Tron~ uie ~ll' M; mach do, Inl11Q Iluii d6m! lllaeh Cltd kho,i Itr(jll~ li!le: \(~~i ~_ , ' ~ L, • L gia Iri 1: \',1 1r<.ll1g Ih;il ngat mach tHong lrng \'l1i glei In O. Theo c,i.ch quy d!lIh gi,i trj IlhH Irl:ll, ph~p tmin tU)'(:1I e6 Ih~ duq'c hiC::u dl\':n nhu doall m,lch tn.ie ~t1ng ,~()ng cae ki1O,[, trong khi d6 ph~p toan h()i ~e tu'o'ng ling \'6'i dO,lll Ille.tell mill' nt)i ti(p dc kho,i. Vl'"1i «ic qlly Ul'"1C d6. tien d~ hai ella dai ~6 Boo] C() the: dUne bi~u chcn thitnh dc me.tch wong dunng nilu "au: aT b. c (a+b),(a' cJ IIlnh .2.2 SCi rl() mach dtlllg !lgal 1ll111h hoa dlO linn chXI ph:1Il plHil S~r tuong dU'ung gili'<l hai sO' do lmwh khmi nilY co th~ duqc kicl11 c!ll'rllg h,ing cac h,ing chan I)' tuong ung. Dl!<l \,:\0 ~~r tlro'ug thlch giCi'a dc bieu Ihuc iClgie \'01 dc S0 (1<) mach khmi ta co Ihe t'.IO ra dc m',lch khoa tuong duong de hiel! thCre \()glC \'t1'i nhlt'llg dC) rhll'c qlp kh,\c nhall dl/a \"ilO c[te pher hl(':'n dC;i trong m,.tcll di~n, I\-'lac lItl \,,~y eo m()t so 1m,Ii m'.leh kh(mg the hi~u dicn dU"l,K 1!J;l11h h,'t hop giCi'<l de t() h(.ip mach ~()ng song \'il n6i li':p, troog nhu'ng tIlJ'i'.ing hnp (kl la ph,ii x{ry dtfll~ mach dl,fa theo h,ing ch,tn Iy hm~e su dt.mg pi1Llnng rhiip li.IO t')\1 h(,1P c[ic h('ll kel. Chllllg ta xci tnro-ng h0P m',lch theo lien kcI lll'll'h crill: de (tLfl1!lg (1i Ill' ph:in phia Ir,ii 111<,leh "ang pilfin phla phiil 11l'.leh bao g()1l1 uic dtrong (ab, aed. ecb. cd \.Phll'(mg rh{tp t'.IO t~\P hop de lien ke't the hi¢n nhu' "au: nell Ircn bilt e d IIlnh 2.3 /l.t<1(11 du. 1<5' du'o-og di IiI' pll<i,n ben Ir[ii maeh du , ,<log philO ben ph'l! ta dClng I{II cit de kho,i, kill do mach dien se U trall" thai MJll!.! ebn .• . ~ L' trang tru'o-ng Iwp trcn moi duo'ng C('l it nhAt I11Qt khOii 111(1 thl 11I' lch s0 l1 t . 'l~ th,-ii mil'. Vi d~L ne'u trcn ducil1g (lcd ta dt'lllg 1:\1 c:t de khOii a, c, d m~lCh se d tr~\llg thai dl'lng. Nhu \'~\y d6i \'6i nli,leh Irtn, bieu thCl'e IClgic tLfO'llg c1uong ~C Ia: 21 a.b+a.e .d+c.e. h+c.d Khi SlJ' dl.m,!! 1~' lhuy0'l chuyc'n mach Irong hieL! dien die hi~u thClC iClgic, de dinh 15' ella d, \i ,,6 chlly~n mi tell C() th~ rut ra ilf elc lien de. nguyen \)' d6i ng:hl etla I)' thuyc'\ mach uj \'<li \r() kllti quail In~ng. ]\;<ru\'t'n 1\' dlii 11(l~u: JEW)'ell 1<' d6i ngilU dua Ircll cu w'5 X[tV dIm" uk '"'. • '"' L • L. ., _ c bi~u thCre d6i ng'\u. D{)i llg:1ll eLla llH)1 bi2u th{t'c di.1i s6 Bool 1:1 IlH)t hi2u lIllIe ](\gic lrong d6 de b16n u elta bic'u thli'C ban (Ull duqc thay him,!! (/"."J' lhay bimg '(r, '(i' thay hllllg 'I'. ph':p tuy01l chroe lhay btlllg phcp Il()!, phcp h(ll lhH/C Ihay him,!! pllcp tuyen. Kill th~l!lh hlp bicll thCre d6i ngilu la piJ.'ti su' dung de d:iu ngo(1C de d.ltn IXlO Irat II! tinh 1O in bieu there. Nguyen 15' dui ng{\ll: gi;i Lrj eLla 1l1l)t hieu 111l1-c (h.1l s6 Bool bllllg bll ella gia Ir! bbieu thlic doi ng,\u Wong u'ng. Ta e6 lh~ chung minh nguyen I)' dCii ngtIU htlllg plu(o'ng phiip quy m,lp tmin hqc va Sli ch;Jllg cae djnh I)' de Morgan. 1. ('{u: h:'U11 Wgic \:1 khlii n-chieu HilIl1 16gie /I bi~ll throe djnh nghTa la anI! xa k.h6ng gian I/-cl!icu B" \'ao II: B" > II il={O,l) (2.16) t<cu x; li\ phun Itt Ihu9C B. khi do x=(x/"",xI/J Iii m91 \'ccto ella kh6ng gian veeto' /I chicu nil. Hum logic trell kh6ng gian n" ({j Ihe duqe viet dU'{l'j dang .J(r/ r"J. KhCmg gian veeto' un eh(ra 2/1 diem \'a m(lt yceW x Ihu(K kh6ng glan B" chtQ'e hieu di~n iii mot trang s6 2" diem clla kh6ng gian 1/ ehi~ll nil. C{IC diem nay e6 the' dUQ'e d~lt {ltong (rng \'oi cae dinh eua llH)t hlnh kh6i e6 2" dinh trong kh6ng gian B". HilIll logie.!L\·/ . x,,.! SC ct~lt tuang ling m6i dinh clla hlnh kh6i \'(.'~i cae gia trj () hO(lc I. 22 rrong llu11 \,lfe I111e'l k~ clie m'.Kh ~6 khllllg nhil't thle-L ph' lI cli,ll ~)_I' l Ln () ho:)c J eho 1:11 e<t Cite dinh ella khoi - dIe \"l'elO' x. i\NI dch ehillh de IHin. 1121111 \(lglcj"dude djnh Ill:!hTa Ul ,Inh).a eua t(tp con A cLla kh(lllg t'lan B" \'ito B. r A c;; 13" + B B={O,l) (.~.17) Khi la x01 m(lt 1(lp con Ac Bn \;1 kim f ct'l l111en x,le dill!. 1;1 A.llhLl'Ill:! diem IhU(lC W \,;1 kh6ng thu(lc A l XE un \ A } 121 lilllh ta kh0ng quail 1~1111 L6i \'il h.~, hi~u 121 l/inh-d; Cite dinh Ihu()e ,\ d. IU'l~lIg ling \'('jl de gi<,i Ifi () ho:1C J se JlIrrc gqi Ian IUCft lit i/illlr-O \,,\ d//lII-I. Vdi Ciich h.~' Im;u UIC dlllh ella hinh kh6i 2" (tinh nhu' \'ZI)'. lll()\ hill11 16giefsc ,inh X'.I de di(:llltrOlII; khll11g glan B" \';\0 I~IP h(,1"p /().J.d)· I(x) : il"-> (O.l.dl 121N) '[\\?U 1l1l)1 dinh ella hinh kh6i II-ehieu bi':u (hell du6i d~lllg !lieu thCre I(ltlie. la nh(\I1 dLrqc nn)1 bi':u thCre h()i. VI dl,l trong khong gian 3-ehicll. dinh Olls0 tlXong Crng \'(ii bi':u Ihl'rc 16gie XI_\"_'X;, Bi(:L1 thCfe Il)glC e6 Ihe nh~11l ctWK ILl' cae dinh cLld hlllh kh6i 2" cillCU kll1g deh gbi .\, llC'U gi<.'t Ir! eua 10,.1 d() thu- i wong ling 1i\ . r hO[le x, lle'u gi,\ 11'\ 10,1 d(l tlHi' I 1:1 'if. Bi~u thtre h()1 chi nhtll1 gl<l Ir\ '1' \<.Ii m()t diem x du)' J1h~it. I\llll' \'il)' 10,.1 do- ella dinh s0 tu'O'llg Ling w1i n1lit phfin ILl' Irong bi~u Ihlk h()j. \'l()t deh I()ng qual La cllra \,;\0 k.h,ii nil:lll kh6'i e,fp III nhl1 Sillt: M(ll \'ccto' " chicu C:: le/ . i't}. c,.::{O.l.xf. "<II dul,l"c gOl 1;1 "I/lii" \'~\ m(,)1 kIll)/" ClllLI III ~Y'Il.r \- ~c gqi 1;\ Utili' nil} III. Vfdu: (01 I) !akh6ictlpO: (Olx)-kh6ieiip I: Kh;ll nl~'m I.fili'j EI h'lng qual h6a eLla klHii IHcm diuh eua kh6i lI-elllcLI. Kh//i nij) //I 1:1 kh6i cnn III ehlcLI co ch(ra 2'" lImh. Vi d\1 kh6i (0/.\) ehtra hai dmh (O/()) ";1 (OJ j). Noi m()t deh kh;ic k~' It!.r ('6 Ih': nktn gi,i II"! '()' 110.)c '1'. Mi)l kh6i ecip rn Il(Ong ling \'(11 bi(:'u link hl')i L'ln'ril 1/-111 blCil. Blc'u link h(li n21),' e6 th': nh:111 (hf(.K nell ghi x, khi gl;1 If! Ililing trng ella 19<1 d(l thCr i 1;1 '()' \'~\ .\', khl gi,i Ir\ tqa d() d6 EI '1' Lmng dt'J \'\ tIi cua k) 11.1 l kh(lIlg clune tinh dell. 1\, t\ deh kh;lc ky I~r .r c6 Ihe nh'Jn gi;l tf! '{)'.' r \'21 k.h6i dp 11/ ,0 lUring dUO'll::; \'6'i bi0'u Ilnlc i6gie gum (II-ill) bic'n . .\1()1 k116i dp () sc 1l1O'ng Ling \'6i dinh cluy nh{11 cua kll6i II-ehicu, kh6i dip 11/ ,'it: biel! elien 2'" dinh, nhlr \'[IY m()1 khoi dp til sc bao pill! 2'" kh6i b[lC o. M{)I dch 16ng quat. v6i hai !flp I1qp cae khoi C I V~\ C 2 • nc'u I~p hQ'p dinh 0,\0 phll 1116i kh6i Ihu(lc nhom (' I 1~1 1(lp con ella ItlP hop !.tinh hao phll de kh6i eLla C 2 , khi d6 [a nOI r;'mg C 2 bao phu C I • 2. Cae d~ng chU<ln tile eua ham 16gic T\'cu h;\111 i(lgie.l khLing ehCra ll/o/l-d, khi do dC:' xae dinh ham la din dlra ra I<)p h(.~p de d/II/I-I V;'I dilllr-O. Do kh6ng ehu:a cal' d/llh-d ncn 1(lp hop dc d/nh-I(\ "l{)) Y~I !(Ip hl)"P dc d/nh-O (Vi!)) htl nhau. \'1(11 dinh (" (kh6i cAp ()) !Lru"llg CfIlg y(ri m()t bieu [hCre hOI F(cj. khi d6 h:ll11 .f(x) sc dU(iC hi~u (hen lh6ng qua I(lp hop de dr'IIIi-1 \! /f) nhtl" -;au: H\) = V Pic) (2.19) P(c) gqi la bi~u thCre h(li ct.rc tieu hay ng{m gqll l~\ tfeh eLfc lieu. Nlur [r~n oJ. de c)p 1(1i, m()[ kh6i eAp m d~\i di~n eho 2'" dinh. hiju thCl"e h(li ct.re li(:u : i: luang (rng \'6i -;6 Iwmg cuc li2u de dillh Iham gia \,;)0 hi~u Ih(rc h(li V~I st:: nh,}n gi,i Iri 1 1'.Ii nhil'ng dinh nay. Ricu thCre (2.19) IiI hieu thu:c tuyen ella de hieu Ihue h(li eve [iell \"a c!Lrq'e gQi Ja d' mg ehu,ln I,k luycn ella hi0u Ih(rc logic. Ta x5e dinh hi:tnl hCI I(lgie ( goi t<"11 I~I ha111 Otl )eu<l m(l[ ham f I~I m(ll hi:lln nh,:111 gi<.l Iri '()' 1<.li nhung dillli 1l1~1 f nhtlll gi,i Ir! 'I' Y;I nh,-}n gia lri 'J' I'.li nl11.1ng dinh lll~lfllh,}n gi5 tri '()'; h~\ln nilY chrqe ky- hi~u h,\ng I" - I 1\) ~ 1 ffJ/lx) 12.20) Ta ell the nh,}n dUQ'C h;Ull I neu tlMY V1(P(e) hitng V1,(P(c») lrong bi~u therc (2.19). I(x) = v P(c) (2.21 ) TLf do ta co the nh,)n dWJC hietl thefC eua J(x) bflllg d.ch ,ip dl.lIlg cong 1111.re: I~ I . j(.\) = /\ S(c) (2.22) eci ;,(,fJ Oil' bi2'u thlre S(c) nh(1ll dUQ'C tu pre) h,\ng lhay X J Ih~lIlh x J ' x, th;lllh \ \,;:1 thay phcp h(li (/\) thanh phcp tuyen (v). Ta nh(m IhAy S(C) sc tuung (rug J' -" vO'i (]" -1) dinh ngo~li trLL' dinh tuong ung voi Pic) \'~I duqc gqi Ii bicu tlltrc tuy~n c~rc d'.li. Ctch bicu e1ien hiun logic fthong qua pIH~P h,?i cua d.c bi6u thtTC tuycn qrc d'.li g<.li EI d"ll1g chui;n lAc h()i. Tiep Iheo de co the kh,io s:it l11¢t d'.lng chu,1n lftc nif;.l, chung ta dinh I1ghla h~lm lo,~i trir 16gic XOR nhu' sau: XOR la phep 101m bai ng6i cho gia Iri 'J' nC:u chi l11(lt Hong hai IOcln hang llh~ln ,~ia Ir! 'I' va nh,:m gia Ir! '()' lrollg nhiIng tnri:mg hop C('H\ lai. xEBy= XyA.\.)' 0.23) Trong bi~'Ll Ihtrc trl:n til thft)' n(u x ho;\c )' lu6n nh,)ll gi,i tq Ihi blc'u tlILk :-;0 nhfm gla Irj tu'ong ((JIg theo )' hO;:lc theo x, co nghlu 1;1 I EB.r::::: x. Them vilO <.16 phep tmln XOR EB, cling gi6ng nhu phep u)ng. til(Sa ll1iln Illlh ch;'il giao ho;.tn, kGt IH)'p \'~l phfll1 piloi va-i phep I1h,ln. DL.ra \,~IO cae tinh eh,-lt Iren la co thc dlIa ra d',\Ilg chuftn tftc Iheo phcp ta,in XOR nlILr suu. M()t hilln 16gic hfll k5 1 e6 the (hrq'c hicu dicn theo h¢ therc sau: ./Ix}. '\c' x, ! :::::.J(x}, x" X •. }. (), _I',+}, .• X.,) x, +/(\'} . .r 2 • •. r,. [. (J. '\.1' . x)x, cer tiep It,lC nlur \'(1)' ta sc nh':.in dl!'()'C: /(.1'" Xc' x".J -:::::. RO . () j.Y I Xc x., vI( I. (). ". Oi XIX, XII V v 1(1,1, ,1),,,,,,, x". (2.241 Mi1i s() IWng Irong bi(:u th(fC Iren IiI m(lt bieu th(rc IH)i Cl.rc IICU do d6 chi co th0 co 1l1l?1 ,tinh nhfll1 gi,i If'! • j' d6i \'('ii tifng h() gl[l In clla _\,._L, ,_1'/" Thay ph~p loan v btmg phcp toan EB V~l \'Cii mbi x lhay bang tl\l:'u tlllJ-c (It:£; \',) ta nh(lll dU\K hicu IhlfC 16gic ILf(ing (hwng chi chtJ'a phcp h~li \'~\ phcp Im.li Irtf l6gic. j(,,,, .c,., x,) =j((J, ,(J)(lCBx,)(lffix.) (lffix,,1 ffi /!I, (), . ., 0) xl(lEBxc) (lEBx,,) @ I(£)/(1.1" I)X I _I,' \·". (1.25) M6 UtC dtiu [\go~\e vi] ,ip dl.ll1g cac tfnil chilt k6t h{)"p, philn ph()i \'(ii phcp IH)i. Iu nh(ll1 dLIQ'C: '/(1'1 . .r~, x,,) :::::U n EB (I/xiEB l,-~r! EB '" EEl lInX., CB (/I.'X}X! CB (lUXIX" EB EB 1I}.,.\"1.\'" EB lI]J.\,_r, EB EB lI rH _}J,,\'I/'}.\''' EB (2.26 ) eic h¢ s() II", (/1, ''', (II:. "co thl~' tim duae h'-tng deh tLrtlll,';! lrn~ ute bicu thLK C.25) \,<'1. (2.26). t)c bic'u (iien cAe he so a, til' U 1_' I, \a dlfa \'~to kh{li nl~!l1 vi ph,ln clia ham logiC, Ta dinh llghl<l vi ph:m logic I~I biC:lI thue: 0,27) Do ph~p lO,ill ffi hleU di~n dong thai phcp lAy tonf! ":1 pilcp 1(1)' hi~lI nen hiC:u thut tren C(')I1 goi \;1 sai ph[m \6gic, Vi rh,ln 16gie e6 cae t[nh eh,Yt eua toan Itr tuyen tinh. Them vao d6 kct qu,:.i eua phep Imin l:iy vi ph;lrI th~o hie'n .\, clla ll1<lt lich logic sc bfmg '()' n~u bien x, khong Iham gia vilo tfeh d ,,2 hhllg ehinh Ilch logiC IO~li Irtf tli x, ne'u.r, Iham gia ":1.0 bien Ihve. Vidu: o X 1 X:.X,X 1 ==X 1 X 1 1.\'-1 ffiXI,\)h:1 =X 1 X,'\1 U\', ?' x 1 x:>x,x-1 ==x1x:,x.;"'"-I ffiXIX:>X;XI :=-0 C'(, SII' dl,mg k.h,ii ni(:111 vi phtm 16gie, de h¢ so (/",,, , tI", Irong d6 lX, E[l "n] ella hie'u Ihue (2,26) sc dlrqe "iC:I (hr6i d<',lt1g I=l m , (:(, E[l.""n] (2.2~) Tit bie:'u Ihve n~ly suy ra h¢ s6 qXI(X, (t", hiing I()ng 10i,li tru logic eLla de gia tr! 1',11 de dinh baD phu m(lt khoi til ehieu, Kh6i tIl-chicu n;ly clltia.\ I~.ll e{le \'j Iri wong ung \,(ji (I, va '()' h,li Irtt ea d.c "i tr{ con 1<,li. 20 § 2.3. Toi thicil twa de bi(:u thlll' hJgic :"J(l] dung ella Il1UC ni:'y hilll ltfi CtC pinfo'ng phap 16i tlHCU bo,1 de bl~u Ihlie h'lgic CO' b,in khi Ihie'l ki':' c,i.c m;.lch ~6. Vi~c 16i 11Ii":u hoa CtC bicu IhL"I'C h'lgie l(lIl1 de hi":u Ihue u6 tr/l' nen don gl<-ln hGn. Dicu d61all1 g]~lIll bell thuoe \'i\ tiing hi0u Jl<ing lu,.teh ( Irell phuGng di~n Ih6i gi:Jn In~ \'01 thi)"i gian th~fe hi¢n ) dW)"e I(;ng IH/p. NgOi)J ra d(}i \'6i \"i¢e ph;11 hi¢ll 10J d tlill' nghi¢m tillel boY. nhiJng ll1,.\Ch dllg (ton gielll eho phcp gi,11l1 IhlJj gian t1m de tl() gi:t Iri Ihir nghi¢m \'il giellll Ihb"i gJan ph;il hien i/il. Do d() hlli loan t6i IhicLl hOel dc blCU Illl'I'c 16gJC d6ng \'ai In) quan Irong trollg qua Irltlh \r)ng hqp \'il [eli Lru m:.lch. Cho h~lln 16gJC II hienfrx l , .r" x,,) bicu dien dU(l'i di,lllg ellu,"in [i\C [uYCtl (t6ng dlc 1feh i(lgIC). T6i thicll hO:1 s6 ilf</ng eile lfch logic clla hll1l1{ \,;1 s6 Iwmg de hie'n logic lrong lung tich 16gJC dan t6i \'i~c Ull11 giellll gici Ih:lIlh the ch6 1',10 mi.lelI: so luqng c<.\c phtin tlr co' S(1, s6 [If(;mg celc (f{\u \"ilO ella lll'.Kh \,;1 d.C Jail vao Cll:J cae phtin Ilr mach. di¢n Ifeh ella mach. gi;'ul1 thbi gi,m kiclll nghi~l1l m'.leh. H;un I duVe bl~U dicn blri 1(lp hop cae d/llh-/ V I (f) "ll t,-!P hqp de IIIIIII-d Vi/), GiiJa Gie Ifeh qre tiC"u ella hilIll (Yll de kh6i e6 m6i lUling quan m(lI m(lt: kh6i cAp III SC luong ung \'(Ji nH)t lieh It'lgle (hJeu IhLIe h(li) g{')lll (II-III) Im':'I\. D!nh nghli.l tic" t6i giall : khlii c dLH!C gQi 1:1 Ifeh uii gi,11l eLJd !J;ul1In\Yu: T~lp hqp V(c) ella de dinh ( kh6i h;k 0 l. Vic) n V,ifI" 0: Vic) c V,(O U ViI): :::3 khoi c': V(C') c V1(O u V,,(f) \';1 V(e) (;; V(e'). Vi du: Gi,-i slr ta co ho1l11.f(x"x 1 .·i',i) e6 t~IP hqp dinh Vo(j), V M) \,;'1 V,M). j c o ~ 1 o 1/ d { L{ Vo(t) ~ : (0.0.0).11.0.11) J V I (I) ~ : (0.1.0).(0.1.1 ).( 0.0.1 ).1 1.1.0)J V 2 (1) ~ (1.0.1).(1.1.1)J IIlllh 2.5 Lie tich tlii gi,ill (,r Lr) \ il (,u 1 J. 27 . ehCra ll/o/l-d, khi do dC:' xae dinh ham la din dlra ra I<)p h(.~p de d/II/I-I V;'I dilllr-O. Do kh6ng ehu:a cal' d/llh-d ncn 1(lp hop dc d/nh-I( "l{)). X 1 X:.X,X 1 ==X 1 X 1 1.&apos ;-1 ffiXI,)h:1 =X 1 X,'1 U', ?' x 1 x:>x,x-1 ==x1x:,x.;"'"-I ffiXIX:>X;XI : =-0 C'(, SII' dl,mg k.h,ii. hi~u 121 l/inh-d; Cite dinh Ihu()e , d. IU'l~lIg ling '('jl de gi<,i Ifi () ho:1C J se JlIrrc gqi Ian IUCft lit i/illlr-O ,, d//lII-I. Vdi Ciich h.~'