Aljibhra Izimiso eziphusile. Aljibhra logic. Elements logic zezibalo

Kuleli zwe lanamuhla thina ziya ziba mandla ekusebenziseni ezihlukahlukene imishini nasezintweni zobuchwepheshe zamuva. Kepha akusikho lokho kuphela uma kubalulekile ukuba sisebenzise amandla ngokoqobo angaphezu kwawabantu: ukuhambisa umthwalo ukukhulisa yona ukuphakama, umbe umsele eside futhi ijulile, njll Izimoto namuhla ukuqoqa amarobhothi, ukudla iphekwe Multivarki futhi aphansi izibalo izibalo ukukhiqiza yokubala ... Ziyanda Sijwayele ukuzwa inkulumo ethi "algebra Izimiso eziphusile". Mhlawumbe isikhathi sesifikile ukuba siqonde indima zabantu kudalwa amarobhothi futhi imishini ikhono ukuxazulula zezibalo hhayi kuphela, kodwa futhi izinkinga okunengqondo.

logic

Ngo logic Greek - uhlelo ngaphambilini yokucabanga ukuthi kudala ubudlelwano phakathi izimo enikeziwe ikuvumela ukuba wenze okuqondiswayo ngokusekelwe ekucabangeleni kanye nokuqagula. Kaningi, sikucela nomunye: "Kunengqondo ukuba" Impendulo kuqinisekisa esikucabangayo noma ugxeka isitimela yokucabanga. Kodwa inqubo akagcini lapho: siqhubeka ukukhuluma.

Ngezinye izikhathi inani nemibandela (okokufaka) mkhulu kangaka, futhi ubuhlobo phakathi kwabo kuyadida futhi uyinkimbinkimbi ukuthi ubuchopho bomuntu akakwazi "inhlabamkhosi" ngesikhathi esisodwa. Ungase udinge inyanga ezingaphezu kweyodwa (nesonto, unyaka) ngokuba ukuqonda okwenzekayo. Kodwa kusukela epilweni yangamalanga yesimodeni akasinikeli kwalezi zikhathi kuyisikhathi sokwenza izinqumo. Futhi thina iphendukele lwama-computer. Futhi yilapha lapho kukhona algebra futhi logic, nemithetho yayo kanye nezimpahla. Emva ukulanda yonke idatha yokuqala, sivumela computer ukuqaphela zonke ubudlelwano, ukuqeda ukungaboni ngaso futhi ukuthola isixazululo egculisayo.

Mathematics kanye logic

Famous Gotfrid Vilgelm Leybnits washayelwa welithi "logic zezibalo", okuyinto imisebenzi Kwakulula ukuqonda kuphela indilinga lencane lezazi. Sinesithakazelo ngokukhethekile yilona isiqondiso akuzange kumenze, futhi phakathi XIX leminyaka logic zezibalo aziwa ambalwa.

I isithakazelo esikhulu abantu abasemkhakheni wezesayensi kuye kwabangela impikiswano lapho eyiNgisi Dzhordzh Bul wamemezela inhloso yakhe ukusungula legatsha wezibalo, hhayi kokuba nhlobo ukusetshenziswa esiwusizo. Njengoba sazi kusuka emlandweni, ngalesi sikhathi ngenkuthalo ekuthuthukiseni ukukhiqizwa yezimboni, sisungule zonke izinhlobo imishini elisizayo, t. E. Zonke okutholwe yisayensi baye baba orientation esisebenzayo.

Lapho ubheka ikusasa, sithi algebra Izimiso eziphusile - esetshenziswa kakhulu emhlabeni namuhla yingxenye wezibalo. Ngakho-agumenti yakho Buhl elahlekile.

Dzhordzh Bul

Ubuntu umbhali ufanelwe ukuqashelwa okukhethekile. Ngisho unikezwa iqiniso lokuthi abantu esidlule wakhula phambi kwethu, namanje-ke kufanele kuqashelwe ukuthi iminyaka engu-16 kaJohane. Buhl wayefundisa esikoleni sendawo, futhi eminyakeni kuya ku-20 wavula isikole sakhe siqu ngokungcweliswa Lincoln. Sezibalo yingcweti ngokuphelele ngezilimi ezinhlanu lesinye isizwe, futhi ngesikhathi sakhe sokuphumula, wayefunda imisebenzi Newton kanye Lagrange. Futhi konke lokhu - endodaneni isisebenzi ojayelekileko!

Ngo-1839, Buhl wathumela amaphepha wakhe wokuqala yesayensi kule Cambridge Mathematical Journal. Usosayensi neminyaka engu-24 ubudala. umsebenzi Boole sika amalungu ngakho nesithakazelo yi-Royal Society, ngo-1844 lapho ngithola indondo ngegalelo lakhe ekuthuthukiseni izingabunjalo. A amaphepha ambalwa eshicilelwe lapho amalunga logic nezibalo, isayensi yezibalo wavumela abasha ukuba okuthunyelwe Uprofesa e College Kakhokho County ichazwe. Khumbula ukuthi lapho uJeremiya imfundo Boole kakhulu wayengekho.

umqondo

Ngomqondo onabile, algebra Izimiso eziphusile silula. Kukhona izitatimende (elinengqondo izinkulumo) ukuthi, kusukela umbono wesayensi yezibalo, kungenziwa kuphela kuchazwa ngamagama amabili: "iqiniso" noma "amanga". Ngokwesibonelo, izihlahla eziqhakaza entwasahlobo - iqiniso, ehlobo ke snows - amanga. Ubuhle mathematics ukuthi akusiyo okutlhogekako ukusebenzisa izinombolo kuphela. Ukuze izahlulelo algebra ngempela ulingana noma isitatimende incazelo ehlukile.

Ngakho, i-algebra logic ingasetshenziswa ngokoqobo nomaphi: ku ukuhlela nekubhala imfundo, ukuhlaziya ulwazi engqubuzanayo ngezenzakalo ukuzimisela ukulandelana yezenzo. Into ebaluleke kakhulu - ukuqaphela ukuthi akunandaba ukuthi sithola iqiniso noma falsity izitatimende. Kulezi "indlela" futhi "kungani" udinga indiva. Okusemqoka kuphela namazwi aveza iqiniso: iqiniso amanga.

Yiqiniso, uHlelo imisebenzi ebaluleke kakhulu-algebra logic elotshwe ngezibonakaliso ezifanele kanye nezimpawu. Futhi ufunde kubo - kusho ukuthini ukufunda ulimi olusha angaphandle. Akukho okungenakwenzeka.

imiqondo Basic nezincazelo

Ngaphandle kokuya ukujula, ake ukubhekana amagama ka. Ngakho, algebra Izimiso eziphusile kusikisela:

• izitatimende;
• imisebenzi okunengqondo;
• imisebenzi kanye nemithetho.

Izitatimende - yiliphi ngoyebo ukuthi kungahunyushwa ezimbili zenani. Zilotshiwe nezinombolo (5> 3) noma washayelwa emagama latayelekile (indlovu - esincelisayo esikhulu kunazo zonke). Kulokhu, inkulumo ethi "entanyeni indlulamithi akuyona" Ubuye ilungelo khona, kuphela Aljibhra Izimiso eziphusile bakuchaza ngokuthi "amanga."

Zonke izitatimende kufanele kube ocacile, kodwa zingase zibe eziyisisekelo noma kwakwakhe. ukusetshenziswa kwakamuva besigaxa okunengqondo. E. Ngo izitatimende algebra izahlulelo kwakwakhe ezakhiwe kwalokho aphansi logic imisebenzi.

Izimiso eziphusile algebra imisebenzi

Sesiyazi khumbula ukuthi imisebenzi ku-algebra yezahlulelo - okunengqondo. Njengoba nje algebra izinombolo besebenzisa imisebenzi izibalo ukuze kungezwe, kukhishwe, noma nokumadanisa iinomboro, izakhi logic zezibalo ukuvumela ukuba zenze izitatimende eziyinkimbinkimbi, ukunqabela noma ukubala umphumela wokugcina.

Logic imisebenzi ngenxa semthethweni kanye lula esashiwo ifomula, ajwayelekile kithi izibalo. Properties of Izimiso eziphusile algebra zibalo enze kube nokwenzeka ukuba alobe futhi abale engaziwa. imisebenzi enengqondo kuvame esalotshwa etafuleni iqiniso. izakhi zayo ichaze amakholomu kanye nokusebenza Computing okuyinto eyenziwa kuwo, futhi imigqa ukukhombisa umphumela izibalo.

logic eziyisisekelo isenzo

Okuvame kakhulu kule Izimiso eziphusile algebra imisebenzi kukhona kuphika (HHAYI), kanye okunengqondo KANYE futhi NOMA. Ngakho kungenzeka ukuchaza cishe zonke izinyathelo e-algebra izahlulelo. Safunda ngokuningiliziwe ngamunye imisebenzi emithathu.

I kuphika (cha) lisetshenziswa elementi eyodwa kuphela (we-). Ngakho-ke, ukusebenza ibizwa ngokuthi kuphikisa unary. Ukurekhoda welithi "hhayi A" usebenzisa izimpawu ezifana: ¬a, a noma A !. In ifomu tabular kubukeka sengathi lokhu:

Umsebenzi ukuphika isimo ejwayelekile ye lasho amazwi anjalo: uma okokuqala kuyiqiniso, khona-ke A - kungamanga. Ngokwesibonelo, inyanga lumayelana Earth - iqiniso; Umhlaba uzungeza inyanga - amanga.

ukubuyabuyelela enengqondo futhi ngaphezu

Elinengqondo KANYE ukusebenza ibizwa ngokuthi ngokuhlangana. Kusho ukuthini? Okokuqala, ukuthi singasetshenziswa operands ezimbili, okusho ukuthi, mina - .. operation Binary. Okwesibili, kuphela endabeni ka-iqiniso kokubili operands (kokubili A no B) ziyiqiniso futhi le nkulumo ngokwayo. Isaga, "Isineke Ngomzamo omncane nje," kubonisa ukuthi izici ezimbili kuphela engasiza umuntu wakwati kubeketelela bumatima.

yezimpawu ezisetshenzisiwe zokuqopha: A∧B, A⋅B noma A && B.

Isihlanganiso iyafana ukubuyabuyelela kuzibalo. Ngezinye izikhathi bathi - ukubuyabuyelela okunengqondo. Uma uphindaphinde tincenye imigqa etafuleni, sithola yalokho efana ukucabanga okunengqondo.

Disjunction kuyinto okunengqondo NOMA msebenzi. Kuyinto TRUE uma okungenani eyodwa izitatimende kuyiqiniso (noma A noma B). Kulotshiwe kanje: A∨B, A + B noma A || B. etafuleni iqiniso misebenzi yilezi:

Disjunction efanayo kwalokho izibalo. Ngaphezu kwalokho ukusebenza okunengqondo has mngcele owodwa kuphela: 1 + 1 = 1. Kepha siyakhumbula ukuthi ngendlela ngesakhiwo digital kukhawulwe logic zezibalo 0 no-1 (lapho 1 - iqiniso, 0 - amanga). Ngokwesibonelo, amazwi athi "in the Museum ungabona wobuciko noma ukuthola inkampani ezinhle" kusho ukuthini lokho ungabona imisebenzi yobuciko, futhi kungenzeka ukuhlangabezana nomuntu ezithakazelisayo. Ngesikhathi esifanayo, musa siqambe kungenzeka ekugcwalisekeni ngesikhathi esisodwa kokubili izenzakalo.

Imisebenzi nemithetho

Ngakho, thina vele ukuthi ukusebenza okunengqondo usebenzisa algebra Izimiso eziphusile. Imisebenzi ukuchaza zonke izakhiwo izici logic nezibalo, futhi zisisiza ukuba lula izitatimende kwakwakhe eziyinkimbinkimbi. I ecacile kakhulu futhi elula kubonakala kwenqatshwa impahla wona lama-derivatives imisebenzi. Ngu-derivatives ziqondwa xor, basuke beqonde futhi kulingana emiholweni. Njengoba sesifundile kuphela imisebenzi eyisisekelo, bese impahla futhi kuphela ababheke.

Associativity kusho ukuthi amazwi anjengokuthi, "kokubili A no-B, no-B 'ukulandelana nohlu operands akunandaba. Ifomula kulotshiwe kanje:

(A∧B) ∧V = A∧ (B∧V) = A∧B∧V,

(A∨B) ∨V = A∨ (B∨V) = A∨B∨V.

Njengoba ubona, lokhu akuyona into engavamile ukuba ngokuhlanganyela kodwa disjunction.

Commutativity uveza ukuthi umphumela we-ngokuhlanganyela noma disjunction akuxhomekile okuyinto nto kubhekwe ekuqaleni:

A∧B = B∧A; A∨B = B∨A.

Distributivity ivumela ukudalula kubakaki ngamazwi eziyinkimbinkimbi okunengqondo. Imithetho iyefana abakaki mbobo ukubuyabuyelela kanye kwalokho e-algebra:

A∧ (B∨V) = A∧B∨A∧V; A∨B∧V = (A∨B) ∧ (A∨V).

Unit izakhiwo kanye ekuqaleni, okungaba omunye operands nazo afana ukubuyabuyelela algebraic ngo-zero noma eyodwa, futhi ngaphezu a unit:

A∧0 = 0, A∧1 = A; A∨0 = A, A∨1 = 1.

Idempotency lisitshela ukuthi uma kuqhathaniswa operands amabili alinganayo umphumela ukusebenza okufanayo, ungakwazi 'uphonse "ngokweqile izinkinga ukucabanga operands. Futhi ngokuhlangana, futhi disjunction kuyinto imisebenzi idempotent.

B∧B = B; B∨B = B.

Ukuthola Kusinikeza nethuba lula equation. Absorption uthi lapho inkulumo sisetshenziswa kulowo we-, kwenye operation nge element efanayo umphumela we- is iyavuleka msebenzi.

A∧B∨B = B; (A∨B) ∧B = B.

ukulandelana imisebenzi

Kulandzelana kwalokucuketfwe lokuhleliwe imisebenzi kubaluleke kakhulu. Empeleni, njengoba i-algebra, kukhona umsebenzi kuqala olusebenzisa algebra Izimiso eziphusile. Amafomula kungenziwa lula kuncike ukubaluleka imisebenzi kuphela. Usezingeni we abalulekile kakhulu azinakwa, sithola ukulandelana elandelayo:

1. Ukungakwamukeli.

2. Isihlanganiso.

3. disjunction, xor.

4. Empeleni, basuke beqonde, kulingana emiholweni.

Njengoba ubona, kuphela kuphikisa we ngokuhlangana futhi akudingeki igunya alinganayo. A eziseqhulwini disjunction futhi xor bayalingana, kanye eziseqhulwini ayesho ukuthini futhi kulingana emiholweni.

Imisebenzi ka yini esiyishoyo esikisela futhi kwefrakshini

Njengoba sishilo, ngaphezu imisebenzi eyisisekelo okunengqondo, logic zezibalo kanye imfundiso yokuziphendukela algorithm usebenzisa amagama ahlobene. Kuyinto avame ayesho futhi kulingana emiholweni.

Empeleni, basuke beqonde noma umphumela onengqondo - lesi sitatimende, lapho isenzo esisodwa yisimo, kanye nezinye - umphumela kusebenta kwaso. Ngamanye amazwi, kulesi siphakamiso nge kwebhaxa "uma ... ngaleso sikhathi." "Ngemva idina kufika lokwahlulela." E. Ngoba ukushayela ukuze some egqumeni isihlibhi. Uma kungekho isifiso sokuthuthela esehlile entabeni, bese hudula isihlibhi akudingekile. Kulotshiwe ngakho: A → B noma A⇒B.

Kulingana emiholweni lisikisela ukuthi Umtselela lophelele kwenzeka kuphela uma kokubili operands ziyiqiniso. Ngokwesibonelo, ngobusuku usuthathelwa usuku ke (bese kuphela), lapho ilanga liphuma lé. Olimini logic zezibalo kwalesi sitatimende kulotshiwe njengoba A≡B, A⇔B, A == B.

Eminye imithetho algebra Izimiso eziphusile

I-Algebra ukwahlulela eba, futhi ososayensi abaningi abathakazelayo ukuba zisungule imithetho emisha. Isenzakalo esidume kunazo zonke ezibhekwa postulates sezibalo Scottish O. De Morgan. Waqaphela futhi wanikeza incazelo izakhiwo ezifana kuphikisa abaseduze, ngaphezu nemibi double.

Vala ekuphikeni kusikisela ukuthi ngaphambi abakaki Akunakuphikwa: hhayi (A noma B) = hhayi A noma B. HHAYI

Lapho we- kwenqatshiwe, kungakhathaliseki ukubaluleka kwayo, uthi mayelana kwalokho:

B∧¬B = 0; B∨¬B = 1.

Futhi ekugcineni, kuphika double ngokwayo ivala. okungukuthi ngaphambi noma kuphikisa we- ubalekela noma uhlala eyodwa kuphela.

Indlela ukuxazulula ukuhlolwa

Logic kusho yokuncishiswa predetermined zibalo. Njengasemhlabeni algebra Amanga, kubalulekile ukuba Ubuningi bamahora lula isimo sokuqala (ukuqeda imisebenzi yokufaka eyinkimbinkimbi nabo), khona-ke ukuqala ufuna impendulo efanele.

Yini okufanele ukwenze ukuze lula? Guqula wonke amagama ahlobene e ukuhlinzwa nje okulula. Khona-ke, wambule zonke kubakaki (noma okuphambene nalokho, ukwenza kubakaki ukunciphisa lesi sici). Isinyathelo esilandelayo kufanele kube ukusebenzisa Izimiso eziphusile Aljibhra izakhiwo umkhuba (ukumuncwa izakhiwo zero futhi omunye, futhi t.).

Ekugcineni, ezothando kufanele libe inombolo okungenani angaziwa, kuhlangene imisebenzi elula. Indlela elula ukubheka isixazululo, uma wenza inani elikhulu ezingezinhle eduze. Khona-ke impendulo ngeke vela sengathi ngokwalo.