Binary kev sib raug zoo thiab lawv lub zog

Ib tug ntau ntawm cov kev sib raug zoo rau Piv txwv li poob lawm nrog los ntawm ib tug loj tus naj npawb ntawm cov tswv yim txij thaum lawv txhais cov ntsiab lus thiab analytical tsom xam ntawm xaus paradox. Ib tug ntau yam ntawm cov ntsiab lus sib tham nyob rau hauv qhov tsab xov xwm nyob rau hauv lub teeb mus ib txhis. Txawm tias thaum tham txog lub dual hom, los ntawm qhov no yog meant ib binary kev sib raug zoo ntawm ob peb ntau yam. Thiab kuj ntawm cov khoom los yog utterances.

Raws li ib tug txoj cai, lub binary kev sib raug zoo yog qhia los ntawm R, uas yog, yog tias xRx rau tej nqi ntawm x nyob rau hauv lub teb ntawm R, xws li ib tug vaj tse yog hu ua reflexive, qhov twg x thiab x - yog ua khoom ntawm kev xav, thiab R yog ib tug kos npe rau ntawm ib co daim ntawv ntawm kev sib raug zoo ntawm cov neeg . Nyob rau tib lub sij hawm, yog hais tias tus express los yog xRy® yRx, nws hais txog symmetry xeev uas ® - lub implication kos npe rau, zoo ib yam li lub union ntawm "yog hais tias ... ces ..." Thiab thaum kawg, deciphering inscriptions (xRy Ùy Rz). ®xRz qhia txog transitive kev sib raug zoo, nrog rau cov kos npe rau ntawm u - qhov no yog ib tug tau.

Ib tug binary kev sib raug zoo uas yog ob qho tib si reflexive, yam, thiab transitive yog hu ua ib tug equivalence kev sib raug zoo. Qhov ratio ntawm f - ib tug muaj nuj nqi, thiab ntawm Î f thiab Î f implies lub koob pheej ntawm lawv y = z. Tej yam yooj yim binary muaj nuj nqi yuav muab tau yooj yim thov mus rau ob yooj yim cov lus sib cav teem nyob rau hauv ib tug txiav txim, thiab tsuas yog nyob rau hauv cov ntaub ntawv no, nws muab ib tug nqi rau nws, qhia cov ob kab zauv, noj nyob rau hauv ib tug cov ntaub ntawv.

Nws yuav tsum hais tias f maps x rau y, Yog hais tias f yog ib tug muaj nuj nqi ntawm lub cheeb tsam txhais cheeb tsam tseem ceeb x thiab y. Txawm li cas los, thaum extrapolates f x rau y, thiab y kuv z, ces qhov no ua rau lub fact tias f qhia tau hais tias nyob rau hauv x z. Ib tug piv txwv yooj yim: Yog hais tias f (x) = 2 x yog siv tau rau haum arbitrary integer x, ces peb hais tias f maps ib tug kos npe rau teeb ntawm tag nrho cov zauv paub rau ntau ntawm cov tib tag nrho, tab sis lub sij hawm no txawm tus xov tooj. Raws li hais saum toj no, cov binary kev sib raug zoo uas ib txhij reflexive, yam, thiab transitive, yog txoj kev sib raug ntawm equivalence.

Raws li cov saum toj no, cov kev sib raug zoo ntawm equivalence txiav txim los ntawm lub thaj chaw ntawm binary kev sib raug zoo:

• reflexivity - lub ratio (M ~ N);
• symmetry - yog hais tias koob pheej ntawm lawv M ~ N, yuav muaj N ~ M;
• transitivity - yog hais tias ob tug sib luag thiab M ~ N N ~ P, qhov tshwm sim M ~ P.

Thaum pom tias daim ntawv thov thaj chaw ntawm binary kev sib raug zoo nyob rau hauv ntau yam. Reflexivity - yog ib tug ntawm cov yam ntxwv ntawm ib co mus, qhov twg txhua lub caij ntawm lub xeem poob lawm yog nyob rau hauv no koob pheej ntawm lawv tus kheej. Piv txwv li, nyob nruab nrab ntawm cov zauv a = c thiab a³ nrog - reflexive kev sib txuas lus, vim hais tias muaj yog ib txwm ib tug = c = c, thiab a³, s³ nrog. Nyob rau tib lub sij hawm, qhov ratio ntawm inequality ib tug> c - antireflexive vim hais tias ntawm cov impossibility ntawm cov inequality ib tug> a. Lub axiom ntawm cov cuab yeej no yog kho cim: aRc® ara U CRC, ntawm no lub cim ® qhia lo lus "implies" (los yog "implies") thiab koj kos npe rau - es los ntawm "thiab" (los yog tau). Los ntawm daim ntawv no nws raws li nram no uas yog hais tias qhov tseeb ntawm ib lub taub Tawvv raws li muaj tseeb thiab Arc qhia ara thiab CRC.

Symmetry entails lub hav zoov ntawm cov kev sib raug zoo thiab yog hais tias lub hlwb khoom ntxeev, xws li ib tug symmetrical kev sib raug zoo rearrangement ntawm cov khoom tsis ua rau tus transformation ntawm lub daim ntawv "binary kev sib raug zoo." Piv txwv li, lub kev sib raug zoo ntawm koob pheej ntawm lawv ib tug = c yog symmetrical vim lub equivalence piv c = ib; kuj Attendance a¹s thiab kev txiav txim, raws li nws muaj raws li qhov kev sib txuas lus s¹a.

Transitive teeb - nws yog ib tug nyob rau hauv uas tau raws li cov nram qab no yuav tsum tau: ntawm Î x, z Î y ® z Î x, qhov twg ® ua raws li ib tug kos npe rau hloov cov lus: "yog hais tias ... ces ...". Hais lus mis li nyeem raws li: "Yog hais tias ywj siab ntawm x, z yog y, z raws li muaj nuj nqi ntawm x."