•(p V •(q /\. p)).

1c:;

~

.HCTJ1HMTOCHa BpeAHOCT ¢opMJ7Ie y KOjoj ce nojaBJDyjy HCKa3Ha CJIOBa 3a:BJKH O;J; JfCTMHMTOCHMX BpeAHOCTM Koje cy AOAeJDeHe HIM CJIOBHMa.

Bpe;:i;HOCT I13pa.3a X--0-)' •

fipHMep2. O.n;pe.n;:wMo

=3a.BUOI

o:r Bpe;:rn0<..""TI!

Koje cy :ro.:re.fbelle

MC.TMHMTOCtty

upo.MeIL"bHBHM x,

Bpe.n;HOCT ¢opM)'lle

y JJ .::.

F: •(pV• q) => pAq aKO je -r(p) =TM -r(q) =l.. Cne.n;etm aHanomjy ca 8pojeBHJ1M 1'13pa3HMa, 3aKJD)"IyjeMo Aa je -r(F) jeAffaKO UCTMHMTOCHOj BpegHOCTJ.1 ¢opM)'lle •(TV •1.) => TA l. Koja je .n;o8:wjeHa M3 ¢opMYJie F 3aMeHOM cnoBa p 1'1 q o.u;roBapajyhnM nonf'IKJru KOHCTaHTaMa: p ca T M q ca l.. )eAHOCTaBHO ce .n;o8Mja Aa je -r(F) = T.

+

3ap;aTaK O,i:i;pe.n;M MCTMHMTocey BpeAHOCT cpopMYJie F H3 npen:o,JJ;Hor npm1epa, a.Ko je: -r(p)=TH-r(q)=T; - -r(p)=l.H-r(q)=T; -r(p) = .L:~u(q) = l.. ..'Tl
eljMBalbe BpeAHOCTH HeKe cpopt.ryne 3a CBaKO .n;o.n;eJDJIBalbe HCTJIBJITOCIDlX zpep;HOCTJII HCKa3H1'1M CJIOBHMa Hajje,qHOCTaBHJ.tje je IIpHXa3aTit }' ObIDfh.'Y raOe,""Ie KOjy ea.3HBaMO HCTHHHTOCHa TaOim::a;a opMYJie.

_ IlpHMep3. 3~)CTe MCTJIIHJIITOCHe Ta8~e ¢op?.iyne

•(p => q)Ap -=DPMHpaMO y CKJia.n;y ca Ha'IBHOM Ha KOjH H3paqyttaBaMO Bpe;J;HOCT OBe cpopMpie 3a neKpeTHe spe.n;HOCTH MCKa3HJIIX CJIOBa p H q.

p

I

T

9. T

I I

1!. => 9. T

T

l.

l.

l. l.

T

T

'

l.

T

I I I I I

•([!_ => 9) '. -.(p_ => 9) A(!_ l. l. I T

I

T

l. l.

~

l. l.

j·

•

3a)J;aTaK :::>opMHpaj HCTMHHTOCHY TabJIHI.J;y cpopMYJie p =>

·(p V •(q Ap)).

15

•

ilo~npse

qenqm KOJIOHe HCTHHHTOCHe TablIJ.file qiopM)'Jle y Kojoj ce

ITpm1.11Ko1\1-.tp'opM.11palha .11cT.11H.11TocHe Ta8mme HeKe HCKa3He cpoMYJie, KOJIOHe y Koj.11Ma HaBO,ll;HMO J1CTJ1HJ1TOCHe Bpep;HOCT.11 J1CKa3HHX CJIOBa IIOII)'lbaBaMO CMCTeManrrno ,;::i;a He fotCMO J130CTaB.11JIJ1 HeKy MoryhHOCT. 0IIIIITJ1 IIOCTynaK ITOrryH>aBalba KOJIOHa KOj e O,Il;fOBapajy .11CKa3H.l1M CJIOBJ1Ma MO:>Ke ce BM,n;eT.11 y Hape,n;HOM rrp.11Mepy r,n;e je cpopMJrpaHa Tafourr~a cpopMy11e ca TPM MCKa3Ha cnoBa.

nojaBJhyjy qeTHpH CJIOBa.

p

q

T T

T T T T .l .l .l .l T

T

T T T T T .1 .l l_

.l .l .l .l .l

T

T T 1 .l .l

i

r

T .l .1

T J_

.1 T T .1 .1 T T .l .l

IlpHKep4.

s

O;:i;pe;un10 3a Koje j e ttcTIIBJrrocHe Bpe,n;HoCTM .11cKa3HMX c110Ba p, •r => p v q) => l. Ta'Illa, aKO TaKBe Bpe,n;HOCT.11 yorrnITe ITOCToje.

.1 T .1 T .1 T .1 T

Jla illl IIOCTOje Tpa:>KeHe HCTIDrnTOCHe Bpe,n;HOCTH

Ji

q .11 r cpopMyna

Koje cy, OTKpMheMO 113

ncnnnrrorne rnfoum;e ,n;aTe cpopMYJie.

i T .l T .l

,

p T T T T l. l.

I

l. .l

q l I I

T T .l

I I

l.

I f

I l

T T l. l.

I I

1

r T .l T .l T .l T l.

.l

T l. T l. T l.

•r => p V q ; T T T T

pvq T T T T T

T T

T l.

T

T .l

l.

(•r => p V q) => l. l. .l l. l. .l .l l. T

I I

I I

H3 ra8mm;e BIWtMo ,n;a je T((•r => pv q) => .l) = T 3a T(p) =T(q) =T(r) =.l, Kao .11,n;a je TO j eAJIBo pemeH>e IIOCTaBJbeHor 3ap;aTKa.

+

J1cKa3Ha cpop~tyna je TayroJionrja aKo je TaqHa 3a 8m10 Koje .11cTMH.l1TOCHe spe,n;nocn1 HCKa3HHX cnoBa Koja ce y IhOj rrojaBJbyjy. Peq wayillo;wfuja je rp'll
H

obpa30BaHa j e o;:i peq}{

iiiay woc =

UCTO lf iWlOC

= p33JM, 3aKOH, p331IOf.

IIpHMep 5. .[{a j e cpopMYJia •(p V q)

<;>

•P /\ •q rayT011or.11ja, 3aKJbyqyjeMo .113 lheHe MCTMHMTOCHe

Ta8mn~e.

T l. l.

.l T l.

T .l

l.

l.

T

l.

l.

T T

.l

l.

T

T

T

3aµ:aTaK YfCIIHTaj .u:a IDl cy rne,n;ehe cpopMyne rayTOnornje:

, p=>(pVq); 3 p => (qVr);

"' p=>(p/\q); 4 p => (p => qVr).

T T T

•

E a J1CKa3IDIX ¢opMyrra HacTOj11Mo ,n;a 3ao8111)eMo ¢opW1palbe 11>eHe Ta8mru;e K~a ro,n; je TO Moryhe. y Hape,n;HOM rrp11Mepy orrncaheMO je,n;aH MHOfO e¢11KaCmrjH IIOCT}'IlaK AOKa3JIBa11>a ii;a je HeKa ¢opMyrra TayTOrrornja. Peq je o MeTOAJI cBol)e11>a Ha ancypp;.

~le1:0:xa CBol)eu.a

H.aancyp;i;

A ikypg je pe-q ~

narnHcKor nopeKJia

IlpHMep6.

Jil

,UOKa)KJ.1MO MeTO,ll;OM CBol)elba Ha arrcypp; ,n;a je ¢opMYJia pA(p => q) => q TayTorror11ja. IlpemocTaB11Mo ,n;a ,n;aTa ¢opMYJia HHje TayTOrrornja, Tj. ,n;a je -r(p A (p => q) => q) = .L Y311Majyn11 y ofo11p Ta8rrmzy 11MIIID1Kaq:Kje, 3aKJbyqyjeMo .l(a je Ta,IJ,a -r{pA{p => q))=T J1T(q)=l.. Ha ocHoBy Ta8rrm.i;e 3a KOHjymru;"Hjy ,n;aJbe ,n;ofofja.Mo ,n;a je -r(p) = T, -r(p => q) = T J1 -r(q) =l.. Mel)yT11M, ca,n;a J13 -r(p) = T J1 -r(q) = l., crrep;M ,n;a je -r(p => q) = l., IIITO HJ1je Moryne jep cMo paH"Hje 3aKJbyqJ1JIH ,n;a je -r(p => q) = T. ,UaKrre, ,n;aTa ¢opMyrra jecTe TayTOrrornja.

+

3Ha
JIOfU1'Ka

HeMolyh1uJciU.

;IoKa3IIBaJbe na je

cpopMyiia Tayrorrornja MeTO:IOM CBOf}elha

Ha ancyp~ CC.CTOjH ce yrove !lace rrpenIOCTaBH rra

;i;aTa cpopM)Jla ru1je Tayronornja, a 3aTI1M ce 113 Te rrpeTIIOCTaBKe,

3a,u;aTaK MeTo,n;oM CBol)elba Ha arrcyp,n; p;OKa)KJ.1 .z:1a cy crre,n;ene ¢opMyrre Tayrnrror11je: p=>(q=>p); 2 (p=>q)A•q=>•p; 3 (p => q) => (•q => •p); 4J (p => q)V(q => p).

ocnalbalbeM Ha HCTHHHTOCHe Ta8n111i;e, H3BOW'f HeMoryha rnTy~ja - Ha npHMep, .n;a je.D;Ha

ncra cpopMpa HMa

IlpHMep7.

•

IloTrrytto attaJIOfHO rrpeTXO}:\HOM IIpKMepy ,D;OKa3yjeMO J1 CJie.n;ehe OIIIIITifje TBpl)eibe. AKo cy A 11 B 811Jlo Koje 11cKa3He ¢opMyrre, ott,n;a je A A (A => B) => B Tayrnnornja. IlpemocTaB11MO ,n;a ,n;aTa ¢opMyrra HJ1je rayTOrrornja, Tj. ,n;a je -r(A A (A => B) => B) = .L. Y311Majyn11 y ofo11p Ta8mru;y JfMIIJIKKaIJ;11je 11 KoHjyma.i;Mje, 3aKJbyq)rje~m ;:i;a je -r(A) = T, -r(A => B) = T 11 -r(B) =l.. Me1JYT11M, 113 -r(A) = T 11 -r(B) = .L, crrep;M ,n;a je -r(A => B) = .L, IIITO mrje Morybe jep je -r(A => B) =T. Ilp11MeT11Mo ,n;a cMo Ha oBaj Ha'IJffi ,n;oKasaJIH ,n;a cy cBe ¢opMyrre .r1arnr 08m1Ka (Kojnx JfMa 8ecKoHaqHo MHoro) TayTOrrornje. Ha rrpHMep, Tayronornje cy (pv q)A(pv q => •r) => •r [A =pV q, B=•r] •pA(•p => (p => q)) => (p => q) [A=-p,B=p => q] 11 TaKo ,n;aJbe.

+

;i:se ncnnnrro...-iie T H .L

Bpe;:mocm, K

&

3a,u;aTaK AKo cy A 11 B HeKe 11cKasHe ¢opM)17Ie, JJ;OKa)KJ.1 ,n;a je (•A => _,B => B => A. Tayro,-ronrja. 1

17.

Tayrorrol'Hje H

3aKOHH 3aKJh"Y'IHBalha

:IBe HCKa3He ¢opM}17Ie cy JIQI'J{q](J{ eKBKBaneHTHe aKO HMajy HCTe HCTHHHTOCHe Bpe:(HOCTJI 3a bJGIO KOjH U36op :HCTHIDITOCIDIX BpeAflOCTH CJIOBa Koja ce y lbHMa noja.BThyjy. Ilp1D1ep 1. op f}J:Ie ..,p v •q H ·(p A q) jecy JIODf'IIOi eKB:HBaJieHTHe. y TaqHOCT OBe TBp,n;H>e ce )"BepaBaMO axo <}>opMMpaMO JiCTIIBHTOCHe Ta6JI:11~e OBJllX cpopM)'Jla.

p

Tay-rmontje q ~-p ,q

Ea3UBajy ce lle ~ fopraHOBH 33.KOHH.

! •P I •q : •P v •q )

I I

.l

I

.l

I

T

!

}{

pq~-p..,q

q

T T .l

I

T

.l

.l

.l

.l

.l

T

T T

.l

T T T

I

T

I .l

T

.l

.l

.l

.l

T T

•

llirje TeIII.Ko yo'Ilmi: ~a cy HCKaJHe cpopM)'Jle A H B nornqKH eKBHBaJieHTHe aKo H caMo axo je ¢opMY71a A ~ B Tayronornj~ IlpHMep 2. J1crnrrajMo KOjH o~ CJieAelrn:x HCKaJa cy JIOI'H'IKJ1 eKBHBaJieHTHH HCKa3y c5uiUu

3alby8mett je uoiUpec5aH yCJ108 3a cpeliy. 1. AKo je HeKo 3aJby6JDeH, oH,n;a je OH cpeb.att; -· AKo je HeKo 3aJby6JDeH, oH,n;a HHje cpenaH; 3. AKo HeKo HMje cpeb.aH, OH,D;a HHje 3aJDy6JDeH; 4. AKo HeKo IDrje 3aJby6JDeH, OH,D;a HHje cpenaH; 5. HeKo je 3aJby6JDeH HJIH HMje cpeb.att. HeKaje p: 6JITH JaJby6JDeH,

q: 6HTH cpenaH.

Ta,ua je ucKaJ c5uiUu 3albyDlbeH je uoiUpeDaH yc1wB J1CKaJu 1-5 cy Tap;a pe,n;oM 06JI:11Ka: ~

2 3 4 5 OpM}'lle p ~

q,

..,q~•pH-p~'q

jecy norK'll01 eK3HBa."IeHTHe.

18

3a cpeliy, 3anpaBo HMilJIHKa~Hja q :::::> p.

p :::::> q; p :::::> •q; •q :::::> -,p; •P :::::> •q; pV•q.

llirje TeIII.Ko rrp0Bep:11rn ,n;a cy MeJ)y cpopM)'JlaMa :. (q :::::> p) ~ (p :::::> q);

2. (q :::::> p) ~ (p :::::> •q); 3. (q :::::> p) ~ (•q :::::> ·p); 4. (q :::::> p) ~ (-,p :::::> •q); 5 (q=:>p)~(pV•q), Ta ~ononrje cpopMyne 4 H 5, IlITO 3HaqJ1 ,n;a cy ,n;aTOM HCKa3y eKBHBaJieHTHH HCKa3H rro,n; 4n5. + .

E onorn:je cy JfCKa3He cpopMYJie Kojmrn ce 01rncyjy BruKHJi 3aKOIDI MHIIIJbeIDa.

3axoH 11aJ1>yqee.a ·

- ~ KOjJf Hal.{JfH cy TayTOJIOrJfje cacTaBHJf p;eo CBaKOr KOpeKTHOr 3aKJbYl.IJfBalba,

rpeher

£::""'CTpoBaheMO y Hapep;HJfM rrpHMepmrn.

·pv-.p pv-p eHJfl\a p;a je p v ·P TaYTOJIOrJfja, 3Ha'lltl p;a je CBaKJ1 CJIO)KeH JfCKa3 OBOr o5JIHKa yaeK :qaH 5e3 ofoHpa o KOM JfCKa3y p je peq. Ta,rne cy CBe peqeHJ1.l\e: Aila JfMa 16 rop;HHa JfJIJf ~a HeMa 16 rowma; Tpoyrao ABC je rrpaBoyrIDI J1ID1 Tpoyrao ABC HHje rrpaBoynrn;

1,41 je pernelhe je,n;Hat.Urne x2 =2

•

!

J1ID1

1,41 mtje pernel:be jeAffa'Dme x2 =2.

+

IlpHMep4 .

Je::raH op; Hajjep;HOCTaBHJfjHX Jf Hajqerrtlie KopHrrillema 3aKOHa 3aKJb)"IlIBaI:ba 5a3JtpaH · e s a TayrnnornjH p /\ (p

*

q)

*

Modus ponens

q (BH,ll;Jf rrpHMep 6. Ha cTpaim 17).

0Ba TayTOJIOrJfja ce Ha cprrcKOM je3JfKY MO)Ke HCKa3aTH Ha CJiep;elIB Ha'iHH:

pA(p => q) => q

a1w je p u u3 p cnegyje q, oHga je q. IfaJOie, Jf3 rrpeTIIOCTaBKM p Jf p

* q Jf3BO,IJ;JfMO 3aKJhyqaK q.

p p => q

q

Ca:la HJfje TeIIlKO OTKpJfTH IIOMettyTY Tayrnnomjy y CJiep;eheM 3aKJhYl.IJfBalhy. IlpernocTaBKa p: AHa )KMBJf y Eeorpap;y. IlpeTrrocTaBKa p * 3a:'VbyqaK

q:

q: A.Ko AHa )i(J1BJf y

AHa )l(JfBJ!

r Cp5ujJf.

Eeorpap;y, OH}J;a .-ma )KJiBJf y Cpfo1j u .

•

..fodns tollens

IlpHMep 5. Ha ocHoBy rnyrnnomje

(p

KOMe Jf3 rrpeTIIOCTaBKM p

::::>

*

q) /\ •q

*

•p, p;o5Hja.Mo rrpaBuno 3a.KJD)"IJIBaH>a rrpe~i:a

q H •q H3BO,ll;HMO 3aKJhyqaK •p.

p => q 1\ -iq => - p

p => q -:q I

=:>

p.

H aBop;HMO Jf jep;Ho KOHKpeTHO 3a.KJbyqHBaIDe 5a3HpaHO aa oBoj rnyronornjH.

p

*

q: A Ko je qernopoyrao ABCD KBa;war, oll}J;a ce y .!her:: : !:;~:~e yrmcant Kp)T.

'P

..., : 'qernopoyrao ABCD ce ae ~m)Ke ~7JKZ.:'~. :~p:::-. 3 aKJhyqaK • p: qernopoyrao ABCD HHje KBap;paT. TpaH31ITlIBHOCT

IIpnMep6. Tay rnnomja(p

10IJTIDD(al1Jrje

* q) /\ (q * r) * (p * r)

crnju ma rne;::i;eher jeAHOCTaBHor

3a.KJbyqHBalha. IlpernocTaBKa p IlpeTrrocTaBKa 3aKJhyqaK p

*

*

q*

q: A.Ko tteKo )KHBJf y Eeorpa.'.ly, 011;i;a OH )KUBU y Cp511j tt. r: A.Ko HeKo )KHBH y CpC1!1f, OH}J;a OH )f(JIBH y Eep01m.

p => q

/>,

q => r => p => r

p => q q => r p => T

r: A.Ko HeKo )f(JfBH y EeorpaJiy, OBJl.a OH )KUBU y EBponu.

AHanortty rrpHMetty, HapoqJfTO y MaTeMaTii'iKJf.M TBpl}eID1wa, Jrna Tayrononrja ::::op;

(p ~ q) /\ (q ~ r)

*

(p ~ r).

+

19

1

3a.1mHH xmrrpanoo

Ilp1D1ep 7. Tayron:onrja •q => -p) ~ (p => q) ce BeoMa qecTO cpebe y MaTeMaTW:IKMM p;oKa3MMa. 0Ha OMoryha11a :i;a ce J'MeCTO HMIIJIJfl q )l;OKruKe l1MIIJUtKau;11ja •q => ·P· Ta.Ko, aKO ;Kt!J"IIDIO ,;:i;a AOKIDKeMo H.MIIIDfKaI\Hjy: Axo ·e n 2 HenapaH 5poj , o:ep;a 1e., HenapaH 5poj, npeMa OBOj TayrononrjH pteCTO lbe MO:>KeMO )l;OKa3aTJ1 .IDOj eKBHBaJieHTHY HMIIJIJIKarofjy: AKo - mrje HerrapaH -F ·, o:ep;a '1 2 HHje ttenapaH 5poj.

0Ba .n;pyra l1MIIJIJfl(an;Mja je CBaKaKO jep;HOCTaBHHja 3a p;oKa3HBalbe. 3aHCTa, aKO n HHje tterrapaH, ott.n;a je rrapaH, Tj. n = 2k, 3a HeKo k, rra je n 2 = 4k2, op;aKJie cnep;u .n;a je n 2 napaH, mTo 3Haqu .n;a HHje HenapaH.

+

IlpHMep8. HajjeAHOCTaBIDtjH. :40Ka3 .n;a je 3a CBaKM peanaH 5poj x TaqHa 11MnJI11Kan;11ja 1

l -3x:t:O =>

x=t=-

3

jecTe OHaj y KOMe ce npH.Me.IDyje Tayrnnomja ( •q => Tpa:>KeHa IDITIIDfl
•P)

~

(p => q). IlpeMa IDoj,

1

x=- => l -3x=O,

3

a OBY 11MnJI11Kan;11jy je je.n;HocTaBHO p;oKa3aTH jep Tpe5a caMo np0Bep11T11 .n;a JIH je TaqHa jep;HaKOCT

1 1-3·-=0.

3

0Ba je.n;ttaKocT je Ta'IHa, q}fMe je p;oKa3 p;aTe l1.MIIJIH.Kau;11je 3aBprneH.

Cuol)elbe Ha ancypp; (reductio ad absurdum)

• • 20

+

JlpHMep9. AKo H:3 •P cnep;H q M •q (Tj. HeKa p;Ba npoT11BpeqHa rnpl)eIDa), OHp;a M3BOp;MMO 3aKJb)"laK p. 0Bo npaBMO 3aKJb)"IHBaIDa 5a311paHo je Ha Tayrnnomj11

(op => (q"' •q)) => p. HajCTapitju np1wep npH:MeHe oBor 3aKoHa 3aKJb)"IMBaIDa je qyBeHM p;oKa3 .n;a je.n;HaquHa

x2 = 2 HeMa pan;}lOH(lJIHa pernelha. 0Baj p;oKa3 heMO J13JIO:>KMTM KaCHMje Ha CTpaHM 63. 0BOM npl1.JI}fi(0M heMO npHKa3aTl1 p;oKa3 jep;HOCTaBHMjer rnpl)elba p: 5poj 1,41 tt11je perneIDe jep;HaquHe x2 = 2. Heran;Hja oBor rnpl)eIDa je •p: 5poj 1,41 jecTe perneIDe jep;HaquHe x2 = 2. AKo npeTIIOCTaBMMO •p, oHp;a 5ucMo MMaJIM p;a je q: 1,41 2 =2, a JiaKo ce npoBepaBa .n;a je ~aKJie, rnqHo je p.

+

3aKOH Habpajan.a

IlpHMep IO.

H,

TwrrwqaH rrpwMep rrpttMeHe Ta}'TOrror.wje (p ~ r) /\ (q =:. r) =:. (p V q =:. r) rrpe,a;cTaBJba ,a;oKa3 TBpl)elba ,a;a je 3a CBaKJ.i npwpo,a;aH 8poj n npmnBOJl n(n + 1) rrapaH 5poj. HeKaje p: n je rrapaH 8poj, q: n je HerrapaH 5poj, r: n(n + 1) je rrapaH 5poj. Hwje TernKo ,a;oKa3aTM MMTIJIHK~e p =:. r (aKo j e n rrapaH 8poj, OHJJ;a je n(n + 1) napaH 5poj) J1 q =:. r (aKo je n Herrapatt 8poj, OH,a;a je n(n + 1) rrapaH 8poj) . Ha ornoBy Ha.Be;i;rne TayTOrrornje ,a;o5wjaMo ,a;a J13 n je rrapaH 8poj MJIJ1 n je HerrapaH 8poj crre,w.1 ,a;a je n(n + 1) napaH 8poj, O,!l;HOCHO, ,a;a je 3a cBaKM rrpwpo,a;aH 8poj n rrpo113BO,a; n(n + 1) napaH 8poj.

p

~ r

~

p. q ~ r

q~

A

y

~

"

I IlpHMep 11.

,IJ;e MoprauoBH 3aKOHH

3a CBaKa ,a;Ba peaJIHa 5poja x 11 y TaqHe cy eKBMBarrem~wje x · y=O ~ x=OVy=O J1 x2 +y2 =0 ~ x=O/\y=O. Ha ocHoBy Tayrorrornja

•(p A q)

¢>

•(p V q)

¢>

-p V •q -p A •q

(p ~ q) ~ (•p ~ •q), •(p/\q) ~ •pV•q Jf

·(p v q)

~

-p /\ •q

3aKJbyqyjeMo ,a;a cy TaqHe l1 crre,a;eb.e eKBMBaJieHI.\}tje x · y;tO ~ x;tO/\y=O J1

'"Rl1Ha

-:3.

x2+y2;t0 ~ x;tOVy;tO. 3a.HcTa ttMaMo ,a;a je (x · y=O ~ x=OVy=O) ~ (-,x · y=O ~ -,(x=OVy=O)) ~ (x · y;tO ~ •x=O/\ •y=O) ~ (x · y;tO ~ x;tO/\y:t:O }, J1 (x2 +y2 =0 ~ x=O/\y=O) ~ (•x2+y2=0 ~ -,(x=O/\y=O)) ~ (x2+y2;t0 ~ •X=OV•y=O) ~ {x2+y2:t:O ~ x:t:OVy;tO) . 0BaKBO 3aIIJ1CJ1Balbe eKBJ1BaJieHl.V'fja y HJ.f3)', O,!l;HOCHO, q>0pIDq>3.Ibe TaK03BaHOr ,-umua eKBUBalleHu,uja, ,a;o3BOJbeHo je Ha ocHoBy rnyrorrontje (p ~ q) /\ (q ~ r) =:. (p ~ r . 11pen;tt3HMje, rrpeMa oBoj Ta)'TOJiontjH 3aKJhyqyje~10 ;xa pmmnm cy eKB1ma.aeHIBe CBaKe ,D;Be cyce,a;He qiopMYJie y namry eKB1ma.n:e1muja, moa cy eKB1ma,1eHTHe n rrpBa ~I rrocrre,a;Iba qiopM)'Jla y TOM namzy.

+

T paH3IITllBHOCT eirnHBan:emutje

_p¢>q • ~

C¢>r

~

'.'¢>T

r,<:>n ....

p

c<>.,, .. ¢> r

21

A Iloj.MOBH cxyn u cxynoBHa JIPHil3.AHOCT cy OCHOBHH MaTeMan1qKJ1 nojMOBH. IloMony ce }BO;::J,e am ~yrn MaTeMaTif'IKH 08je1cr11 norryr OpojeBa, pemu~Hja, cp)'HKI.J;Hja, reoMeTPujcrmx cpnrypa ITTp;.

H>JIX

J hrrylmmHo, ocynoBn Hacrajy oxyrunalhe.M HeKaKBHX CTBapH y u;emrny. CBaKa o,n; CTBapn xoje qirae ry n;emrny, rj. CKYII, Ha3HBa ce eJieMeHT Tor cKyna. ,Ua j e x e,,e~ieHT CKJIIa S 3anucyje ce cpopM)'JIOM

x E S H qJ!ITa ce ,,HKC je eJieMeHT cKyna

ec" nJn1 ,,1mc np1mall,a acyny ec". Ua x mrje eJieMeHT acyna

S 3arn1cyje ce cpopMYJIOM x
c:Kyna ec" mTII .,Jf.KC He rrpuna.n;a CKYIIJ ec':

,n;eqnrnminja

CBaJGI Ch.'JTI j e IIOTIIYHO o.n;peljeH rnojKM eneMeHTHMa. )J;pyrHM peqHMa, ,n;Ba cKyna

cy jeA,HaKa aKo 11 ca.Mo aKO J.IMajy HCTe eJieMeHTe.

BeHOB .n;11jarpaM cKyna

B ={J,R,P,G}

-----

B

Hajje,n;HoCTaBHHjH rrpHMepH CKynoBa cy OHM KOjH MMajy ,,MaJIH" Opoj eJieMeHaTa H Koje 30or Tora MO)l(eMO 3a,n;aTH HaBoljel:beM CBMX l:befOBMX eJieMeHaTa H3Meljy BHTHqacTMX 3arpa,n;a { H} (o,n;BajajynH eJieMeHTe 3ape3HMa YKOJIHKo MX HMa BHIIIe o,n; je,n;Hor). MeljyTMM, OBaKBe cKynoBe neMO HaBO,n;HTH caMO y HeKMM eJieMeHTapHHM rrpHMepHMa.

j E B,R E B,P E B,G E B

y MaTeMaTMI.J;H ce yrJiaBHOM pa3MaTpajy cKynOBH Koje He MO)l(eMO 3a,n;aTH ,,eJieMeHT IIO eneMeHT" (qaK H Kap;a cy OHM KOHaqHH, a HMajy Bem1KH Opoj eJieMeHaTa). 0BaKBe CICJIIOBe 3a,I{ajeMO HaBoljeIDeM CBojciUBa IIOMony Kojer H3p;BajaMO CBe oOjeKTe KOjH HMajy TO CBOjCTBo. AKo ca S 03Ha'I:HMo CBojcTBO, a ca S(x) qHJ:beHHu;y p;a ,,x HMa CBOjCTBo

S', OIIp;a 3aIIJ1COM

I

S={x S(x)}

p;ecpuIDrmeMO CK)'II

~: = fi-.2~3,4, ...}

a..-yrr rrpnpo:urnx bpojeBa

s qajH cy eJieMeHrn CBH o8jeKrn KOjH HMajy CBOjCTBO s.

IlpHMep 1. CKyn

A qJ!ljH cy jep;MHH eJieMeHrn 1, 2 H 3 oITMcyjeMo ca A= {1,2,3}.

IlpKMern ,n;a je A = {3,1,2}, Kao HA ={x Ix= 1 V x=2 V x=3}.

=

= { ..., -2,-1,0,1,2,... }

0.1LI ne.'IHX

8pojeBa

~=[~ jp E :=,q E ~I} aq'Il pau:nomrnmi:x

CKyn

A npBHX 1 000 000 000 npHpop;HMX OpojeBa jecTe KOHaqaH, aJIH je ,n;HcKyTaOHJIHO

p;a JIM CMO y CTa.IbY p;a HaBe,n;eMo cBe IDeroBe eJieMeHTe. IlpMMepa pap;H, aKo OHCMO MOrJIM ;:i,a 3anMcyjeMo no jep;aH npHpop;aH Opoj y CeKJH,n;H, 3a HaOpajal:be CBHX

A rpe5ano OH HaM cKopo 32 ro,n;HHe. J InaK, CK)11I A jep;HOCTaBHO 3ap;ajeMo jep;HaKomhy A = {x x E N /\ x ::; 109}.

en:e 1eHaTa CK)'Ila

I

bpojeBa Ilpa3aH CK)'II (cKJII KojH HeMa eJieMeHTe) MO)l(eMo oITHcaTH je,n;ttaKoIIIny 0 = {x Ix ;t:x}. Ilpm.1eTH p;a cpopMYJia x ;t: x HHje Ta'!Ha HM 3a jep;Ho

- -:

- I 0 h2

;: 4,5

CKYil pea.LHirx: 6pojeBa

cr
x, rra oHa 3aHcTa orrHcyje rrpa3aH

vilOB

y

_-..io cy cKynoBH A 11 B je,n;HaKH, Tj. aKo je A =B, oH,n;a je 3a CBaxo x Ta'!Ha ....,,1rnanemi;11ja: xEA ~ xEB.

'3a

eKBHBaJieHI.J;Hja je KOHj)7HKI.J;Hja ABe o8parne KMIIJilfKaI.J;Hje:

(x E A

*

x E B) t\ (x E B

*

x E A ). :C:;rna 11Mnn11kar.i;11ja ,,KaJKe" ,n;a je CBaKH eJieMeHT CKyrra A Taxol)e H eJieMeHT cryna B, a :::r::r.ra ,n;a je Ta'IHO 11 o8paTHO, ,n;a je CBaKH eJieMeHT CKyrra B yje,n;Ho H eJieMeHT CKyrra A. CKyn A je no,n;cicyn cKyrra B, y 03Hall;J1 A h B, axo K ca.Mo axo je CBaKK eneMeHT cKyna A yje,n;Ho 11 eJieMeHT cKyrra B, Tj. aKo 11 ca..'do axo je 3a CBaKo x Ta'!Ha KMIIJIKKaI.J;lfja:

I'

~eqJJlHlfl!H

x E A * x E B. CKyn A je npaBH (CTpom) no,n;cicyn CKyrra B, y 03Hall;H A C B, axo K ca.Mo aKo je A h B 11 B ~A (B HKje no,n.cKyn o,n; A ). ~

~---

Pe."IaI.J;Hjy h (,,811T11 rro,n;cKyn") Ha3HBaMo HHKJiy3JrjoM, a penar.i;11jy C cTpor oM IDIKJIY3HjOM. 3a,n;aTaK

&

IlpHMern ,n;a cy npmrn,n;albe (E ) 11 HHKJIY3Hja (b)

lkmnaj HHKJiy311jcKH o,n;Hoc Mel)y CKyrrOBHMa A= {x Ix E N Ar::; 9}, B={x x=2Vx=3} 11 C={2,3,4}.

I

YJHMajyn11 y ofo11p rrpen:om .n;eKHKI.J;Kjy K pa3~£aTPaIDe 11.crrpe,::t H>e, Ja.Klbyqyje~m ,n;a j e CJie,n;ena eKBHBaJieHr.i;11ja rntrna 3a 8KJio Koje CKyrrOBe A 11 B:

cymnmcirn pasJIH1ilITlt o:moat.

s

~

A =B ~ Ah BAB h A.

'-.Y

y Hape,n;HOM TBpljelby HaBO,ll;JiMO OCHOBHa (K O'IID'Jie,n;Ha) CBOj CTBa HHKJIY3Hje J1. crpore

aES, :a; fi S,

KRKJiy3Hj e.

{a1· kS,akS 3a 811no Koje cKynoBe A, B H C BaJKJt:

_"lJiQ

*
-. 0h A,

5. A:z0

Ah A, 3 A h B t\ B h A ~ A = B, 4 A h B t\B h C * A h C,

6. A

0 C .A, TOOpeMa

C B * A h B, 8. A C Bt\B C C * ACC. ~. A

IIapTHTHBHH cicyn .n;aTOr CKyrra S, y 03Hau;H P 5'1, jeCTe CI>.JII CBH:X H>eroBKX no.n;cKyrroBa. }J;aKJie, Pl S) ={X I X h S}.

.:::em~mn

--

=x} . ...ilf

IlpHMep2. O.n;pe,n;HMo P({l,3,5}). Je,n;HOqJiami no,n;CKyrrOBH o.n; {l,3,5} cy {l }, {3} u {5}. ,:::{Bo'IJiaHJ1 IIOJJ;CK}"IIOBJ! o;:i: {l,3,5} cy {1,3}. {1,5} K {3,5}. KottatIBo, 0 h {l,3,5} u {l ,3,5} h {l ,3,5}, na JI.Memo ~a j e

P ( {l,3,5}) = {0,{l},{3},{5},{l,3},{l ,5},{3,5},{ l ,3,5}}.

23

y 03Hal.{H A n B, ,n;eqnrn1nneMO ca An B={x Ix EA Ax EB}.

IIpeceK CKynoBa A H B,

Ymtjy Ch.')'IIOBa A n B, y 03Hau;11 A U B, ,n;eqmmnneMo ca AU B={x Ix E AVx EB} .

IIPECEK •-1 B

Pa3J1JIKf Ch.-yIIOBa A tt B, y 03Hau;H A \ B, ,n;eqmHttIIIeMo ca A B={xjxEAAxftB}. CKfIIOBH AH B cy AJltjyHKTHH aKO je A

YHHJA • -i U B

IlpKMep HeKa jeA

n B =0.

3.

= {l,2,3,4,6}, B = {3,4,7,8,9} H C={4,5,6,8,9,10}. n C).

Oµpe;urno (AU B) \ (B

Au B={l,2,3,4,6,7,8,9},

B n C={4,8,9},

(Au B) \ (B n C)={l,2,3,6,7}

AU PA3/UIKA .-1. B

A

tD

•

B\.-1.

u; KO~

fl11IE:\!EHT Ch'YIIA

3a aTaK KopHcTetrn: CKYIIOBHe orrepal.{Hje, OIIJfillf oceHqeHe ,n;enoBe BeHOBHX ,n;HjarpaMa.

2

1

5)

4)

6)

@® ®®@ c

c

c

c

c

c

3a~aTa"'

AKo je A farno KOjH CKJII, o,n;pe,D;H A

n 0, A u 0, A \ 0, 0 \ A.

AKo je A ~ S, oH,n;a ce pa3llfl
A. U.n·=S A('

(.-l."

,-i..·=Z =.-J.

s. A.Ko je jaCHo 0 KOM CKJIIY sje peq, OH,D;a ce KOMrrneMeHT rro,n;cKyna A (O,D; S) 03HaqaBa ca Ac.

)J;11peKTHO 113 .o;e11H11u;11je cKyrrOBHHJtx onepa.I.{Hja AOb11ja..'1:0: xEAUB~xEAVxEB,

x EA

n B ~ x EA Ax EB,

xEA \ B~xEAA•xEB.

= B. axo je 3a CBaKo x Ta'llia eKBKBai,ernrnja:

.n.

x Erie x EB.

IIp11MeHoM OBHX eKBHBaneHu;Hja CKYIIOBHH HAeHTHTeTH ce cBo,ue Ha TaYTonon1je. 1

IlpHMep4.

H,!IeunrreTe Kojn ce

)J;oK(l)K11MO Aa je 3a cBe cKyrroBe A, B 11 C Ta'llia jeAHaKOCT

A\ (BU C)=(A \ B)

n

(A C).

)J;a bl1CMO AOKa3anH OBaj cKyrrOBHH Hp;eHnfTeT, )J;OBOlbHO je JIOKa3aTI1 na je 3a CBa.KO x TaqHa eKBl1BaneHu;Hja B U C ~ x E (A \ B) n (A \ C.. E IIpeMa .o;e11H11u;11jaMa cKynOBHHX oneparµtja, K.\fCU!O ;i;a je

E

~

B v

_r1

~

o.:moce Ha HeKa IJ>H

npon3BOlbHa cxyna Moxe~m

~

x E A A • (x E B U C) E _A-, E.JVxE C)

Kao 11

x E (A\ B)

n (A\ C) ~ x EA\ BAx EA\ C ~

c

(x E AA•x E B)A(x E AA•x EC).

)J;aKne, Tpe5a p;oKa3aTH p;a je 3a cBaxo x rnqHa eKBHBaneHu;11ja x E . A~ x E 3 .r E C ~ x E 11. A •x E B) A (x EA fl. •XE C).

A . (EU C)

HeKaje

@

p: x E A, q: x E B, r: x E C. IIocneAlha eKBHBaneHu;Hja, y3 YBep;eHe 03HaKe, nocraje

p

-,

'1

•

~ pA-.q)A(pA•r~ .

OcTaBlbaMO KopHCHHKY oBor )'l,16eHKKa p;a p;OK(l)Ke .ua je oBa opM)7la rayro;iornja. T11Me he p;oKa3 p;aTor 11.p;eHTHTera 811r11 3aBpmeH.

+

rrpoBepaBarn

cegqelbeM. rnmrmor rprupK"IXor npax.a3a HeKa TJ>H CK"'Ila

c

c

IlpHMepS. HeKa cy A 11 B npoH3BOlhHH nop;cKynOBJi op; S. )J;OKa_)f(JfilO ]le Mopra.HoBy j e.::maxocr

s

(An B)' =N U B'. cnep;H p;a je A n B ~ S, Kao

IIp11MeTHMO p;a H3 A, B ~ H ;i;a je peq 0 KOMIL"IeMeHTIIMa y op;Hocy Ha cKyrr S. YIMajyhH Ha YMY .o;eHHJlu;Hjy KOMnne~ieHTa, ;tOBOlhHO _ie ;tOKa3an1 p;a je 3a CBaKO x J13 Ta'IHa eKBHBaneHu;Hja x E (A n B . ~ x E A' u B', 0.!IHOCHO •XE An B ~ x E NVx EB', rj.•(x E Afl.x EB)~ •x E AV•x EB. AKo CTaBHMO p;a je p: x E A, q: x E B, nocnep;J:ba eKBHBane1m:uj a nocraie •(p A q) ~ •P V •q. 0Ba qiopMY71a jecre rayrnnornja ~II03Hara xao J;e ~fopra.HoB 3aKoH), op;aKne cnep;M p;a je Tp(l)KeHa jep;ttaxocr Taq}fa..

s

+

3a:-.aTaK

~

c

'--./

/c

(,.

~ "---../ c

)J;OK(l)KH p;a cy 3a bMno Koje CKYIIOBe A, B H C Ta'!He je!Ulaxocru:

(A\ B) \ C=A \(BU C), [(pA •q) A• r ~ pA•(qVr)J A\ (B \ C) =(A\ B) U (An C). [pA•(qA • r) ~ (pA• q) V(p Ar)]

25

~

--------

----

A

- ------

Ypel)eHH nap. ,il;eKapToB npoH3BO)J; YpeljeBJI nap (a,b) je .u;BO'IJiaIDI Hl13 'Ilfj11 je rrpB11 'lliaH (rrpBa Koop,[{11HaTa) a 11 'lltju je ;i:p)Tif 'IJiaH (APyra Koop}:{HHaTa) b. 3a ypel)eHe rrapoBe BaJKH (a,b) =(c,d) ~a =c/\ b =d.

Pa3'"IJ

;:i;BO'L"IaIDi

crcyn o;i ype!>eHor rrapa. {l.2} =
AxB _ _ _ __

(a,: ) (b,l ) (c, l )

.,

J];eKaprns rrpoH3BO.J; CK}'IlOBa A 11 B, y 03Har.µf AxB, jecTe CK)'II 'rnj11 cy eJieMeHTH ypel)eHH napoBH 'Ilija npBa KoopwrnaTa rrpl1Ila,[{a CK)'Ila A, a Apyra KOOPAHHaTa CK)'IIY B, O,[{HOCHO, AxB={(a,b) I a E A/\b EB}.

I I I I

IlpHMep l. HeKa je A = { 1,2} H B = { 0, 1,2}. 0ApeAHMO cKynoBe AxB 11 BxA:

AxB = { (l,0),(1,l),(l,2),(2,0),(2,l),(2,2) }; BxA = {(O,l),(1,l),(2,l),(0,2),(1,2),(2,2)}.

(a;:) (bl2) (c.2) . I .

~

.-\~

CKynOBJ.I AxB n BxA

cy pa3IDf'IJfm H HHKJIY3HjcKH tteynopeAHBH, Tj . AxB ~BxA 11BxA ~AxB .

IlpKMenrno

Jf

.£\a je

(AxB)

n

(BxA) = { (1,1),(2,2)}.

BxA (:,c) 2,c)

Ey;lyb..H ,'.la npa3a.H CKyn HeMa eneMeHTe, HHje Moryhe opM11pam ypel)ette rrapoBe ca KoopwmaTaMa 113 rrpa3Hor cKyna. 0,[{aB,[{e cneAH ,[{a cy 3a CBaKH cKyn A TaqHe je,[{HaKOCT11

0xA=Ax0=0. •

.. CCD .

3a;:i;aTaR

")

B

HeKa je A= {l,2,3} 11 B = {1,2}. 0APeAJ1:

(AxA)

n

(BxA);

2 (BxA) \ (AxB); 3 (A \ B)x(A

n B). JR

y ·~

I ...

)J.eKapTOB npoH3BO~ CKynOBa ce YBOAH no y3opy Ha rrpaBoyrJIH KOOpAJIBaTIDi CHCTeM (Koj11 ce Ha3JIBa H )J.eKapTOB KOOpA.J1HaTH.l1 CHCTeM).

_,

E

...-

A(l 2)

,

AKo Koop~aTHe oce rrpaBoyrnor

.I.

D

KoopwmaTHOf CJICTeMa CXBaTIIMO Kao rpaJiqKe np.m<.a3e CKyna peammx 8pojaBa R Ta,[{a ce

-

paBa.H y Kojy je IIOCTaBJbeH KOOpA.J1HaTH11 CJ1CTeM MO)l(e CXBaTJ1TJ1 Kao )J.eKapTOB IIpOH3BOA ypel)emIM rrapoBHMa peaJIHHX 8pojeBa.

26

0

'

J.

JRxR

3ffKYjeMo ca

~ ~

-2

c

£

-J

B(2 -2)

'

x

E 'HMep2. MMO ,n;a je 3a me cKynoBe A, B M C Ta'Il!a jeAJiaKOCT

e

Ax(B U C) = (AxB) U (AxC). :J>HO je ,!l;OKa3aTM ,n;a 3a CBaKO x My Ba)KJ.f eKBJIBaJie~a x,.1 E A B l..J C ~ (x,y E A .B· U -.4. .C .

c

{l ,e) (2,e)

Ic1Jn cd1)

eHOM ,n;eqmmn_\Mje ,[J;eKapToBor npmrnBo,n;a M y1rn:je MM~m

x,y)EAxBUC

~xEAAyEBUC ~

x E

.~. I

•• :

E B - E C) B

~

(x,y) E (AxB) U (AxC)

(x,y) E AxBV (x,y) E AxC ~ (x E AAy E B)V .x E AAy EC).

CTaBMMO ,n;a je

p: x E A, q: y E B,

r: y E C,

a TaYTOJIOflfjH p A (q v r) ~ (p A q) v (p Ar) (,n;OKa)l(J.fTe JW je OBa cl>opM}'lla 3aJfCTa OJIOrHja) 3aKJbyqyjeMO ,n;a je ,n;arn CKyrrOBHa je,n;HaKOCT Ta'IHa . •

p;aTaK _ ;::::.;...ll(lf ,n;a je 3a me cKyrroBe A, BM

Ypel)ena TpojKa (a,b,c)

c Ta'Il!a jeAJiaKOCT Ax(B n

C)

je Tpo'l7Irurn HM3 'Ilfjn je rrpBu 'IJiaH

=(AxB) n (AxC).

a, ~yrn q;1aH b n

TpehH "lJiaH c. 3a ypeljeHe rpojKe BIDKH

(a 1,a2,a3 )

=(b 1,b2,b3)

~

a 1 =b1 Aa1 =b1 Aa3 = b3•

Cnwrno ce .n;ecl>murnIY ypeljeHe 'leTBopKe, ypeljeHe ueTopKe nr.:1. µeqmmurn!::i

y 03HaI.{Jt AxBxC, jecre Ch.JII 'Iltju cy eJieMeHTH ypeljeHe TpojKe 'Ilfja npBa KoopwmaTa uplil~a CKyna A, :ipyra

,[J;eKapTOB npoM3Bo,n; CKyrrOBa A, B M C,

Koop,n;MHaTa CKyrry B, a rpelia Koop.mrnarn CK}'IIY C, OllHOCHO,

CKJIIOBH .-b<..-l, Ax.AxA, ITT:I. Ha3Jraajy

AxBxC={(a,b,c) I a E AAb E BAc EC}. Cn11qtto ce .n;ecl>11HMrne ,IJ;eKaprnB upon3Bo,n; B1rme o;r -rpn CKJ'll
--- ---- ------

..... xJR ·~~o j e A= {a,b}, ott,n;a je

A.xAxA = { (a,a,a),(a,a,b ),(a,b,a),(a,b,b ),( b,a,a),(b,a,b ),(b,b,a),(b,b,b)i .

JR

ce _ilexaproBM creneHH cxy:rra A K .,-ecro ce o&."I~jype.aoM .-\!, A3 JIT:L

fipHMep3.

x

'

+

3ap;aTaK AKo je A= {a,b,c}, B = {O,l}, C = {l,2}, o,qpe,n;M:

• AxBxC; 2 AxBxB; "" (A xBxC) n (AxCxB);

3 BxBxB; 5_ (AxBxC)

n (BxAxC).

3ap;aTaK AKo je A= {l,2,3},

1 AxBxC;

B = {a,b,c,d}

2 AxAxA;

H C= {O,l,2,3}, o.o;pe,::i;H c5poj ei:eMeHara CK}TIOBa:

3. BxBxC.

27

~

-

~---

A

Euuapue perra:q11je Pe,TarnrjaMa ycrrocraB;i,a.-..m Be3e H3.Mel)y o8jeKaTa (6pojeBa, reoMeTpHjcKmc q)Jl[rypa HT.n;. Koje IIpO.f'laBaMO.

fomapHa pe."Ia.IVlja acyna A je forno Koj11 nop;cKyII )J;eKapToBor rrp o:H3Bo.n;a AxA.

Yoblf'laje1m je :ra ce pe.-iamtje oaHatraaajy YaJ"'Illll cnoBnMa rpqxor a.""Icpac5eTa.

-------

HeKa je A= { a,b,c} 11 p no.n:cKyII op; AxA p;aT ca p = {(a,a),(a,b),(a,c),(b,b),(c,a)}. ,/J;aKJie, p je jep;Ha 5:HHapHa peJiaJ.\11ja CKyIIa A. 0By pen~11jy CJI:HKOB:HT:Hje MO)f(eMo npmca3arn Ha p;Ba HaqHHa: rpa¢oM 11 TabJIJ11.\0M. BepyjeMo .n;a cy Hape.n;He CJIHKe ,n;oBOJbHe p;a ce yoq:Jll KaKo ce ¢opM11pa Ta5JI:HJ.\a, OAHOCHO rpa¢ penaqHje.

a,

~(B ~,,

(c,

c

e

b)

I I

a b c

II a

b

c

.l

T T

.l

T

.l

.l

T

T

:

•

IIpw1rn:KOM cpopMwpalba rpacpa penaQwje HeMa rroTpec5e HaBOAHTH ypeljeHe rrapoBe. Mw CMO HX HaBo.n;wnw caMo .n;a 5wcMo oc5jacHHnH 3Haqelhe cTpeJIHQa.

p aeKa bHHapaa pen~Hja cKyna A, Tj. aKo je p k AxA, yofo.1qajeHo je p;a ce (x,y) E p mmre x p y 11 q:HTa ,,:HKC je y pen~11jH po ca 11nc11noH': YMeCTO (x,y) ~ p IDfCalleMO xp y (p je npeqpTaHO CJIOBO po).

AKo je

~xecro

TaKol)e, y aacraBJcy hel'.m fornapHe pemu~Hje 3BaTH Kpabe penau,ujaMa.

IlpHMep2. Tafoume u rpacpoBe peilaIIJfja cpopmi:p~m ~O y Cty'Iajy ~a cy Te pe.i"IaIIHje Ha KOHa:ciffiBf ~)IlOBHMa.

Pen~tje ~ H

< Mel)y peaJIHJiM 8pojem1Ma Ba)f(HH cy rrp1.rnepH penaq11ja Ha

(8eCKOHaqaoM) CKJIIY ~. 08ojeIDl p;eJIOBJll KOOpp;:HHaTHHX paBHJ1, CXBa!iemt: Kao IlO,ll;CKyIIOBJf OA -::xR, 3anpaBO cy rpa¢WIKJ1 npl1Ka3J1 OBHX penaq11ja. I1p11MeTJ1Te .n;a Ha cmrrµt p;one }:\ecHo Ta'Il
• 28

fornapHa pena1v,1ja p cicyna A je: • pecpJieKCHBHa aKo je x p

x 3a cBaKo x Jll3 A; x H3 A HMje x p x; CHMeTpH11Ha aKo je 3a CBe x Hy Jll3 A yaqHa JL'>mmfl y p x; aHTHCHMeTpH'IHa aKo je 3a cBe x Hy H3 A Ta'Illa J1MlIJIJtKau;J1ja xpyAypx=>x=y; JIHHeapHa aKo je 3a CBe x J11 y H3 A Ta'Illa wicjyma.\J1ja x p y Vy p x; TpaH3HTHBHa aKo je 3a cBe x, y J11 z. 113 A Ta'Illa 1n.1nJrnKaI.V1ja

• HpecprreKCHBHa aKo 3a cBe • • • •

,neqmml.Wf b

x p y A y p z => x p z. ·---~

IlpHMep

3.

J1cm1TajMo Koje O,ll; oco8J11Ha HaBe)l;eIDtX y rrpeTXo)l;Hoj )l;eqmmn:prju rum peJia.QJrja p cKyna A= {a,b,c} H3 rrpHMepa 1. J1crrocTaBHlie ce )l;a oBa penaa;nja HeMa mrje}lH)' op; IIOMeHyna OCObJllHa. PenaQHja p HHje peqlJieKCHBHa, jep HHje TaqHo )l;a cy CBH eneMeHrn: cKyna ca coboM:

A y penaQ11j11

~

(~ ~\..

c

a p a, b p b J11 c p c.

PenaQHja p HHje 11peqmeKCHBHa, jep HHje Ta'Illo )l;a CBH eneMeHTH cKyrra A HHcy y penaQHjH ca coboM:

a p a, b p b H c p c.

)J;a penaQHja p HHje HH CHMe-rpH'Illa HM aHTHCHMeTpM'llia, rroKasyjy Hape:uie .J;Be nbene y Koj11Ma cy )l;eTalbHO HCilHTaHe CBe MOryilHOCTJ1.

x a a a

-Y

b b b c c c

x

x

a

T

T

T

b c

T

..L

..L

T

T

a

..L

T

T

b c

T

T

T

..L

..L

a

T

T

T

b c

..L ..L

..L ..L

T

I I

T

I

•

I

I

T

I

II I

T

x

I

a a a

I I

b I b I b I t c c I c I

'

l'. a

b c

i xel'. ; l'.ex T I T

I

!

T T .1

a

• b I T c a b

1:

c

•

i I

! I

I

..L T T T

..L

.1

T ..L ..L

T f

.1 ..L

X=J.' xe l'. Al'. ex=>X= l'_ T T ..L T ..L .1 ..L T T I T .1 II T .1 .1 .1 T T T I

;

~'

I

'

I

J1aKo cy rrpeTxop;He Tabene BeoMa HHop~rnnmHe, EhUXOBO opMl!paH>e :\IOJKe ,!la rrpe,ll;CTaBJba 3aMopaH rrocao aKo je cKyn KOHaqaa u HMa Belill bpoj eneMeHaTa, ;::i:oK y cnyqajeBHMa becKoHaqHHX cKynoBa HHje HH Morrhe ¢op:\mpan1 osa.KBe rabe.ae. IloKyrnajMo )l;a OTKpHjeMo be3 Tabena )l;a JIH je penaa;Jrja p mrneapHa u ~a m1 j e TpaH3J1TJ1BHa. AKO II(l)Klby ycMepHMO Ha ,!1;e1rnHQHjy penaQHje p, O,ll;HOCHO Ra IbeH rpa, bp30 he:\IO

b pc 11 c p b, Ti. - lb pc\ c p b). )J;a penaQHja p HMje -rpaH3HTIIBRa, cnew1 H3 qJ.flbeHHQa: c p a, a p b :1 c p b, Ti. Jl3

yoq11TH )l;a penaa;Hja p HHje JIJraeapaa, jep je, Ha rrpHMep, qm1>eHHQe )l;a Hlije Ta'Illa JlMIIJIJIKaa;Jfja c p

a A a p b =>

cp

b.

+

29

-------

- -- - - - --------

E

Pena~je

-------------

nopeTKa H eKBJIBarreH~Hje

HeK.a je p pe;ramtja CKJIIa A. Penan;:Mja p je perran;Hja nopeTKa (ypel)efbe) aKo je peqmeKCJIBHa, aHTlfCJ.IMeTpI1'IHa M 'YpaH3MTHBHa. i.......--

IlpBKep 1. Y Hape)lHoj -rabem1 cy mrne.n;eee TPH BaJKHe penan;:11je rropeTKa. Penamrja ~ na CK}'Il}' ? (3a CBe x, , z n3

)

Penan;:Mja I (.n;eJhHBOCTM) Ha cKyrry N (3a cBe n, m, k M3 N)

V1HKJIY3Mja k Mel)y CKyrrOBMMa (3a CBe cKyrroBe A, B, C)

n m/\m n=>n=m n m/\m k => n k

A k B/\B k A=> A =B A k B/\B k C =>A k C

AkA $

x => x=

IlpnMenrno .n;a je penan;:Jrja $ ea CKyrry lR JIMHeapHa, jep je x $ y MJIM y $ x 3a bMJIO Koja p;Ba 8poja x n y. PenaI{Jtja I (.n;CJbJIBOCrn) Ha CI
+

3a;J;aiaK )J.a JDf je penan;:Mja cicyrra A= {a,b,c,d} rrpHKa3atta rpacl>oM Ha cmf:ql1 ,qeCHo penan;:Mja nopeTKa? ,L:{a JDf je OBa perran;:Mja mrneapHa?

Perran;:Mja p je penan;:Mja JIHHeapHor nopeTKa (n11HeapHo ypel)eH>e) aKo je peqmeKCHBHa, aHTMCMMeTpl1'!Ha, JIJiHeapHa M TpaH3MTHBHa. Pena.iµrja p je perran;:Mja cTporor nopeTKa (cTporo ypel)eH>e) aKo je Hpecl>neKCMBHa H TpaH3MTMBHa. Peaawrja $ Ha cxyrry ? je mrneapHo ypel)eH>e.

L-IlpHMep2. CBaKoj penan;:nj11 nopeTKa p HeKor cKyrra A o.n;roBapa je.n;Ha penan;:Mja cTporor rropeTKa

a (TaKol)e Ha A) TaKBa .n;a cy 3a cBe x My H3 A rnqHe eKBMBaJieHn;:ttje: x p y @ x a y v x = y, O):l;HOCHO, x a y @ x p y /\ x =/:- y. TaKo penan;:MjH $ Ha CKYJlY JR op;roBapa penan;:11ja <, jep je 3a CBaKo x, y M3 JR: x $y @x
k

op;roBapa CTpora MHKJIY3Hja. 3a cBe CKyrroBe A

cy eKBJmarre~je A k B @ A C B VA =B, O):l;HOCHO, A C B @ A k B /\A=!:- B.

30

+

MB

TaqHe

----

-··----

Perra.Q11ja p je pena1µ1ja eKBHBanemµrje aKO je pe¢rreKCHBHa, CHMeTJ>WIHa 11 TJ>aH3MTIIBHa.

~

µeif>nuuUHja

IIpHMep3. Ha Hape,n;Hoj CJIM.QH rrpHKa3aH je rpa¢ je,D;He perra.Q11je eKBHBanemµtje CK}'Ila

S = {a,b,c,d,eJ}. H 11je TernKo yoq11T11 ,n;a je CK}'Il S OBOM perra.QMjoM 113,n;eJbeH Ha TPH rro,n;cKyrra KOjH cy Mel)ycobHO ,D;Mcj)'HlCTIDI TaKO ,n;a j e CBaKH erreMeHT 811rro KOr IIO,lJ;CK)'Ila IIOBe3aH ca bHJIO KOjHM erreMeHTOM 113 HCTOf IIO,lJ;CKyrra (HapaBHO H ca caMnM coboM). CBaKM o,n; OBHX no,n;CicyrroBa Ha3HBa ce KJiaca eKB11Barremi;11je.

0

0•

(l;.~ '

\\,'I•

j)

lloja IIpeI.J;H3H11je, KJiace eKB11BaJieH.QMje cy rro,n;cKyrroB11: {a,b,c}, {d} 11 {eJ}.

••

+ JeAJiaKOCT ( = ) je TJ1IDf'IaH

J1crrocTaBJba ce ,n;a CBaKa perraQMja eKBHBaJieH.QMj e HeKor cKyrra ,n;err11 Taj cKyrr Ha Meljyco8Ho ,n;11cjyttKTHe Henpa3He rro,n;cKyrroBe qlfja j e ynHj a j e,D;HaKa CKyily Ha KOMe j e perraQ11ja ,n;e¢11H11caHa. CBaKM o,n; Till no.z:i;cKyrroBa Ha3KBa ce xnaca eKBKBanemµrje yoqeHe perraQ11je.

rrpHMep pemuµrje eKB1manem.urje.

IIpHMep4.

0

Pe
a

I1pmra3 pen:aunje ·e:.uraxocn1 Mel)y peaamrn 6pojeBIDt:a

PerraQ11ja p cKyrra .Qemrx 8pojeBa Z ,n;e¢uHMcana j e Ha cne~elrn: Ha'IKH: x p y aKKo je x -y _::\eJbJIB ca 2. ,[J;oKa)KHMO ,n;a je p penaQ11ja eKBHBaJieH.QMje.

~

rr /.

11

it /

CuMeiUput-tHocm. IIpernocTaBHMO ,n;a cy x Hy .QeJIM 8poj eB11 TaKBH ,n;a j e x p y . Ta,n;a j e x-y p;elbHBO ca 2. MeljyTHM Tap;a je 11 y-x p;elbHBO ca 2 (j ep j e y - x =-(x- y)) , rra je TIL.'de HY p x.

o

10

o

1

x o) ""'.(

2 x-

/ -~ fpa je:maxocrn cKyna "--i. = 1a,!7,c,d}

~~~~~ 0 . 0 . 0 •. 0 . 0 .

c

r9_'-\

~ -B8~

0

CB·

e CKyrr Z je OBOM peJiaQHjoM nogelbeH Ha ]{Ba ;::i;ncj~'THa no;:iacyna: aqn Henapma: Qemuc 8pojeBa { ..., -5, -3, -1,1,3,5, ... } u cxyn napmcx ne-IKX 8poj eBa { ... -4, -2,0,2,4,6 --l·

~

o

-2 - 1

TpaH3UiUU8Hocm. IIpernocTaBHMO p;a cy x, y Hz n;em1 8poj eBu TaKBH p;a j e x p y 11 y p z. Ta,n;a cy x-y 11 y-z ,n;eJbHBH ca 2. KaKo je x-z= (x- y) + (x- z ), 3a1Gh)-qyjeMo ,n;a je x-z ,n;eJbHBO ca 2, Tj. x p z. :1
~

+

8 31 I

A )'H.KI{Jfje cy je~a o.n; HajBa_)f(HMjmc Be3a J13Mel)y MaTeMaTMKe J1 CBeTa OKO Hae. IlO,\IOilY IMfX MO,ll;emt:paMo J1 npaTMMO pa.n; MexaHJ1qKHJC, bMOJIOIIIKHJC J1 pa3HHJC .n;pyrttx CHCTeMa. 11HryJITHBHO, cpymavrjaMa ce ycnOCTaBlbajy Be3e H3Mel)y eJieMeHaTa ,ll;Ba CKyna TaKO IIITO ce CBaKOM eJieMeHTY je.n;1rnr cKyna rrptt.n;p)')Kyje TaqHo je.n;aH eneMeHT .n;pyror cKyna.

IlpKMep 1. Ha Hape~ cmt:KaMa rrpHKasaHo je jeww IIpH,ll;p)')KMBalbe Koje cBaKoM eneMeHTY CKJIIa A= {a,b,c,d} .n;o,z:i;elbyje Taqiio je.n;att eneMeHT cKyrra B = {l,2,3}. ,ll;pyrHM peq11:Ma, ,z:i;ecpHHHCaHa je je)l;Ha cpyttKU;Hja H3 A y B.

AxB

A

AClIIj:> YoqeHy cpyttK~Hjy CXBaTaMO Kao CKyn

J= {(a,l),(b,2),(c,2),(d,3)}. +

CKynfje y111a:i;Hja M3 A y B, y 1 j<;;, AxB11:

03Ha~11:f: A~

B, aKo B(l)l(e cne,z:i;etrn ycnoBH:

2. 3a CBaKo x H3 A rrocTojH TaqflO je.n;aH eneMeHT y M3 B TaKaB .n;a (x,y) E

,neqmmi.1.1uja

f

AKo f: A ~ B, CKJII A ce Ha3HBa µ;oMeH HJIH obnacT µ;e<)>u:Hu:caHocTu:, a cKyn B KO}J;OMeH J1ill[ obJiaCT Bpeµ;:HOCTH cpyttKU;Mje

f

A

L-

IJpHMep2. HeKa je A= {a,b,c,d} H B ={l,2,3} (Kao y rrpeTxo,z:i;HoM rrpttMepy). CKJII g= {(a,l),(b,2),(d,2)} HHje cpyttKU;Hja H3 A y B, jep eneMeHT c H3 A Httje rrpBa KOOp,Il;HHaTa HHje)l;Hor rrapa J13 g, Tj. eJieMeHTY c HHje rrptt,z:i;py)KeH HMje,z:i;aH eneMeHT J13 B.

h = {(a,l),(b,2),(c,1),(c,2),(d,3)} Httje cpyttKU;ttja H3 A y B, jep p;Ba pa3JIJ1qurn napa H3 h MMajy HCTY rrpBy Koopp;irnaTy c, Tj. eneMeHTy c HHje rrpH,z:i;py)KeH caMo je.n;aH eneMeHT H3 B. CKyn

e= {(a,l),(b,l),(c,l),(d,2)} J1 s = {(a,3),(b,2),(c,l),(d,2)} jecy cpyttKU;Hje tt3 A y B. +

CKyrIOBM

32

f murreMo fix) = y, a rroHeKa;a;a :u x -f

AKo f A ~ B, )'MeCTO (x,y) E

I

y.

&

IlpHMep3.

AKo

jef {a,b,c,d}

~

{l,2,3} ¢y1uai;:uja :113 npHMepa l, oH,qaje j{a) = 1,j{b) =2,j{c) =2,j{d) =3,

rrp:11 'IeM)' cy OBe je;a;ttaKOCTH ,qpyra'Ilfje 3arrncaH:11 o;a;ttoc:11

(a,l) E f, (b,2) E f, (c,2) E f, (d,3) E

f

Pa;a;:11 8oJbe rrperrre;a;HOCTH Kop:ucTHMO H rne;a;etrn 3arr:11c:

b d).

f=(a c 1223

0BaKaB 3arr:11c je rroro;a;aH 3a ;a;e¢JffiJ1caH>e ¢~ja J13Mel)y ;a;Ba KOHaqJia cxyna ca MaJIHM 8pojeM eJieMeHaTa.

+

'

flpHMep4.

Ka;a;a cy y Ill1TaH>Y ¢YHKl..l;J1je J13Mel)y KOHa'IHHX CK}'IIOBa ca "BeJIHKJru" 8pojeM eJieMeHaTa HJIH 8ecKOHa'IHJ1X CK}'IIOBa, He MO)l(eMO HX ;a;e¢HHHCaTJ1 Kao y rrpeTXOAHJfM rrpttMepttMa. Y oBaKBHM CJI}"IajeBHMa ¢YHKr.i;Hje ce ttajqernne 3a;a;ajy HaBol)eH>eM rrpaBHJia no KOMe ce eneMeHTHMa je;a;irnr cKyrra ;a;o;a;eJbyjy eJieMeHTH ;a;pyror cKyrra. BeoMa Ba)l(He ¢YHKr.i;Hje Mel)y cKyrroBHMa 8pojeBa rrpe;a;crnBJbajy otte Koje cy ;a;e¢ttHHCaHe HeKHM H3pa30M ca je;a;ttoM rrpoMeHJbHBOM. Ha rrpHMep, H3pa30M 2x + 1 ;a;e¢HHHCaHa je ¢YHKl..l;Hja f JR --+ JR Koja CBaKOM peaJIHOM 8pojy x npH,qp}')l(Yje o;a;roBapajyny Bpe;a;ttocT 2x + 1: f

f

f

f

3 - 7; 2,5 - 6; 0 - l; -1 - -1, ... Je;a;HoCTaBHO, Ka)l(eMo ;a;a je ¢YHKl..l;l1ja f :R --+ ~ ;a;e¢nmicaHa ca

j{x)=2x+l. CacBHM je rrpttpo;a;Ho ;a;a OBY ¢YHKr.i;Hjy rpa¢J1'IKJf npm<:IDKeMo y npaBoyrnoM Koop;a;HHaTHOM CHCTeM)'. •

JR

f

JR 3

YI 3

2

2

1

1,. (0,1)

0

--<

Rx~

:/ 1

11t'.'laH

01

-1

1

2

3 x

-1

-2 -39 - - - - - - -9-3

;,h =2xT I

l (l,3)

-2 (-2 ,-3) I

•

-3

33

1

I

ilpJ1Mep 5. IloCdarpaj~IO ;we .pyai<:qMje,

f: {a,b,c,d} -+ {1,2,3} Hg: { 1,2,3} -+ {t,u, v, w}, p;aTe ca

t=(a b Cd) 1122

KoMII03lmJi° a

~e ymanrje je

:te$mrnca.aa c.aMO aKo je ;rOMeB je::me $yma:m ·e e:tlUK Ko:tmi:eHJ ;1pyre cpymaurje.

ITp11Menrre ;:race j" g cpym::mrja H3 rrp1rnepa 1.

31·

tvw

Ilpm1ern.Mo :4a je ;i;oMeH cpyrua:v.1je g HCTH Kao Kop;oMeH cpy1ua~.aje f .Je
h(b) =g(f(b)) =g(l) =t, h(d) =g(f{d)) =g(2) =V.

h(a) = g(f(a)) = g(l) = t, h(c) =g(f(c)) =g(2) =v, jlaKJie,

d)·

h =(ab c ttvv

KOMI103UIOlja

He MO)l(e .:ieqnnmcanL

Hg=( 1 2

f

A

g

c

A

c

ymcu:ttja h HCl3JIBa ce KOMII03HI.V·fja cp~Hja f Hg. • KoMH03HIJ;:uja cpym<:qaja f: A -+ B Hg: B -+ C je cp)'H~Hja h: A -+ C, .n;ecpHHHCaHa ca eqmmum ~a

h(x) =g(f(x)). h ce 03HaqaBa ca g o f )J;aKJie, (g o j) (x) = g(j(x)).

IlpHMep 6.

EKCIL"DIIIJITHO (ilaT.

explidtus) 3Ha'IH jarno, He;:i;BOGiltCileHO,

ornopeHo.

(j o g)(x) =fig(x)) =fi3x + 1) =-2(3x + 1) -1 =--Ox-3 (go j)(x) =g(f(x)) =g(-2x- l) =3(-2x-l) + 1 =--Ox-2

(j o j)(x) = j(f{x)) = fi-2x- l) =-2(-2x- l) -1=4.x+1 (go g)(x) =g(g(x)) =g(3x+ 1) =3(3x+ 1) + 1 =9x+4

+

~Hjef? ·~ ::.l 11g: R -+ IR .n;ecpHHHcaHe cy ca fix) =x-1.ag(x)=1-x. Op;pep;M:j o g, go f,f o f,g o g, (j o j) of

34

\':\~

~

i

)7HKL\J'fja f A -+ B je: • l-I-y1uo~Hja (
;::e¢HBJIUl!Ja

x1 :t:x2 ~fix) :f: fix), O)J;HOCHO, l1MIIJIJ1KaQ11ja (Koja je IIO 3aKOHY KOHTpaII03Hlll1je eKBHB3.JJ.eHTHa npeTXOp;Hoj) fix1) = fix2) ~ x 1=x1; Ha-YHKD;Hja, aKO 3a CBaKO y H3 B IIOCTOjH x H3 A Ta.KO L{a je fix' = y; OHjeKD;Hja, aKo je 1-1-YHKI.\Hja H tta-¢YHKI.\Hja. 1-1

AKo je f: A

f: A

Ka

-+

-+

B 1-1-YHKI.\Hja, IIHilleMo f: A

-+

B, aKo je Ha-¢)'HKI.\Hja rnrmeMo

3axoH KOliTpaII03JlUHje

(..,q => -p

~

p => q

A

HHje

1-1

A

Hlilje

1-1

1-1

B, a aKo je 811jeKI.\»ja, IIHilleMo f: A -;;: B.

1IpHMep7. ,ll;OKYHKI.\Hja f: IR.

-+

IR. p;e¢HHJ1CaHa ca fix)=-2x+l

811jeKQ11ja. ,ll;OKYHKI.\"ja. HeKa cy xi " x2 peaJIHl1 opojeBH TaKBl1 p;a je fix 1) =fix). Tap;a je, npeMa p;e¢11HHQ»j11 ¢YHKI.\»je f, -2xl + 1=-2x2+ 1, op;aKJie np11MeHOM II03HaTHX csojcTaBa ca811palha H MHO)l(eiba peaJ1Hu:x opojeBa 3aKJbyqyjeMo p;a je x 1 =x2. ,ll;oK
b---4

.

r

c

A

B

X=y-1 Tap;ajefix) =-2x+ 1

=-2(Y_~ )+1 =y. 1

I~

-2 • I~

AKo f: A -;;: B, ottp;a nocTOjH 811jeKI.\1tja g: B-;;: .A. TaKBa .ua je

j{g(y)) = y, 3a cBaKo y 113 B yjy

"

g(fi.x)) =X, 3a CBaKO x H3 .A..

)7HKQ11ja g H3 npeTXop;He TeopeMe HaJJIBa ce 1msep3Ha ymmuja ¢ymamje f u oOeJie)f(aBa ce ca J-'.

IIpHMep8. Y npeTxop;HoM np11Mepy CMO noKa3aJJ.Jt na je ¢ymcuuj a f: ~ -+ ~ naTa ca fix) =-2x + 1 811jeKI.\1tja. Y p;oKaJY L{a je f Ha-¢puau1ja o.:q>e!ULlll CMO, 3anpaBo,

HHBep3tty ¢YHKI.\Hjy !:< -+ ~- Osa ¢ymcu11ja j e ne¢HHHcatta ca J-1(x) =- x - l . IlposepHTe p;a JIH cy 3a csaKo x »3 ::C Ta'IBe je~HaKOCTil flJ- 1(x)) =x u J-1(f{x)) =x.

J-1:

+

35

A ~

0

0 D 0 G ~

0

0 G 0 G

~

~

0 0 0

[2J

D

0

CD -0 0 [?]

G GJ 0

0

[2J

G

KaMfomarnpHKa je MaTe~ranr'IKa WK~a Kaja ce yrmrnHaM 8aBl1 KaHa'IHl1M CK)'IlaBHMa l1 rrpofo1eMH.."'da op;pel)i.rnaIDa 8paja eneMeHaTa TaKBHX cKynaBa. Ilp11 TaMe, BeaMa qeCTa Hl1je Maryhe aneM, Ben pa3MMIIIJhaibeM, IIITa heMa mrycTpaBaTl1 HapeWIJ™ rrpHMepHMa.

IlpHMep 1. Oµ;pe;nrua Opaj eiieMeHaTa CK)'Ila A •mj11 cy eJieMeHTl1 CBl1 ,o;BaIJ,mppeHl1 8pajeBl1 3armcaHH naMahy HenapH11X IJ,11¢apa: 1, 3, 5, 7, 9. CK)7II A je a
JtlrraK, 811na 811 BeaMa ttarraptta HaBeCTl1 CBe lheraBe

eneMeHTe u rrpe8pojaTI1 HX (rrarne,o;ajTe CJ111KJ Ha Maprntt11). IlaCTyn11heMa Ha ,o;pyr11 Ha'Ilm. CBaKH ,[\Bal.l,11¢peH opaj je 3anpaBa je,o;aH ,[\BaqJiaHJ1 Hl13 IJ,l1¢apa. CBaKl1 eneMeHT

S ,n;afarjaMo ¢apMMpajyl'rn: H1if3 a,o; ,o;Be IJ,l1¢pe, 811pajyh11 IJ,11¢pe l13 {l,3,5,7,9}. 3a rrpso ~IeCTa (cneBa) 11.MaMa neT MaryhHaCTJ1, a 3a CBaKy a,o; THX neT MaryhHaCTl1 nacroju no neT MoryliHacTJ1 3a p;pyry IJ,11¢py. ,Il;aKJie, cKyn S l1Ma 25 eneMettaTa. CKJIIa

J1CIIOCTaBJha ce ):\a 8paj eJieMeHaTa CK)'Ila A HJ1je MHara BeJIJ1KJ1 l1 ,o;a CMa MOfJIJ1 CBe ,o;a l1X rrpe8pajHMa cncTeMaTJ1qttHM 3alll1:Cl1BalheM. Mel)yTHM, Ha aBaj HaqMH npaKTJ1qHa mrje 1'mryhe a.n;pep;wrn, Ha np11Mep, 8paj CBMX ce,o;MoIJ,11¢peHMX 8pojesa 3anMcaHMX 1P1¢pa.Ma 1, 3, 5, 7, 9. Pa3Ml1IIIJhajyha Kao y cnyqajy ,o;BaIJ,11¢peHMX 5pojesa, 8p3a AOJia3J.fMa .n;o pe3ynTaTa: ce,o;MaIJ,11¢pettl1X 8pajeBa 3an11caHMX naMahy IJ,M¢apa l, 3, 5,

7, 9 YKJIIHO Jh'>fa 57 =78 125 . • IlpHMep2. HeKa je B CK)'II CBHX IleTa!p1¢peHJ1X 8pajeBa y KOjl1Ma ce IJ,l1¢pe He naHaBJhajy. 0Bara

IIJTil HeMa arpaHWieiba Koje n;11¢pe ce Mory najaBJbl1BaTl1, IIITa 3HaqJ1 ,o;a cy ,o;o3BOJbeHc CBe l.l,11¢pe ,o;eKa;utor cHcTeMa: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. AKa 3aMMCJIMMa neTaIJ,M¢peHe 8pajese Kao rreTO'lllaHe H1if3aBe IJ,l1¢apa, ,o;ana3l1Ma ,o;a cne,n;el'rn:x 3aKJbyqaKa: • Ha rrpBa MeCTa (cneBa) MOJKeMa naCTaBl1Tl1 bl1JIO Kajy n;11¢py pa3JIJ1qJ1TY a,o; Hylle, IIITO 3Ha'llf ,o;a HMaMa ,o;eBeT MaryhttaCTl1, • Ha p;pyra MeCTa MOJKeMa nacTaBl1Tl1 0l1Jia Kojy a,o; IJ,H¢apa, aCHM aHe Kaja je nacTaBJheHa Ha npBO MeCTa, IIITa 3HaqJ1 ,o;a naHaBO HMaMa ,o;eBeT MaryhHaCTl1, • Ha Tpebe MeCTa, HaKaH l130apa rrpBe ,IJ;Be l.l,11¢pe, MaJKeMa yn11caTJ1 je,o;Hy a,o; acaM npeocTaJIMX IJ,M¢apa, • Ha qernprn Mecrn, HaKoH l138apa npBe TPl1 IJ,l1¢pe, yn11cyjeMa je,o;Hy o,o; npeacTamn: ce,o;aM IJ,l1¢apa, • J1

Ha Kpajy, Ha rreTO MeCTa, HaKaH H38apa npBe qeTl1Pl1 IJ,l1¢pe, ynMcyjeMa je,o;Hy a,D.

rrpeacTaJIJ1X rnecT IJ,11¢apa. ,Il;aKJie, CK)'II B KMa 9 · 9 · 8 · 7 · 6 =27. 216 eneMeHaTa.

+

3a;:i;aTaK Kalll1Ka HMa meCTOD;l1:¢peHHX 5pojeBa y Kajl1Ma ce IJ,l1¢pe He naHaBJhajy?

36

. ~-·

·. ··.\'11(

IlpHMep 3. A

Ha IIIaxOBCKOM TYPHHPY je yqecTBOBano neT uuuttcra: Depa, MHKa, Jla3a, CMM:a u An:a.. Kom1Ko rrapTHja je YKYTIHO op,wrpatto, aKo ce HrpaJio no rn:cre~ry CBaKO ca CBa.KJt:M? CBaKH o,n; rreT yqecHHKa Typttwpa je O.n;Hrpao ca rrpeocTamfM TaKMJ.PiapJn.rn rro jeAHY

TI~

:

AC

<

rraprn:jy, O,Il;HOCHO, YKyrrHO qeTHpH rrapTHje. Mel)yrHM, yi
+

~1

,1

3a;:i;aTaK

1) KOJIHKO ,n;y)KH o8pa3yje ce,n;aM Tat.Ia.Ka Meljy Kojmia He rrocrnje TJ>H Koje rrpKIIa~ajy je,n;ttoj rrp aBoj?

2) Ha KOJIHKO Hat.IHHa ce o,n; 20 yqeHHKa je,n;ttor op;elbe11>a MO)Ke K3a8pan1 }J;BO'Ulatta ,n;erreraQHja?

3) Korr11Ko 11Ma ,n;BoqnaHHX rro,n;CKynoBa cKyrra Koj1111Ma 100 erreMeHaTa? rreHMep4. KorrHKO cKyn o,n; 100 erreMeHaTa HMa rreTot.IJiaH11x rro,n;cKyrroBa? Epoj rreTot.IJiaHMX HH30Ba Mel)yco8Ho pa3JIM'IMTI1X erreMeHaTa ,n;aTor cKyrra o,n;pel)yjeMo pa3MHIIIJbajytrn Kao y rrpHMepy 2 - HMa I1X 100 · 99 · 98 · 97 · 96 = 9 034 502 400. IleTOt.IJiaHHX rro,n;cKyrroBa MMa 3HaTHO Malhe Hero neTO'UlaHI1X HJ130Ba. Hwje TeIIIKO 3aKJb)"IMTJ1 ,n;a o,n; CBaKOr neTO'Ulattor no,n;cKJila p;aTor CKJIIa MO)Ke~10 cpopmtpaTH

5 · 4 · 3 · 2 · 1 = 120 pa3JI.11'IBTI1X IIeTOt.{JlaHJIX HJ1.30Ba y KojnMa ce eneMeHTJ! He " rrottaBJbajy. 3aKJbyqyjeMo ,n;a rreTO'UlaIDf.X HH30Ba pa3JIJf'Umcx eJieMeHaTa .:i,aror CKyila J1Ma 120 rryTa BHIIIe Hero rreTO'UlaHHX IIOACKJIIOBa OBOf CI
9 034 502 400: 120 =75 287 520 . • 3a11;aTaK Korr11Ko 11Ma ce,n;MO'UlaHMX rro,n;CK:yrroBa CKyrr KOjH ~'1a

39 eneMeeara?

,IJ;HpHXJieOB npHH~ )J;HpJ1XJ1eOB npHHQHII je BeOMa KOpHCHO TBpljelbe 'Dlja j e q.mp MJ'.'Iarvrja cacBJrn

n · m + 1 npe~ern Tpeoa pacrrope;nmr y n Kyntja, omra ce HajMalhe y je,n;ttoj Kyrttj H Mopa HaJia3HTI1 Sap m +l rrpem.reT.

je,n;HocTaBHa: aKo

~

Hajje,n;uoCTaBHKje noCJie,n;HQe cy:

" y CKyrry o,n; TPH npttpOWia opoj a oap ,n;Ba cy .HCTe n apHOCTH; " y CBaKOj rpyrrK OA jep;attaecT OCOOa lhI1X meCT !>IOpajy foITH JICTOf IIOila n ca. 3ap;aTaK.

j

AKo

y IUKOJIK HMa 735 yqeeHJCa, OHAa ttajMalhe TPH yqemtKa CJJ.a.Be pol)enmrn ncror

,n;aua. 3amrn?

37

KBa.HrmfmxaTopu, nope.rr rroruqxux Be3HJfKa, urpajy BaJKHY ynory rrp11n11KoM .popM}'llamrje -;BpljeH>a Koja ce op;Hoce Ha crynoBe, rrocec5Ho KaAa je peq o beCKOHa'IHJrn CK}IIOBKMa

YHIIBep3a.""IHOM KBaHTIHpuxarnpy OAJ'OBapa peq ,,CBaKU" (c511no Koju, rrpon3Bo;i,aa). Ymmep3aJIHU KBattTHcpl1KaTop ce oc5eJie)l(aBa c11Mc5onoM V (obpHJTO A). £r3HCie~aJIIIOM KBaHTIIc}>mcaTOpy OAJ'OBapa peq ,,IIOCTOj11" (HeKU, 5ap jeAaH). £r3ucTe~jaJIHU KBaHnt:cpm
3 (obpHyTO E).

Ynorpec5y KBaHrncpl1KaTOpa Hajrrpe heMo J1JIYCTpoBaT11 pa3MaTpajyh11 peqeHmW Koje HJIC}" MaTeMaTWIKe IIpHpOAe.

Ilpniep I. IlocMarpajMo bUHaptty pen~jy ,,Bone·rn" Ha CKJIIY CBHX JbYAH - 03Haq11Mo je ca V. Axo

cy x

Ji y

HeKe ococ5e, OHAa V(x,y) 3HaqJ1 ,,X BOJIJ1 y':

3ammrn:Mo peqeHIDzy:

AKo MuKa aonu Majy, 011.ga u Maja aonu MuKy, yrroTpeboM

nornqxux orrepal.\Hja V(M11Ka, Maja) => V(Maja, M11Ka). PeqeHHIW KojoM ce TBPAH Aa CBaKa oco5a Y3Bpaha JbybaB ococ511 Koja je BOJIJ1 MO)l(eMo l13pa3l1TJ1 jeAHOCTaBHO IIOMohy yttl1Bep3aJIHOI' KBaHT11cp11KaTopa

(Vx)(Vy)(V(x,y) => V(y,x)).

y peqeHJ.fI\Jl ,,Maja BOJIYl HeKora" Kp11je ce

er3l1CTeHI~J1jaJIHJ1 KBaHT11cp11KaTop: ,,IIOCTO

eeKo Kora Maja BOJIYl': op;Hoceo

(3x) V(Maja,x). HaBOAJL'AO jom HeKOJlJ!lKO rrpHMepa: CBaKO BOJIYl HeKora - (Vx)(3y) V(x,y); HeKo BOJIYl CBaKora - (3x)(Vy) V(x,y); HeKo He BOJIYl HHKora - (3x)(Vy)--, V(x,y); CBaKO ee BOJIYl HeKora - (Vx)(3y)--, V(x,y).

+

3aii;aTa IlopeA peJiaJ.\Hje

v 113 11peTXOAHOI' 11p11Mepa IIOCMaTpajMo jorn ABe ococ511He 1bYAl1:

,,fot:Tii M)7IllKOr 11ona" 11 ,,c511T11)l(eHcKor11ona': HeKa M(x) 3Haq11 ,,x je MYIIIKor 11ona", a

Z(x) 3Haq11 ,,x je )l(eHcKor 11ona': IlpeBeAHTe Ha cp11cK11 je3HK 3a1111ce: _ (Vx)(M(x) => (3y)(Z(y) A V(x,y))); 2 (3x)(Z(x) A (Vy)(M(y) => V(y,x))); 3 (3x)(Z(x) A (Vy)(M(y) => --., V(x,y))); '±J (3x)(M(x) A (Vy)(Z(y) => V(x,y))).

38

..,

..- ...

-~;:::....··~

-

-,

:.; -.liill..,11,

1111

:I

Jlfaa KBaHTHcpHKaTopa o5aBe3HO cne,n;w rrpoMeHJbHBa Kaja Y3HMa Bpe,n;HOCTH H3 HeKor, yrnaBHOM, yttanpe,n; 3a,n;aTOr cKyna. l..JecTO ce y cpopMynaMa wcTwre H Taj cKyn, Tj. KOpHCTe ce TaK03BaHJ1 orpaHJ.fqeHH KBaHTHcpHKaTopw:

Vx E A - ,,3a cBaKH eJieMeHT x cKyna A" lf

3x E A - ,,rrocTOjH eneMeHT x CKyna A~

A

IIpnMep2. ).

IlpeBe,D;HMO HeKOJIHKO MaTeMaTJ.flIKMX TBp,ll;H>H Ha je3Jfl( cpopl'.ryna rrpm!eHOM KBaHTHcpHKaTopa. 38wp 8wno Kor npwpo,n;Hor 8poja H ttyne je,n;HaK je TOM 8pojy:

(Vx E N)(x+O=x). KBa,n;paT rnaKor peaJIHor 5poja je HeHeraTHBaH:

(Vx E JR)(.x2

~

O).

Je,n;HaqwHa x2-6x + 5 = 0 HMa pernelba y cKyny u;enwx 8pojeBa:

(3x E Z)(.x2-6x+5 =0). Je,n;HaqwHa x2 = 2 HeMa pernelba y cKyny pau;woHaJIHHX 8pojeBa:

•(3x E Q)(.x2 = 2). He nocTOjH peaJiaH 8poj KOjH je Betrn o,n; CBKX ocTaJIKX peaJIHKX 8pojeBa:

-,(3x E JR)(Vy E JR)(y < x). • IlpHMeTHMO ,n;a ce orpaH}fqeHJ!I KBaHTHcpKKaTopH Mory eJIJ1MJ1HJ1Cant CJieAeIDL\i 0

eKBHBaJieHu;wjaMa:

(Vx E A)F ~ Vx(x EA=> F)

(3x E A)F ~ 3x(x E AAF),

H

npw qeMy je F HeKa rrpoH3BOJbHa cpopM)'Jla.

Be3e Me9y KBaHTHcpHKaTopm.rn ,n;ajy TaK03BaHJI ,Ue MopraHOBH 3aKOHlf:

-,(Vx)F ~ (3x)• F w -,(3x)F ~ (Vx)• F.

IIpnMep3. IlpeMa ,Ue MoprattOBHM 3aKOHHMa 3a KBaHTHcpmraTope, Herarµtja peqemme ,,CBaKO BOJIH HeKora" je ,,HeKO He BOJIJ!I HHKOra":

-,(Vx)(3y) V(x,y) ~",a

~

(3x)(Vy)• V(x,y).

TaKo9e, npeMa HCTHM 3aKOHHMa, peqeHmi;a ,,Jep;Ha'Ilma x2 = 2 HeMa pemeJha

y cxyrry x je

pau;woHaJIHKX 8pojeBa" jecTe eKBHBaJieTHa peqeHYfl.l;K ,,3a csaim pammHCL"laH c5poj

.x2;t:2": -,(3x E Q)(x2=2)

~

(Vx E Q)x2=2.

PeqeHHu;a ,,He IIOCTOjH peaJiaH 8poj KOjH je Betrn o,n; CBHX OCTCLTIIX pea;nmx 8poieBa· wcTo TBP,n;H IIITO H peqemui;a _,, O,n; cBaKor peJIHor 8poja nocToj~1 sebn c5pot:

-,(3x E IR)(Vy E ::i)(y < x) ~ (Vx E JR)(3y E R)(x ~ y.

+

39

HnycrpyjMo, ea 1q>ajy, seo~rn 8m1CJ
IlpK11ep4. Ilol)n:MO OI.J; rrpenIOCTIIBKH:

1 HeKH rrpocpecopH cy MaTeMaTwiapH. ~ Csu MaTeMaTWiapJt

cy AObpH JbYAJf:.

Koja o;:i. rne;::retrn:x peqeirn:qa je HCIIpaBaH 3aKJbyqaK oBe ,n;Be rrpeTIIOCTaBKe?

A) CBJ1 rrpocpecopH cy AObpH JbYAJf:.

E) HeIOI rrpocpecopH cy AObpu lbJAH. B) HeKH rrpocpecopH HHCY AObpH JbYAff. IIpuMeTHMo Hajrrpe Aa ce AaTe rrpernocTaBKe o,n;Hoce Ha TPff BpcTe lhy,n;ff: • rrpocpecope - 03Haqlf:Mo ca P CKyn rrpocpecopa, MaTeMaTWiape - 03HatfHMo ca M cKyn MaTeMaTffqapa, • AObpe lbJAe - 03Ha'IHMo ca D cKyn AObpmc lbJAff, ff ;:i;a, Y3 OBe 03HaKe, rrpernocTaBKe K~ ,n;a je p

n M 7-0 ff ,n;a je M

IlocToje ca.Mo ABa Moryha OAffOCa CKynOBa P, M ff

~

D.

D ff OHff cy rrpffKa3aHff cne,n;etmM

BeHOBHM AJijarpa.MHMa.

D

IlpecpopMYJIHIIIBMO peqe~e A), E) ff B) Ha je3ffK cKynoBa:

A) P ~ D (CBff rrpocpecopff cy ,n;o8pff lbJAff.) E) P n D 7-0 (HeKH rrpocpecopff cy AObpff lhY,Il;ff.) B) P \ D 7- 0 (HeKH rrpocpecopff Hffcy ,n;o8pff lb JAff.) Y3~1Majylrn y ofoHp rrpeTXOAffe CJIJ1Ke, 3aKJbyqyjeMO ,n;a je,n;ffHO

p

n D 7-0 Mopa bffTff

JfCrryH>eHO. ,UaKJie, HcrrpasaH 3aKJbyqaK AaTHX rrpeTrrocTaBKff je peqettm.w E). HeKo he MO)l(,Il;a rrpoTecrnoBaTff 3bor yse,n;ettmc rrpeTrrocTaBKff rnp,n;ehff ,n;a OHe HffCY Taqee. Mel)yrHM, MK OBOM rrpmrnKOM He ,n;ffcKyTyjeMo o ffCTffHffTOCTff rrpeTrrocTaBKff, Ben 0 HCIIpaBHOCTff 3aKJbyql1Balba. HaBOAJf.MO jom ABa IIpffMepa 3aKJbyq>1Balha no ffCTffM rrpffHD;ffITffMa.

llpeiUuociUaBKa: HeKff rrapanenorpaMff cy KBap;paTff. IIpeiUuociUaBKa: CBff KBa,n;paTff ffMajy je,n;HaKe AHjaroHaJie. 3aKJby11aK: HeKff rrapanenorpaMff ffMajy jeAHaKe ,n;ffjaroHane. IIpeiUuociUaBKa: HeKff CJIOHOBff cy ITffHTBffHff. IIpeiUuociUaBKa: CBff IIHHTBHHff neTe. 3aKlbJ1-tllK: HeKH CJIOHOBff neTe.

+

40 " -

~~--~~~~~~-

'----

3ap;an;H Jlon.PIKH Be3HJIIJ;lf ·~ :-pe,n;M 3a Koje cy je,n;Ho1v11cppeHe rrpHpo,n;He 8pojeBe

r- S::7Ax>3;

A

2, x=1Vx2'.'.7; 3 •x2=4=:>•x=2; 5 x=1Vx=2<=>(x-l)(x-2)=0.

. 2'.'.3<=>x>3; -

x TaqHe crre,n;ene cpopMyrre:

cy MCKa3lll: p: Teno je 3,n;paBo; q: ,!J;yx je 3,n;paB. Harrm1rn peqemuw Koja o.n;roBapa lllCKa3y p ::::> q. - O,n;pe,n;lll rromqKy CTPYKTYPY peqeH~e JI.a 5u gyx 5uo 39pa8, goBOJbHO je ga .'...!.aTlf

iUeno

....; 5yge 39pa80. ~ I

p

J1cKa3He <}>opMyrre. TayT011on1je

q

T T T ; T

r I F

T T

T

l..

l. ! T

T

..1

l.

T T

T ! .1

4. O,;we,n;M 8ap je,n;Hy cpopM)71Y F qJl{ja je lllCTJffiHTOCHa Ta8IDfi\a ,n;aTa Ha Maprmm.

.1 .1

5 ~ieTO,n;oM CBol)elha Ha arrcyp,n; ,n;oK3.)l(JI{ ,n;a cy crre,n;ene cpopM)71e Tayrnrromje.

.l .l .l ' .l

3 opMMpaj lllCTlllHlllTOCHe Ta8~e cpopM)71a lll o.n;pew1 Koj e cy Tayrorrornj e. p Aq ::::> p; 2) q ::::> pV q; 3 pAq ::::> (p <=> q); -1 pV•q ::::> • (pAq); ::::> q) A (r ::::> q) ::::> q V •(p V r); 6 p V •q ::::> •(p A •r).

C 'XP

~ (pA•q

*

•r)

*

(pAr

::::>

q);

2) ((p <=> q)A(r * s))A(pAr)

*

T l ..l ..l I T

qVs.

r AKo cy A, B M C rrpolll3BOJbHe lllCKa3He cpopMyrre, ,n;oK
•(AAB) *•A V•B;

..l

2 (A* B)A (B ::::> C) * (A::::> C).

E

TayT011or11je H 3aKOHH 3aKJ1>yq1ma11>a ...,_ ,!J;arn je cpopMyrra (p ,n;aTOj cpopMyrrlll:

1) (pA•q)

*

r;

*

2) p

q) V r. Koja je o,n; crre,n;eillrx cpopM)'Jla JIOIWIKJ{ eKBJIBaJ"IeHTHa ::::>

(•q

*

r);

3 p

*

(•r

*

q)?

8. Kojl!l cy o,n; crre,n;eillrx lllCKa3a rrorwna1 eKBirnaneHnrn HCKa3Y " nt

BJ!Ol

y rrpaBy mrn

A

caM ja rry,n;"? A) AKo rn Tlll y rrpaBy, oH,n;a caM ja rry,n;; B) AKo caM ja rry,n;, oH,n;a rn HMClll y rrpaBy; B) Tl!l HlllClll y rrpaBy lll ja ID1CaM rry,n;; r) AKo ja HlllCaM rry,n;, oH,n;a Clll rn y rrpaBy; A. AKo ja HHCaM rry,n;, oH,n;a TH HlllClll y rrpaBy.

9. Pa3rOBapajy TPH ,n;pyra. IIepa: ,,Jla30, aKo ,n;o8Hjem rreT lll3 MaTeMaTlllKe, ,n;o8Hnern H rreT lll3 cplt3JIKe." MHKa: ,,Jla30, aKo ,n;o8Hjem neT Jt3 cpn3J1Ke, ,n;o8Hhern H rreT H3 MaTeMannce:' Jla3a: ,,CBaKaKo he 8HTl!l Ta'ffia H3jaBa 8ap jemior o,n; Bae ,n;Boj m.~e." ,!J;a JIH je Jla3HHa lll3jaBa Ta'ffia?

10. ,!J;OK3.)l(JI{ ,n;a je 3a CBaKJ!l rrpHpOAaH 8poj n Ta'ffia JrunJIJ1KaI.J;Jllj a:

E

2

aKO je n napa.H 8poj, mma j e n napaH 8poj.

'1 1

CKynoBH. OcHOBHe onepan;uje ca cKynoBHMa

A

l l . .UoK
(A U B)' =A' n B'. YuyiUciUBo. ITorneAaj npHMep 5. Ha cTpaHH 25. Koj e cy OA cJieAenJIIX cpopMyna TaqHe 3a forno Koje cKynoBe A H B?

: A 3 A 5) A l) A

kB=> A n B =A; k B => A \ B = 0 ; n B =0 =>A \ B =A; \ 0=0 => A =0;

2) An B =B =>A kB; 1 A k B => B \A = 0; A n 0 =A=> A =0; AU B=0 => A=0AB=0.

Ypel)eHH nap. )];eKapTOB npoH3BO)J; l . ,[(oK
(A \ B) xC=(AxC) \ (BxC) .

E

l:

Koj e cy OA cnegehJ!IX cpopMyna TaqHe 3a bHJIO Koje cKynoBe A H B?

A k B => AxA k BxB; 2 A k B => AxA k AxB; 3 A n B =0 => (A xA ) n (BxB) =0.

BHHapHe perran;Hje 1..... Pena1.vrjy ::::; cKyna JR qecTo nocMaTpaMo H Ha HeKJ1M nogcKynoBHMa OA R

A b

• Hal.{pTaj rpacp H cpopMHpaj Tabnmw penaQHje::::; Ha cKyny A= {l,2,3}. 2 H~pTaj rpacp H cpopMHpaj TabJIMQY penaQHje < Ha cKyny A= { 1,2,3}.

17. Ha CKyny S = {a,b,c,d,e } AecpHHHCaHa je bMHapHa penaQHja a rpacpoM KojM je rrpHKa3aH Ha MaprHHJ1. _ Pen~Hjy a npeACTaBM Kao nogcKyn OA SxS Hasol)elheM lheHMX eneMeHaTa. 2 ,[(a Jrn j e penaQMja a pecpneKCMBHa, HpecpneKCHBHa, CJ1MeTpJ1IqHa, aHTJIICJ1IMeTpJ1IqHa,JIJ1HeapHa,TpaH3JIITJIIBHa?

-

l..... I1pHK cKyna R • ··:_ ·1..,.. c - Bo. BHAM rrpMMep 2. Ha cTpaHH 28.

l S HaseAJ!I 8ap jeAaH npHMep cKyna H bMHapHe penaQMje Ha lheMy Koja je TpaH3JIITJIIBHa, aJIJII HHje pecpneKCJIIBHa HJIITJII je CJIIMeTpJ1qHa.

2

r

.

,,

..--

20. Ha rraprnnrnHoM cKyrry cKyrra rrpHpo,n;muc 8pojeBa P(N ) ,n;eqmHHCaHa je 8HHapHa perraQHja p Ha crre,n;ehH HaqJ.1H: A pBaKKoA n B=0. ,D;a JIJ.1 je perraQHja p peq)JieKCHBHa, Hpeq)JieKCHBHa, CliMeTpHqHa, aHTJ.1CJ1MeTpJ.1qHa, JIJ.1HeapHa,TpaH3J.1TJ.1BHa?

E

PeJian;Hje nopeTKa H eKBHBaJieHn;Hje OCT

21. O,n;pe,n;H 8ap je,n;aH rrernqrraH rro.n;cKyrr o,n; N Ha KOMe je pemU.\Hja ,n;eJbHBOCTH JIHHeapHo ypel)elbe. 22. HajMalbMM 8pojeM CTperrHQa ,n;orryttHTH rpacp rrpHKa3aH Ha CID!I.{J.1 ,n;eCHo TaKO ,n;a OH 8y,n;e rpacp perraQHje eKBHBarreHQHje Ha CKyrry {a,b,c,d,e }, a 3aTHM o,n;pe,n;H KJiace Te perraQHje. 23. Ha cKyrry rrpHpo,n;HHX 8pojeBa N ,n;ecpHHHCaHa je bHHapHa perraQMja p ea crre,n;el:rn: Haq11H: mp n aKKO m + n je rrapaH 8poj. ,lJ;oKa:>KH ,n;a je p perraQHja eKBHBaJieHQHje.

( 0~e•

•c

24. Ha cKyrry QeJIHX 8pojeBa Z ,n;ecpHH11caHa je 811HapHa perraQHja p Ha crre,n;ehH HaqHH: n aKKo ,lJ;oKa)l
mp

lml =lnl.

YJIKD;Hje 25. O,n;pe,n;H cBe cpyttKQHje J.13 cKyrra {1,2} y cKyn {l ,2,3} H cBe cpyttKQJfj e m { l,.2,3} y {1,2}. 26. yHKQHja 1: IR _.,,. l) l(O); 2) .f(-1);

IR ,n;ecpHHHcatta je ca j(x) = O,Sx - 2. O,n;pe,n;H: 3 j(0,5);

'!

j(--0,5);

5 j(f()) ); 15 flft.3 )).

A

27. yHKQHje f: {l,2,3,4} _.,,. {l,2,3} Hg: {l,2,3} _.,,. {l ,2,3} cy ,n;aTe ca

l = ( 1234) Hg= (123). O,n;pe,n;H KOMIT03HQHJe. g of, g o g. 2212

321

28. HaBe,n;H HeKa ,n;Ba cKyrra A H B H cpyttKQ11jy 113 A y B MJIH J.13 By A Koj a j e: 1) 1-1-yHKQHjaf: {l,2,3,4} _.,,. {a,b,c,d} ,n;aTa j e cal=(

123 4

b cad

,D;a JIH je OB a cpyttKQHja 8HjeKQaja? AKo jecTe, o,n;pe,n;H

).

B

1-'.

30. yttKQHje f: IR _.,,. IR Hg: R _.,,. R cy p;ecpmrncatte ca l(x) =2x-1 Hg(x) =-3x+ 1. 1) ,lJ;oKa:>KH ,n;a cy l Hg 8ajeKIµtje " O;:qJe;:p11-1 u g -1• 2) O,n;pe,n;H l o g, ,n;oKa:>KH ,n;a je OBa KOMII03unuja forjeKQMj a H o,n;pe,n;n 3) ,lJ;OKa:>KJ.1 je,u;rrnKOCT (j 0 g)-1 = g-1 0 1•

E

(j" g)-1•

1-

A.3

EneMeHTH KOMbHHaTopHKe 31. l) Ha KOIDIKO Ha'llHa ce MO)l(e ff3a8parn: rnmppa

(password)

o,n; rreT 3HaKoBa Kojy

'Ilme ca.Mo Mana rnoBa eHrnecKe a8eu;e,n;e (lhHX 26) ff u;ffcppe ,n;eKa,n;Hor c11cTeMa (IbHX 10) ? TipM TOMe IIIHcppe ce Mory cacTOjaTff ff Ca.MO o,n; CJIOBa ffJIJ1 caMO o,n; 8pojeBa, a 3H~ yttyrap IIIHcppe ce Mory rroHaBJhaTM. ~ Ha KOJIMKO HatfHHa ce MO)l(e ff3a8paTff IIIHcppa o,n; rreT 3HaKOBa TaKO ,n;a CBff 3Hau;J1

8y,n;y Mel)yco8Ho pa3nwrnrn?

32c Ha CJIMQM JieBo je IIpHKa3aHO rnecT rronyrrpaBHX ca 3aje,n;HJftIKJ1M rroqeTKOM. KoJIHKO KOHBeKCHHX yrnoBa o,n;pel)yjy oBe rronyrrpaBe? .:(~

Y rpy1rn

o,n;

10 oco8a HaJia3e ce HffHa ff An;a. Ha KOJIMKO Haq11Ha ce Mory ff3a8paTM

qeTl1pff oco8e, rro,n; yrnoBOM ,n;a aKo je ff3a8paHa H11Ha Mopa 8ffTff M3a8paH 11 Au;a?

KBaHTH4>HKaTopH

B

3.,. Koj e cy o,n; cn e,n;ehKx cpopM)71a TatfHe? : (Vx E N)(3y E N)(x::::; y); 2 (3x E Z )(3 · x=-2); 3) (Vx E Q)(3y E Q)(x · y= l); 4) (Vx E R)(x > 0 ~ -x < O); 5) {3x E Q)(x < O!\ .x2 =9); 6) (3x E R )(x > 3!\3x-l =0).

35 )J,aTa j e cpopM)'Jla • (3x E R )(x < 1 !\x > 2). Kaja je o,n; cne,n;en11x cpopMyna eKBMBaneHTHa ,n;aTOj cpopM)71ff? ~ (Vx E IR)(x > 1 !\x < 2); 3) (Vx E R)(x ~ 1Vx::::;2);

44

2) (3x E R )(x ~ 1 V x::::; 2); 4 \ (3x E R )(x < 1 V x > 2).

PEAJIHJ1 EPOJEBJ1 eijy

:Je~aJIHH 3anJ1CH 6pojeBa JI onepa.Q.Hje ca lhJrna ( 46)

A Ilep110,n;Hqm-1 H KOHaqHH ,n;eu;HMaJIHH 3armrn: 6pojeBa; ocHOBHe orrepau;Hje ca ,n;eu;HMaJIHHM 6pojeBHMa; yrroTpe6a KaJIKynaTopa; OCHOBHa CBOjCTBa orrepau;Hja ( 46) .:::Ie.DJIBOCT npJ1p0,!1;HHX JI n;eJDIX 6pojeBa (52)

A JJ;eJI?HBOCT rrpHpo,n;Hmc 6pojeBa (52) E EyKJIH,n;oB amopHTaM; pacTaBJbaJt.e rrpHpOAHHX 6pojeBa Ha rrpoCTe qHHHou;e (53)

B EpojeBHe 6a3e (56)

CKfII pan;J10HaJIHHX 6pojeBa (58) A IIepHo,n;HqHH ,n;eu;HMaJIHH 3amtCJI H pa3JIOMD;H; paqyH ca pasJIOMD;HMa (58)

CKfII peaJIHHX 6pojeBa (62) A EecKoHaqHH ,n;eu;HMaJIHH 3arn101 H Hpau;HoHaJIHH 6pojeBH; 6pojeBHa rrpaBa (62) E OcHOBHe oco6HHe orrepau;J1ja ca peaJIHHM 6pojem1Ma (66)

B Ilorne,n;Hu;e ornoBHHX ocoforna onepau;Hja H rropeTKa peaJIHHX 6poj eBa (67) AncoJiyTHa Bpe,a;HocT peaJIHor 6poja (69) A IIojaM arrconyTHe Bpe,qHOCTH (69) E OrnoBHe oco6HHe arrconyTHe Bpe,qHOCTH (70) Ilpu:6JIJ1:arne Bpe,n;HOCTif peaJIHHX 6pojeBa (71)

A IlpH6mo1rna Bpe,n;HOCT peaJIHor 6poja; rrpaBHJia 3aOKp)71ThH.BaH>a; arrconyTHa H penanrnHa rpemKa; orrepau;Hje ca rrpH6m DKIDrn Bpe,n;HOCTHMa (71)

Matematika sa zbirkom zadataka prvi razred gimnazije

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