Definition:Power Set

Definition
The power set of a set $S$, denoted $\mathcal P \left({S}\right)$, is the set defined as follows:


 * $\mathcal P \left({S}\right) := \left\{{T: T \subseteq S}\right\}$

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:
 * $T \in \mathcal P \left({S}\right) \iff T \subseteq S$

Axiomatic Set Theory
The concept of the power set is axiomatised in the Axiom of Powers in Zermelo-Fraenkel set theory:
 * $\forall x: \exists y: \left({\forall z: \left({z \in y \iff \left({w \in z \implies w \in x}\right)}\right)}\right)$

Also known as
The rendition powerset is frequently seen.

Some sources do not use the term power set, merely referring to the term set of all subsets.

Variants of $\mathcal P$ are seen throughout the literature: $\mathfrak P, P, \mathrm P, \mathbf P$, etc.

Another significant notation is:
 * $2^S := \left\{ {T: T \subseteq S}\right\}$

This is used by, for example,.

The relevance of this latter notation is clear from the fact that if $S$ has $n$ elements, then $2^S$ has $2^n$ elements‎.

Also see

 * Cardinality of Power Set