Power set


In mathematics, the power set of any set is the set of all subsets of, including the empty set and itself, variously denoted as, ?, ℘,,, or, identifying the powerset of with the set of all functions from to a given set of two elements,. In axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set.
Any subset of is called a family of sets over.

Example

If is the set, then the subsets of are
and hence the power set of is.

Properties

If is a finite set with elements, then the number of subsets of is. This fact, which is the motivation for the notation, may be demonstrated simply as follows,
Cantor's diagonal argument shows that the power set of a set always has strictly higher cardinality than the set itself. 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.
The power set of a set, together with the operations of union, intersection and complement 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.
The power set of a set forms an abelian group when considered with the operation of symmetric difference and a commutative monoid when considered with the operation of intersection. It can hence be shown that the power set considered together with both of these operations forms a Boolean ring.

Representing subsets as functions

In set theory, Function #Function space| is the set of all functions from to. As "2" can be defined as , is the set of all functions from to. By identifying a function in with the corresponding preimage of, we see that there is a bijection between and, where each function is the characteristic function of the subset in with which it is identified. Hence and could be considered identical set-theoretically.
This notion can be applied to the example above in which to see the isomorphism with the binary numbers
from 0 to with being the number of elements in the set.
In, a "1" in the position corresponding to the location in the enumerated set indicates the presence of the element. So.
For the whole power set of we get:
SubsetSequence
of digits		Binary
interpretation		Decimal
equivalent

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.

Relation to binomial theorem

The power set is closely related to the binomial theorem. The number of subsets with elements in the power set of a set with elements is given by the number of combinations,, also called binomial coefficients.
For example, the power set of a set with three elements, has:
Using this relationship we can compute using the formula:
Therefore, one can deduce the following identity, assuming :

Recursive definition

If is a finite set, there is a recursive definition of.
In words:
The set of subsets of of cardinality less than or equal to is sometimes denoted by or, and the set of subsets with cardinality strictly less than is sometimes denoted or. Similarly, the set of non-empty subsets of might be denoted by or.

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 as the set of subsets of 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 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, 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 = 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 multigraphs. Given two multigraphs and, a homomorphism consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set of homomorphisms from to 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 are in bijection with the graph homomorphisms from to the multigraph definable as the complete directed graph on two vertices augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of as the multigraph, called the power object of.
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 of vertices and of edges, and has two unary operations giving the source and target vertices of each edge. An algebra all of whose operations are unary is called a presheaf. Every class of presheaves contains a presheaf 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 that is closed and has an object, called a subobject classifier. Although the term "power object" is sometimes used synonymously with exponential object, in topos theory is required to be.

Functors and quantifiers

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.