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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 ...
, specifically
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 ...
, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all
ordered pair 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 h ...

ordered pair
s where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of
set-builder notation In 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. Although any ...
, that is : A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of ''n'' sets, also known as an ''n''-fold Cartesian product, which can be represented by an ''n''-dimensional array, where each element is an ''n''-
tuple 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). ...
. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an
indexed family 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 h ...
of sets. The Cartesian product is named after
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

René Descartes
, whose formulation of
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
gave rise to the concept, which is further generalized in terms of
direct productIn 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 ...
.


Examples


A deck of cards

An illustrative example is the
standard 52-card deck The standard 52-card deck of French-suited playing cards French-suited playing cards or French-suited cards are playing cards, cards that use the French Suit (cards), suits of (clovers or clubs ), (tiles or diamonds ), (hearts ) ...
. The standard playing card ranks form a 13-element set. The card suits form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards. returns a set of the form . returns a set of the form . These two sets are distinct, even disjoint.


A two-dimensional coordinate system

The main historical example is the
Cartesian plane A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

Cartesian plane
in
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
. In order to represent geometrical shapes in a numerical way, and extract numerical information from shapes' numerical representations,
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

René Descartes
assigned to each point in the plane a pair 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, called its
coordinates In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

coordinates
. Usually, such a pair's first and second components are called its ''x'' and ''y'' coordinates, respectively (see picture). The set of all such pairs (i.e., the Cartesian product , with ℝ denoting the real numbers) is thus assigned to the set of all points in the plane.


Most common implementation (set theory)

A formal definition of the Cartesian product from set-theoretical principles follows from a definition of
ordered pair 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 h ...

ordered pair
. The most common definition of ordered pairs, Kuratowski's definition, is (x, y) = \. Under this definition, (x, y) is an element of \mathcal(\mathcal(X \cup Y)), and X\times Y is a subset of that set, where \mathcal represents the
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 an ...
operator. Therefore, the existence of the Cartesian product of any two sets in
ZFC
ZFC
follows from the axioms of
pairing In mathematics, a pairing is an ''R''-Bilinear map#Modules, bilinear map from the Cartesian product of two ''R''-Module (mathematics), modules, where the underlying Ring (mathematics), ring ''R'' is Commutative ring, commutative. Definition Let ''R ...
, union,
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 an ...
, and
specification A specification often refers to a set of documented requirements to be satisfied by a material, design, product, or service. A specification is often a type of technical standard A technical standard is an established norm (social), norm or require ...
. Since
functions 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 ...
are usually defined as a special case of
relations Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
, and relations are usually defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is necessarily prior to most other definitions.


Non-commutativity and non-associativity

Let ''A'', ''B'', ''C'', and ''D'' be sets. The Cartesian product is not
commutative 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 ...
, : A \times B \neq B \times A, because the
ordered pair 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 h ...

ordered pair
s are reversed unless at least one of the following conditions is satisfied: * ''A'' is equal to ''B'', or * ''A'' or ''B'' is 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 ...

empty set
. For example: : ''A'' = ; ''B'' = :: ''A'' × ''B'' = × = :: ''B'' × ''A'' = × = : ''A'' = ''B'' = :: ''A'' × ''B'' = ''B'' × ''A'' = × = : ''A'' = ; ''B'' = ∅ :: ''A'' × ''B'' = × ∅ = ∅ :: ''B'' × ''A'' = ∅ × = ∅ Strictly speaking, the Cartesian product is not
associative 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). ...
(unless one of the involved sets is empty). : (A\times B)\times C \neq A \times (B \times C) If for example ''A'' = , then .


Intersections, unions, and subsets

The Cartesian product satisfies the following property with respect to intersections (see middle picture). : (A \cap B) \times (C \cap D) = (A \times C) \cap (B \times D) In most cases, the above statement is not true if we replace intersection with union (see rightmost picture). : (A \cup B) \times (C \cup D) \neq (A \times C) \cup (B \times D) In fact, we have that: : (A \times C) \cup (B \times D) = A \setminus B) \times C\cup A \cap B) \times (C \cup D)\cup B \setminus A) \times D/math> For the set difference, we also have the following identity: : (A \times C) \setminus (B \times D) = \times (C \setminus D)\cup A \setminus B) \times C Here are some rules demonstrating distributivity with other operators (see leftmost picture):Singh, S. (August 27, 2009). ''Cartesian product''. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/ : \begin A \times (B \cap C) &= (A \times B) \cap (A \times C), \\ A \times (B \cup C) &= (A \times B) \cup (A \times C), \\ A \times (B \setminus C) &= (A \times B) \setminus (A \times C), \end : (A \times B)^\complement = \left(A^\complement \times B^\complement\right) \cup \left(A^\complement \times B\right) \cup \left(A \times B^\complement\right)\!, where A^\complement denotes the
absolute 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 ...

absolute complement
of ''A''. Other properties related with
subset 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 a ...

subset
s are: :\text A \subseteq B \text A \times C \subseteq B \times C; :\text A,B \neq \emptyset \text A \times B \subseteq C \times D \!\iff\! A \subseteq C \text B \subseteq D.


Cardinality

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 ...
of a set is the number of elements of the set. For example, defining two sets: and Both set ''A'' and set ''B'' consist of two elements each. Their Cartesian product, written as , results in a new set which has the following elements: : ''A'' × ''B'' = . where each element of ''A'' is paired with each element of ''B'', and where each pair makes up one element of the output set. The number of values in each element of the resulting set is equal to the number of sets whose Cartesian product is being taken; 2 in this case. The cardinality of the output set is equal to the product of the cardinalities of all the input sets. That is, : , ''A'' × ''B'', = , ''A'', · , ''B'', . In this case, , ''A'' × ''B'', = 4 Similarly : , ''A'' × ''B'' × ''C'', = , ''A'', · , ''B'', · , ''C'', and so on. The set is
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...
if either ''A'' or ''B'' is infinite, and the other set is not the empty set.


Cartesian products of several sets


n-ary Cartesian product

The Cartesian product can be generalized to the ''n''-ary Cartesian product over ''n'' sets ''X''1, ..., ''Xn'' as the set : X_1\times\cdots\times X_n = \ of ''n''-tuples. If tuples are defined as nested ordered pairs, it can be identified with . If a tuple is defined as a function on that takes its value at ''i'' to be the ''i''th element of the tuple, then the Cartesian product ''X''1×⋯×''X''''n'' is the set of functions : \.


n-ary Cartesian power

The Cartesian square of a set ''X'' is the Cartesian product . An example is the 2-dimensional
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
where R is 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: R2 is the set of all points where ''x'' and ''y'' are real numbers (see the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
). The ''n''-ary Cartesian power of a set ''X'', denoted X^n, can be defined as : X^n = \underbrace_= \. An example of this is , with R again the set of real numbers, and more generally R''n''. The ''n''-ary Cartesian power of a set ''X'' 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 ...

isomorphic
to the space of functions from an ''n''-element set to ''X''. As a special case, the 0-ary Cartesian power of ''X'' may be taken to be a
singleton 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 calculus, change (mathematical analysis, analysis). It ...
, corresponding to the
empty function 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 ...
with
codomain 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 ...

codomain
''X''.


Infinite Cartesian products

It is possible to define the Cartesian product of an arbitrary (possibly
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

infinite
)
indexed family 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 h ...
of sets. If ''I'' is any
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a Set (mathematics), set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. Th ...
, and \_ is a family of sets indexed by ''I'', then the Cartesian product of the sets in \_ is defined to be : \prod_ X_i = \left\, that is, the set of all functions defined on the
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a Set (mathematics), set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. Th ...
such that the value of the function at a particular index ''i'' is an element of ''Xi''. Even if each of the ''Xi'' is nonempty, the Cartesian product may be empty if the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
, which is equivalent to the statement that every such product is nonempty, is not assumed. For each ''j'' in ''I'', the function : \pi_: \prod_ X_i \to X_, defined by \pi_(f) = f(j) is called the ''j''th
projection map In mathematics, a projection is a mapping of a Set (mathematics), set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for function composition, mapping composition (or, in other words, which is idem ...
. Cartesian power is a Cartesian product where all the factors ''Xi'' are the same set ''X''. In this case, : \prod_ X_i = \prod_ X is the set of all functions from ''I'' to ''X'', and is frequently denoted ''XI''. This case is important in the study of cardinal exponentiation. An important special case is when the index set is \mathbb, the
natural numbers 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 ...

natural numbers
: this Cartesian product is the set of all infinite sequences with the ''i''th term in its corresponding set ''Xi''. For example, each element of : \prod_^\infty \mathbb R = \mathbb R \times \mathbb R \times \cdots can be visualized as a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
with countably infinite real number components. This set is frequently denoted \mathbb^\omega, or \mathbb^.


Other forms


Abbreviated form

If several sets are being multiplied together (e.g., ''X''1, ''X''2, ''X''3, …), then some authorsOsborne, M., and Rubinstein, A., 1994. ''A Course in Game Theory''. MIT Press. choose to abbreviate the Cartesian product as simply ×''X''''i''.


Cartesian product of functions

If ''f'' is a function from ''A'' to ''B'' and ''g'' is a function from ''X'' to ''Y'', then their Cartesian product is a function from to with : (f\times g)(a, x) = (f(a), g(x)). This can be extended to
tuple 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 and infinite collections of functions. This is different from the standard Cartesian product of functions considered as sets.


Cylinder

Let A be a set and B \subseteq A. Then the ''cylinder'' of B with respect to A is the Cartesian product B \times A of B and A. Normally, A is considered to be the
universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development ...
of the context and is left away. For example, if B is a subset of the natural numbers \mathbb, then the cylinder of B is B \times \mathbb.


Definitions outside set theory


Category theory

Although the Cartesian product is traditionally applied to sets,
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 ...
provides a more general interpretation of the product of mathematical structures. This is distinct from, although related to, the notion of a
Cartesian square 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 ...
in category theory, which is a generalization of the
fiber product 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 d ...

fiber product
.
Exponentiation Exponentiation is a mathematical 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) ...
is 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 the Cartesian product; thus any category with a Cartesian product (and a
final object In category theory, a branch of mathematics, an initial object of a category (mathematics), category is an object in such that for every object in , there exists precisely one morphism . The dual (category theory), dual notion is that of a te ...
) is a
Cartesian closed category In category theory, a category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories' ...
.


Graph theory

In
graph theory 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 gen ...
, the Cartesian product of two graphs ''G'' and ''H'' is the graph denoted by , whose
vertex Vertex (Latin: peak; plural vertices or vertexes) means the "top", or the highest geometric point of something, usually a curved surface or line, or a point where any two geometric sides or edges meet regardless of elevation; as opposed to an Apex ( ...
set is the (ordinary) Cartesian product and such that two vertices (''u'',''v'') and (''u''′,''v''′) are adjacent in , if and only if and ''v'' is adjacent with ''v''′ in ''H'', ''or'' and ''u'' is adjacent with ''u''′ in ''G''. The Cartesian product of graphs is not a product in the sense of category theory. Instead, the categorical product is known as the
tensor product of graphs In graph theory, the tensor product ''G'' × ''H'' of graphs ''G'' and ''H'' is a graph such that * the vertex set of ''G'' × ''H'' is the Cartesian product ''V''(''G'') × ''V''(''H''); and * vertices (''g,h'') and (''g',h) are adjacent in ''G'' ...
.


See also

*
Binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
* Concatenation of sets of strings *
Coproduct 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 d ...
*
Cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

Cross product
*
Direct product of groups 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 ...
*
Empty product 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 ...
*
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
*
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 ...
*
Finitary relation 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 ...
* Join (SQL) § Cross join * Orders on the Cartesian product of totally ordered sets *
Axiom of power set Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of the set order theory, ordered with respect to Inclusion (set theory), inclusion. In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of ...
(to prove the existence of the Cartesian product) *
Product (category theory) 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 d ...
*
Product topology Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...
*
Product type In programming language A programming language is a formal language comprising a Instruction set architecture, set of instructions that produce various kinds of Input/output, output. Programming languages are used in computer programming to impl ...
*
Ultraproduct The ultraproduct is a mathematics, mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structure ( ...


References


External links


Cartesian Product at ProvenMath
*

{{Set theory Axiom of choice Operations on sets