# power set

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

## Properties

1. 1.

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

2. 2.

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

3. 3.

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

## Example

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

## Related definition

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

## Remark

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