## You are here

Homepower set

## Primary tabs

# 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*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

## Corrections

Minor style things by mathwizard β

2^X is okay for any set by mps β

example? by alozano β

Missing set braces by almann β