Euler umzobo. Euler idayagramu - imizekelo ubuhlakani

Leonhard Euler (1707-1783) - edumileyo yezibalo Swiss kwaye Russian, ilungu le-St Petersburg Academy of Sciences, uninzi lobomi bakhe eRashiya. Yeyona yaziwa apha kuhlalutyo zemathematika, amanani, ulwazi, kwaye logic ithathwa isangqa Eulerian (Euler-weVenn umzobo) osetyenziswa ukubonisa imihlaba iingqiqo kunye nezinto yeeseti.

Dzhon weVenn (1834-1923) - sobulumko IsiNgesi kunye logician, intsebenziswano nombhali imizobo Euler-weVenn.

iingqiqo ziyahambisana engahambelaniyo

Ingqiqo gama libhekisela uhlobo yokucinga, ebonisa iimpawu ezibalulekileyo udidi izinto ezifanayo. Bona babonakala omnye okanye iqela lamagama, "imephu yehlabathi", "Le kvintseptakkord lwiimi", "ngoMvulo", kunye nabanye.

Kwimeko apho iziqalelo umthamo ingqikelelo ephethwe ngokupheleleyo okanye ngokungaphelelanga umthamo enye, sithetha izinto ezihambelanayo. Ukuba naliphi na elementi umthamo echazwe ingqiqo asisiso kumda enye, sibe nendawo kunye concepts ezingahambelaniyo.

Ngenxa yoko, ngamnye iintlobo kwengqiqo sinengqokelela yaso ulwalamano kunokwenzeka. iyahambisana ngenxa yala magama alandelayo:

• yesazisi (ukulingana) miqulu;
• ekudibaneni (ziyangenana) miqulu;
• ngaphantsi (ngaphantsi).

Ukuze engahambelaniyo:

• ngaphantsi (uququzelelo);
• umahluko (contraries);
• bethana (kontradiktornost).

Ngokwalo, ulwalamano phakathi kwala magama ekuqiqeni nga kutyunjwe ngokusebenzisa izangqa ka Euler-weVenn.

ubudlelwane ukulingana

Kulo mzekelo, ingqikelelo athetha into enye. Ngokunjalo, isixa iingqiqo data ziyafana. Umzekelo:

A - uSigmund Freud;

In - umsunguli psychoanalysis.

nokuba:

A - square;

B - an uxande alinganayo;

C - irhombus equiangular.

Lisetyenziswa ukubhekisela izangqa twatse ngokupheleleyo Euler.

Kuhlangana (ukudlulela)

Olu didi lubandakanya ingqikelelo ukwabelana izinto rhoqo afumaneka ngokunxulumene ekudibana kuzo. Oko kukuthi, isixa-mali omnye iikhonsepthi engagqibekanga ifakiwe kwenkululeko kwenye:

A - utitshala;

B - bomculo.

Njengoko sibonile kulo mzekelo, umthamo kweekhonsepthi ezilala: Ootitshala iqela elithile kunokuba abathandi umculo, kunye vice versa - phakathi fans umculo kuba nabameli kumsebenzi wokufundisa. A Umlinganiselo efanayo baya kuba kwimeko apho kukho ingqiqo A lenza, umzekelo, "umhlali" kwaye njengoko B - "autodriver".

Ukungeniswa (mncane)

Ngokwalo eziboniswe njengezehlelo ezahlukeneyo scale Euler umzobo. Ubudlelwane phakathi kwala magama kulo mzekelo luphawulwa kukuba ingqiqo owayamileyo (lokunyusa ivolyum encinci) is fully yinxalenye subordinating (unyuse ivolyum ngaphezulu). Kulo mzekelo, ikhoboka ifikelele ingqiqo sithobela ngokupheleleyo.

Umzekelo:

A - emthini;

B - pine.

Ingcinga iya kuba ilawulwe ingqiqo A. Ekubeni pine isebenza imithi, elithi A iba subordinating kulo mzekelo, "ukufunxa" umthamo Ingcamango V.

Ngaphantsi (uququzelelo)

Ratio ibonisa amagama amabini okanye ngaphezulu ngokufanayo, kodwa endingowakhe apho ecacisiwe ekwabelwana uluhlu generic. Umzekelo:

A - ऑप्टीमायझर isilungiseleli;

In - guitar;

C - violin;

D - a isixhobo somculo.

Ingcamango of A, B, C abayi idlulela ngokuphathelele omnye komnye, Noko ke, bonke baya kuba kudidi zezixhobo zomculo (ingqiqo D).

I malungana (contraries)

Abachaseneyo ubudlelwane phakathi kwala magama kwengqiqo data relatedness ayivani sodidi efanayo. Ngoko omnye iikhonsepthi uye iipropati ezithile (iimpawu), lo gama ezinye zazo aphike ngokubuyisela i echasayo umlinganiswa. Ngenxa yoko, nathi sisebenzisana kunye nezichasi. Umzekelo:

A - i ezincinanana;

B - giant.

Euler isangqa kwi ubudlelwane malunga phakathi imiqathango yahlulwe ngokwamacandelo ezintathu, owokuqala okuhambelana ingqiqo A, okwesibini - ekwahluleni, kwaye eyesithathu - iingqiqo lokuphumla kunokwenzeka.

Impikiswano (kontradiktornost)

Kulo mzekelo, zombini iingqiqo izimvo uhlobo olufanayo. Njengokuba bekunjalo kumzekelo wokuqala, omnye iikhonsepthi ibonisa iimpawu ezithile (iimpawu), lo gama enye waphika kubo. Nangona kunjalo, xa kuthelekiswa sengqondo esahlukileyo, owesibini, isigama esahlukileyo, indawo impahla wakhanyela enye. Umzekelo:

A - umsebenzi onzima;

B - umsebenzi olula (non-A).

Evakalisa kwenkululeko iingqiqo hlobo, isangqa Euler yahlulwe kubini - wesithathu, umlamleli kulo mzekelo akakho. Ngoko ke, iingqiqo nazo ezi izichasi. Kulo mzekelo, omnye kubo (A) uba entle (avume naziphi ebonisa) kunye yesibini (B okanye A) - okubi (bemkhanyela umqondiso efanelekileyo), "IPhepha leNgcaciso" - "ayikho iphepha emhlophe", "imbali yesizwe" - "imbali amazwe," njalo-njalo ...

Ngoko ke, umlinganiselo umthamo kweekhonsepthi ngokunxulumene omnye luphawu oluphambili isigqibo Euler izangqa.

Relations phakathi iiseti

Kufuneka kananjalo kwahlule phakathi kwezakhi kunye nobuninzi umthamo nenika Euler izangqa. Ingcinga waboleka ukusuka nobuninzi yenzululwazi lweMathematika kwaye ebanzi ngokwaneleyo. Imizekelo ye-logic nemathematika ukubonisa njengento iseti ethile yezinto. Izinto ngokwabo izinto iseti. "A lot abe kakhulu, njengoko enokuthelekelelwa" (Georg Cantor, umseki kwethiyori iseti).

iiseti Isikhundla olwenziwe yi oonobumba A, B, C, D ... njl, izinto yeeseti - nabancinci: .. A, b, c, d ... njl Imizekelo iseti abafundi zibekwe eklasini enye, iincwadi bemi. kwi kwithala ethile (okanye, umzekelo, zonke iincwadi kwithala leencwadi ethile), amaphepha kwi kwidayari, amaqunube kwi Glade ehlathini, njalo njalo. d.

Kwelinye icala, ukuba kukho iseti ethile ayiqulathanga naluphi na iimpawu, ngoko ke kuthiwa umqondiso blank and bonisa Ø. Umzekelo, kukho ezininzi iingongoma ekudibaneni yemigca parallel, kukho ezininzi izisombululo lenxaki x ² = -5.

Ukwanelisa mngeni

Ukusombulula inani elikhulu lemisebenzi ezisetyenziswa Euler umzobo. Imizekelo babonise ingqiqo yonxibelelwano imisebenzi logic wabeka theory. Isebenzisa ingqiqo inyaniso iyidandalazise. Umzekelo, isangqa zibonakaliswe Igama a domain inyaniso. Ngenxa yoko, le ndawo ngaphandle kwesangqa iya kuba bubuxoki. Ukuze ubone indawo yetshathi ukusebenza logic kufuneka lingaqandusela imimandla ekuchazeni Euler umzobo apho imilinganiselo yayo iziqalelo A no-B iyinyaniso.

Ukusebenzisa izangqa Euler wafumana ngokubanzi isicelo iluncedo mveliso eyahlukeneyo. Umzekelo, kwimeko nge ukhetho yobungcali. Ukuba isifundo esichaphazelekayo ngokukhetha umsebenzi elizayo, inokubuyiselwa elikhokelwa ziikhrayitheriya zilandelayo:

W - into othanda ukuyenza?

D - endikuzuzayo?

P - kunokuba ndingenza imali elungileyo?

Simela oku ngokohlobo imizobo: Euler umzobo (imizekelo ubuhlakani - umlinganiselo kuhlangana):

Umphumo iya kuba ezo imisebenzi eziya kuba ekudibaneni kwe kwizangqa ezintathu.

indawo eqhiwukileyo alandelayo Euler-weVenn ndibala kwimathematika (theory iseti) ekubaleni lwendibaniso kunye neepropati. Euler idayagramu nobuninzi yomfanekiso izinto livalelwe uxande ebonisa isethi jikelele (U). Endaweni nezangqa nazo zinokusetyenziswa amanye amanani ezivaliweyo, kodwa undoqo uhlala okufanayo. Amanani yendlela yokwenza omnye komnye, njengoko iimeko ingxaki (kwimeko ezininzi jikelele). Kwakhona, amanani data zizakuboniswa ngokufanelekileyo. Njengoko izinto phantsi kunokucingwa benze iiseti amanqaku ibekwe ngaphakathi inxalenye ezahlukeneyo mzobo. Ngokusekelwe kulo kungaba umthunzi kwindawo ethile, ngokwenjenjalo ukukhomba isethi elalisandul 'ukusekwa.

Kwiintlobo data kuvumelekile ukwenza imisebenzi ezisisiseko zemathematika: ukongeza (sum of iiseti lweziqalelo), ukuthabatha (umahluko), uphinda-phindo (imveliso). Ukongeza, ngenxa imizobo Euler-weVenn unako ukwenza imisebenzi nangamaxesha kuthelekiswa nenani izinto zabo wabavoti woluntu, zamadoda kuzo.