Power set

If is the set , then all the subsets of are • (the empty set, also denoted \varnothing or \empty) • • • • • • • and hence the power set of is . == Properties ==

If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact, as well as the reason for the notation denoting the power set , are demonstrated below. : An indicator function or a characteristic function of a subset of a set with the cardinality is a function from to the two-element set , denoted as , and it indicates whether an element of belongs to or not; If in belongs to , then , and otherwise. Each subset of is identified by or equivalent to the indicator function , and as the set of all the functions from to consists of all the indicator functions of all the subsets of . In other words, is equivalent or bijective to the power set . Since each element in corresponds to either or under any function in , the number of all the functions in is . Since the number can be defined as (see, for example, von Neumann ordinals), the is also denoted as . Obviously holds. Generally speaking, is the set of all functions from to and . 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 , together with the operations of union, intersection and complement, is a σ-algebra over 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 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 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, is the notation representing the set of all functions from to . As "" can be defined as (see, for example, von Neumann ordinals), (i.e., ) is the set of all functions from to . As shown above, and the power set of , , are considered identical set-theoretically. This equivalence can be applied to the example above, in which , to get the isomorphism with the binary representations of numbers from 0 to , with being the number of elements in the set or . First, the enumerated set is defined in which the number in each ordered pair represents the position of the paired element of in a sequence of binary digits such as ; of is located at the first from the right of this sequence and is at the second from the right, and 1 in the sequence means the element of corresponding to the position of it in the sequence exists in the subset of for the sequence while 0 means it does not. For the whole power set of , we get: Such an injective mapping from to integers is arbitrary, so this representation of all the subsets of is not unique, but the sort order of the enumerated set does not change its cardinality. (E.g., can be used to construct another injective mapping from to the integers without changing the number of one-to-one correspondences.) However, such finite binary representation is only possible if can be enumerated. (In this example, , , and are enumerated with , , and respectively as the position of binary digit sequences.) The enumeration is possible even if has an infinite cardinality (i.e., the number of elements in is infinite), such as the set of integers or rationals, but not possible for example if 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 –elements combination from some set is another name for a –elements subset, so the number of combinations, denoted as (also called binomial coefficient) is a number of subsets with elements in a set with elements; in other words it's the number of sets with elements which are elements of the power set of a set with elements. For example, the power set of a set with three elements, has: • subset with elements (the empty subset), • subsets with element (the singleton subsets), • subsets with elements (the complements of the singleton subsets), • subset with elements (the original set itself). Using this relationship, we can compute using the formula: \left|2^S \right | = \sum_{k=0}^ \binom{k} Therefore, one can deduce the following identity, assuming : \left |2^S \right| = 2^n = \sum_{k=0}^{n} \binom{n}{k} == Recursive definition ==

If is a finite set, then a recursive definition of proceeds as follows: • If , then . • Otherwise, let and ; then . In words: • The power set of the empty set is a singleton whose only element is the empty set. • For a non-empty set , let e be any element of the set and its relative complement; then the power set of is a union of a power set of and a power set of whose each element is expanded with the element. == Subsets of limited cardinality ==

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 concept of the power set of as the set of all subsets of 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. 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 , there is no guarantee that a class of algebras contains an algebra that can play the role of 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 (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 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 (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 that plays the role for subalgebras that 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 , 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 ==

There is both a covariant and contravariant power set functor, and . The covariant functor is defined more simply as the functor which sends a set to and a morphism (here, a function between sets) to the image morphism. That is, for . Elsewhere in this article, the power set was defined as the set of functions of into the set with 2 elements. Formally, this defines a natural isomorphism . The contravariant power set functor is different from the covariant version in that it sends to the preimage morphism, so that if . This is because a general functor takes a morphism to precomposition by h, so a function , which takes morphisms from b to c and takes them to morphisms from a to c, through b via h. 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. == See also ==