Power set

Encyclopedia

In mathematics

, the

(

s of

.

Any subset

of is called a

and hence the power set of is

on

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 countably infinite

set is uncountably infinite. For example, the power set of the set of natural number

s can be put in a one-to-one correspondence

with the set of real number

s (see cardinality of the continuum

).

The power set of a set

, intersection

and complement

can be viewed as the prototypical example of a Boolean algebra. In fact, one can show that any

The power set of a set

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 laws) that the power set considered together with both of these operations forms a Boolean ring

.

In set theory,

s from

s from

between 2

This notion can be applied to the example above in which to see the isomorphism with the binary numbers

from 0 to 2

In

element. So {

For the whole power set of

. The number of sets with elements in the power set of a set with elements will be a combination

also called a binomial coefficient

.

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

Define the operation

In English, return the set with the element added to each set in .

In other words, the power set of the empty set is the set containing 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.

There are other more efficient ways to calculate the power set. For example, use a list of the

positions of

, the same can be done with the power set 2

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 (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 do enjoy both 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

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

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

as a category

that is closed

(and moreover cartesian closed

) and has an object Ω, called a subobject classifier

. Although the term "power object" is sometimes used synonymously with exponential object

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the

**power set**(or**powerset**) of any set*S*, written ,*P*(*S*), ℘Weierstrass p

In mathematics, the Weierstrass p , also called pe, is used for the Weierstrass's elliptic function. It is occasionally used for the power set, although for that purpose a cursive capital, rather than lower-case, p is more widespread...

(

*S*) or 2^{S}, is the set of all subsetSubset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

s of

*S*, including the empty set and S 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 setAxiom of power set

In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:...

.

Any subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

of is called a

*family of sets*

overFamily of sets

In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets...

*S*.## Example

If*S*is the set {*x*,*y*,*z*}, then the subsets of*S*are:- {} (also denoted , the empty setEmpty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

) - {
*x*} - {
*y*} - {
*z*} - {
*x*,*y*} - {
*x*,*z*} - {
*y*,*z*} - {
*x*,*y*,*z*}

and hence the power set of is

## Properties

If*S*is a finite set with |*S*| =*n*elements, then the power set of*S*contains elements. This can be proved by mathematical inductionMathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

on

*n*.Cantor's diagonal argument

Cantor's diagonal argument

Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural...

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

Cantor's theorem

In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...

shows that the power set of a countably infinite

Countable set

In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...

set is uncountably infinite. For example, the power set of the set of natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

s can be put in a one-to-one correspondence

Bijection

A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

with the set of real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s (see cardinality of the continuum

Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by |\mathbb R| or \mathfrak c ....

).

The power set of a set

*S*, together with the operations of unionUnion (set theory)

In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

, intersection

Intersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

and complement

Complement (set theory)

In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

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

*S*forms an Abelian groupAbelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

when considered with the operation of symmetric difference

Symmetric difference

In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....

(with the empty set as the identity element and each set being its own inverse) and a commutative monoid

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

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

Boolean ring

In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists only of idempotent elements....

.

## Representing subsets as functions

In set theory,

*X*^{Y}is the set of all functionFunction (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s from

*Y*to*X*. As "2" can be defined as {0,1} (see natural number), 2^{S}(i.e., {0,1}^{S}) is the set of all functionFunction (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s from

*S*to {0,1}. By identifying a function in 2^{S}with the corresponding preimage of 1, we see that there is a bijectionBijection

A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

between 2

^{S}and , where each function is the characteristic function of the subset in with which it is identified. Hence 2^{S}and could be considered identical set-theoretically. (Thus there are two distinct notational motivations for denoting the power set by 2^{S}: the fact that this function-representation of subsets makes it a special case of the*X*^{Y}notation and the property, mentioned above, that |2^{S}| = 2^{|S|}.)This notion can be applied to the example above in which to see the isomorphism with the binary numbers

from 0 to 2

^{n}−1 with n being the number of elements in the set.In

*S*, a*1*in the position corresponding to the location in the set indicates the presence of theelement. So {

*x*,*y*} = 110.For the whole power set of

*S*we get:- { } = 000 (Binary) = 0 (Decimal)
- {
*x*} = 100 = 4 - {
*y*} = 010 = 2 - {
*z*} = 001 = 1 - {
*x*,*y*} = 110 = 6 - {
*x*,*z*} = 101 = 5 - {
*y*,*z*} = 011 = 3 - {
*x*,*y*,*z*} = 111 = 7

## Relation to binomial theorem

The power set is closely related to the binomial theoremBinomial theorem

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

. The number of sets with elements in the power set of a set with elements will be a combination

Combination

In mathematics a combination is a way of selecting several things out of a larger group, where order does not matter. In smaller cases it is possible to count the number of combinations...

also called a binomial coefficient

Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

.

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

- set with 0 elements
- sets with 1 element
- sets with 2 elements
- set with 3 elements.

## Algorithms

If is a finite set, there is a recursive algorithm to calculate .Define the operation

In English, return the set with the element added to each set in .

- If ,then is returned.
- Otherwise:
- Let be any single element of .
- Let , where '' denotes the relative complementComplement (set theory)In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

of in . - And the result: is returned.

In other words, the power set of the empty set is the set containing 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.

There are other more efficient ways to calculate the power set. For example, use a list of the

*n*elements of*S*to fix a mapping from the bitBit

A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...

positions of

*n*-bit numbers to those elements; then with a simple loop run through all the 2^{n}numbers representable with*n*bits, and for each contribute the subset of*S*corresponding to the bits that are set (to 1) in the number. When*n*exceeds the word-length of the computer, typically 64 in modern CPUs but greater in modern GPUs, the representation is naturally extended by using an array of words instead of a single word.## Subsets of limited cardinality

The set of subsets of*S*of cardinality less than κ is denoted by or Similarly, the set of non-empty subsets of*S*might be denoted by## Topologization of power set

Since any family of functions*X*^{Y}from*Y*to*X*might be topologized establishing the so-called function spaceFunction space

In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

, the same can be done with the power set 2

^{S}identified as {0,1}^{S}. This particular type of function space is often called a**hyperspace**and the topology on the power set is referred to as**hypertopology**.## 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*X*as the set of subsets of*X*generalizes naturally to the subalgebras of an algebraic structureAlgebraic structure

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

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

Lattice (order)

In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...

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 do enjoy both 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

Multigraph

In mathematics, a multigraph or pseudograph is a graph which is permitted to have multiple edges, , that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge....

s. Given two multigraphs

*G*and*H*, a homomorphism*h*:*G*→*H*consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set*H*^{G}of homomorphisms from*G*to*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*G*are in bijection with the graph homomorphisms from*G*to the multigraph Ω definable as the complete directed graphComplete graph

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...

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

*G*as the multigraph Ω^{G}, called the**power object**of*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

*V*of vertices and*E*of edges, and has two unary operations*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 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 toposTopos

In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...

as a category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

that is closed

Closed category

In category theory, a branch of mathematics, a closed category is a special kind of category.In any category , the morphisms between any two given objects x and y comprise a set, the external hom...

(and moreover cartesian closed

Cartesian closed category

In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in...

) and has an object Ω, called a subobject classifier

Subobject classifier

In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. As the name suggests, what a subobject classifier does is to identify/classify subobjects of a given object according to which elements...

. Although the term "power object" is sometimes used synonymously with exponential object

Exponential object

In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories...

*Y*^{X}, in topos theory*Y*is required to be Ω.