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.