power set

DefinitionIf $X$ is a set, then the power set of $X$, denoted by $𝒫(X)$, is theset whose elements are the subsets of $X$.

Properties

1. If $X$ is finite, then $|𝒫(X)|=2^{|X|}$.

2. The above property also holds when $X$ is not finite.For a set $X$, let $|X|$ be the cardinality of $X$.Then $|𝒫(X)|=2^{|X|}=|2^X|$,where $2^X$ is the set of all functions from $X$ to $\{0,1\}$.

3. For an arbitrary set $X$, Cantor’s theorem states:a) there is no bijection between $X$ and $𝒫(X)$, andb) the cardinality of $𝒫(X)$ is greater than the cardinality of $X$.

Example

Suppose $S=\{a,b\}$. Then $𝒫(S)=\{\mathrm{\varnothing },\{a\},\{b\},S\}$.In particular, $|𝒫(S)|=2^{|S|}=4$.

Related definition

If $X$ is a set, then the finite power set of $X$, denoted by $ℱ(X)$, is theset whose elements are the finite subsets of $X$.

Remark

Due to the canonical correspondence between elements of $𝒫(X)$ and elements of $2^X$, the power set is sometimes also denoted by $2^X$.

Title	power set
Canonical name	PowerSet
Date of creation	2013-03-22 11:43:46
Last modified on	2013-03-22 11:43:46
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	23
Author	matte (1858)
Entry type	Definition
Classification	msc 03E99
Classification	msc 03E10
Classification	msc 37-01
Synonym	powerset
Related topic	PowerObject
Related topic	ProofOfGeneralAssociativity
Defines	finite power set
Defines	finite powerset