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power set

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Definition If is a set, then the power set of , denoted by $\mathcal{P}(X)$ , is the set whose elements are the subsets of .

If is finite, then $\vert\mathcal{P}(X)\vert=2^{\vert X\vert}$ .
The above property also holds when is not finite. For a set , let $\vert X\vert$ be the cardinality of . Then $\vert\mathcal{P}(X)\vert=2^{\vert X\vert}=\vert 2^X\vert$ , where is the set of all functions from to $\{0,1\}$ .
For an arbitrary set , Cantor's theorem states: a) there is no bijection between and $\mathcal{P}(X)$ , and b) the cardinality of $\mathcal{P}(X)$ is greater than the cardinality of .

Suppose $S=\{a,b\}$ . Then $\mathcal{P}(S)=\{\emptyset, \{a\}, \{b\}, S\}$ . In particular, $\vert\mathcal{P}(S)\vert=2^{\vert S\vert}=4$ .

is a set, then the finite power set of

, denoted by $\mathcal{F}(X)$ , is the set whose elements are the finite subsets of

Due to the canonical correspondence between elements of $\mathcal{P}(X)$ and elements of

, the power set is sometimes also denoted by

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"power set" is owned by matte. [ full author list (4) | owner history (2) ]

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Other names:

powerset

Also defines:

finite power set, finite powerset

Keywords:

Set, Power, Cardinality

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Cross-references: canonical, bijection, Cantor's theorem, functions, cardinality, finite, subsets
There are 71 references to this entry.

This is version 15 of power set, born on 2001-10-06, modified 2007-07-25.
Object id is 136, canonical name is PowerSet.
Accessed 16748 times total.

Classification:

AMS MSC:	03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
	03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)

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