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:


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

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.

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

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

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.