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In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2, is the...

In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2, is the set of all subsets of S. In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. Any subset F of is called a family of sets over S. If S is the set {x, y, z}, then the complete list of subsets of S is as follows: and hence the power set of S is If S is a finite set with |S| = n elements, then the power set of S contains elements. Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. For example, the power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see cardinality of the continuum). The power set of a set S, together with the operations of union, intersection and complement can be viewed as the prototypical example of a Boolean algebra. In fact, one can show that any