power set
Definition If $X$ is a set, then the power set^{} of $X$, denoted by $\mathrm{\pi \x9d\x92\xab}\beta \x81\u2019(X)$, is the set whose elements are the subsets of $X$.
Properties

1.
If $X$ is finite, then $\mathrm{\pi \x9d\x92\xab}\beta \x81\u2019(X)={2}^{X}$.

2.
The above property also holds when $X$ is not finite. For a set $X$, let $X$ be the cardinality of $X$. Then $\mathrm{\pi \x9d\x92\xab}\beta \x81\u2019(X)={2}^{X}={2}^{X}$, where ${2}^{X}$ is the set of all functions from $X$ to $\{0,1\}$.
 3.
Example
Suppose $S=\{a,b\}$. Then $\mathrm{\pi \x9d\x92\xab}\beta \x81\u2019(S)=\{\mathrm{\beta \x88\x85},\{a\},\{b\},S\}$. In particular, $\mathrm{\pi \x9d\x92\xab}\beta \x81\u2019(S)={2}^{S}=4$.
Related definition
If $X$ is a set, then the finite power set of $X$, denoted by $\mathrm{\beta \x84\pm}\beta \x81\u2019(X)$, is the set whose elements are the finite subsets of $X$.
Remark
Due to the canonical correspondence between elements of $\mathrm{\pi \x9d\x92\xab}\beta \x81\u2019(X)$ and elements of ${2}^{X}$, the power set is sometimes also denoted by ${2}^{X}$.
Title  power set 
Canonical name  PowerSet 
Date of creation  20130322 11:43:46 
Last modified on  20130322 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 3701 
Synonym  powerset 
Related topic  PowerObject 
Related topic  ProofOfGeneralAssociativity 
Defines  finite power set 
Defines  finite powerset 