Power set

In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. 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. The powerset of S is variously denoted as P(S), 𝒫(S), P(S), ${\displaystyle \mathbb {P} (S)}$, ${\displaystyle \wp (S)}$, or 2^S. The notation 2^S, meaning the set of all functions from S to a given set of two elements (e.g., {0, 1}), is used because the powerset of S can be identified with, is equivalent to, or bijective to the set of all the functions from S to the given two-element set.

Power set

The elements of the power set of {x, y, z} ordered with respect to inclusion.
Type	Set operation
Field	Set theory
Statement	The power set is the set that contains all subsets of a given set.
Symbolic statement	${\displaystyle x\in P(S)\iff x\subseteq S}$

Any subset of P(S) is called a family of sets over S.