Taurangi Boolean. taurangi o arorau. Elements o arorau pāngarau

I roto i te ao i teie mahana piki e whakamahi tatou i te whānuitanga o ngā mīhini me gadgets. A e kore e anake i te mea e tika ana ki te tono kaha mo'oni mahulu: neke te kawenga ki te whakaara i te reira ki te tiketike, keri awa roa me te hohonu, me ētahi atu Cars tenei ra kohikohi Robots, tunua kai ko Multivarki me tātaitanga tauhanga māmā hua tātaitai ... Ētahi atu a ake pinepine rongo tatou i te parau "taurangi Boolean". Pea kua tae mai te wa ki te mahino te tūranga o te tangata i roto i te hanga o lōpoti me ngā mīhini te kaha ki te whakaoti pāngarau kore anake, engari ano hoki raruraru arorau.

arorau

I roto i te arorau Kariki - he pūnaha whakahau o te whakaaro e hanga te hononga i waenganui i nga tikanga i homai, me te taea ki a koe kia waeua i runga i ngā whakapae, me whakatau tata. Tino maha, ui tatou tahi i te tahi: "Ko reira arorau ki" te whakautu fakapapau'i tatou whakapae ranei faaino te tereina o whakaaro. Otiia e kore e te tukanga mutu i reira: tonu tatou ki te kōrero.

I te tahi taime te mea nui te maha o ngā āhuatanga (tāuru), me te hononga i waenganui i a ratou he pera huanane me matatini e kore e taea e ki "ngaungau" katoa i kotahi te roro tangata. Kia hiahia koe ki ētahi atu i te kotahi marama (wiki, tau) mo te māramatanga o te mea kei te tupu. Otiia e kore e ora hou hoatu tatou enei wā wa ki te hanga whakatau. A huihui matou ki te āwhina o rorohiko. A ko reira i konei e reira ko te taurangi, me te arorau, me ona ture, me ngā āhuatanga. I muri i te tikiake katoa o nga raraunga taketake, tukua matou te rorohiko ki te mohio ki te whanaungatanga katoa, ki te whakakore ngā whakahēnga me ki te kitea te otinga pai.

Pāngarau me te arorau

whakatakoto Famous Gotfrid Vilgelm Leybnits te ariā o "arorau pāngarau", i nga ngāwari ki te matau anake te porohita iti o ngā pūkenga mahi. O ngā moni ko te aronga kihai i meinga, me ki te waenganui o te rau tau XIX o arorau pāngarau mohiotia e torutoru.

Kua meinga e te moni nui i roto i te hapori pūtaiao he tautohe i roto i i whakapuakina i te Ingarihi Dzhordzh Puru tona whakaaro ki te whakatū i te peka o te pāngarau, e kore he tino kore whakamahi mahi. Ka rite ki matau ana tatou i te aamu, i tenei wa kaha whakawhanake production ahumahi, whakawhanakehia matou nga mea katoa o mihini pŭpŭ tauturu, kua i t. E. āhuatanga pūtaiao katoa he takotoranga mahi.

Titiro mua, mea tatou e te taurangi Boolean - te whakamahia tino i roto i te ao nei i teie mahana wahi o pāngarau. Heoi ngaro koutou tautohe Buhl.

Dzhordzh Puru

Ko te tuakiri o te kaituhi tika tokanga makehe. Ahakoa homai te meka e i roto i tupu te iwi mua ake i mua ia tatou, tonu kia tuhia te reira e i roto i nga tau 16 o Hoani. Buhl whakaako i te kura kainga, a ki te 20 tau ka puaki te kura ake i roto i Lincoln. Mathematician maitai taukei rima reo ke, me i tona wa e manawapa, i te tai'oraa i te mahi o Newton me Lagrange. Na tenei katoa - i runga i te tama a te kaimahi noa!

I roto i te 1839, ka tonoa Buhl ana pepa pūtaiao tuatahi i roto i te Cambridge Mathematical Journal. tahuri Scientist 24 tau. mahi a Boole ko ngā mema na hiahia o te Royal Society, i roto i te matahiti 1844 ua farii e ia he mētara mo tona takoha ki te whanaketanga o te tātari pāngarau. i whakaahuatia te torutoru pepa whakaputaina i roto i ai nga mea timatanga o te arorau pāngarau, pāngarau whakaaetia te taitamariki ki te tango i te pou o te ahorangi i te College o County Cork. A haamana'o na e, i te mātauranga tino Boole kihai i.

whakaaro

I roto i te parau tumu, he tino māmā taurangi Boolean. He tauākī (arorau kīanga) e, i te mata o te tirohanga o te pāngarau, anake e taea te tautuhi i roto i nga kupu e rua: "pono" ranei "teka". Hei tauira, rakau i roto i te puna pua - te pono, i roto i te raumati hukarere - he teka. Ko te ataahua o te pāngarau e kore ko reira tino tika ana ki te whakamahi i ngā tau anake. Hoki te whakaritenga taurangi tino pai i tetahi tauākī ki tikanga ahurei.

Ko te kupu, e taea te whakamahi i te taurangi o te arorau mooni nga wahi katoa: i roto i te whakaritenga me te tuhituhi ako, te tātaritanga o te taupatupatu o ngā kōrero e pā ana ki te ngā, me te faaotiraa o te raupapa o ngā mahi. Ko te mea tino nui - ki te ite e kore e mea faufaa te reira e founga tatou whakatau i te pono mahaki ranei o tauākī. Mai i "pehea" me "te aha" e hiahia ana koe enei ki te tau'a ore. E mea mahuinga ko te tauākī anake o meka: te pono ko te teka.

O te akoranga, papatono nga mahi tino nui o te taurangi o te arorau e kua oti te tuhituhi ki te tohu, me ngā tohu e tika ana. A ako ratou - te tikanga reira ki te ako i te reo ke hou. Kahore e taea.

ariā me ngā whakamāramatanga Basic

Te kore te haere ki te hohonu, mahi matou ki kupu. Na, presupposes taurangi Boolean:

• tauākī;
• ngā mahi arorau;
• mahi me ture.

Tauākī - tetahi faaiteraa faaiteraa e taea te whakaatu ai e rua-utu. kua oti te tuhituhi ratou rite tau (5> 3) ranei whakatakoto kupu maori (arewhana - te kararehe nui rawa). I roto i tenei take, i te parau "e kore he kaki o te tūwhaipapa" ano he tika ki te tīariari, anake taurangi Boolean te tautuhi i te reira rite "he teka."

kia unambiguous tauākī katoa, engari kia waiho ratou taketake pūhui ranei. whakamahi Tata paihere arorau. E. I roto i te pūhui tauākī taurangi whakaritenga i hanga e te tua o ngā mahi arorau pūmotu.

ngā mahi taurangi Boolean

mahara kua tatou e te ngā mahi i roto i te taurangi o whakaritenga - arorau. Just rite te taurangi o tau mā te whakamahi i te ngā mahi tauhanga ki te tāpiri, tango, whakataurite tau ranei, tukua huānga arorau pāngarau ki te hanga tauākī matatini, ki te whakakahore ki te tātai i te hua whakamutunga ranei.

ngā mahi arorau hoki te whakaōkawa, me te tapatahi whakahuatia e te tātai, waia ki a matou i roto i te tātai. Āhuatanga o whārite taurangi Boolean hanga reira taea ki te tuhi me te tātai i te unknown. ngā mahi arorau e te nuinga o tuhia e te tepu pono. Ko tōna huānga tautuhi tīwae me mahi rorohiko whakamana nei ki runga ki a ratou, a ka whakaatu nga rarangi te hua o tātaitanga.

arorau Basic o mahi

Ko te tino noa i roto i te whakahaeretanga o taurangi Boolean he whakakorehanga (NOT), me te arorau ME me RĀNEI. Na ko reira taea ki te whakaahua ngā nga kaupae katoa i roto i whakaritenga taurangi. ako matou i roto i te taipitopito i ia o nga ngā mahi e toru.

tono te whakakorehanga (kore) e ki tetahi anake huānga (operand). Na reira, i huaina te mahi ko te whakakorehanga tohutūmahi. Hei tuhi i te ariā o "kore te" taua tohu te whakamahi i: ¬A, he A ranei !. I roto i te puka ripa rite tenei titiro reira:

Ko te mahi o te whakakāhoretanga angamaheni o te tauākī taua: ki te mea he pono te te, na te - he teka. Hei tauira, faaafaroraa i te marama a tawhio noa te whenua - te pono; faaafaroraa Earth a tawhio noa te marama - he teka.

whakarea Arorau me tua

huaina arorau ME mahi te he taha. Eaha te auraa? Tuatahi, e taea te tono i te reira ki te rua operands, arā, ahau - .. mahi-rua. Tuarua, he reira anake i roto i te take o te pono o e rua tauwhakaarotau (e rua A me B) te pono, me te whakapuaki ano. Te whakatauki, "Kātaki, me te kaha iti" fakahu'unga e taea e rua anake ngā āhuatanga tauturu i akakoromaki i te tangata ki te mau fifi.

E whakamahia tohu mō te tuhi: A∧B, A⋅B ranei A && B.

He rite ki te whakarea i roto i tauhanga taha. I te tahi taime a mea - whakarea arorau. Ki te tini koe nga mea timatanga o te rarangi o te tepu, te tiki matou he hua rite ki te whakaaro arorau.

Ko te arorau mahi RĀNEI Disjunction. Ko reira TRUE ki he i te iti rawa tetahi o nga tauākī pono (rānei A B ranei). A∨B, A + B ranei A || B.: kua rite tenei te tuhituhi te tepu pono mo enei ngā mahi e:

Disjunction tua tauhanga rite. mahi tua arorau kua kotahi anake here: 1 + 1 = 1. Ko mahara tatou e i roto i te hōputu mamati iti te ki arorau pāngarau 0 me 1 (te wahi 1 - te pono, 0 - teka). Hei tauira, te parau "i roto i te whare taonga e taea te kite koe i te faahiahia ranei kitea he kamupene pai" te tikanga o te mea e taea e kite koe i mahi o toi, a ko reira taea ki te whakatau i te tangata ngā. I te wa ano, e kore e whakahaere tikanga i roto i te taea o fakahoko tukutahi o ngā e rua.

Taumahi me ture

Na, e matau kua matou ki ta te mahi arorau te whakamahi i te taurangi Boolean. Functions whakaahua nga āhuatanga katoa o te āhuatanga o arorau pāngarau, ka tukua ki a tatou faaohie tauākī pūhui matatini. Ko te mārama me te ohie tino āhua taonga pato'iraa o te ngā mahi pärönaki. Na roto i te E matau pärönaki XOR, pānga, me te ōrite. Rite tonu ki ta matou i tai'o anake ki te ngā mahi taketake, a ka he ano hoki te taonga whakaaro anake ratou.

tikanga Associativity e i roto i te tauākī pērā i "e kore e rua A ko B, ko B 'raupapa rārangi o nga tauwhakaarotau mea faufaa. te tuhituhi i te tātai te rite e whai ake nei:

(A∧B) ∧V = A∧ (B∧V) = A∧B∧V,

(A∨B) ∨V = A∨ (B∨V) = A∨B∨V.

Ka rite ki a koutou e taea te kite, e kore te mea ahurei ki te taha, engari he disjunction tenei.

tohe Commutativity e kore e te hua o te taha disjunction ranei whakawhirinaki i runga i nei whakaaro tūemi i i te tīmatanga:

A∧B = B∧A; A∨B = B∨A.

Distributivity taea whakaatu taiapa i roto i ngā kīanga arorau matatini. He rite ki te reu tuwhera i roto i te whakareatanga, me te tua i roto i te taurangi Ture:

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

āhuatanga Unit me wahanga, e taea e he hoki tetahi o nga tauwhakaarotau rite ki te whakareatanga taurangi e kore ranei tetahi, me tua o te wae:

A∧0 = 0, A∧1 = A; A∨0 = A, A∨1 = 1.

parau Idempotency tatou e te mea e rua ngā rite tauwhakaarotau te hua o te mahi ko te taua, ka taea e koe "maka" te tauwhakaarotau haafifi i whakaaroaro ana i te taikaha. Na ko idempotent nga ngā mahi te taha me te disjunction.

B∧B = B; B∨B = B.

taea hoki Hoko ki tatou te faaohie i te whārite. kī absorption e ka tono te faaiteraa e ki tetahi operand, tetahi mahi ki te huānga taua o te tauwhakaarotau hua kei te ngongo mahi.

A∧B∨B = B; (A∨B) ∧B = B.

raupapa o ngā mahi

He o faufaa nui te raupapa o ngā mahi. Mau, rite mo te taurangi, i reira te mea he mahi matua e whakamahi ana i te taurangi Boolean. Ka taea te faaohie tātai kaupapa anake ki te hiranga o te ngā mahi. Ao te o te nui tino ki hokiate, whiwhi tatou i te raupapa e whai ake nei:

1. Whakakāhoretanga.

2. te taha.

3. Ko te disjunction, XOR.

4. Ko te pānga, ōrite.

Ka rite ki a koutou e taea te kite, anake te whakakorehanga o te taha, me te kore e whai matua rite. Ko tētahi kaupapa o te disjunction me XOR e rite, kia rite ki te pai kia rite ki te kaupapa matua o te pānga, me te ōrite.

Functions o pānga me te ōritenga

Rite ki ta matou, i roto i te tua ki te whakahaeretanga o arorau taketake, arorau pāngarau me te ariā o te hātepe te whakamahi i pärönaki. Ko reira tino maha te pānga, me te ōrite.

Te pānga ranei hopearaa arorau - tenei tauākī, i roto i nei kotahi mahi ko te huru, me te tahi atu - te hua o tona whakatinanatanga. I roto i te mau parau te tahi atu, tenei tono ki te take o "ki te ... ka". "I muri i te tina mai te tauanga." E. hoki peia ki te kia tightened i runga i te maunga reti. Ki te kahore he hiahia ki te neke iho i te maunga, a ka toia te reti e kore te mea e tika ana. Kua oti te tuhituhi pera: A → B A⇒B ranei.

Ōritenga uhinga e puta te pānga kupenga anake ina e pono tauwhakaarotau e rua. Hei tauira, te po homai ara ki te ra ka (a ka anake), ina ara ake te ra ki runga ki te pae. I roto i te reo o te arorau pāngarau o tenei tauākī te tuhituhi rite A≡B, A⇔B, He B. ==

Ētahi atu ture o te taurangi Boolean

whakawhanake Algebra whakawa, a he maha kaipūtaiao hiahia ki te whakatakoto ture hou. E whakaaro ana te rongonui tino mau marôraa mathematician Scottish O. De Morgan. kite ia a ka hoatu he whakamāramatanga o taua āhuatanga rite whakakorehanga tata, tua me te negative rua.

fokotu'u mai Katia whakakāhoretanga e te aroaro o te reu he kahore whakakahore: e kore (A ranei B) A = kore B. NOT ranei

No ka whakakahore te tauwhakaarotau te, ahakoa o tona uara, mea atu e pā ana ki tua atu:

B∧¬B = 0; B∨¬B = 1.

Na rawa, te whakakorehanga rua iho tikanga ai. ie i mua i rānei whakakorehanga tauwhakaarotau riro tonu kotahi anake ranei.

Me pēhea te ki te whakaoti whakamātautau

Logic titau te whakangāwaritanga tomu'a whārite. Just rite i roto i te taurangi Takoto, ko reira e tika ana ki te whakahaere i maximally huru tuatahi (ki te pehea e ngā mahi tāuru uaua, a ki a ratou), ka tīmata titiro mo te whakahoki tika.

Aha ki te mahi ki te faaohie? Tahuri nga pärönaki katoa i roto i te mahi ohie. Na ka hurahia nga taiapa katoa (ranei hurihia, ki te hanga i te taiapa ki te whakaiti i tēnei huānga). kia te taahiraa i muri mai ki te whakamahi i ngā āhuatanga taurangi Boolean i roto i te mahi (ngā āhuatanga absorption kore me kotahi, me t.).

I te pae hopea, e ngā te whārite o te maha mōkito o taurangi, ngā ki ngā mahi ohie. Ko te huarahi māmā ki te titiro mo te otinga, ki te hanga koe i te maha nui o kino tata. Na ka pakū te whakahoki ake me te mea i tahaki.