In

_{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

Power set Algorithm

in C++ {{Set theory Operations on sets

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the power set (or powerset) of a set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

is the set of all subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

s of , including the empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...

and itself. In axiomatic set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

(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 , , , $\backslash mathbb(S)$, 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
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, set fami ...

'' over .
Example

If is the set , then all the subsets of are * (also denoted $\backslash varnothing$ or $\backslash empty$, theempty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...

or the null set)
*
*
*
*
*
*
*
and hence the power set of is .
Properties

If is a finite set with thecardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

(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
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...

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 ''Icardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

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
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...

set is uncountably infinite. The power set of the set of natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...

s can be put in a one-to-one correspondence with the set of real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s (see 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 \mathfrak c (lowercase fraktur "c") or , \math ...

).
The power set of a set , together with the operations of union, intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...

and complement, can be viewed as the prototypical example of a Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...

. In fact, one can show that any ''finite'' Boolean algebra is isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

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
In mathematics, 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 the order in which they are written. That is, the group operation is comm ...

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
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
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

Inset theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

, 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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

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 calledbinomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

) 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 $\backslash left,\; 2^S\; \backslash $ using the formula:
$$\backslash left,\; 2^S\; \backslash right\; ,\; =\; \backslash sum\_^\; \backslash binom$$
Therefore, one can deduce the following identity, assuming $,\; S,\; =\; n$:
$$\backslash left\; ,\; 2^S\; \backslash \; =\; 2^n\; =\; \backslash sum\_^\; \backslash binom$$
Recursive definition

If $S$ is afinite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. T ...

, then a recursive definition of $P(S)$ proceeds as follows:
*If $S\; =\; \backslash $, then $P(S)\; =\; \backslash $.
*Otherwise, let $e\backslash in\; S$ and $T=S\backslash setminus\backslash $; then $P(S)\; =\; P(T)\backslash cup\; \backslash $.
In words:
* The power set of the empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...

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
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is ...

; 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 ofcardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

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 analgebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...

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 = 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
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

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 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
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...

(and moreover cartesian closed
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 mat ...

) 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

Incategory theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...

and the theory of elementary topoi, the universal quantifier
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...

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.Saunders Mac Lane
Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg.
Early life and education
Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...

, Ieke Moerdijk
Izak (Ieke) Moerdijk (; born 23 January 1958) is a Dutch mathematician, currently working at Utrecht University, who in 2012 won the Spinoza prize.
Education and career
Moerdijk studied mathematics, philosophy and general linguistics at the Un ...

, (1992) ''Sheaves in Geometry and Logic'' Springer-Verlag. ''See page 58''
See also

* Cantor's theorem *Family 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, set fami ...

* Field of sets
In mathematics, a field of sets is a mathematical structure consisting of a pair ( X, \mathcal ) consisting of a set X and a family \mathcal of subsets of X called an algebra over X that contains the empty set as an element, and is closed unde ...

* Combination
References

Bibliography

* * *External links

* * *Power set Algorithm

in C++ {{Set theory Operations on sets