In mathematics, the **power set** (or **powerset**) of a set *S* is the set of all subsets of *S*, including the empty set and S itself.^{ [1] } 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.^{ [2] } The powerset of S is variously denoted as P(S), 𝒫(*S*),^{ [3] }*P*(*S*), (*S*), ℘(*S*) (using the "Weierstrass p"), or 2^{S}. The notation 2^{S} is used because given any set with exactly two elements, the powerset of *S* can be identified with the set of all functions from *S* into that set.^{ [1] }

- Example
- Properties
- Representing subsets as functions
- Relation to binomial theorem
- Recursive definition
- Subsets of limited cardinality
- Power object
- Functors and quantifiers
- See also
- References
- Bibliography
- External links

Any subset of P(S) is called a * family of sets * over *S*.

If S is the set {*x*, *y*, *z*}, then the subsets of S are

- {} (also denoted or , the empty set or the null set)
^{ [3] } - {
*x*} - {
*y*} - {
*z*} - {
*x*,*y*} - {
*x*,*z*} - {
*y*,*z*} - {
*x*,*y*,*z*}

and hence the power set of S is {{}, {*x*}, {*y*}, {*z*}, {*x*, *y*}, {*x*, *z*}, {*y*, *z*}, {*x*, *y*, *z*}}.^{ [4] }

If *S* is a finite set with |*S*| = *n* elements, then the number of subsets of *S* is |P(*S*)| = 2^{n}. This fact, which is the motivation for the notation 2^{S}, may be demonstrated simply as follows,

- First, order the elements of S in any manner. We write any subset of
*S*in the format {γ_{1}, γ_{2}, ..., γ_{n}} where γ_{i}, 1 ≤*i*≤*n*, can take the value of 0 or 1. If γ_{i}= 1, the i-th element of*S*is in the subset; otherwise, the i-th element is not in the subset. Clearly the number of distinct subsets that can be constructed this way is 2^{n}as γ_{i}∈ {0, 1} .

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 *S*, 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 *S* forms an abelian group 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, *X*^{Y} is the set of all functions from Y to X. As "2" can be defined as {0,1} (see, for example, von Neumann ordinals), 2^{S} (i.e., {0,1}^{S}) is the set of all functions from *S* to {0,1}. By identifying a function in 2^{S} with the corresponding preimage of 1, we see that there is a bijection between 2^{S} and P(S), where each function is the characteristic function of the subset in P(S) with which it is identified. Hence 2^{S} and P(S) 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 *S* = {*x*, *y*, *z*}, to get 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 enumerated set { (*x*, 0), (*y*, 1), (*z*, 2) } indicates the presence of the element. So {*x*, *y*} = 011_{(2)}.

For the whole power set of *S*, we get:

Subset | Sequence of digits | Binary interpretation | Decimal equivalent |
---|---|---|---|

{ } | 0, 0, 0 | 000_{(2)} | 0_{(10)} |

{ x } | 0, 0, 1 | 001_{(2)} | 1_{(10)} |

{ y } | 0, 1, 0 | 010_{(2)} | 2_{(10)} |

{ x, y } | 0, 1, 1 | 011_{(2)} | 3_{(10)} |

{ z } | 1, 0, 0 | 100_{(2)} | 4_{(10)} |

{ x, z } | 1, 0, 1 | 101_{(2)} | 5_{(10)} |

{ y, z } | 1, 1, 0 | 110_{(2)} | 6_{(10)} |

{ x, y, z } | 1, 1, 1 | 111_{(2)} | 7_{(10)} |

Such a bijective mapping of *S* to integers is arbitrary, so this representation of subsets of *S* is not unique, but the sort order of the enumerated set does not change its cardinality.

However, such finite binary representation is only possible if *S* can be enumerated. This is possible even if *S* has an infinite cardinality, such as the set of integers or rationals, but not for example if *S* is the set of real numbers, in which case we cannot enumerate all irrational numbers to assign them a defined finite location in an ordered set.

The power set is closely related to the binomial theorem. The number of subsets with k elements in the power set of a set with n elements is given by the number of combinations, C(*n*, *k*), also called binomial coefficients.

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 using the formula:

Therefore, one can deduce the following identity, assuming :

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 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.

The set of subsets of *S* of cardinality less than or equal to κ is sometimes denoted by P_{κ}(*S*) or [*S*]^{κ}, and the set of subsets with cardinality strictly less than κ is sometimes denoted P_{< κ}(*S*) or [*S*]^{<κ}. Similarly, the set of non-empty subsets of *S* might be denoted by P_{≥ 1}(*S*) or P^{+}(*S*).

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

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.^{ [5] }

In abstract algebra, a **Boolean algebra** or **Boolean lattice** is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra.

In mathematical analysis, the **Weierstrass approximation theorem** states that every continuous function defined on a closed interval [*a*, *b*] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform.

In the mathematical field of order theory, an **ultrafilter** on a given partially ordered set (poset) *P* is a certain subset of *P,* namely a maximal filter on *P*, that is, a proper filter on *P* that cannot be enlarged to a bigger proper filter on *P*.

**Universal algebra** is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study.

In order theory, a field of mathematics, an **incidence algebra** is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity. Subalgebras called **reduced incidence algebras** give a natural construction of various types of generating functions used in combinatorics and number theory.

In combinatorics, a branch of mathematics, a **matroid** is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice.

In mathematics, a **Heyting algebra** is a bounded lattice equipped with a binary operation *a* → *b* of *implication* such that ≤ *b* is equivalent to *c* ≤. From a logical standpoint, *A* → *B* is by this definition the weakest proposition for which modus ponens, the inference rule *A* → *B*, *A* ⊢ *B*, is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by Arend Heyting (1930) to formalize intuitionistic logic.

In mathematics, a **distributive lattice** is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets.

In mathematics, a **partition of a set** is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset.

**Order theory** is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.

This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:

A **lattice** is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum and a unique infimum. An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.

In mathematics, a **closure operator** on a set *S* is a function from the power set of *S* to itself that satisfies the following conditions for all sets

In mathematical order theory, an **ideal** is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.

In mathematics, a **field of sets** is a mathematical structure consisting of a pair consisting of a set and a family of subsets of called an **algebra over ** that contains the empty set as an element, and is closed under the operations of taking complements in finite unions, and finite intersections.

In mathematics, a **complete Boolean algebra** is a Boolean algebra in which every subset has a supremum. Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra *A* has an essentially unique completion, which is a complete Boolean algebra containing *A* such that every element is the supremum of some subset of *A*. As a partially ordered set, this completion of *A* is the Dedekind–MacNeille completion.

In mathematics, in the branch of combinatorics, a **graded poset** is a partially ordered set (poset) *P* equipped with a **rank function***ρ* from *P* to the set **N** of all natural numbers. *ρ* must satisfy the following two properties:

Boolean algebra is a mathematically rich branch of abstract algebra. Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1. Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under zero or more operations satisfying certain equations.

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.

- 1 2 Weisstein, Eric W. "Power Set".
*mathworld.wolfram.com*. Retrieved 2020-09-05. - ↑ Devlin 1979 , p. 50
- 1 2 "Comprehensive List of Set Theory Symbols".
*Math Vault*. 2020-04-11. Retrieved 2020-09-05. - ↑ Puntambekar 2007 , pp. 1–2
- ↑ Saunders Mac Lane, Ieke Moerdijk, (1992)
*Sheaves in Geometry and Logic*Springer-Verlag. ISBN 0-387-97710-4*See page 58*

- Devlin, Keith J. (1979).
*Fundamentals of contemporary set theory*. Universitext. Springer-Verlag. ISBN 0-387-90441-7. Zbl 0407.04003. - Halmos, Paul R. (1960).
*Naive set theory*. The University Series in Undergraduate Mathematics. van Nostrand Company. Zbl 0087.04403. - Puntambekar, A. A. (2007).
*Theory Of Automata And Formal Languages*. Technical Publications. ISBN 978-81-8431-193-8.

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