power set

Definition If X is a set, then the power setMathworldPlanetmath of X, denoted by 𝒫⁒(X), is the set whose elements are the subsets of X.


  1. 1.

    If X is finite, then |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 |𝒫⁒(X)|=2|X|=|2X|, where 2X is the set of all functions from X to {0,1}.

  3. 3.

    For an arbitrary set X, Cantor’s theoremMathworldPlanetmath states: a) there is no bijection between X and 𝒫⁒(X), and b) the cardinality of 𝒫⁒(X) is greater than the cardinality of X.


Suppose S={a,b}. Then 𝒫⁒(S)={βˆ…,{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 the set whose elements are the finite subsets of X.


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

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