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where $2^N$ is the power set with $n$ elements (subsets).

Does it hold true to any set or just the power set $2^N$?

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Do you know what $\subseteq$ means and what "antisymmetric" means? – Chris Eagle Nov 22 '12 at 19:23
actually i was wondering if it were true on the power set 2^n. – Jane Nov 22 '12 at 19:36
up vote 3 down vote accepted

By definition, two sets $A$ and $B$ are equal if and only if $A\subseteq B$ and $B\subseteq A$. Now a relation $R$ on a set $X$ is antisymmetric if $aRb$ and $bRa$ implies $a=b$ for all $a,b\in X$. Does this help you see why $\subseteq$ is antisymmetric?

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but isn't that true for any set? – Jane Nov 22 '12 at 19:35
yes, it is. The subset relation is antisymmetric because of the definition of set equality. This is true for all sets, so it is true for the power set of a set. – Holdsworth88 Nov 22 '12 at 19:37

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