TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the power set (or powerset) of a set is the set of all
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s of , including the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

and itself. In
axiomatic set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although ob ...
(as developed, for example, in the axioms), the existence of the power set of any set is postulated by the
axiom of power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. The powerset of is variously denoted as , , , $\mathbb\left(S\right)$, (using the "
Weierstrass p In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as p-functions and they are usually denoted by the symbo ...

"), 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 Set theory is the branch of that studies , 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 , is mostly concerned with those that are ...
'' over .

# Example

If is the set , then all the subsets of are * (also denoted $\varnothing$ or $\empty$, the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

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

# Properties

If is a finite set with the
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(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 Image:Indicator function illustration.png, A three-dimensional plot of an indicator function, shown over a square two-dimensional domain (set ): the 'raised' portion overlays those two-dimensional points which are members of the 'indicated' subs ...
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 ''IA'': ''S'' → , and it indicates whether an element of ''S'' belongs to ''A'' or not; If ''x'' in ''S'' belongs to ''A'', then ''IA''(''x'') = 1, and 0 otherwise. Each subset ''A'' of ''S'' is identified by or equivalent to the indicator function ''IA'', 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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, ''XY'' is the set of all functions from ''Y'' to ''X'' and .
Cantor's diagonal argument 250px, An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above. In set theory, Cantor's diagonal argument, also cal ...
shows that the power set of a set (whether infinite or not) always has strictly higher
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
than the set itself (or informally, the power set must be larger than the original set). In particular,
Cantor's theorem In elementary set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. ...
shows that the power set of a
countably infinite In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
set is
uncountably In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many Element (mathematics), elements to be countable set, countable. The uncountability of a set is closely related to its cardinal number: a set ...

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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s can be put in a
one-to-one correspondence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

with the set of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s (see
Cardinality of the continuum In set theory Set theory is the branch of that studies , 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 , is mostly concerned with those that ar ...
). The power set of a set , together with the operations of union,
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
and
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
, can be viewed as the prototypical example of a
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. In fact, one can show that any ''finite'' Boolean algebra is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). 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
subalgebraIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of a power set Boolean algebra (see
Stone's representation theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
). The power set of a set forms an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
when it is considered with the operation of
symmetric difference In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(with the empty set as the identity element and each set being its own inverse), and a
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
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 ringIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

# Representing subsets as functions

In
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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, i ...
, is the notation representing the set of all
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s from to . As "2" can be defined as (see, for example, von Neumann ordinals), (i.e., ) is the set of all
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s from to . As shown above, and the power set of , , is considered identical set-theoretically. This equivalence can be applied to the example , in which , to get the
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...
is closely related to the power set. A –elements combination from some set is another name for a –elements subset, so the number of
combination In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of t ...

s, denoted as (also called
binomial coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
) 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, then a
recursive definition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of $P\left(S\right)$ proceeds as follows: *If $S = \$, then $P\left(S\right) = \$. *Otherwise, let $e\in S$ and $T=S\setminus\$; then $P\left(S\right) = P\left(T\right)\cup \$. In words: * The power set of the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

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 (mathematics), set , often denoted by (or ), are the Element (mathematics), elements not in . When all sets under consideration are considered to be subsets of a given set , the absolute complement of is t ...

; 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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
multigraph In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s. Given two multigraphs and , a
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
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 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 In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be th ...
. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

as a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
that is closed (and moreover
cartesian closed In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dir ...
) and has an object , called a
subobject classifierIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
. Although the term "power object" is sometimes used synonymously with
exponential object In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, in topos theory is required to be .

# Functors and quantifiers

In
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
and the theory of elementary topoi, the
universal quantifier In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
can be understood as the
right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of ...
of a
functor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

between power sets, the
inverse image In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

functor of a function between sets; likewise, the
existential quantifier In predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order l ...
is the
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of ...
.
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ...
,
Ieke Moerdijk Izak (Ieke) Moerdijk (; born 23 January 1958) is a Netherlands, Dutch mathematician, currently working at Utrecht University, who in 2012 won the Spinoza prize. Education and career Moerdijk studied mathematics, philosophy and general linguistic ...

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

*
Cantor's theorem In elementary set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. ...
*
Family of sets In set theory Set theory is the branch of that studies , 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 , is mostly concerned with those that are ...
*
Field of sets In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
*
Combination In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of t ...

* * *