Matematika sa zbirkom zadataka prvi razred gimnazije

He6ojma l1Kop;irnoB1tli

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

~

MaTeMaTHK.a Yr;r6eHUK ca 36upKo r 3JfX crpyqirnx mKOJia

IlpBO Jl3:ICUbe

Ayrnp: ;i,p He6ojma Jlfxo~momm Jfuycrpaqaje: ~.\H;lpej BojKoBJID, MnnaH llparojrroBJID oI-Orpaqmje: IIO.U:a.IUI y KOnoeIDI 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

2009. ro~me) u 3a Kpaj

cpe,n;Iher o6pa3oBalha (Kojw cy y rrpwrrpeMH). CTaH,n;ap,n;H rrpeJJ;CTaBJhajy cyrnnmCKa 3Ha.Iba, BeIIITime H yMelba Koje yqeHwn;w Tpe6a ,n;a rroce,n;yjy Ha Kpajy o,n;pel)eHor D;HKJIYCa o6pa3oBalba H nocraBJbeHH cy Ha TPH HHBOa rrocnrrttyha: OCHOBHM, cpe,n;H>H

Jt[

Harrpe,n;HH.

Ilwcalbe OBe KlbHre BOQeHO je H JIWilU1M y6e1)elbeM ayTopa ,n;a MJia,D;HM lhY,D;HMa yrnaBHOM He rrpe,n;cTaBJba rrpo6JieM illTa Tpe6a ,n;a Hayqe, Beh KaKO he TO IIOCTHhH. EpojHa HCTp(l)KHBalba y o6JiaCTH MeTO,ll;HKe HaCTaBe MaTeMaTHKe rroKa3yjy ,n;a cy MJia,n;H Jb}7v-1 cnoco6HM ,n;a Hayqe H pa3YMejy MHoro BHme o,n; oHora IIITO ce yrnaBHOM O'feKyje, }'KOJIHKO HM ce rrpe,n;CTaBH Ha O)l;rOBapajyhM Ha'Il1H. Beh ,n;pKe BpeMe, y CBeTy ce HacTaBHH rrporpaMH caMo o6orahyjy,

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

Jt[

IDIBOa o6pa30Ba.J:b3. yonnrre.

Y3HMajyhw y ofowp pa3JIHKe y ca,n;p)l(ajnMa noMeHYTJt:x nporpaMa Kao u TPJf mmoa 3Ha.J:b3. Koje he rrpe,n;Bwl)aTH cTaH,n;ap,n;w 3a Kpaj cpe,ll;Iber o6pa30B3.Iba, ca,n;p)l(ajH oBe KIDure

cy no~e.n.em1 Ha TPH mmoa - A,

E H B. ,ll;aTy rro,n;eny Tpe6a cXBaTHTH opwjeHTan;HoHo H qmeKrn6HJIHo, jep nocraB.lbalbe jacmrr rpamma Mel)y rrporpaMHMa

Jt[

HHBOHMa IIOCTurnyha HMTH je Moryhe HHTH je ,n;o6po 3a o6pa30BaH.e.

CyrecTwje H rrpwMe,n;6e qwTanan;a oBe KH>Hre YBeK cy ,n;o6po,n;omJie.

Ayrop

3

~

- KaKo f'IHTH H3 oBe KlbHre

YBOJ Y ITOMI.TPH)Y ..__ _ _ _ _ __,

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----

HacTaBHe TeMe cy o6pat)eHe y rroce6HHM maBaMa. HacnoBH ne~ja y OKBHpY TeMe HCTaKHJTH cy MaCHHM OIOBHMa. Mcrro,n; HacJioBa JieKIJ;Hje HaBe,n;eHo je KojHM HHBOMMa rroje.u;mrn H>eID1 ,n;enoBH rrpHrra,n;ajy. CBaKH HHBO nparn H ollJic op;roBapajyher ca,n;p)l(aja. KacHHje, y OKBHPY caMHX ne~ja, :HCTaKHJTH cy CaMO H:HBOH, ,Il;OK cy OIIHCH HJIH H30CTaBJbeH:H HJIJi cy Ha MaprHHaMa HCTaKHYTe IMiXOBe CKpaheHe Bep3Hje.

·--~mww~•- -

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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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Mapr~HaMa, aJIB Ji y rroce6HHM 6rroKOBH~a ,n;a~e cy

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BaJKHe HarroMeHe Koje rrpaTe OCHOBHH TeKCT, HCTaKffYTa cy HajBa>KIDt:ja rnpt)e1ba, Kao Ji CJIHKe KOje HJIYCTpyjy TeKCT H omucmaBajy cHaJiaJKelbe y KlhH3H. 3HaK 3a orracHocT (-rpoyrao ca Y3BWIID1KOM) rro3HBa caMo Ha rrojaqaHy IIaJKihY rrpHJIHKOM qJiTalba, KaKO 6Ji ce rrpe,n;yrrpe,n;HJie HeKe TJiIIMqHe He,n;oYM~e Ji HejacHohe.

Y rroce6HHM 6110KOBHMa ,n;aTH cy Heo6aBe3HH ca,n;p)l(ajH KOjH cy y ,ll;HpeKTHOj Be3H ca OCHOBHHM TeKCTOM. 0BH KpaTK:H TeKCTOBH ,ll;OHOCe HeKe MaTeMaTJiqKe 3aHHMJbJ1BOCTH, a IIOHeKa,n;a Ji ,n;eTaJbHHja Ji rrpen;H3HHja o6jalllbelha rrojMoBa H KOHn;errarn KOjH no ca,n;p)l(ajy rrpeBa3HJia3e OKBHpe rrporpaMa.

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llxBeapae aejep,Ha'llClre ca ~ow a.mrolln'Olll

~)l(aju KH>HTe cy rrogeJheHu Ha rpu mmoa - A. E 11 B. 3a CBaIGf op;elbaK mm ~eo OJ;eJh.Ka UCTalGIJT j e mrno y CI~

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Ca}W)l(JI Haj cno)Kemtje .nenoBe rpamrna. Cai:q>2Kaj n

-IS4T). •E"..-~

Aiia ..-- a--~ll-1>~

OBor ,n;ena Mory ce HHTeprrperuparn H Kao ca,ri;pxi.>J re[~-f

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cy 03Haqem1 ca o aacra.Bn.ajy ce Ha ,n;enoBe KOjH cy 03Haqem1 ca A u ca H>nMa rrpe;JCTa.Bmajy HOBY n;emray y OKBJtpy KH>JITe. ,[{eo B ce Ha;lOBe3}'j e Ha µ;erroBe 03HaqeHe ca A lf Ii n

...,, _,,__ ot _• ~

: ~opMyne. 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 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)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 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. •.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 rronffIKeMO 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-' 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 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(pVq); 3 p => (qVr);

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

T T T

•

E 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 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 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 KHKI.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

* 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} =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.mY ¢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 npmymcu:ttja h HCl3JIBa ce KOMII03HI.V·fja cp~Hja f Hg. • KoMH03HIJ;:uja cpymH 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 CIa 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 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 8m1CJeHO. ,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'.'.3x>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 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 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)