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

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

3.

# 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 $2^{X}$.

## Mathematics Subject Classification

03E99*no label found*03E10

*no label found*37-01

*no label found*

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