In

_{1}, ..., ''X_{n}'' as the set
: $X\_1\backslash times\backslash cdots\backslash times\; X\_n\; =\; \backslash $
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
: $\backslash .$

^{2} is the set of all points where ''x'' and ''y'' are real numbers (see the ^{''n''}.
The ''n''-ary Cartesian power of a set ''X'' is

_{i}''. Even if each of the ''X_{i}'' is nonempty, the Cartesian product may be empty if the _{i}'' are the same set ''X''. In this case,
: $\backslash prod\_\; X\_i\; =\; \backslash prod\_\; X$
is the set of all functions from ''I'' to ''X'', and is frequently denoted ''X^{I}''. This case is important in the study of cardinal exponentiation. An important special case is when the index set is $\backslash mathbb$, the _{i}''. For example, each element of
: $\backslash prod\_^\backslash infty\; \backslash mathbb\; R\; =\; \backslash mathbb\; R\; \backslash times\; \backslash mathbb\; R\; \backslash times\; \backslash cdots$
can be visualized as a

_{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 at ProvenMath

*

{{Set theory Axiom of choice Operations on sets

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 ...

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
: $A\backslash times\; B\; =\; \backslash .$
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 ...

, 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 thestandard 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 theCartesian 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'') ...

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 ...

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 ...

. 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 ofordered 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 ...

. The most common definition of ordered pairs, Kuratowski's definition, is $(x,\; y)\; =\; \backslash $. Under this definition, $(x,\; y)$ is an element of $\backslash mathcal(\backslash mathcal(X\; \backslash cup\; Y))$, and $X\backslash times\; Y$ is a subset of that set, where $\backslash mathcal$ 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 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 notcommutative
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 ...

,
: $A\; \backslash times\; B\; \backslash neq\; B\; \backslash times\; A,$
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 ...

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 ...

.
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).
: $(A\backslash times\; B)\backslash times\; C\; \backslash neq\; A\; \backslash times\; (B\; \backslash times\; C)$
If for example ''A'' = , then .
Intersections, unions, and subsets

The Cartesian product satisfies the following property with respect to intersections (see middle picture). : $(A\; \backslash cap\; B)\; \backslash times\; (C\; \backslash cap\; D)\; =\; (A\; \backslash times\; C)\; \backslash cap\; (B\; \backslash times\; D)$ In most cases, the above statement is not true if we replace intersection with union (see rightmost picture). : $(A\; \backslash cup\; B)\; \backslash times\; (C\; \backslash cup\; D)\; \backslash neq\; (A\; \backslash times\; C)\; \backslash cup\; (B\; \backslash times\; D)$ In fact, we have that: : $(A\; \backslash times\; C)\; \backslash cup\; (B\; \backslash times\; D)\; =;\; href="/html/ALL/s/A\_\backslash setminus\_B)\_\backslash times\_C.html"\; ;"title="A\; \backslash setminus\; B)\; \backslash times\; C">A\; \backslash setminus\; B)\; \backslash times\; C$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 ...

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 ...

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

Thecardinality
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

`n`-ary Cartesian power

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: RCartesian 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
: $X^n\; =\; \backslash underbrace\_=\; \backslash .$
An example of this is , with R again the set of real numbers, and more generally Risomorphic
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 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 ...

''X''.
Infinite Cartesian products

It is possible to define the Cartesian product of an arbitrary (possiblyinfinite
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 ...

) 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 $\backslash \_$ is a family of sets indexed by ''I'', then the Cartesian product of the sets in $\backslash \_$ is defined to be
: $\backslash prod\_\; X\_i\; =\; \backslash left\backslash ,$
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 ''Xaxiom 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 ...

, which is equivalent to the statement that every such product is nonempty, is not assumed.
For each ''j'' in ''I'', the function
: $\backslash pi\_:\; \backslash prod\_\; X\_i\; \backslash to\; X\_,$
defined by $\backslash pi\_(f)\; =\; f(j)$ 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 ''Xnatural 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 ...

: this Cartesian product is the set of all infinite sequences with the ''i''th term in its corresponding set ''Xvector
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 $\backslash mathbb^\backslash omega$, or $\backslash mathbb^$.
Other forms

Abbreviated form

If several sets are being multiplied together (e.g., ''X''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 : $(f\backslash times\; g)(a,\; x)\; =\; (f(a),\; g(x)).$ This can be extended totuple
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 $B\; \backslash subseteq\; A$. Then the ''cylinder'' of $B$ with respect to $A$ is the Cartesian product $B\; \backslash times\; A$ of $B$ and $A$. Normally, $A$ is considered to be theuniverse
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 $\backslash mathbb$, then the cylinder of $B$ is $B\; \backslash times\; \backslash mathbb$.
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 ...

.
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

Ingraph 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 ...

* 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