Wp/lus/Khawnkhâwm

Chanchin kimchang : Khawnkhawmzirna

Khawnkhâwm

Khawnkhâwm tih hi chhiarkawpa thil hman nasat ber leh chhiarkawp innghahna bulthüm taksa a ni a. He thuziakah hian a awmze tlangpui kan sawi dawn a ni.

Sawifiahna

Khawnkhâwm kan tih hi eng pawh, mumal taka thliar hran theih, a huhova chiang taka ngaihtuah khâwm theih leh mimal ngaihdana innghat lo hi a ni. Entirnan, ka kezungpui zawng zawng tih hi thil, mumal taka chiang taka ngaihtuah theih a ni. Chuvangin ka kezungpui zawng zawng chu khawnkhâwm pakhat a lo ni ta a ni. Amaherawhchu, Aizawla nula hmelṭha zawng zawng lo ti ta ila, thil mumal lo, thliar fel hleih theih loh a ni; mi hrang hrang ngaihdàna innghat mai mai a nih avangin khawnkhâwm a ni thei lo.

Entirna dang lehah, nambar, 13 aia lian zawng zawng lo ti leh ta ila, khawnkhâwm kan siam leh a ni. A chhan chu, nambar 13 aia lian hi chhiar vek thei lo mah ila, thil mumal tak, mimal ngaihdana innghat ni lo, a ni. Chu kan khawnkhawm siam chu $X$ tiin lo vuah ta ila, heti hian kan ziak: $X=\lbrace a \mid a> 13\rbrace$. Hetah hian nambar, 13 aia lian tawh phawt chu X chhungkhung tiin kan sawi. Entirnan, 17 chu X chhungkhung a ni. A chhinchhiahnain, "$17 \in X$" tiin kan ziak; 7 erawh chu 13 aia a têt avangin X chhungkhung a ni ve lo. A chhinchhiahnain "$11\notin X$" tiin kan ziak.

Amaherawhchu nambar, 13 aia lian hlei lutuk lo, ti leh ta ila, khawnkhâwm kan siam lo a ni. A chhan chu lian lutuk lo tih hi thu awmze bulbal nei lo a ni; mimal ngaihdan leh kan thil sawi azira danglam mai mai a ni. (Entirnan, vansang chanchin sawi nan chuan, "thla hi lei aṭang a hla lutuk lo" ti ila a dik a, mahse thla leh lei inkar chu kèa kal tùr sawi nan chuan "a hla lutuk lo" tih chi rual a ni lo leh thung.)

Entirna dang lehah chuan, ka mit dinglam tih hi khawnkhawm a ni a, chhungkhung pakhat a nei, chu a chhungkhung chu ka mit dinglam a ni. Heti hian a ziah leh theih:

$X$={ka mit dinglam}.

Khawnkhâwm hi thu chauh ni lo, chhinchhiahna hlang ringawt hmang pawhin a sawifiah theih. Entirnan,

"$X=\lbrace a, aw, b, ch, ..., z\rbrace$ lo ni ta se."

kan tih chuan X hi Mizo a-aw-b zawng zawng khawnkhâwm a ni a,

"$X=\lbrace x\in\mathbb{R}\mid x\geq 10\rbrace$ lo ni se"

kan tih chuan, X chu nambar, 10 aia të lo (10 leh a chunglam zawng) zawng zawng khawnkhâwm a ni.

Chhungkhung-nihna

Kan sawi tawh angin, kan khawnkhawma awmte kha a chhungkhung kan ti a. X hi kan khawnkhâwm ni se, x hi a chhungkhung a nih chuan tihian kan ziak:

$x\in X$

$y$ hi a chhungkhung a nih loh erawh chuan tihian kan ziak:

$y\notin X$

Khawnpéng

Chanchin kimchang : Khawnpéng

A hi B k=khawnvpéng a ni.

$A$ leh $B$ hi khawnkhâwm ve ve lo ni ta se, $A$ chhungkhung apiang hi $B$ chhungkhung a nih ve zël chuan, $A$ hi $B$ khawnpéng tiin kan sawi. Entirnan, $B$ hi ka kezung zawng zawng lo ni ta se, $A$ hi ka kezungpui dinglam tih lo ni thung se, $A$ hi $B$ khawnpéng a ni. A chhan chu, ka kezungpui dinglam ($A$ chhungkhung neih chhun) chu ka kezung zawng zawng ($=B$) zinga a tel ve vang a ni. Hetiang dinhmunah hian

"$A \subset B$"

tiin kan ziak.

A hi eng khawnkhawm pawh lo ni se, ama khawnpéng a ni ngei ngei. Chumi awmzia chu,

$A\subset A$

tih hi a dik ngei ngei a ni. A chhan chu, A chhungkhung tawh phawt chu A chhungkhung a nih vek vang a ni!

A hi B khawnpéng a nih a, an intluk loh thung erawh chuan heti hian kan ziak:

$A\subsetneq B$

.

Thulâkna

• Halmos, Paul R., Naive Set Theory, Springer-Verlag, 1960
• Hausdorff, Felix, Grundzuge der Mengenlehre, AMS Chelsea Publishing, New York, 1978.