Yuav ua li cas mus piv rau cov zajlus kom kab zauv: muaj nuj nqi, kev cai thiab cov piv txwv

Niaj hnub no peb yuav kawm ua ke mus piv rau cov zajlus kom kab zauv, peb tau txais ntsib nrog qhov yooj yim cov kev cai thiab xyuas qhov tseeb rooj ntawm logic zog.

Yuav pib nrog, yog vim li cas no. Muaj koj puas tau pom yuav ua li cas tham? Thov nco ntsoov tias peb hais lus thiab tej yam yeej ib txwm raug cov kev cai ntawm logic. Nyob rau hauv thiaj li yuav paub lub sij hawm ntawm tej kev tshwm sim thiab tsis txhob raug daig, kawm yooj yim thiab meej cov kev cai ntawm logic. Lawv yuav pab koj tsis tau tsuas tau txais ib tug zoo qib nyob rau hauv computer science los yog xav tau ntau tshaj cov khoom nyob rau hauv lub unified lub xeev kev xeem, tab sis ua nyob rau hauv lub neej yog tsis random.

ua hauj lwm

Yuav kom kawm tau li cas mus piv rau logic kab zauv, koj yuav tsum paub:

• Yuav ua li cas nta tsis Boolean algebra;
• Yuav txo tau thiab hloov dua siab tshiab txoj cai kab zauv;
• qhov kev txiav txim ntawm kev ua hauj lwm.

Tam sim no peb saib nyob rau hauv cov teeb meem nyob rau hauv zoo meej. Peb pib nrog cov neeg ua haujlwm. Lawv yog cov zoo nkauj yooj yim yuav tsum nco ntsoov.

1. Tus thawj tshaj plaws peb nco ntsoov cov zajlus kom sib npaug, nyob rau hauv cov ntawv nyeem nws yog hu ua ib tug tau lub lag luam. Yog hais tias tus mob yog sau nyob rau hauv daim ntawv ntawm cov kev qhia, lub lag luam uas qhia los ntawm ib tus inverted lub zuam, npaug kos npe rau, los yog "&".
2. Cov neeg uas nquag siv zog - zajlus kom tas li ntawd los yog disjunction. Nws cim zuam los yog ntxiv rau kos npe.
3. Ib tug heev ib qho tseem ceeb feature yog lub negation los yog inversion. Nco ntsoov yuav ua li cas nyob rau hauv lub Lavxias teb sab lus koj raug rho tawm prefix. Graphically, lub inversion yog qhia los ntawm ib tug prefix ua ntej cov kev qhia, los yog cov kab rov tav kab saum toj no nws.
4. Cov zajlus kom tsim nyog tau (los yog implication) qhia los ntawm ib tug xub los ntawm tus nqi ntawm cov kev tshawb nrhiav. Yog hais tias peb xav txog lub lag luam los ntawm tus taw tes ntawm view ntawm Lavxias teb sab lus, nws sau raws nkaus Ii mus rau lub hom kab lus qauv: "yog hais tias ... ces ...".
5. Tom ntej no yog lub equivalence, uas yog denoted los ntawm ob-txoj kev xub. Nyob rau hauv Lavxias teb sab, lub lag luam yog raws li nram no: "tsuas yog".
6. Sheffer stroke cais ob tug kab zauv ntawm lub ntsug bar.
7. Pierce xub, piv Sheffer stroke, shares qhia ntsug xub taw downwards.

Nco ntsoov kom nco ntsoov tias cov neeg ua haujlwm yuav tsum tau ua nyob rau hauv nruj ib theem zuj zus: negation, lej khu, tsis tas li ntawd, thiaj li, tus equivalence. Rau kev ua hauj lwm "Sheffer stroke" thiab "cov zajlus kom tsis" muaj yog tsis muaj txoj cai ua ntej. Yog li ntawd, lawv yuav tsum tau ua nyob rau hauv qhov kev txiav txim nyob rau hauv uas lawv sawv ntsug nyob rau hauv ib tug complex qhia.

qhov tseeb cov lus

Piv cov Boolean qhia thiab txua qhov tseeb lub rooj rau nws ntxiv kev txiav txim siab yog tsis yooj yim sua yam uas tsis tau paub txog lub rooj ntawm yooj yim ua hauj lwm. Tam sim no peb muaj kom tau raws li nrog lawv. Nco ntsoov tias qhov tseem ceeb yuav tau yog ib tug yeej muaj tseeb los yog cuav nqi.

Rau cov tau ntawm lub rooj yog raws li nram no:

Rooj disjunction lag luam rau:

negation:

qhov tsim nyog tau:

equivalence:

Barcode Schiffer:

Pierce xub:

simplification ntawm cov kev cai

Nyob rau cov nqe lus nug ntawm yuav ua li cas mus piv rau logic kab zauv nyob rau hauv computer science, yuav pab peb nrhiav tau cov lus teb yooj yooj yim thiab meej cov kev cai ntawm logic.

Peb pib nrog cov nyuaj kev cai lij choj ntawm contradiction. Yog hais tias peb muab cov lus rov tswv yim (A thiab Nea), ces peb tau txais ib cov lus dag. Nyob rau hauv cov ntaub ntawv ntawm tas li ntawd ntawm opposite tswv yim, peb tau txais qhov tseeb, raws li txoj cai yog hu ua "cov kev cai lij choj ntawm lub cais nruab nrab." Feem ntau nyob rau hauv Boolean algebra muaj kab zauv nrog ib tug muab ob npaug rau negation (tsis Nea), ces peb tau txais ib lo lus teb A. Tseem muaj ob los ntawm txoj cai ntawm de Morgan:

• yog hais tias peb muaj lub negation ntawm cov zajlus kom tas li ntawd, peb muab tau cov npaug ntawm ob kab zauv nrog ib tug inversion (tsis (A + B) = * Nea Neuve);
• zoo xws li cov ua, thiab lub thib ob txoj cai, peb noj tsis kam ntawm kev ua lej khu, peb tau txais mus ntxiv ob qhov tseem ceeb rau lub inversion.

Heev heev Individuals, tib tus nqi (A los yog B) tsim los khoo ua ke. Nyob rau hauv cov ntaub ntawv no, raws li txoj cai ntawm repetition (= A * A + B los yog A = B). Muaj cai thiab acquisitions:

• A + (A * B) = A;
• A * (A + B) = A;
• A * (Hea + B) = A * B.

Muaj ob tsevneeg cai:

• (A * B) + (A * B) = A;
• (A + B) * (A + B) = A.

Piv zajlus kom kab zauv yog ib qho yooj yim yog tias koj paub cov kev cai ntawm Boolean algebra. Txhua yam uas sau tseg nyob rau hauv qhov seem ntawm tus kev cai lij choj cov khoom yuav raug kuaj empirically. Rau lub hom phiaj no peb qhib cov nkhaus raws li cov kev cai ntawm kev kawm txog zauv.

QAUV 1

Peb tau kawm txog tag nrho cov nta ntawm simplifying zajlus kom kab zauv, nws yog tam sim no tsim nyog los sau cov tshiab kev txawj ntse rau hauv kev xyaum. Peb xav kom koj ua kom tawm ua ke peb piv txwv los ntawm lub tsev kawm ntawv qhov kev pab cuam thiab daim pib ntawm lub unified lub xeev ntsuam xyuas.

Nyob rau hauv thawj Piv txwv li, peb yuav tsum tau mus piv rau qhov qhia: (P * E) + (C * nws). Ua ntej, peb tig peb lub xim rau lub fact tias nyob rau hauv ob lub thawj thiab thib ob nkhaus muaj tib lub variables nrog muaj yuav ua rau nws tawm ntawm lub nkhaus. Tom qab peb tau ua los ntawm manipulating cov kev qhia: C * (E + nws). Nyuam qhuav pib peb ntsia txoj cai ntawm lub cais nruab nrab, thov nws nrog hwm rau kev qhia. Tom qab nws, peb yuav hais tias E + = 1 nws yog yog li ntawd peb qhia yuav siv sij hawm rau hauv daim ntawv: C * 1. Cov uas ua kev qhia, peb yeej tseem yuav yooj yim zog los ntawm kev paub txog hais tias C 1 = C *.

QAUV 2

Peb tom ntej no hauj lwm yuav tsum: dab tsi yog tseem muaj ib tug yooj yim to taub Boolean qhia yog tsis (C + nws) tsis + (C + E) + C * E?

Thov nco ntsoov nyob rau hauv qhov kev piv txwv yog lub negation ntawm complex kab zauv, qhov no yuav tsum tau tshem ntawm, ua los ntawm txoj cai ntawm De Morgan. Thov lawv, peb muab tau cov nram qab no qhia: * E + Nes Nes * nws + C * E. Ib zaug ntxiv peb pom lub repetition ntawm ib tug nce mus nce los nyob rau hauv ob nqe lus, kom nws tawm ntawm lub nkhaus: HEC * (E + nws) + C * E. Ib zaug ntxiv, thov lub Cais Act: HEC * 1 + C * E. Peb rov qab hais tias cov lus "Nes * 1" sib npaug Nes: Nes + C * E. Peb kuj muaj siv distributive kev cai lij choj: (HEC + C) * (HEC + E). Peb thov cov kev cai lij choj ntawm lub cais nruab nrab: HEC + E.

QAUV 3

Koj twb pom tias yog tiag heev yooj yim mus piv rau lub Boolean qhia. Piv txwv li №3 yuav pleev xim rau nrog tsawg nthuav dav, sim ua koj tus kheej.

Piv qhov qhia: (D + E) * (D + F).

1. D * D + D * F + E * D + E * F;
2. D + D * F + E * D + E * F;
3. D * (1 + F) + E * D + E * F;
4. D + E * D + E * F;
5. D * (1 + E) + E * F;
6. D + E * F.

Raws li koj tau pom, yog tias koj paub cov kev cai ntawm simplifying complex zajlus kom kab zauv, ces txoj hauj lwm no yeej yuav tsis ua rau koj cov teeb meem.