Algebra boolean. algebra ekuqiqeni. Elements ekuqiqeni zezibalo

Kweli hlabathi thina ngakumbi usebenzisa iindidi oomatshini kunye elektroniki. Kwaye hayi kuphela xa oko kuyimfuneko ukusebenzisa amandla ngokoqobo angaphaya: ukuhambisa umthwalo ukuyinyusa kubude, embe umsele elide yaye zinzulu, njl Cars namhlanje ukuqokelela robots, ukutya okuphekiweyo Multivarki kunye izibalo izibalo ezinesiqalelo ukuvelisa izibali ... Kakhulu kwaye amaxesha amaninzi sisiva ibinzana elithi "algebra seBhulin". Mhlawumbi lifikile ixesha lokuba ukuqonda indima njengabantu ekudalweni robhothi kunye noomatshini ukukwazi ukusombulula izibalo nje kuphela, kodwa iingxaki ubhalo.

logic

Ubuhlakani Greek - inkqubo wayalela ingcamango kudala ubudlelwane phakathi iimeko anikwe likuvumela ukuba wenze intelekelelo ezisekelwe kuqikelelo kunye noqikelelo. Lidla sibuza omnye komnye: "Kuyinto kusengqiqweni" Impendulo ingqina iingqikelelo zethu okanye ugxeka uloliwe izimvo. Kodwa inkqubo akayeki khona; siqhubeka ukuthetha.

Ngamanye amaxesha mkhulu ngoko ke inani iimeko (input), kwaye ubudlelwane phakathi kwabo edidayo Untsonkothe ukuba ingqondo yomntu ayikho nako "ukugaya" zonke ngaxeshanye. Kusenokufuneka ngenyanga ngaphezu kwesinye (ngeveki, unyaka) ukwenzela ukuqonda ukuba kwenzeka ntoni. Kodwa ke ubomi mihla akasiniki la mathuba ixesha ukwenza izigqibo. Kwaye babhenele kusetyenziswa ikhompyutha. Kwaye ke apha ukuba kukho sezibalo logic, kunye nemithetho kunye neempawu zayo. Emva kokuba uyilayishe ezantsi yonke idatha yoqobo, sivumela ikhompyutha ukuqaphela zonke ubudlelwane, ukuphelisa ephikisanayo kunye ukufumana isisombululo esamkelekileyo.

Mathematics and logic

Famous Gotfrid Vilgelm Leybnits yaqulunqa ingcamango "logic zemathematika", apho imisebenzi lula ukuqonda kuphela isangqa encinane zabaphengululi. Of umdla ethile le ulwalathiso ingabi mhlophe, kwaye phakathi XIX kwinkulungwane ekuqiqeni lweMathematika eyaziwa ngabambalwa.

Umdla omkhulu ekuhlaleni lwezenzululwazi wabangela imbambano apho eliNgesi Dzhordzh Buli wavakalisa injongo yakhe ukuseka isebe yemathematika, ndingenabo obam ngokupheleleyo akukho ukusetyenziswa aluncedo. Njengoko sisazi kwimbali, ngeli xesha ngokubonakalayo ekuphuhliseni imveliso mveliso, sisungule zonke iintlobo zoomatshini abancedisayo, t. E. Zonke izinto ezifunyaniswe zizazinzulu ziye orientation practical.

Xa phambili, sithi ukuba algebra seBhulin - yeyona ndlela isetyenziswa kakhulu ehlabathini namhlanje inxalenye lwemathematika. Ngoko ingxabano yakho Buhl ebilahlekile.

Dzhordzh Buli

Ubuntu umbhali kufuna ingqalelo ekhethekileyo. Nokuba banikwe yokuba abantu elidlulileyo bakhula phambi kwethu, kunjalo kufuneka kuqatshelwe ukuba kule minyaka 16 uYohane. Buhl wafundisa esikolweni lali, yaye iminyaka ukuya ku-20 wavula isikolo sakhe Lincoln. Sezibalo wazazi ngokugqibeleleyo iilwimi ezintlanu zasemzini, kwaye ngexesha lakhe nanceba, ilesa incwadi imisebenzi Newton kunye आयडेंटीफिकेशन iSazisi. Kwaye konke oku - unyana ngumsebenzi oqhelekileyo ngayo!

Ngowe-1839, Buhl wathumela amaphepha yakhe yokuqala zenzululwazi e Cambridge Mathematical Journal. Scientist wajika ngonyaka 24. umsebenzi Boole likaThixo amalungu anomdla of the Royal Society, ngowe-1844 wafumana imbasa ngegalelo lakhe kuphuhliso uhlalutyo zezibalo. A amaphepha ashicilelwe ezimbalwa apho izinto ekuqiqeni zemathematika, imathematika wavumela aselula ukuba athathe kwisithuba unjingalwazi kwi College of Yazo County bachazwa. Khumbula ukuba imfundo Boole kakhulu wayengekho.

umbono

Enyanisweni, algebra Boolean ilula kakhulu. Kukho iingxelo (ubhalo amabinzana) ukuba, ukususela kwinqanaba umbono imathematika, singachazwa kuphela ngamagama amabini: "bokwenyaniso" okanye "obuxoki". Umzekelo, imithi ibiselityatyambile spring - inyaniso, ehlotyeni nekhephu - ubuxoki. Ubuhle yemathematika ukuba akukho kokwaneleyo ukusebenzisa amanani kuphela. Ngenxa yezigwebo algebra kakhulu ilingane naziphi na iingxelo kunye nentsingiselo ekhethekileyo.

Ngoko ke, lo algebra ekuqiqeni zingasetyenziswa ngokoqobo kuyo yonke indawo; ekucwangcisweni kunye nokubhala uqeqesho, uhlalutyo ulwazi ezingquzulanayo malunga iziganeko kunye nokuzimisela bokulandelelana yezenzo. Eyona nto ibalulekileyo - ndaqonda ukuba akukho ayinamsebenzi indlela ukugqiba inyaniso okanye falsity kweengxelo. Ukusuka kwezi "kutheni" kufuneka "nendlela" ngoyaba. Into ebalulekileyo ingxelo kuphela Enyanisweni, inyaniso ubuxoki.

Kakade ke, nodweliso lwenkqubo i misebenzi ibaluleke algebra ekuqiqeni ebhalwe ngemiqondiso efanelekileyo kunye neesimboli. Kwaye ufunde kubo - oko kuthetha ukuba ukufunda ulwimi olutsha lwasemzini. Ayikho into engenakwenzeka.

nezimaphambili ezingundoqo kunye neengcaciso

Ngaphandle kokuba ibe nobunzulu, siya kujongana kwamagama. Ngoko ke, algebra Boolean busenza ukuba:

• iingxelo;
• imisebenzi ubhalo;
• imisebenzi kunye nemithetho.

Iingxelo - nayiphi na imbonakalo affirmative aokusetyenziswa ukuchazwa eyaziswa ezimbini-. Bona zibhalwe amanani (5> 3) okanye amagama aqhelekileyo aliqulunqileyo (elephant - esiyimamali enkulu). Kulo mzekelo, ibinzana elithi "entanyeni indlulamthi alikho" Kwakhona unalo ilungelo lokuba khona, kuphela algebra Boolean bakuchaza nje "ubuxoki."

Zonke iingxelo kufuneka ibe ayintsonkothanga, kodwa basenokuba ezisisiseko okanye ezimbaxa. ukusetyenziswa Recent umqulu ubhalo. E. In iingxelo algebra izigwebo ezimbaxa kuyilwa ngokuthi kongezwe imisebenzi logic sabaqalayo.

imisebenzi algebra Boolean

Thina sele khumbula ukuba imisebenzi kwi algebra izigwebo - ubhalo. Kanye njengokuba algebra ngamanani besebenzisa imisebenzi yokalo emileyo ukongeza, uthabathe, okanye uthelekise amanani, izinto logic lwemathematika kuvumela ukwenza iingxelo ezintsonkothileyo, ukukhanyela okanye ukubala eziphumo zokuphela konyaka.

imisebenzi Logic ngokuba semthethweni kunye nokusebenziseka oluvakaliswe ifomula, wazana nathi arithmetic. Iimpawu zibalo algebra Boolean kwenza kube lula ukubhala nokubala okungaziwa. imisebenzi ubhalo ngokuqhelekileyo abhalwe yi inyaniso itafile. sezulu ukuchaza imihlathi kunye nokusebenza computing leyo yenzelwa phezu kwabo, kwaye imiqolo zibonisa ngenxa izibalo.

logic Basic amanyathelo

Olona hlobo imisebenzi algebra Boolean ezi kuphika (NA), kunye enengqondo KUNYE kunye OKANYE. Ngoko ke kunokwenzeka ukuba ukuchaza phantse onke amanyathelo izigwebo algebra. Safunda iinkcukacha ngamnye ezintathu.

Le kuphika (hayi) lisetyenziswa element enye kuphela (we-). Ngoko ke, lo msebenzi ibizwa ngokuba kuphika unary. Ukurekhoda ingqiqo ka "hayi A" esebenzisa obunjalo: ¬A, A okanye A !. Xa uhlobo tabular ngayo sijongeka njengoku:

Umsebenzi ukuphika wendlela mazwi: Ukuba A kuyinyaniso, ngoko A - bubuxoki. Umzekelo, inyanga nendawo eyayisaya kuba Umhlaba - inyaniso; Umhlaba kugxile nenyanga - ubuxoki.

ukuphinda acwangciswe nokudityaniswa

Ukuyondelelaniswa KUNYE umsebenzi kuthiwa isihlanganisi. Kuthetha ukuthini? Okokuqala, ukuba kunokusetyenziswa operands ezimbini, oko kukuthi, mna - .. operation Binary. Okwesibini, nguye kuphela kwimeko lenyaniso zombini operands (zombini A no-B) liyinyaniso kwaye ibinzana ngokwayo. Lo mzekeliso, "Nomonde kwaye inzame omncinane" libonisa ukuba ababini kuphela izinto ezinokunceda umntu ukumelana neenzima.

iisimboli kusetyenziselwa ukurekhoda: A∧B, A⋅B okanye A && B.

Beqhuba uyafana phindo in arithmetic. Maxa wambi bathi - phindo ubhalo. Ukuba phinda iziqalelo ze imiqolo itafile, sifumana isiphumo esifanayo ukusebenzisa ukucinga.

Disjunction i ubhalo OKANYE umsebenzi. Yinyani ukuba ubuncinane enye kwezi ngxelo ziyinyani (mhlawumbi A okanye B). Kubhaliwe kwathiwa ngolu hlobo: A∨B, A + B okanye A || B. inyaniso itafile kule misebenzi zezi:

Disjunction Ukongeza efanayo arithmetic. Ukongeza operation ubhalo na umqathango omnye kuphela: 1 + 1 = 1. Kodwa ke khumbula okokuba ngendlela yedijithali kungaphelelanga logic mathematical 0 no 1 (apho 1 - inyaniso, 0 - bobuxoki). Umzekelo, ibinzana 'kwi imyuziyam ungabona wobugcisa okanye ukufumana inkampani elungileyo "kuthetha ntoni ungabona imisebenzi yobugcisa, kwaye kusenokwenzeka ukuba ukudibana nomntu umdla. Kwangaxeshanye, musa kulawula naziphi na inzaliseko ngaxeshanye zombini ziganeko.

Imisebenzi kunye nemithetho

Ngoko, kakade usebenzisa algebra Boolean into msebenzi enengqondo. Imisebenzi ukuchaza zonke iimpawu eziqalelo ekuqiqeni kwemathematika, kwaye ukuvumela ukuba lula iingxelo ezimbaxa nezintsonkothileyo. I ecacileyo kakhulu kwaye elula kubonakala impahla ukwaliwa imisebenzi yezabelo. Ngu ahlobene ziqondwa XOR, isiphumo kunye nokufana. Njengoko sele ufunde kuphela imisebenzi esisiseko, kwaye ke lo mhlaba kwakhona ingqalelo kuphela kwabo.

Associativity kuthetha ukuba iingxelo ezifana "bobabini A no-B, kunye B 'ukulandelelana uluhlu lweempawu operands Akukhathaliseki. Ifomula ibhalwe ngolu hlobo lulandelayo:

(A∧B) ∧V = A∧ (B∧V) = A∧B∧V,

(A∨B) ∨V = A∨ (B∨V) = A∨B∨V.

Njengoko ubona, oku ngale ndlela isihlanganisi kodwa disjunction.

Commutativity uthi iziphumo isihlanganisi okanye disjunction akuxhomekekanga apho nto kuye kwaqwalaselwa ekuqaleni:

A∧B = B∧A; A∨B = B∨A.

Distributivity ivumela liveze kwezibiyeli amabinzana ezinzima zilandelelana. Imithetho ezifanayo ipharenthensisi yokuvula kwi phindo kunye Ukongeza kwi Algebra:

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

iipropati Unit kunye ekuqaleni, leyo yenye operands nayo iyafana-phindo aljibra ngu zero okanye omnye, kunye nokongezwa iyunithi:

A∧0 = 0, A∧1 = A; A∨0 = A, A∨1 = 1.

Idempotency isixelela ukuba xa kakhulu operands ezimbini ngokulinganayo ngesiphumo msebenzi iyafana, unako "ukulahla" i operands bazixakekisa nokuqiqa kwemfuneko. Kwaye imisebenzi isihlanganisi kunye disjunction zizinto idempotent.

B∧B = B; B∨B = B.

Kwakhona Ukufunyanwa ikuvumela ukuba lula inxaki. Absorption ithi ukuba xa ibinzana lisetyenziswa we- omnye, omnye umsebenzi kunye nto olufanayo ngesiphumo we- nto zikwazi ukusebenza.

A∧B∨B = B; (A∨B) ∧B = B.

nolandelelwano lwemisebenzi

Ulandelelwano imisebenzi kubaluleke kakhulu. Eneneni, njengoko ngenxa aljebra, kukho umsebenzi eziphambili usebenzisa algebra Boolean. Iifomula kungenziwa lula phantsi kuphela intsingiselo imisebenzi. Ranking eziphawuleka kakhulu ukuba okungenamsebenzi, siye sifumane le ndlela ilandelayo:

1. Ukuphika.

2. beqhuba.

3. Le disjunction, XOR.

4. Ingongoma, ukulingana.

Njengoko ubona, kuphela kuphika kwe isihlanganisi kwaye kuza kuqala ngokulinganayo. A ophambili disjunction kunye XOR bayalingana, kwakunye eziphambili kukusulela kunye nokufana.

Imisebenzi okubandakanya kunye nokufana

Njengokuba besesitshilo, ukongeza kwimisebenzi esisiseko ubhalo, logic zemathematika kunye nethiyori ye ubuchule besebenzisa yezabelo. Kubaluleke kakhulu amaninzi kukusulela kunye nokufana.

Ingongoma okanye isiphumo ubhalo - le ngxelo, apho inyathelo elinye yimeko, kunye nezinye - ngenxa zokuphunyezwa kwayo. Ngamanye amazwi, esi sindululo kunye benze 'ukuba ... emva koko ". "Emva isidlo kufike sokubala." E. ngokuba uqhuba ukuba some phezu kwentaba ebusika. Ukuba akukho umnqweno ukuba bayehla entabeni, uze utsale ebusika ayikho imfuneko. Ibhalwe njalo: A → B okanye A⇒B.

Ukulingana kuthetha ukuba isiphumo net kwenzeka kuphela xa omabini operands ziyinyani. Umzekelo, ubusuku unika indlela eya suku ke (kwaye ngoko kuphela), xa ilanga liphuma lé. Xa ulwimi ekuqiqeni zezibalo yale ngxelo ibhalwe njenge A≡B, A⇔B, A == B.

Neminye imithetho algebra Boolean

Algebra umgwebo esivelayo, yaye izazinzulu ezininzi abanomdla ukuqulunqa imithetho emitsha. Edume kakhulu zithathwa postulates yezibalo Scottish O. De Morgan. Waphawula wanika inkcaza iipropati ezifana kuphika kufutshane, ukudibanisa elibi kabini.

Vala Ukuphika kubonisa ukuba phambi kokuba ipharenthesis akukho bekhanyela: hayi (A okanye B) = hayi A okanye B NOT

Xa we- asamkelwa, nokuba ixabiso layo, uthini Ukongeza:

B∧¬B = 0; B∨¬B = 1.

Ke ekugqibeleni, lo kuphika kabini ngokwalo ihlawula. ie phambi nokuba kuphika ochaza liyalahleka okanye uhlala enye kuphela.

Indlela yokusombulula iimvavanyo

Logic ithetha lula kwangaphambili zibalo. Kanye kule algebra Lala, kuyimfuneko ukuba kakhulu lula imeko lokuqala (ukuba ukuphelisa imisebenzi igalelo ezintsonkothileyo, kunye nabo), ngoko iqala ukufuna impendulo echanekileyo.

Kufuneka wenze ntoni ukuze lula? Guqulela yezabelo kwi operation elula. Emva koko ungabutyhili zonke kwizibiyeli (okanye ngenye indlela, ukuze kwizibiyeli ukunciphisa eli element). Inyathelo elilandelayo kufuneka ibe ukusebenzisa iipropati algebra seBhulin in practice (iipropati nokufunxwa zero omnye, kunye t.).

Ekugqibeleni, inxaki kufuneka luqulathe igama ubuncinane inani angaziwa, zidibene imisebenzi elula. Indlela elula ukukhangela isisombululo, ukuba wenze inani elikhulu negatives close. Ngoko ke impendulo izakuveliswa ngokungathi ngokwalo.