Euler daim duab: piv txwv thiab lub sij hawm

Kev kawm txog zauv no yeej tseem zoo paub daws teeb science, Yog hais tias koj tsiv mus deb ntawm lub tswv yim. Yog li, ib khub triple apples yuav graphically depict qhov yooj yim ua hauj lwm uas yog lub hauv paus ntawm kev kawm txog zauv, tab sis raws li sai raws li lub dav hlau ntawm kev ua si expands, cov khoom no yog tsis txaus. Leejtwg sim portray rau apples haujlwm rau infinite poob lawm? Qhov tseeb ntawm qhov teeb meem yog hais tias tsis muaj. Qhov ntau complex cov ntsiab, uas ua hauj lwm rau hauv lub lej nyob rau hauv nws kev txiav txim, qhov ntau problematic nws ciali lawv pom tau qhov sib qhia, uas yuav tsum tau tsim los pab txhawb kev nkag siab. Txawm li cas los, nyob rau hauv kev zoo siab li niaj hnub kawm ntawv, thiab science nyob rau hauv Feem ntau, tau tau rho nram qab no Euler, piv txwv thiab lub sij hawm uas peb sib tham txog hauv qab no.

Ib me ntsis keeb kwm

Plaub Hlis Ntuj 17, 1707 muab lub ntiaj teb no qhov kev kawm Leonarda Eylera - koj paub txog uas nws tau pab rau kev kawm txog zauv, physics, shipbuilding thiab txawm music theory tsis tau overestimated. Nws tej hauj lwm yeej pom thiab nyob rau hauv kev thov rau qhov no hnub nyob ib ncig ntawm lub ntiaj teb no, txawm lub fact tias science tsis sawv tseem. Tshwj xeeb yog txaus luag yog qhov tseeb hais tias Mr. Euler twb ncaj qha kev koom tes nyob rau hauv txoj kev loj hlob ntawm Lavxias teb sab lub tsev kawm ntawv ntawm kev kawm leb siab, qhov ntau li vim hais tias cov yuav ntawm txoj hmoo, nws ob zaug rov qab mus rau peb lub xeev. Tus kws tshawb fawb tau ib tug tshwj xeeb muaj peev xwm los tsim kom tau pob tshab nyob rau hauv nws logic algorithms, txiav tawm tag nrho cov ruaj thiab nyob rau hauv tsis muaj sij hawm mus los ntawm cov kev mus rau lub meej. Peb yuav tsis ntsuas tag nrho nws cov hauj lwm zoo, raws li nws yuav siv sij hawm ib tug txiav txim nqi ntawm lub sij hawm, thiab qhia rau peb rov qab mus rau qhov kev kawm ntawm tsab xov xwm. Nws yog nws leej twg pom tias kev siv ntawm ib tug graphical sawv cev ntawm kev ua hauj lwm nyob rau hauv poob lawm. Euler daim duab tov rau ib yam, txawm lub feem ntau nyuaj npaj, tau qhia visually.

Yuav ua li cas yog lub essence?

Nyob rau hauv kev xyaum, cov nram qab no Euler daim duab uas yog qhia hauv qab no yuav siv tau tsis tsuas nyob rau hauv kev kawm txog zauv, raws li lub tswvyim ntawm "poob lawm" yog tsis cim rau cov kev qhuab qhia. Yog li ntawd, lawv muaj tau ntse thov nyob rau hauv kev tswj.

Lub tswvyim qhia tau hais tias cov saum toj no kev sib raug zoo teev A (ib tug irrational tooj), B (rational zauv) thiab C (natural numbers). Voj qhia tias lub teeb yog muaj nyob rau hauv lub teeb B, ces lub teeb A tsis tshuam nrog lawv. Ib qho piv txwv ntawm ib tug yooj yim, tab sis kom meej meej piav txog cov specifics ntawm "kev sib raug poob lawm" uas yog ib yam nkaus thiab paub daws teeb rau ib tug tiag tiag kev sib piv hais tias tsuas yog vim hais tias ntawm lawv cov infinity.

logic algebra

Qhov no cheeb tsam ntawm mathematical logic ua hauj lwm cov nqe lus, uas yuav ua tau ob qho tib si yeej muaj tseeb thiab cuav ua cim. Piv txwv li, los ntawm cov elementary: tus xov tooj 625 yog divisible los ntawm 25, tus xov tooj 625 yog divisible los ntawm 5, tus xov tooj 625 yog yooj yim. Cov thawj thiab thib ob kev pom zoo - qhov tseeb, thaum lub sij hawm tom kawg - ib tug dag. Ntawm cov hoob kawm, nyob rau hauv kev xyaum nws yog ntau yooj yim, tab sis lub point yog qhia kom meej meej. Thiab, ntawm chav kawm, qhov kev txiav txim dua muab kev koom tes Euler daim duab, cov piv txwv ntawm lawv cov kev siv yog ib yam nkaus thiab yooj yim thiab intuitive yuav las mees lawv.

Ib me ntsis ntawm kev tshawb xav:

• Cia lub teeb A thiab B ua ib ke thiab yog tsis khoob, ces rau cov kev tshuam lag luam yog cov nram qab no txhais lub koom haum thiab negation.
• Kev tshuam ntawm poob lawm A thiab B muaj hais tias yuav mus rau tib lub sij hawm raws li tus txheej A thiab teem B.
• Ob peb ua ke ntawm A thiab B muaj hais tias yuav mus rau lub teeb ib tug los yog teem B.
• Ib tug negation ntawm lub teeb - ib tug txheej uas muaj cov ntsiab uas tsis yog lub teeb A.

Tag nrho cov no yog dua portrayed raws li Euler daim duab nyob rau hauv logic, raws li nrog lawv txhua ua hauj lwm, tsis hais txog ntawm cov neeg kawm ntawv kev nyuaj yuav khees thiab pom.

Axioms ntawm algebra ntawm logic

Xav hais tias hais tias 1 thiab 0 yog txhais thiab muaj nyob rau hauv ib tug ntau yam ntawm A, ces:

• Ib tug negation ntawm lub negation ntawm lub teeb yog lub teeb ntawm ib tug;
• Ib tug plurality ntawm union nrog ne_A yog 1;
• Ib tug plurality ntawm union 1 yog 1;
• Ib tug union ntawm lub teeb nrog nws tus kheej yog lub teeb A;
• Association of A 0 yog lub teeb A;
• Ib tug plurality ntawm kev tshuam nrog ne_A yog 0;
• Ib tug plurality ntawm cov kev tshuam nrog nws tus kheej yog lub teeb A;
• kev tshuam ntawm A 0 yog 0;
• kev tshuam ntawm A 1 yog set A.

Lub ntsiab thaj chaw ntawm lub algebra ntawm logic

Cia lub poob lawm A thiab B ua ib ke thiab yog tsis khoob, ces:

• rau kev tshuam thiab union ntawm poob lawm A thiab B ua commutative cai lij choj;
• rau kev tshuam thiab union ntawm poob lawm A thiab B ua associative cai lij choj;
• rau kev tshuam thiab union ntawm poob lawm A thiab B ua distributive kev cai lij choj;
• tsis kam muab cov kev tshuam ntawm A thiab B yog cov kev tshuam ntawm negations of A thiab B;
• tsis kam muab cov union ntawm poob lawm A thiab B yog lub union ntawm negations ntawm A thiab B.

Hauv qab no yog qhia raws li nram no Euler tshuam piv txwv thiab txuas lub poob lawm A, B thiab C.

prospects

Cov hauj lwm Leonarda Eylera rightly suav hais tias yog lub hauv paus ntawm niaj hnub kawm txog zauv, tab sis tam sim no lawv ntse siv nyob rau hauv cov cheeb tsam ntawm tib neeg kev ua si uas yog tus tshiab, coj tsawg kawg yog neeg coj: Euler daim duab, piv txwv thiab cov kab kos piav qhia txog cov mechanisms ntawm txoj kev loj hlob qauv, seb Lavxias teb sab los yog Anglo-American version .