Editing Power set (section)

{{Short description|Mathematical set of all subsets of a set}}
{{For|the search engine developer|Powerset (company)}}
{{Infobox mathematical statement
| name = Power set
| image = Hasse diagram of powerset of 3.svg
| caption = The elements of the power set of {''x'', ''y'', ''z''}  [[order theory|ordered]] with respect to [[Inclusion (set theory)|inclusion]].
| type = [[Set (mathematics)#Basic operations|Set operation]]
| field = [[Set (mathematics)|Set theory]]
| statement = The power set is the set that contains all subsets of a given set.
| symbolic statement = <math>x\in P(S) \iff x\subseteq S</math>
}}

In [[mathematics]], the '''power set''' (or '''powerset''') of a [[Set (mathematics)|set]] {{mvar|S}} is the set of all [[subset]]s of {{mvar|S}}, including the [[empty set]] and {{mvar|S}} itself.{{sfn|ps=|Weisstein}} 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]].{{sfn|ps=|Devlin|1979|page=50}} 
The powerset of {{mvar|S}} is variously denoted as {{math|{{itco|{{mathcal|P}}}}(''S'')}}, {{math|{{itco|𝒫}}(''S'')}}, {{math|''P''(''S'')}}, <math>\mathbb{P}(S)</math>, or {{math|2<sup>''S''</sup>}}.{{efn|The notation {{math|2<sup>''S''</sup>}}, meaning the set of all functions from {{math|''S''}} to a given set of two elements (e.g., {{math|{{mset|0, 1}}}}), is used because the powerset of {{mvar|S}} can be identified with, is equivalent to, or bijective to the set of all the functions from {{mvar|S}} to the given two-element set.{{sfn|ps=|Weisstein}}}}
Any subset of {{math|{{itco|{{mathcal|P}}}}(''S'')}} is called a ''[[family of sets]]'' over {{mvar|S}}.