In mathematics, the **power set** of a set *x* is the set of all subsets of *x*, which in set-builder notation can be represented as follows:

P(x) = {y|y⊆x}

In Zermelo-Frankel set theory the axiom of powerset ensures that for any set x there exists a set y which consists of all the subsets of x. In certain formulations this is equivalent to "the powerset of a set x always exists" but in others it's more broad, instead meaning "a set with *at least* all the subsets of x exists for all x". In the latter case the axiom of specification is needed to narrow down the set to a strict power set.

## Properties

- Since ∅⊆x and x⊆x,then ∅∈P(x)∧x∈P(x). Consequently, ¬∃x(P(x)=∅).
- The cardinality of the power set of
*x*given*x*is finite is equal to 2^{|x|}, where*|x|*is the cardinality of*S*. Additionally, the powerset of any set is strictly greater in cardinality than the set itself, meaning there is no possible bijection between elements of a set and it's powerset, even if the initial set is infinite.

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