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Definition If $X$ is a set, then the <</SPAN>#76#>power set of $X$ , denoted by $\powset{X}$ is the set whose elements are the subsets of $X$
- If $X$ is finite, then $|\powset{X}|=2^{|X|}$
- The above property also holds when $X$ is not finite. For a set $X$ let $|X|$ be the cardinality of $X$ Then $|\powset{X}|=2^{|X|}=|2^X|$ where $2^X$ is the set of all functions from $X$ to $\{0,1\}$
- For an arbitrary set $X$ Cantor's theorem states: a) there is no bijection between $X$ and $\powset{X}$ and b) the cardinality of $\powset{X}$ is greater than the cardinality of $X$
Suppose $S=\{a,b\}$ Then $\powset{S}=\{\emptyset, \{a\}, \{b\}, S\}$ In particular, $|\powset{S}|=2^{|S|}=4$
If $X$ is a set, then the <</SPAN>#77#>finite power set of $X$ , denoted by $\mathcal{F}(X)$ is the set whose elements are the finite subsets of $X$
Due to the canonical correspondence between elements of $\powset{X}$ and elements of $2^X$ the power set is sometimes also denoted by $2^X$
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"power set" is owned by matte. [ full author list (4) | owner history (2) ]
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Cross-references: canonical, bijection, Cantor's theorem, functions, cardinality, finite, subsets
There are 81 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 23554 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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Pending Errata and Addenda
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