Definition:Power Set

Definition
The power set of a set $S$ is the set defined and denoted as:


 * $\powerset S := \set {T: T \subseteq S}$

That is, the set whose elements are all of the subsets of $S$.

Note that this is a set all of whose elements are themselves sets.

It is clear from the definition that:
 * $T \in \powerset S \iff T \subseteq S$

Axiomatic Set Theory
The concept of the Definition:Power Set|power set]] is axiomatised in the Axiom of Powers in Zermelo-Fraenkel set theory:

Also see

 * Subset is Element of Power Set
 * Cardinality of Power Set of Finite Set


 * Definition:Set of All Mappings


 * Definition:Power Structure