Power set

In mathematics, the power set of a set S, written as ${\displaystyle P(S)}$ or ${\displaystyle {\mathcal {P}}(S)}$,^[1] is the set of all subsets of S. In terms of cardinality, a power set is larger than the set it originates from. If S is a finite set with n elements, then ${\displaystyle P(S)}$ would have ${\displaystyle 2^{n}}$ elements.^[2][3]

Examples

• The power set of ${\displaystyle \{2,5\}}$ is ${\displaystyle \{\{\},\{2\},\{5\},\{2,5\}\}}$.
• The power set of ${\displaystyle \{3,4,10\}}$ is ${\displaystyle \{\{\},\{3\},\{4\},\{10\},\{3,4\},\{3,10\},\{4,10\},\{3,4,10\}\}}$.

Related pages

• Zermelo–Fraenkel set theory, which includes an axiom on power set