Ntau zaus hauv kev kawm lej muaj ntau qhov nyuaj thiab cov lus nug, thiab ntau cov lus teb tsis meej ib txwm. Tsis muaj kev zam yog ib lub ntsiab lus xws li cardinality ntawm cov teeb. Qhov tseeb, qhov no tsis muaj dab tsi ntau tshaj li tus lej qhia ntawm tus lej ntawm cov khoom. Hauv kev nkag siab dav dav, ib txheej yog axiom; nws tsis muaj lus txhais. Nws yog nyob ntawm tej yam khoom, los yog theej lawv cov teeb, uas yuav yog khoob, finite los yog infinite. Tsis tas li ntawd, nws muaj cov lej lossis cov lej, matrices, sequences, ntu thiab kab.
Hais txog qhov hloov pauv uas twb muaj lawm
Ib qho tsis muaj lossis khoob uas tsis muaj tus nqi tsim nyog yog suav tias yog lub ntsiab lus tseem ceeb vim tias nws yog ib qho subset. Kev sau ntawm tag nrho cov subsets ntawm ib qho tsis yog qhov khoob S yog ib pawg ntawm cov teeb. Yog li, lub hwj chim ntawm ib lub teeb muab suav hais tias yog ntau, xav tau, tab sis ib leeg. Cov txheej no yog hu ua lub zog ntawm S thiab yog qhia los ntawm P (S). Yog tias S muaj N cov ntsiab lus, ces P(S) muaj 2^n subsets, txij li lub subset ntawm P(S) yog ∅ los yog ib tug subset uas muaj r ntsiab los ntawm S, r=1, 2, 3, … Uas ntawm txhua yam infiniteteeb M yog hu ua lub zog kom muaj nuj nqis thiab yog lub cim qhia los ntawm P (M).
Cov ntsiab lus ntawm kev teeb tsa txoj kev xav
Qhov kev paub no yog tsim los ntawm George Cantor (1845-1918). Niaj hnub no nws yog siv nyob rau hauv yuav luag tag nrho cov ceg ntawm kev ua lej thiab ua haujlwm raws li nws qhov tseem ceeb. Nyob rau hauv cov txheej txheem kev xav, cov ntsiab lus tau sawv cev rau hauv daim ntawv teev npe thiab muab los ntawm hom (cov txheej txheem khoob, singleton, finite thiab infinite sets, sib npaug thiab sib npaug, universal), union, kev sib tshuam, qhov sib txawv, thiab ntxiv cov lej. Nyob rau hauv lub neej niaj hnub, peb feem ntau tham txog ib phau ntawm cov khoom xws li ib pawg ntawm cov yawm sij, ib pab yaj, ib pob ntawv, thiab lwm yam. Nyob rau hauv lej qib 5 thiab tshaj, muaj natural, integer, prime, thiab composite tooj.
Cov khoom hauv qab no tuaj yeem txiav txim siab:
- natural tooj;
- tus tsiaj ntawv;
- thawj qhov tsis sib xws;
- daim duab peb sab sib txawv.
Nws tuaj yeem pom tau tias cov piv txwv no yog cov khoom txhais tau zoo. Xav txog ob peb yam piv txwv ntxiv:
- tsib tus kws tshawb fawb nto moo tshaj plaws hauv ntiaj teb;
- xya tus ntxhais zoo nkauj hauv zej zog;
- peb tus kws phais zoo tshaj plaws.
Cov piv txwv cardinality no tsis yog cov khoom sau tau zoo, vim hais tias cov qauv "ntau tshaj", "zoo nkauj tshaj", "zoo tshaj" txawv ntawm ib tus neeg mus rau lwm tus.
sets
Tus nqi no yog tus lej txhais tau zoo ntawm cov khoom sib txawv. Xav tias:
- wordset yog cov lus sib dhos, sib sau ua ke, chav kawm thiab muaj cov ntsiab lus;
- khoom, cov tswv cuab sib npaug;
- sets feem ntau txhais tau los ntawm cov tsiaj ntawv loj A, B, C;
- set ntsiab yog sawv cev los ntawm cov ntawv me a, b, c.
Yog tias "a" yog lub ntsiab lus ntawm lub teeb A, ces nws tau hais tias "a" belongs rau A. Cia peb qhia cov kab lus "tseem" nrog cov cim Greek "∈" (epsilon). Yog li, nws hloov tawm tias a ∈ A. Yog tias 'b' yog lub ntsiab lus uas tsis yog ntawm A, qhov no yog sawv cev raws li b ∉ A. Qee qhov tseem ceeb uas siv hauv qib 5 lej yog sawv cev siv peb txoj hauv qab no:
- apps;
- registries los yog tabular;
- txoj cai tsim kev tsim.
Ntawm kev soj ntsuam ze dua, daim ntawv thov kev pab yog raws li hauv qab no. Nyob rau hauv cov ntaub ntawv no, ib tug meej piav qhia ntawm lub ntsiab ntawm lub teeb yog muab. Lawv tag nrho cov nyob rau hauv curly braces. Piv txwv li:
- set ntawm tus lej khib tsawg dua 7 - sau li {tsawg dua 7};
- cov lej ntau dua 30 thiab tsawg dua 55;
- ntau tus menyuam kawm ntawv hauv chav kawm uas hnyav dua tus kws qhia.
Nyob rau hauv daim ntawv sau npe (cov lus), cov ntsiab lus ntawm cov txheej txheem tau teev nyob rau hauv ib khub ntawm pawg {} thiab sib cais los ntawm commas. Piv txwv li:
- Cia N qhia txog cov txheej txheem ntawm thawj tsib tus lej. Yog li ntawd, N=→ register form
- Txhua tus lej ntawm cov tsiaj ntawv Askiv. Li no V={a, e, i, o, u, y} → register form
- Txhua tus lej khib yog tsawg dua 9. Yog li ntawd, X={1, 3, 5, 7} → daim ntawvnpe
- Set of all letters in lo lus "Math". Yog li ntawd, Z={M, A, T, H, E, I, C, S} → Registry Form
- W yog qhov teeb tsa ntawm plaub lub hlis dhau los ntawm lub xyoo. Yog li ntawd, W={September, October, November, December} → registry.
Nco ntsoov tias qhov kev txiav txim uas cov ntsiab lus tau teev tseg tsis muaj teeb meem, tab sis lawv yuav tsum tsis txhob rov ua dua. Ib daim ntawv tsim los ntawm kev tsim kho, nyob rau hauv ib rooj plaub, ib txoj cai, qauv los yog tus neeg teb xov tooj yog sau nyob rau hauv ib tug khub ntawm brackets kom lub teeb yog raug txhais. Hauv daim ntawv tsim tsim, txhua lub ntsiab lus yuav tsum muaj tib yam khoom los ua tus tswv cuab ntawm tus nqi hauv nqe lus nug.
Nyob rau hauv daim ntawv teev lus sawv cev, ib lub ntsiab lus ntawm cov teeb tau piav qhia nrog tus cim "x" lossis lwm yam kev hloov pauv uas ua raws li cov nyuv (":" lossis "|" yog siv los qhia). Piv txwv li, cia P ua tus lej suav tau ntau dua 12. P nyob rau hauv daim ntawv tsim tsim tau sau ua - { suav tau tus lej thiab ntau dua 12}. Nws yuav nyeem nyob rau hauv ib txoj kev. Ntawd yog, "P yog ib pawg x cov ntsiab lus xws li x suav tau thiab ntau dua 12."
daws piv txwv siv peb txoj kev sawv cev: tus lej ntawm tus lej ntawm -2 thiab 3. Hauv qab no yog cov piv txwv ntawm ntau hom kev teev:
- Ib qho khoob lossis null teeb uas tsis muaj cov ntsiab lus thiab yog cim los ntawm lub cim ∅ thiab nyeem ua phi. Hauv daim ntawv teev npe, ∅ sau {}. Cov txheej txheem finite yog khoob, txij li tus naj npawb ntawm cov ntsiab lus yog 0. Piv txwv li, cov txheej txheem ntawm cov lej suav tsawg dua 0.
- Pom tseeb yuav tsum tsis txhob muaj <0. Yog li ntawd, qhov nonpliag set.
- Ib lub teeb uas tsuas muaj ib qho sib txawv yog hu ua singleton set. Tsis yog yooj yim los yog compound.
Finite set
Ib lub teeb uas muaj qee yam ntawm cov ntsiab lus hu ua qhov kawg lossis tsis kawg. Empty hais txog thawj. Piv txwv li, ib txheej ntawm tag nrho cov xim nyob rau hauv zaj sawv.
Infinity yog ib txheej. Cov ntsiab lus hauv nws tsis tuaj yeem suav sau. Ntawd yog, muaj cov hloov pauv zoo sib xws yog hu ua infinite set. Piv txwv:
- lub zog ntawm qhov teeb tsa ntawm txhua lub ntsiab lus hauv dav hlau;
- set ntawm txhua tus lej tseem ceeb.
Tab sis koj yuav tsum nkag siab tias tag nrho cov cardinalities ntawm lub koom haum ntawm ib pawg tsis tuaj yeem hais tawm hauv daim ntawv teev npe. Piv txwv li, tus lej tiag, vim lawv cov ntsiab lus tsis sib haum rau ib qho qauv tshwj xeeb.
Tus naj npawb cardinal ntawm ib lub teeb yog tus naj npawb ntawm cov ntsiab lus sib txawv hauv qhov muab ntau A. Nws yog denoted n (A).
Piv txwv:
- A {x: x ∈ N, x <5}. A={1, 2, 3, 4}. Yog li ntawd, n (A)=4.
- B=teeb tsa cov ntawv hauv lo lus ALGEBRA.
Sib npaug rau kev sib piv
Ob daim cardinalities ntawm lub teeb A thiab B yog xws li yog tias lawv tus lej cardinal zoo ib yam. Lub cim rau qhov sib npaug yog "↔". Piv txwv li: A ↔ B.
Sib npaug: ob daim cardinalities ntawm pawg A thiab B yog tias lawv muaj tib lub ntsiab lus. Txhua tus coefficient ntawm A yog qhov sib txawv los ntawm B, thiab txhua tus B yog tus nqi ntawm A. Yog li ntawd, A=B. Qhov sib txawv ntawm cov koom haum cardinality thiab lawv cov ntsiab lus tau piav qhia siv cov piv txwv muab.
Qhov tseem ceeb ntawm kev txiav txim siab thiab infinity
Dab tsi yog qhov sib txawv ntawm lub cardinality ntawm ib lub finite set thiab ib tug infinite teeb?
thawj tus nqi muaj lub npe hauv qab no yog tias nws yog khoob lossis muaj tus lej kawg ntawm cov ntsiab lus. Nyob rau hauv ib tug finite set, ib tug txawv txawv yuav teev tau yog hais tias nws muaj ib tug txwv. Piv txwv li, siv lub ntuj naj npawb 1, 2, 3. Thiab cov txheej txheem teev npe xaus ntawm qee qhov N. Cov naj npawb ntawm cov ntsiab lus sib txawv suav hauv cov finite set S yog denoted los ntawm n (S). Nws tseem hu ua kev txiav txim lossis cardinal. Symbolically denoted raws li lub hauv paus ntsiab lus. Yog li, yog tias lub teeb S yog cov tsiaj ntawv Lavxias, ces nws muaj 33 yam. Nws tseem ceeb heev uas yuav tsum nco ntsoov tias lub ntsiab lus tsis tshwm sim ntau dua ib zaug hauv ib lub teeb.
Infinite in the set
Ib txheej yog hu ua infinite yog tias cov ntsiab lus tsis tuaj yeem suav. Yog hais tias nws muaj ib tug unbounded (uas yog, uncountable) natural tooj 1, 2, 3, 4 rau tej n. Ib txheej uas tsis muaj qhov kawg yog hu ua infinite. Tam sim no peb tuaj yeem sib tham txog cov piv txwv ntawm tus lej muaj nuj nqis nyob rau hauv kev xav. Kev xaiv tus nqi kawg:
- Cia Q={natural tooj tsawg dua 25}. Tom qab ntawd Q yog ib qho finite teeb thiab n (P)=24.
- Cia R={ tus lej nruab nrab ntawm 5 thiab 45}. Ces R yog ib tug finite set thiab n (R)=38.
- Cia S={ntaus modulo 9}. Tiam S={-9, 9} yog ib tug finite set thiab n (S)=2.
- Txhua tus neeg.
- Ntau tus noog.
piv txwv tsis kawg:
- tus lej ntawm cov ntsiab lus uas twb muaj lawm ntawm lub dav hlau;
- tus naj npawb ntawm tag nrho cov ntsiab lus hauv ntu ntu;
- txheej ntawm cov lej zoo sib faib los ntawm 3 yog qhov tsis kawg;
- txhua tus lej thiab tus lej.
Yog li, los ntawm cov laj thawj saum toj no, nws paub meej tias yuav ua li cas thiaj paub qhov txawv ntawm qhov kawg thiab tsis kawg.
Lub zog ntawm qhov txuas txuas ntxiv
Yog tias peb sib piv cov teeb thiab lwm yam txiaj ntsig uas twb muaj lawm, ces qhov sib ntxiv yog txuas rau lub teeb. Yog tias ξ yog universal thiab A yog ib feem ntawm ξ, ces qhov sib ntxiv ntawm A yog tus naj npawb ntawm tag nrho cov ntsiab lus ntawm ξ uas tsis yog cov ntsiab lus ntawm A. Symbolically, qhov ntxiv ntawm A nrog rau ξ yog A'. Piv txwv li, 2, 4, 5, 6 yog cov ntsiab lus ntawm ξ uas tsis koom nrog A. Yog li ntawd, A'={2, 4, 5, 6}
Ib txheej nrog cardinality txuas ntxiv muaj cov yam ntxwv hauv qab no:
- ua kom tiav ntawm cov khoom thoob ntiaj teb yog tus nqi khoob hauv nqe lus nug;
- qhov no null teev hloov pauv yog universal;
- nyiaj thiab nws cov khoom ntxiv tsis sib koom.
Piv txwv:
- Cia tus lej ntawm tus lej ntuj yog qhov teeb tsa thoob ntiaj teb thiab A yuav txawm tias. Ces A '{x: x yog ib qho khib nrog tib tus lej}.
- Cia ξ=teeb ntawm cov tsiaj ntawv hauv cov tsiaj ntawv. A=set of consonants. Ces A '=tus naj npawb ntawm cov vowels.
- Qhov ntxiv rau cov txheej txheem thoob ntiaj teb yog qhov khoob khoob. Yuav tau denoted los ntawm ξ. Ces ξ '=Cov txheej ntawm cov ntsiab lus uas tsis suav nrog ξ. Lub khoob khoob φ yog sau thiab denoted. Yog li ntawd ξ=φ. Yog li, qhov ntxiv rau cov txheej txheem thoob ntiaj teb yog khoob.
Hauv kev ua lej, " txuas ntxiv" qee zaum siv los sawv cev ib kab tiag. Thiab feem ntau, los piav txog cov khoom zoo sib xws:
- txuas ntxiv (hauv txoj kev xav) - kab tiag lossis tus lej sib raug zoo;
- linear - ib qho xaj xaj uas qhia qee yam khoom ntawm kab tiag;
- txuas ntxiv (hauv topology) - tsis yog qhov khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khoob khawm
- qhov kev xav tias tsis muaj qhov kawg tsis kawg yog ntau dua cov lej tab sis me dua cov lej tiag;
- lub zog ntawm qhov txuas ntxiv yog tus lej xov tooj sawv cev qhov loj ntawm cov lej tiag.
Qhov tseem ceeb, qhov txuas ntxiv (kev ntsuas), kev xav lossis cov qauv uas piav qhia kev hloov pauv ntawm ib lub xeev mus rau lwm qhov yam tsis muaj kev hloov pauv sai.
Teeb meem ntawm kev sib koom siab thiab kev sib tshuam
Nws paub tias qhov kev sib tshuam ntawm ob lossis ntau lub teeb yog tus lej uas muaj tag nrho cov ntsiab lus uas muaj nyob hauv cov nqi no. Lo lus cov dej num ntawm cov teeb meem tau daws kom tau txais cov tswv yim yooj yim txog kev siv lub koom haum thiab kev sib tshuam ntawm cov khoom. daws cov teeb meem tseem ceeb ntawm cov lus raucov teeb pom kev zoo li no:
Cia A thiab B yog ob qhov kawg. Lawv muaj xws li n (A)=20, n (B)=28 thiab n (A ∪ B)=36, nrhiav n (A ∩ B)
Kev sib raug zoo hauv cov teeb tsa siv Venn daim duab:
- Lub koomhaum ntawm ob pawg tuaj yeem sawv cev los ntawm thaj chaw ntxoov ntxoo sawv cev A ∪ B. A ∪ B thaum A thiab B sib cais.
- Kev sib tshuam ntawm ob pawg tuaj yeem sawv cev los ntawm daim duab Venn. Nrog qhov chaw ntxoov ntxoo sawv cev A ∩ B.
- Qhov txawv ntawm ob lub teeb tuaj yeem sawv cev los ntawm Venn daim duab. Nrog thaj chaw ntxoov ntxoo sawv cev A - B.
- Kev sib raug zoo ntawm peb pawg siv daim duab Venn. Yog hais tias ξ sawv cev rau ib tug universal quantity, ces A, B, C yog peb subsets. Ntawm no tag nrho peb cov khoom sib tshooj.
Summarizing set information
Lub cardinality ntawm ib txheej yog txhais raws li tag nrho cov naj npawb ntawm tus kheej cov ntsiab lus hauv lub teeb. Thiab qhov kawg teev tus nqi yog piav raws li tus naj npawb ntawm tag nrho cov subsets. Thaum kawm txog cov teeb meem no, yuav tsum muaj txoj hauv kev, txoj hauv kev thiab kev daws teeb meem. Yog li, rau cardinality ntawm ib lub teeb, cov piv txwv hauv qab no tuaj yeem ua haujlwm li:
Let A={0, 1, 2, 3}| |=4, qhov twg | A | sawv cev rau cardinality ntawm teeb A.
Tam sim no koj tuaj yeem pom koj lub pob hluav taws xob. Nws yog qhov yooj yim zoo nkauj heev. Raws li twb tau hais lawm, lub hwj chim teeb yog teem los ntawm tag nrho cov subsets ntawm ib tug muab. Yog li ib tus yuav tsum tau txhais txhua qhov sib txawv, cov ntsiab lus thiab lwm yam txiaj ntsig ntawm A,uas yog {}, {0}, {1}, {2}, {3}, {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, { 2, 3}, {0, 1, 2}, {0, 1, 3}, {1, 2, 3}, {0, 2, 3}, {0, 1, 2, 3}.
Tam sim no fais fab tuag P={{}, {0}, {1}, {2}, {3}, {0, 1}, {0, 2}, {0, 3}, { 1, 2 }, {1, 3}, {2, 3}, {0, 1, 2}, {0, 1, 3}, {1, 2, 3}, {0, 2, 3}, { 0, 1, 2, 3}} uas muaj 16 ntsiab lus. Yog li, lub cardinality ntawm lub teeb A=16. Obviously, qhov no yog ib tug tedious thiab cumbersome txoj kev los daws qhov teeb meem no. Txawm li cas los xij, muaj cov qauv yooj yim uas, ncaj qha, koj tuaj yeem paub cov naj npawb ntawm cov ntsiab lus hauv lub hwj chim ntawm tus lej muab. | P |=2 ^ N, qhov twg N yog tus naj npawb ntawm cov ntsiab lus hauv qee qhov A. Cov qauv no tuaj yeem tau txais los ntawm kev siv cov combinatorics yooj yim. Yog li lo lus nug yog 2^11 txij li tus naj npawb ntawm cov ntsiab lus hauv teeb A yog 11.
Yog li, ib lub teeb yog ib qho kev qhia tus lej, uas tuaj yeem ua tau yam khoom tau. Piv txwv li, tsheb, neeg, lej. Hauv kev nkag siab lej, lub tswv yim no yog qhov dav thiab dav dua. Yog tias nyob rau theem pib cov lej thiab cov kev xaiv rau lawv cov kev daws teeb meem raug txheeb xyuas, tom qab ntawd nyob rau theem nrab thiab qib siab dua cov xwm txheej thiab cov haujlwm nyuaj. Nyob rau hauv qhov tseeb, lub cardinality ntawm lub union ntawm ib pawg yog txiav txim los ntawm lub teej tug ntawm cov khoom rau ib pawg. Ntawd yog, ib lub ntsiab yog nyob rau hauv chav kawm, tab sis muaj ib lossis ntau qhov sib txawv.
