powerset

In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. 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.
The powerset of is variously denoted as , , , $\mathbb(S)$, or . The notation , meaning the set of all functions from S to a given set of two elements (e.g., ), is used because the powerset of can be identified with, equivalent to, or bijective to the set of all the functions from to the given two elements set.

Any subset of is called a ''family of sets'' over .

Example

If is the set , then all the subsets of are

* (also denoted $\varnothing$ or $\empty$, the empty set or the null set)
*
*
*
*
*
*
*
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 of the notation denoting the power set are demonstrated in the below.
: An indicator function or a characteristic function of a subset ''A'' of a set ''S'' with the cardinality , ''S'', = ''n'' is a function from ''S'' to the two elements set , denoted as ''I_A'': ''S'' → , and it indicates whether an element of ''S'' belongs to ''A'' or not; If ''x'' in ''S'' belongs to ''A'', then ''I_A''(''x'') = 1, and 0 otherwise. Each subset ''A'' of ''S'' is identified by or equivalent to the indicator function ''I_A'', and as the set of all the functions from ''S'' to consists of all the indicator functions of all the subsets of ''S''. In other words, is equivalent or bijective to the power set . Since each element in ''S'' corresponds to either 0 or 1 under any function in , the number of all the functions in is 2^''n''. Since the number 2 can be defined as (see, for example, von Neumann ordinals), the is also denoted as . Obviously holds. Generally speaking, ''X^Y'' is the set of all functions from ''Y'' to ''X'' 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, 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. 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 "2" 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 , , is 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 a bijective 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 bijective from to the integers without changing the number of one-to-one correspondences.)

However, such finite binary representation is only possible if ''S'' can be enumerated. (In this example, , , and are enumerated with 1, 2, and 3 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 ''S'' 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:

*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 $\left, 2^S \$ using the formula:
$\left, 2^S \right , = \sum_^ \binom$

Therefore, one can deduce the following identity, assuming $, S, = n$:
$\left , 2^S \ = 2^n = \sum_^ \binom$

Recursive definition

If $S$ is a finite set, then a recursive definition of $P(S)$ proceeds as follows:

*If $S = \$, then $P(S) = \$.
*Otherwise, let $e\in S$ and $T=S\setminus\$; then $P(S) = P(T)\cup \$.

In words:
* The power set of the empty set is a singleton whose only element is the empty set.
* For a non-empty set $S$, let $e$ be any element of the set and $T$ its relative complement; then the power set of $S$ is a union of a power set of $T$ and a power set of $T$ whose each element is expanded with the $e$ 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 idea of the power set of as the set of subsets of generalizes naturally to the 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