Power set
In mathematics, the power set (or powerset) of any set S is the set of all subsets of S, including the empty set and S itself. The power set of a set S is variously denoted as P(S), ℘(S) (using the "Weierstrass p"), P(S), ℙ(S), or, identifying the powerset of S with the set of all functions from S to a given set of two elements, 2S. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.
Any subset of P(S) is called a family of sets over S.
If S is the set {x, y, z}, then the subsets of S are:
and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.
If S is a finite set with |S| = n elements, then the number of subsets of S is |P(S)| = 2n. This fact, which is the motivation for the notation 2S, may be demonstrated simply as follows,
...
Wikipedia