Teeb txoj kev xav tam sim no yog ib feem ntawm kev ua lej. Peb txhua tus paub tias nws hu ua teeb ib qho kev sau ntawm cov ntsiab lus sib txawv sib txawv ntawm ib leeg, uas muaj ib tus (lossis ntau dua) tus yam ntxwv sib xws. Teem txoj kev tshawb fawb cov khoom thiab kev sib raug zoo ntawm cov teeb; Cov haujlwm no tau txhawb nqa los ntawm Bolzano thiab Cantor, tom qab ntawd tau ua tiav nyob rau xyoo pua 20th los ntawm lwm tus lej lej, xws li Zermelo thiab Fraenkel.
Nws yog ib qho tseem ceeb uas txhua txheej tau teev tseg zoo kawg nkaus, uas yog, nws tuaj yeem tsim nrog qhov tseeb txawm hais tias muab cov khoom, nws yog lossis tsis yog rau pawg.
- Nyob lej qhov no feem ntau ncaj ncaj. Piv txwv, yog tias teeb tsa tus lej txawm ntau dua 1 thiab tsawg dua 15 tau txiav txim siab, nws yog qhov tseeb tias cov teeb no yuav suav nrog tus lej 2, 4, 6, 8, 10, 12 thiab 14 nkaus xwb.
- Ntawm hom lus, tham txog ib pab pawg tuaj yeem tsis muaj tseeb ntau dua, vim tias yog peb xav tsim pab pawg ntawm cov neeg hu nkauj zoo tshaj plaws, piv txwv li, kev xav yuav muaj ntau yam thiab yuav tsis muaj kev pom zoo kiag tias leej twg yuav yog ib feem ntawm pab pawg no thiab leej twg yuav tsis . Qee qhov tshwj xeeb teeb tsa tsis muaj teeb meem (tsis muaj cov ntsiab lus) lossis teeb tsa ua ke (nrog ib qho nkaus xwb).
Cov cov khoom uas yog ib feem ntawm txheej tau hu ua cov tswv cuab lossis cov khoom, thiab teeb tsa tau sawv cev hauv cov ntawv sau nrog kab zauv: {}. Sab hauv txoj hlua, cov khoom sib cais los ntawm tus lej. Lawv kuj tseem tuaj yeem sawv cev los ntawm Venn daim duab, uas suav nrog cov khoom sib sau ua ke txhua txheej hauv kab thiab kaw kab, feem ntau zoo li lub voj voog. Thaum muaj ob peb ntawm cov kab kaw no, txhua tus ntawm lawv tau muab cov ntawv loj (A, B, C, thiab lwm yam) thiab cov txheej txheem thoob ntiaj teb ntawm cov no yog sawv cev los ntawm tsab ntawv U, uas txhais tau tias yog txheej txheej.
Nrog cov teeb koj tuaj yeem ua tau cov haujlwm; lub hauv paus tseem ceeb yog kev sib koom ua ke, kev sib tshuam, qhov sib txawv, kev sib ntxiv thiab cov khoom Cartesian. Kev sib koom ua ke ntawm ob pawg A thiab B tau txhais raws li txheej A ∪ B thiab qhov no muaj txhua lub ntsiab lus uas muaj tsawg kawg ib ntawm lawv. Cov zauv dav dav uas sawv cev rau nws yog:
- TO= {José, Jerónimo}, B= {María, Mabel, Marcela}; AUB= {José, Jerónimo, María, Mabel, Marcela}
- P= {pear, kua txiv}, C= {txiv qaub, txiv kab ntxwv}; F= {cherry, currant};PUCUF = {pear, kua, txiv qaub, txiv kab ntxwv, cherry, currant}
- M={7, 9, 11}, N={4, 6, 8}; MUN={7, 9, 11, 4, 6, 8}
- R= {pob, skate, duav}, G= {duav, pob, skate}; RUG= {pob, duav, skate}
- C= {daisy}, S= {carnation}; CUS = {daisy, carnation}
- C= {daisy}, S= {carnation}; T= {lub raj mis}, CUSUT = {margarita, carnation, raj mis}
- G= {ntsuab, xiav, dub}, H= {dub}; GUH= {ntsuab, xiav, dub}
- TO={ 1, 3, 5, 7, 9 }; B={ 10, 11, 12 }; AUB={ 1, 3, 5, 7, 9, 10, 11, 12 }
- D= {Tuesday, Thursday}, THIAB= {Wednesday, Friday}; DUA = {Tuesday, Wednesday, Thursday, Friday}
- B= {yoov tshaj cum, muv, hummingbird}; C= {nyuj, dev, nees}; BUC= {yoov tshaj cum, muv, hummingbird, nyuj, dev, nees}
- TO={2, 4, 6, 8}, B={1, 2, 3, 4}; AUB={1, 2, 3, 4, 6, 8}
- P= {rooj, rooj zaum}, Q= {rooj, rooj zaum}; PUQ= {rooj, rooj zaum}
- TO= {khob cij}, B = {cheese}; AUB= {qhob cij, cheese}
- TO={20, 30, 40}, B= {5, 15}; AUB ={5, 15, 20, 30, 40}
- M= {Lub Ib Hlis, Ob Hlis, Peb Hlis, Plaub Hlis}, N= {Kaum Ib Hlis, Kaum Ob Hlis}; MUN= {Lub Ib Hlis, Ob Hlis, Peb Hlis, Plaub Hlis, Kaum Ib Hlis, Kaum Ob Hlis}
- F={12, 22, 32, 42}, G= {a, e, i, o, u}; FUG= {12, 22, 32, 42, a, e, i, o, u}
- TO= {lub caij ntuj sov}, B= {lub caij ntuj no}; AUB= {lub caij ntuj sov, lub caij ntuj no}
- S= {khau khiab, khau khiab, khau khiab}, R= {tsho}; SOUTH= {khau khiab, khau khiab, flip flop, tsho}
- H= {Hnub Monday, Tuesday}, R= {Hnub Monday, Tuesday}, D= {Hnub Monday, Tuesday}; HURUD= {Hnub Monday, Tuesday}
- P= {liab, xiav}, Q= {ntsuab, daj}, PUQ= {liab, xiav, ntsuab, daj}
