Euler daim duab. Euler daim duab - piv txwv nyob rau hauv logic

Leonhard Euler (1707-1783) - nto moo Swiss thiab Lavxias teb sab mathematician, ib tug tswv cuab ntawm lub St. Petersburg Academy ntawm Sciences, feem ntau ntawm nws lub neej nyob rau hauv Russia. Qhov zoo tshaj plaws-paub nyob rau hauv lub zauv tsom xam, statistics, ntaub ntawv, thiab cov logic yog suav tias yog lub vajvoog Eulerian (Euler-Venn daim duab) siv los qhia tias tus uas duav txhua yam ntawm lub ntsiab thiab cov ntsiab ntawm cov poob lawm.

Dzhon Venn (1834-1923) - English philosopher thiab logician, co-neeg sau ntawm lub Euler-Venn diagrams.

Tau tshaj thiab incompatible tswv yim

Lub sij hawm logic hais txog ib daim ntawv ntawm kev xav, xav qhov tseem ceeb nta ntawm ib chav kawm ntawv zoo sib xws cov khoom. Lawv yog muab los ntawm ib tug los yog ib pab pawg neeg ntawm cov lo lus, "lub ntiaj teb no daim ntawv qhia", "Lub hom kvintseptakkord", "Monday" thiab lwm tus neeg.

Nyob rau hauv lub rooj plaub uas lub ntim ntsiab ntawm lub tswvyim ntawm tawg ntho los yog cov uas los ntawm lub ntim ntawm lub lwm yam, tham txog tshaj tswv yim. Yog hais tias tej lub caij uas lub ntim txhais tswvyim tsis mus rau lub Scope ntawm cov lwm yam, peb muaj ib qho chaw uas incompatible tswv yim.

Nyob rau hauv lem, txhua tus lub hom tswv yim nws muaj nws tus kheej set tau kev sib raug zoo. nws yog tsis tau tshaj rau cov nram qab no lub tswv yim:

• qhia tias yog leejtwg (equivalence) tagnrho;
• tshuam (overlap) tagnrho;
• subordination (subordination).

Rau incompatible:

• subordination (kev sib koom tes);
• zoo (contraries);
• contradiction (kontradiktornost).

Schematically, lub kev sib raug zoo ntawm cov ntsiab ntawm logic yuav xaiv siv cov voj voog ntawm Euler-Venn.

equivalence kev sib raug zoo

Nyob rau hauv cov ntaub ntawv no, lub tswvyim ntawm txhais tau tib yam. Raws li, qhov nyiaj ntawm cov ntaub ntawv ntsiab lus yog tib yam. Piv txwv li:

A - Sigmund Freud;

Nyob rau hauv - tus tsim ntawm psychoanalysis.

yog:

A - square;

B - ib tug equilateral duab plaub;

C - equiangular rhombus.

Siv rau xa mus rau tag zoo tib yam voj voog Euler.

Cov kev tshuam (overlap)

Qhov no muaj xws li qeb ntawm lub tswvyim ntawm kev sib koom yam uas hais muaj nyob rau hauv relation mus kev sib tshuam. Hais tias yog, tus nqi ntawm ib qho ntawm cov tswv yim yog cov muaj nyob rau hauv lub Scope ntawm lwm:

A - tus kws qhia ntawv;

B - music kiv cua.

Raws li pom los ntawm qhov no, qhov ntim ntawm lub tswv yim sib tshooj: cov xib fwb tej pab pawg neeg tej zaum yuav music lovers, thiab vice versa - cov suab paj nruag kiv cua yuav ua tau cov neeg sawv cev ntawm kev qhia profession. Ib tug zoo xws li cov piv yuav nyob rau hauv lub rooj plaub uas lub tswvyim A ua, piv txwv li, "pej xeem" thiab raws li B - "autodriver".

Raws (subordination)

Schematically qhia raws li txawv scale Euler daim duab. Cov kev sib raug zoo ntawm cov ntsiab lus nyob rau hauv cov ntaub ntawv no yog yus muaj los ntawm qhov tseeb hais tias ib tug subordinate tswvyim (tsawg heev volume) yog tag nrho ib feem ntawm lub subordinating (loj volume). Nyob rau hauv cov ntaub ntawv no, tus qhev tsis exhaust lub tswvyim tag nrho raws li.

Piv txwv li:

A - tsob ntoo;

B - ntoo thuv.

Lub tswvyim yuav subordinate mus rau lub tswvyim A. Vim ntoo thuv siv rau cov ntoo, lub sij hawm A yuav subordinating nyob rau hauv qhov kev piv txwv, "absorbing" tswvyim volume V.

Subordination (kev sib koom tes)

Piv qhia cov ob los yog ntau tshaj cov ntsiab lus nrog kom, tab sis teej tug nyob rau lub teev qhia generic ntau yam. Piv txwv li:

A - Clarinet;

Nyob rau hauv - guitar;

C - nkauj laus ncas;

D - ib tug suab paj nruas ntsuas.

Lub tswvyim ntawm A, B, C yog tsis overlapping nrog hwm rau txhua lwm yam, Txawm li cas los, tag nrho lawv mus rau cov qeb ntawm cov suab paj nruas seev (lub tswvyim D).

Tus txheem (contraries)

Tawm tsam muaj kev sib raug zoo ntawm cov ntsiab lus ntawm mean relatedness cov ntaub ntawv tswv yim mus rau cov tib genus. Yog li ib tug ntawm cov tswv yim muaj tej yam zog (nta), thaum lawv lwm yam tsis hloov opposite nyob rau hauv cov ua cim. Yog li, peb yuav soj ntsuam nrog txhais ntxeev. Piv txwv li:

A - cov ntsias;

B - giant.

Euler vajvoog nyob rau opposite kev sib raug zoo ntawm cov nqe lus yog muab faib ua peb theem, lub ua ntej uas sau raws nkaus Ii mus rau lub tswvyim A, lub thib ob - nyob rau hauv lub tswvyim, thiab lub thib peb - tus so tau lub tswv yim.

Controversy (kontradiktornost)

Nyob rau hauv cov ntaub ntawv no, tag nrho ob qho tswv yim yog views ntawm tib hom. Raws li nyob rau hauv lub yav dhau los piv txwv li, ib tug ntawm cov tswv yim qhia tej yeeb yam zoo (cwj pwm), thaum lub lwm yam tsis kam muab lawv. Txawm li cas los, nyob rau hauv sib piv rau lub opposite cwj pwm, lub thib ob, cov lus rov tswvyim, tsis yog ib tug muaj kev hloov rau cov khoom teejtug tsis kam muab lwm. Piv txwv li:

A - ib tug yooj yim ua hauj lwm;

B - yooj yim ua hauj lwm (non-A).

Qhia cov uas muaj lub tswv yim ntawm no zoo, lub Euler vajvoog yog muab faib ua ob qhov chaw - ib tug thib peb, intermediary nyob rau hauv cov ntaub ntawv no tsis muaj nyob. Yog li, lub tswv yim kuj yog txhais ntxeev. Nyob rau hauv cov ntaub ntawv no, ib tug ntawm lawv (A) ua zoo (pom tej yam hais txog) thiab lub thib ob (B los yog A) - tsis zoo (tsis tsim nyog qhov kos npe rau), "Dawb Ntawv" - "tsis yog ib tug ntawv dawb", "lub teb chaws keeb kwm" - "txawv teb chaws yav dhau los," thiab lwm yam ...

Yog li, lub ntim piv ntawm cov tswv yim nyob rau hauv kev sib raug zoo rau txhua tus lwm yam yog ib tug tseem ceeb yam ntxwv txiav txim Euler voj voog.

Relations ntawm poob lawm

Peb kuj yuav tsum paub qhov txawv ntawm lub ntsiab thiab lub plurality ntawm volume uas sawv cev rau Euler voj voog. Lub tswvyim borrowed los ntawm plurality ntawm zauv science thiab muaj ib tug txaus sab. Piv txwv ntawm logic thiab lej tso saib nws raws li ib tug tej yam set ntawm cov khoom. Yam khoom lawv tus kheej yog hais ntawm lub teeb. "Ib tug ntau muaj ib tug ntau, raws li ib tug conceivable" (Georg Cantor, qhov tus tsim ntawm txheej kev tshawb xav).

Los poob lawm nqa los ntawm cov niam ntawv loj A, B, C, D ... thiab lwm yam, cov ntsiab ntawm lub poob lawm - lowercase: .. A, b, c, d ... thiab lwm yam piv txwv ntawm lub teeb yuav tsum menyuam kawm ntawv nyob rau hauv tib lub chav kawm ntawv, cov phau ntawv sawv. rau ib lub txee (los yog, piv txwv li, tag nrho cov phau ntawv nyob rau hauv ib tug tsev qiv ntawv), cov nplooj ntawv nyob rau hauv lub chaw muag mis nyuj, berries nyob rau hauv ib lub hav zoov glade, thiab hais txog. d.

Nyob rau hauv lem, yog ib yam set tsis muaj ib yam hais, ces nws yog hu ua ib tug dawb paug kos npe rau thiab qhia Ø. Piv txwv li, ib tug plurality ntawm kev tshuam cov ntsiab lus ntawm thaum uas tig mus kab, ib tug plurality ntawm kev daws rau kab zauv x ² = -5.

Lub rooj sib tham txoj kev sib tw

Yuav kom daws tau ib tug loj tus naj npawb ntawm kev pab raws qib yog lug siv Euler daim duab. Piv txwv qhia cov logic ntawm kev sib txuas lus logic hauj lwm teev kev tshawb xav. Nws yuav siv lub tswv yim ntawm kev tseeb rooj. Piv txwv li, rau lub voj voog denoted Ib lub npe yog ib qhov tseeb no. Yog li, lub cheeb tsam sab nraum lub voj voog yuav tsum yog ib tug dag. Yuav kom txiav txim seb lub cheeb tsam ntawm lub daim ntawv rau lub logic lag luam yuav tsum tau daug lawm cheeb tsam defining Euler daim duab nyob rau hauv uas nws tseem ceeb rau lub ntsiab A thiab B yog muaj tseeb tiag.

Siv Euler voj voog nyob ntau ua daim ntawv thov nyob rau hauv ntau yam industries. Piv txwv li, nyob rau hauv ib qhov teeb meem nrog ib tug kws xaiv. Yog hais tias qhov kev kawm yog muaj kev txhawj xeeb txog kev xaiv ib tug yav tom ntej txoj hauj lwm, nws yuav tsum tau ua los ntawm cov nram qab no cov txheej xwm:

W - kuv nyiam dab tsi ua li cas?

D - hais tias kuv yuav tau txais?

P - tshaj qhov kuv yuav ua zoo nyiaj?

Peb sawv cev no nyob rau hauv daim ntawv ntawm cov kos duab: Euler daim duab (piv txwv nyob rau hauv logic - cov kev tshuam piv):

Cov tshwm sim yuav tsum tau cov neeg professions uas yuav yuav nyob rau hauv cov kev tshuam ntawm lub peb voj voog.

Cais qhov chaw nram qab no Euler-Venn nyob nyob rau hauv kev kawm txog zauv (set kev tshawb xav) nyob rau hauv lub xam ntawm ob peb ua ke thiab zog. Euler daim duab plurality duab ntsiab uas xa tuaj nrog nyob rau hauv tus duab plaub ntawm lub universal set (U). Es tsis txhob hauv lub voj voog yuav kuj yuav siv lwm yam kaw cov nuj nqis, tab sis lub essence tseem zoo li qub. Nuj nqis tshuam nrog txhua lwm yam, raws li tej yam kev mob ntawm qhov teeb meem (nyob rau hauv lub feem ntau kev cov ntaub ntawv). Tsis tas li ntawd, cov ntaub ntawv cov nuj nqis yuav tsum tau sau npe lo rau raws li. Raws li lub ntsiab nyob rau hauv kev saib xyuas tej zaum yuav ua poob lawm cov ntsiab lus nyob hauv qhov sib txawv theem ntawm daim duab. Raws li nws tau ntxoov ntxoo ib cheeb tsam, yog li designating tus tshiab khiv tsim teeb.

Nrog cov ntaub ntawv poob lawm nws yog permissible ua yooj yim xaam ua hauj lwm: Ntxiv (sum ntawm cov poob lawm hais), kev rho tawm (sib txawv), sib npaug (khoom). Nyob rau hauv tas li ntawd, ua tsaug rau lub Euler-Venn diagrams yuav ua tau haujlwm rau lub teeb kev sib piv rau tus naj npawb ntawm lawv cov constituent hais, tsis suav lawv.