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[[Image:Hasse diagram of powerset of 3.svg|thumb|250px|The elements of the power set of the set {''x'', ''y'', ''z''}  [[order theory|ordered]] with respect to [[Inclusion (set theory)|inclusion]].]]

In [[mathematics]], the '''power set''' (or '''powerset''') of any [[Set (mathematics)|set]] {{mvar|''S''}} is the set of all [[subset]]s of {{mvar|''S''}}, including the [[empty set]] and {{mvar|S}} itself, variously denoted as {{mathcal|P}}({{mvar|S}}), 𝒫({{mvar|''S''}}), ℘({{mvar|''S''}}) (using the "[[Weierstrass p]]"), {{math|''P''(''S'')}}, {{math|ℙ(''S'')}}, or, identifying the powerset of {{mvar|''S''}} with the set of all functions from {{mvar|''S''}} to a given set of two elements, {{math|2<sup>''S''</sup>}}. 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]].<ref>{{harvnb|Devlin|1979|page=50}}</ref>

Any subset of {{mathcal|P}}({{mvar|S}}) is called a ''[[family of sets]]'' over {{mvar|''S''}}.

==Example==
If {{mvar|S}} is the set {{math|{{mset|''x'', ''y'', ''z''}}}}, then the subsets of {{mvar|S}} are

* {{math|{{mset}}}} (also denoted <math>\varnothing</math> or <math>\empty</math>, the  [[empty set]] or the null set)
* {{math|{{mset|''x''}}}}
* {{math|{{mset|''y''}}}}
* {{math|{{mset|''z''}}}}
* {{math|{{mset|''x'', ''y''}}}}
* {{math|{{mset|''x'', ''z''}}}}
* {{math|{{mset|''y'', ''z''}}}}
* {{math|{{mset|''x'', ''y'', ''z''}}}}
and hence the power set of {{mvar|S}} is {{math|{{mset|{{mset}},
{{mset|''x''}},
{{mset|''y''}},
{{mset|''z''}},
{{mset|''x'', ''y''}},
{{mset|''x'', ''z''}},
{{mset|''y'', ''z''}},
{{mset|''x'', ''y'', ''z''}}}}}}.<ref>{{harvnb|Puntambekar|2007|pages=1–2}}</ref>

==Properties==
If {{mvar|''S''}} is a finite set with {{math|1={{!}}''S''{{!}} = ''n''}} elements, then the number of subsets of {{mvar|''S''}} is {{math|1={{!}}{{mathcal|P}}(''S''){{!}} = 2<sup>''n''</sup>}}. This fact, which is the motivation for the notation {{math|2<sup>''S''</sup>}}, may be demonstrated simply as follows,
: First, order the elements of {{mvar|S}} in any manner. We write any subset of {{mvar|''S''}} in the format {{math|{γ<sub>1</sub>, γ<sub>2</sub>, ..., γ<sub>''n''</sub> }}} where {{math|γ<sub>i</sub> , 1 ≤ ''i'' ≤ ''n''}}, can take the value of {{math|0}} or {{math|1}}. If {{math|1=γ<sub>''i''</sub> = 1}}, the {{mvar|i}}-th element of {{mvar|''S''}} is in the subset; otherwise, the {{mvar|i}}-th element is not in the subset. Clearly the number of distinct subsets that can be constructed this way is {{math|2<sup>''n''</sup>}} as {{math|γ<sub>''i''</sub> ∈ {0, 1} }}.

[[Cantor's diagonal argument#General sets|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 [[countable set|countably infinite]] set is [[uncountable|uncountably]] infinite. The power set of the set of [[natural number]]s can be put in a [[bijection|one-to-one correspondence]] with the set of [[real number]]s (see [[Cardinality of the continuum]]).

The power set of a set {{mvar|''S''}}, together with the operations of [[union (set theory)|union]], [[intersection (set theory)|intersection]] and [[complement (set theory)|complement]] can be viewed as the prototypical example of a [[Boolean algebra (structure)|Boolean algebra]]. In fact, one can show that any ''finite'' Boolean algebra is [[isomorphic]] to the Boolean algebra of the power set of a finite set. For ''infinite'' Boolean algebras this is no longer true, but every infinite Boolean algebra can be represented as a [[subalgebra]] of a power set Boolean algebra (see [[Stone's representation theorem]]).

The power set of a set {{mvar|''S''}} forms an [[abelian group]] when considered with the operation of [[symmetric difference]] (with the empty set as the identity element and each set being its own inverse) and a [[commutative]] [[monoid]] when considered with the operation of intersection. It can hence be shown (by proving the [[Distributive property|distributive laws]]) that the power set considered together with both of these operations forms a [[Boolean ring]].

==Representing subsets as functions== <!-- this section is referenced forme some other place in this article, do not change its title carelessly -->
In set theory, [[Function (mathematics)#Function space|{{math|''X''<sup>''Y''</sup>}}]] is the set of all [[function (mathematics)|function]]s from {{mvar|Y}} to {{mvar|X}}.  As "2" can be defined as {{math|{{mset|0,1}}}} (see [[Natural number#A standard construction|natural number]]), {{math|2<sup>''S''</sup>}} (i.e., {{math|{{mset|0,1}}<sup>''S''</sup>}}) is the set of all [[function (mathematics)|function]]s from {{mvar|''S''}} to {0,1}.  By identifying a function in {{math|2<sup>''S''</sup>}} with the corresponding [[preimage]] of {{math|1}}, we see that there is a [[bijection]] between {{math|2<sup>''S''</sup>}} and {{mathcal|P}}({{mvar|S}}), where each function is the [[indicator function|characteristic function]] of the subset in {{mathcal|P}}({{mvar|S}}) with which it is identified.  Hence {{math|2<sup>''S''</sup>}} and {{mathcal|P}}({{mvar|S}}) could be considered identical set-theoretically.  (Thus there are two distinct [[set notation#Metaphor in denoting sets|notational motivations]] for denoting the power set by {{math|2<sup>''S''</sup>}}:  the fact that this function-representation of subsets makes it a special case of the {{math|''X''<sup>''Y''</sup>}} notation and the property, [[power set#Properties|mentioned above]], that {{math|1={{abs|2<sup>''S''</sup>}} = 2<sup>{{abs|''S''}}</sup>}}.)

This notion can be applied to the example [[Power set#Example|above]] in which {{math|1=''S'' = {{mset|''x'', ''y'', ''z''}}}} to see the isomorphism with the binary numbers
from 0 to {{math|2<sup>''n''</sup> − 1}} with {{mvar|n}} being the number of elements in the set.
In {{mvar|''S''}}, a "1" in the position corresponding to the location in the enumerated set {{math|1={{mset| (''x'', 0), (''y'', 1), (''z'', 2) }}}} indicates the presence of the element. So {{math|1={{mset|''x'', ''y''}} = 011<sub>(2)</sub>}}.

For the whole power set of {{mvar|''S''}} we get:
{|class="wikitable"
|-
!scope="col"| Subset
!scope="col"| Sequence<br /> of digits
!scope="col"| Binary<br /> interpretation
!scope="col"| Decimal<br /> equivalent
|-
| {{math|1={{mset|                     }} }} || {{math|0, 0, 0}} || {{math|000<sub>(2)</sub>}} || {{math|0<sub>(10)</sub>}}
|-
| {{math|1={{mset| ''x''               }} }} || {{math|0, 0, 1}} || {{math|001<sub>(2)</sub>}} || {{math|1<sub>(10)</sub>}}
|-
| {{math|1={{mset|        ''y''        }} }} || {{math|0, 1, 0}} || {{math|010<sub>(2)</sub>}} || {{math|2<sub>(10)</sub>}}
|-
| {{math|1={{mset| ''x'', ''y''        }} }} || {{math|0, 1, 1}} || {{math|011<sub>(2)</sub>}} || {{math|3<sub>(10)</sub>}}
|-
| {{math|1={{mset|               ''z'' }} }} || {{math|1, 0, 0}} || {{math|100<sub>(2)</sub>}} || {{math|4<sub>(10)</sub>}}
|-
| {{math|1={{mset| ''x'',        ''z'' }} }} || {{math|1, 0, 1}} || {{math|101<sub>(2)</sub>}} || {{math|5<sub>(10)</sub>}}
|-
| {{math|1={{mset|        ''y'', ''z'' }} }} || {{math|1, 1, 0}} || {{math|110<sub>(2)</sub>}} || {{math|6<sub>(10)</sub>}}
|-
| {{math|1={{mset| ''x'', ''y'', ''z'' }} }} || {{math|1, 1, 1}} || {{math|111<sub>(2)</sub>}} || {{math|7<sub>(10)</sub>}}
|}

Such bijective mapping of ''S'' to integers is arbitrary, so this representation of subsets of ''S'' is not unique, but the sort order of the enumerated set does not change its cardinality.

However, such finite binary representation is only possible if ''S'' can be enumerated (this is possible even if ''S'' has an infinite cardinality, such as the set of integers or rationals, but not for example if ''S'' is the set of real numbers, in which we cannot enumerate all irrational numbers to assign them a defined finite location in an ordered set containing all irrational numbers).

==Relation to binomial theorem==
The power set is closely related to the [[binomial theorem]].  The number of subsets with {{mvar|k}} elements in the power set of a set with {{mvar|n}} elements is given by the number of [[combination]]s, {{math|C(''n'', ''k'')}}, also called [[binomial coefficient]]s.

For example, the power set of a set with three elements, has:

*C(3, 0) = 1 subset with 0 elements (the empty subset),
*C(3, 1) = 3 subsets with 1 element (the singleton subsets),
*C(3, 2) = 3 subsets with 2 elements (the complements of the singleton subsets),
*C(3, 3) = 1 subset with 3 elements (the original set itself).

Using this relationship we can compute <math display="inline">\left|2^S \right|</math> using the formula:

<math display="block">\left|2^S \right | = \sum_{k=0}^{|S|} \binom{|S|}{k} </math>

Therefore, one can deduce the following identity, assuming <math display="inline">|S| = n</math>:

<math display="block">\left |2^S \right| = 2^n = \sum_{k=0}^{n} \binom{n}{k} </math>

==Algorithms==
If {{mvar|S}} is a [[finite set]], there is a [[recursive algorithm]] to calculate {{mathcal|P}}({{mvar|S}}).

Define the operation {{math|1={{mathcal|F}} (''e'', ''T'') = {''X'' ∪ {''e''} {{!}} ''X'' ∈ ''T''}}}.

In English, return the set with the element {{mvar|e}} added to each set {{mvar|X}} in {{mvar|T}}.

*If {{math|1=''S'' = { } }}, then {{math|1={{mathcal|P}}({{mvar|S}}) = { { } } }} is returned.
*Otherwise:
:*Let {{mvar|e}} be any single element of {{mvar|S}}.
:*Let {{math|1=''T'' = ''S'' \ {''e''} }} be the set {{mvar|S}} with element {{mvar|e}} excluded (a [[Complement (set theory)|relative complement]] of the [[singleton (mathematics)|singleton]] {{math|{''e''} }} in {{mvar|S}}).
:*And the result: {{math|1={{mathcal|P}}(''S'') = {{mathcal|P}}(''T'') ∪ {{mathcal|F}} (''e'', {{mathcal|P}}(''T''))}} is returned.

In words: The power set of the empty set is the set containing only the empty set and the power set of any other set is all the subsets of the set containing some specific element and all the subsets of the set not containing that specific element.

The following is a [[Python (programming language)|Python]] implementation of the above algorithm as used for lists (note that such lists, opposed to sets, may also hold duplicated enties):

Verbose:
<source lang="python" line="1">
def p(s):
    if s==[]: # base case
        return [s] # if s is empty, then the only sublist of s is s itself
    else:
        e = s[0] # any element from s (in this implementation, we choose the first element)
        t = s[1:] # s with e removed
        pt = p(t) # the list of all sublists of t (note that this is a recursive call)
        fept = [x + [e] for x in pt] # pt with e appended to each sublist
        return pt + fept # the concatenation of all constructed sublists

# example:
print(p(['x', 'y', 'z']))
</source>

Concise form:
<source lang="python" line="1">
def p(s):
    return p(s[1:]) + [x + [s[0]] for x in p(s[1:])] if s else [s]
</source>

==Subsets of limited cardinality==
The set of subsets of {{mvar|''S''}} of [[cardinality]] less than or equal to {{math|κ}} is sometimes denoted by {{math|{{mathcal|P}}<sub>κ</sub>(''S'')}} or {{math|[''S'']<sup>κ</sup>}}, and the set of subsets with cardinality strictly less than {{math|κ}} is sometimes denoted {{math|{{mathcal|P}}<sub>< κ</sub>(''S'')}} or {{math|[''S'']<sup><κ</sup>}}. Similarly, the set of non-empty subsets of {{mvar|''S''}} might be denoted by {{math|{{mathcal|P}}<sub>≥ 1</sub>(''S'')}} or {{math|{{mathcal|P}}<sup>+</sup>(''S'')}}.

==Power object==
A set can be regarded as an algebra having no nontrivial operations or defining equations.  From this perspective the idea of the power set of {{mvar|''X''}} as the set of subsets of {{mvar|''X''}} generalizes naturally to the subalgebras of an [[algebraic structure]] or algebra.

Now the power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the [[lattice (order)|lattice]] of all subsets of some set.  The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an [[algebraic lattice]], and every algebraic lattice arises as the lattice of subalgebras of some algebra.  So in that regard subalgebras behave analogously to subsets.

However, there are two important properties of subsets that do not carry over to subalgebras in general.  First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class, although they can always be organized as a lattice.  Secondly, whereas the subsets of a set are in bijection with the functions from that set to the set {0,1} = 2, there is no guarantee that a class of algebras contains an algebra that can play the role of 2 in this way.

Certain classes of algebras enjoy both of these properties.  The first property is more common, the case of having both is relatively rare.  One class that does have both is that of [[multigraph]]s.  Given two multigraphs {{mvar|''G''}} and {{mvar|''H''}}, a [[homomorphism]] {{math|''h'': ''G'' → ''H''}} consists of two functions, one mapping vertices to vertices and the other mapping edges to edges.  The set {{math|''H''<sup>''G''</sup>}} of homomorphisms from {{mvar|''G''}} to {{mvar|''H''}} can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set.  Furthermore, the subgraphs of a multigraph {{mvar|''G''}} are in bijection with the graph homomorphisms from {{mvar|''G''}} to the multigraph {{math|Ω}} definable as the [[complete graph|complete directed graph]] on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices.  We can therefore organize the subgraphs of {{mvar|''G''}} as the multigraph {{math|Ω<sup>''G''</sup>}}, called the '''power object''' of {{mvar|''G''}}.

What is special about a multigraph as an algebra is that its operations are unary.  A multigraph has two sorts of elements forming a set {{mvar|''V''}} of vertices and {{mvar|''E''}} of edges, and has two unary operations {{math|''s'',''t'': ''E'' → ''V''}} giving the source (start) and target (end) vertices of each edge.  An algebra all of whose operations are unary is called a [[presheaf]].  Every class of presheaves contains a presheaf {{math|Ω}} that plays the role for subalgebras that 2 plays for subsets.  Such a class is a special case of the more general notion of elementary [[topos]] as a [[category (mathematics)|category]] that is [[closed category|closed]] (and moreover [[cartesian closed category|cartesian closed]]) and has an object {{math|Ω}}, called a [[subobject classifier]].  Although the term "power object" is sometimes used synonymously with [[exponential object]] {{math|''Y''<sup>''X''</sup>}}, in topos theory {{mvar|''Y''}} is required to be {{math|Ω}}.

==Functors and quantifiers==
In [[category theory]] and the theory of [[elementary topos|elementary topoi]], the [[universal quantifier]] can be understood as the [[right adjoint]] of a [[functor]] between power sets, the [[inverse image]] functor of a function between sets; likewise, the [[existential quantifier]] is the [[left adjoint]].<ref>[[Saunders Mac Lane]], [[Ieke Moerdijk]], (1992) ''Sheaves in Geometry and Logic'' Springer-Verlag. {{isbn|0-387-97710-4}} ''See page 58''</ref>

==See also==
* [[Set theory]]
* [[Axiomatic set theory]]
* [[Family of sets]]
* [[Field of sets]]

==Notes==
<references/>

==References==
* {{cite book | last=Devlin | first=Keith J. | authorlink=Keith Devlin | title=Fundamentals of contemporary set theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90441-7 | zbl=0407.04003 }}
* {{cite book | last=Halmos | first=Paul R. | authorlink=Paul Halmos | title=Naive set theory | url=https://archive.org/details/naivesettheory00halm | url-access=registration | series=The University Series in Undergraduate Mathematics | publisher=van Nostrand Company | year=1960 | zbl=0087.04403 }}
* {{cite book |last=Puntambekar | first=A. A. | title=Theory Of Automata And Formal Languages | year=2007 | publisher=Technical Publications | isbn=978-81-8431-193-8 }}

==External links==
{{Wiktionary|power set}}
*{{mathworld|urlname=PowerSet|title=Power Set}}
*{{planetmath reference|id=136|title=Power set}}
* {{nlab|id=power+set|title=Power set}}
* {{nlab|id=power+object|title=Power object}}
* [http://www.programminglogic.com/powerset-algorithm-in-c/ Power set Algorithm] in [[C++]]

{{Mathematical logic}}
{{Set theory}}

[[Category:Abstract algebra]]
[[Category:Algebra]]
[[Category:Basic concepts in set theory]]