Definition:Power Set

Definition
The power set (or powerset) 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:
 * $T \in \mathcal P \left({S}\right) \iff T \subseteq S$.

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

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

Alternative notations
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