Power set

In mathematics, the power set (or powerset) of a set Template:Mvar is the set of all subsets of Template:Mvar, including the empty set and Template:Mvar itself.Template:Sfn 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.Template:Sfn The powerset of Template:Mvar is variously denoted as Template:Math, Template:Math, Template:Math, <math>\mathbb{P}(S)</math>, or Template:Math.Template:Efn Any subset of Template:Math is called a family of sets over Template:Mvar.

Example

If Template:Mvar is the set Template:Math, then all the subsets of Template:Mvar are

• Template:Math (also denoted <math>\varnothing</math> or <math>\empty</math>, the empty set or the null set)
• Template:Math
• Template:Math
• Template:Math
• Template:Math
• Template:Math
• Template:Math
• Template:Math

and hence the power set of Template:Mvar is Template:Math.Template:Sfn

Properties

If Template:Math is a finite set with the cardinality Template:Math (i.e., the number of all elements in the set Template:Math is Template:Math), then the number of all the subsets of Template:Math is Template:Math. This fact as well as the reason of the notation Template:Math denoting the power set Template:Math are demonstrated in the below.

An indicator function or a characteristic function of a subset Template:Math of a set Template:Math with the cardinality Template:Math is a function from Template:Math to the two-element set Template:Math, denoted as Template:Math, and it indicates whether an element of Template:Math belongs to Template:Math or not; If Template:Math in Template:Math belongs to Template:Math, then Template:Math, and Template:Math otherwise. Each subset Template:Math of Template:Math is identified by or equivalent to the indicator function Template:Math, and Template:Math as the set of all the functions from Template:Math to Template:Math consists of all the indicator functions of all the subsets of Template:Math. In other words, Template:Math is equivalent or bijective to the power set Template:Math. Since each element in Template:Math corresponds to either Template:Math or Template:Math under any function in Template:Math, the number of all the functions in Template:Math is Template:Math. Since the number Template:Math can be defined as Template:Math (see, for example, von Neumann ordinals), the Template:Math is also denoted as Template:Math. Obviously Template:Math holds. Generally speaking, Template:Math is the set of all functions from Template:Math to Template:Math and Template:Math.

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 (or 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. 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 Template:Math, together with the operations of union, intersection and complement, is a Σ-algebra over Template:Math and can be viewed as the prototypical example of a 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 Template:Math forms an abelian group when it is 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 (with the entire set Template:Math as the identity element). It can hence be shown, by proving the distributive laws, that the power set considered together with both of these operations forms a Boolean ring.

Representing subsets as functions

In set theory, Template:Math is the notation representing the set of all functions from Template:Mvar to Template:Mvar. As "Template:Math" can be defined as Template:Math (see, for example, von Neumann ordinals), Template:Math (i.e., Template:Math) is the set of all functions from Template:Math to Template:Math. As shown above, Template:Math and the power set of Template:Math, Template:Math, are considered identical set-theoretically.

This equivalence can be applied to the example above, in which Template:Math, to get the isomorphism with the binary representations of numbers from 0 to Template:Math, with Template:Mvar being the number of elements in the set Template:Math or Template:Math. First, the enumerated set Template:Math is defined in which the number in each ordered pair represents the position of the paired element of Template:Math in a sequence of binary digits such as Template:Math; Template:Math of Template:Math is located at the first from the right of this sequence and Template:Math is at the second from the right, and 1 in the sequence means the element of Template:Math corresponding to the position of it in the sequence exists in the subset of Template:Math for the sequence while 0 means it does not.

For the whole power set of Template:Math, we get:

Subset	Sequence of binary digits	Binary interpretation	Decimal equivalent
Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:Math	Template:Math

Such an injective mapping from Template:Math to integers is arbitrary, so this representation of all the subsets of Template:Math is not unique, but the sort order of the enumerated set does not change its cardinality. (E.g., Template:Math can be used to construct another injective mapping from Template:Math to the integers without changing the number of one-to-one correspondences.)

However, such finite binary representation is only possible if Template:Math can be enumerated. (In this example, Template:Math, Template:Math, and Template:Math are enumerated with Template:Math, Template:Math, and Template:Math respectively as the position of binary digit sequences.) The enumeration is possible even if Template:Math has an infinite cardinality (i.e., the number of elements in Template:Math is infinite), such as the set of integers or rationals, but not possible for example if Template:Math is the set of real numbers, in which case we cannot enumerate all irrational numbers.

Relation to binomial theorem

The binomial theorem is closely related to the power set. A Template:Math–elements combination from some set is another name for a Template:Math–elements subset, so the number of combinations, denoted as Template:Math (also called binomial coefficient) is a number of subsets with Template:Mvar elements in a set with Template:Mvar elements; in other words it's the number of sets with Template:Math elements which are elements of the power set of a set with Template:Math elements.

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

• Template:Math subset with Template:Math elements (the empty subset),
• Template:Math subsets with Template:Math element (the singleton subsets),
• Template:Math subsets with Template:Math elements (the complements of the singleton subsets),
• Template:Math subset with Template:Math elements (the original set itself).

Using this relationship, we can compute Template: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 Template:Math: <math display="block">\left |2^S \right| = 2^n = \sum_{k=0}^{n} \binom{n}{k} </math>

Recursive definition

If Template:Math is a finite set, then a recursive definition of Template:Math proceeds as follows:

• If Template:Math, then Template:Math.
• Otherwise, let Template:Math and Template:Math; then Template:Math.

In words:

• The power set of the empty set is a singleton whose only element is the empty set.
• For a non-empty set Template:Math, let <math>e</math> be any element of the set and Template:Math its relative complement; then the power set of Template:Math is a union of a power set of Template:Math and a power set of Template:Math whose each element is expanded with the Template:Math element.

Subsets of limited cardinality

The set of subsets of Template:Math of cardinality less than or equal to Template:Math is sometimes denoted by Template:Math or Template:Math, and the set of subsets with cardinality strictly less than Template:Math is sometimes denoted Template:Math or Template:Math. Similarly, the set of non-empty subsets of Template:Math might be denoted by Template:Math or Template:Math.

Power object

{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Ambox }} A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective, the concept of the power set of Template:Math as the set of all subsets of Template:Math generalizes naturally to the set to all subalgebras of an algebraic structure or algebra.

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 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.<ref>Template:Cite journal </ref> 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 Template:Math, there is no guarantee that a class of algebras contains an algebra that can play the role of Template:Math 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 multigraphs. Given two multigraphs Template:Math and Template:Math, a homomorphism Template:Math consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set Template:Math of homomorphisms from Template:Math to Template:Math 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 Template:Math are in bijection with the graph homomorphisms from Template:Math to the multigraph Template:Math definable as the 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 Template:Math as the multigraph Template:Math, called the power object of Template:Math.

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 Template:Math of vertices and Template:Math of edges, and has two unary operations Template:Math 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 Template:Math that plays the role for subalgebras that Template:Math plays for subsets. Such a class is a special case of the more general notion of elementary topos as a category that is closed (and moreover cartesian closed) and has an object Template:Math, called a subobject classifier. Although the term "power object" is sometimes used synonymously with exponential object Template:Math, in topos theory Template:Math is required to be Template:Math.

Functors and quantifiers

There is both a covariant and contravariant power set functor, Template:Math and Template:Math. The covariant functor is defined more simply as the functor which sends a set Template:Math to Template:Math and a morphism Template:Math (here, a function between sets) to the image morphism. That is, for <math>A = \{x_1, x_2, ...\} \in \mathsf{P}(S), \mathsf{P}f(A) = \{f(x_1), f(x_2), ...\} \in \mathsf{P}(T)</math>. Elsewhere in this article, the power set was defined as the set of functions of Template:Math into the set with 2 elements. Formally, this defines a natural isomorphism <math>\overline{\mathsf{P}} \cong \text{Set}(-,2)</math>. The contravariant power set functor is different from the covariant version in that it sends Template:Math to the preimage morphism, so that if <math>f(A) = B \subseteq T, \overline\mathsf{P}f(B) = A</math>. This is because a general functor <math>\text{C}(-,c)</math> takes a morphism <math>h:a \rightarrow b</math> to precomposition by h, so a function <math>h^*: C(b,c) \rightarrow C(a,c)</math>, which takes morphisms from b to c and takes them to morphisms from a to c, through b via h. <ref>Template:Cite book</ref>

In category theory and the theory of 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.Template:Sfn

See also

• Cantor's theorem
• Family of sets
• Field of sets
• Combination

Bibliography

• Template:Cite book
• Template:Cite book
• Template:Citation
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• {{#invoke:citation/CS1|citation

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External links

• Template:PlanetMath
• Template:Nlab
• Template:Nlab
• Power set Algorithm in C++