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[[File:Cartesian Product qtl1.svg|thumb|Cartesian product <math>\scriptstyle A \times B</math> of the sets <math>\scriptstyle A=\{x,y,z\}</math> and <math>\scriptstyle B=\{1,2,3\}</math>]]

In [[set theory]] (and, usually, in other parts of [[mathematics]]), a '''Cartesian product''' is a [[mathematical operation]] that returns a [[set (mathematics)|set]] (or '''product set''' or simply '''product''') from multiple sets. That is, for sets ''A'' and ''B'', the Cartesian product {{nowrap|''A'' × ''B''}} is the set of all [[ordered pair]]s {{nowrap|(''a'', ''b'')}} where {{nowrap|''a'' ∈ ''A''}} and {{nowrap|''b'' ∈ ''B''}}. Products can be specified using [[set-builder notation]], e.g.
: <math>A\times B = \{\,(a,b)\mid a\in A \ \mbox{ and } \ b\in B\,\}.</math><ref>{{cite book|last=Warner|first=S.|title=Modern Algebra|page=6|publisher=[[Dover Publications]]|date=1990}}</ref>

A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product {{nowrap|''rows'' × ''columns''}} is taken, the cells of the table contain ordered pairs of the form {{nowrap|(row value, column value)}}.

More generally, a Cartesian product of ''n'' sets, also known as an '''''n''-fold Cartesian product''', can be represented by an array of ''n'' dimensions, where each element is an ''n''-[[tuple]]. An ordered pair is a [[Tuple#Names for tuples of specific lengths|2-tuple or couple]].

The Cartesian product is named after [[René Descartes]],<ref>{{cite web|title=Cartesian|date=2009|website=Merriam-Webster.com|accessdate=December 1, 2009|url=http://www.merriam-webster.com/dictionary/cartesian}}</ref> whose formulation of [[analytic geometry]] gave rise to the concept, which is further generalized in terms of [[direct product]].

== Examples ==

=== A deck of cards ===
[[File:Piatnikcards.jpg|thumb|Standard 52-card deck]]

An illustrative example is the [[standard 52-card deck]]. The [[Playing cards#Anglo-American|standard playing card]] ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits {{nowrap|{♠, {{color|#c00000|♥}}, {{color|#c00000|♦}}, ♣} }} 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. 

{{nowrap|''Ranks'' × ''Suits''}} returns a set of the form {(A, ♠), (A, {{color|#c00000|♥}}), (A, {{color|#c00000|♦}}), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, {{color|#c00000|♥}}), (2, {{color|#c00000|♦}}), (2, ♣)}.

{{nowrap|''Suits'' × ''Ranks''}} returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}.

Both sets are distinct, even disjoint.

=== A two-dimensional coordinate system ===
[[File:Cartesian-coordinate-system.svg|thumb|Cartesian coordinates of example points]]

The main historical example is the [[Cartesian plane]] in [[analytic geometry]]. In order to represent geometrical shapes in a numerical way and extract numerical information from shapes' numerical representations, [[René Descartes]] assigned to each point in the plane a pair of [[real number]]s, called its 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 {{nowrap|ℝ×ℝ}} with ℝ denoting the real numbers) is thus assigned to the set of all points in the plane.

== Most common implementation (set theory) ==
{{Main article|Implementation of mathematics in set theory}}

A formal definition of the Cartesian product from [[set theory|set-theoretical]] principles follows from a definition of [[ordered pair]]. The most common definition of ordered pairs, the [[Ordered pair#Kuratowski definition|Kuratowski definition]], is <math>(x, y) = \{\{x\},\{x, y\}\}</math>. Under this definition, <math>(x, y)</math> is an element of <math>\mathcal{P}(\mathcal{P}(X \cup Y))</math>, and <math>X\times Y</math> is a subset of that set, where <math>\mathcal{P}</math> represents the [[power set]] operator. Therefore, the existence of the Cartesian product of any two sets in [[ZFC]] follows from the axioms of [[axiom of pairing|pairing]], [[axiom of union|union]], [[axiom of power set|power set]], and [[axiom schema of specification|specification]]. Since [[function (mathematics)|functions]] are usually defined as a special case of [[relation (mathematics)|relations]], 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 {{nowrap|''A'' × ''B''}} is not [[commutative]],
: <math>A \times B \neq B \times A,</math>
because the [[ordered pair]]s are reversed unless at least one of the following conditions is satisfied:<ref name="cnx"/>
* ''A'' is equal to ''B'', or
* ''A'' or ''B'' is the [[empty set]].

For example:
: ''A'' = {1,2}; ''B'' = {3,4}
:: ''A'' × ''B'' = {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)}
:: ''B'' × ''A'' = {3,4} × {1,2} = {(3,1), (3,2), (4,1), (4,2)}

: ''A'' = ''B'' = {1,2}
:: ''A'' × ''B'' = ''B'' × ''A'' = {1,2} × {1,2} = {(1,1), (1,2), (2,1), (2,2)}

: ''A'' = {1,2}; ''B'' = ∅
:: ''A'' × ''B'' = {1,2} × ∅ = ∅
:: ''B'' × ''A'' = ∅ × {1,2} = ∅


Strictly speaking, the Cartesian product is not [[associative]] (unless one of the involved sets is empty).
: <math>(A\times B)\times C \neq A \times (B \times C)</math>
If for example ''A'' = {1}, then (''A'' × ''A'') × ''A'' = { ((1,1),1) } ≠ { (1,(1,1)) } = ''A'' × (''A'' × ''A'').

=== Intersections, unions, and subsets ===
{{multiple image
|align=center
| total_width = 750
|image1=CartDistr_svg.svg
|caption1=Example sets<br>
{{color|#0000c0|''A''}}={''y''∈[[real numbers|ℝ]]:1≤''y''≤4}, <br />
{{color|#c00000|''B''}}={''x''∈ℝ:2≤''x''≤5}, and {{color|#00c000|''C''}}={''x''∈ℝ:4≤''x''≤7}, demonstrating <br />
''A''×(''B''∩''C'') = ({{highlight|''A''×''B''|#FCC6C6}})∩({{highlight|''A''×''C''|#C6FCC6}}), <br />
''A''×(''B''∪''C'') = ({{highlight|''A''×''B''|#FCC6C6}})∪({{highlight|''A''×''C''|#C6FCC6}}), and <br />
''A''×(''B''{{tsp}}\{{hsp}}''C'') = ({{highlight|''A''×''B''|#FCC6C6}}){{tsp}}\{{hsp}}({{highlight|''A''×''C''|#C6FCC6}})
|image2=CartInts_svg.svg
|caption2=Example sets <br />
{{color|#c00000|''A''}}={''x''∈ℝ:2≤''x''≤5}, {{color|#00c000|''B''}}={''x''∈ℝ:3≤''x''≤7}, <br />
{{color|#c00000|''C''}}={''y''∈ℝ:1≤''y''≤3}, {{color|#00c000|''D''}}={''y''∈ℝ:2≤''y''≤4}, demonstrating <br />
{{highlight|(''A''∩''B'')×(''C''∩''D'')|#FCFCC6}} = {{highlight|(''A''×''C'')∩''(B''×''D'')|#FCFCC6}}.
|image3=CartUnion_svg.svg
|caption3={{highlight|(''A''∪''B'')×(''C''∪''D'')|#E0E0FC}} ≠ {{highlight|(''A''×''C'')|#FCC6C6}}∪{{highlight|(''B''×''D'')|#C6FCC6}} can be seen from the same example.
}}
The Cartesian product behaves nicely with respect to [[Intersection (set theory)|intersections]] (see leftmost picture).
: <math>(A \cap B) \times (C \cap D) = (A \times C) \cap (B \times D)</math><ref name="planetmath">{{planetmath reference|id=359|title=CartesianProduct}}</ref>

In most cases the above statement is not true if we replace intersection with [[Union (set theory)|union]] (see middle picture).
: <math>(A \cup B) \times (C \cup D) \neq (A \times C) \cup (B \times D)</math>

In fact, we have that:
: <math>(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:
: <math>(A \times C) \setminus (B \times D) = [A \times (C \setminus D)] \cup [(A \setminus B) \times C] </math>

Here are some rules demonstrating distributivity with other operators (see rightmost picture):<ref name="cnx">Singh, S. (August 27, 2009). ''Cartesian product''. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/</ref>
: <math>A \times (B \cap C) = (A \times B) \cap (A \times C),</math>
: <math>A \times (B \cup C) = (A \times B) \cup (A \times C),</math>
: <math>A \times (B \setminus C) = (A \times B) \setminus (A \times C),</math>
: <math>(A \times B)^\complement = (A^\complement \times B^\complement) \cup (A^\complement \times B) \cup (A \times B^\complement),</math><ref name="planetmath"/>
where <math>A^\complement</math> denotes the [[absolute complement]] of ''A''.

Other properties related with [[subset]]s are:
: <math>\text{if } A \subseteq B \text{ then } A \times C \subseteq B \times C,</math>
: <math>\text{if both } A,B \neq \emptyset \text{ then } A \times B \subseteq C \times D \iff A \subseteq C\text{ and } B \subseteq D.</math><ref>Cartesian Product of Subsets. (February 15, 2011). ''ProofWiki''. Retrieved 05:06, August 1, 2011 from https://proofwiki.org/w/index.php?title=Cartesian_Product_of_Subsets&oldid=45868</ref>

=== Cardinality ===
{{See also|Cardinal arithmetic}}

The [[cardinality]] of a set is the number of elements of the set. For example, defining two sets: {{nowrap|1=''A'' = {a, b}}} and {{nowrap|1=''B'' = {5, 6}.}} Both set ''A'' and set ''B'' consist of two elements each. Their Cartesian product, written as {{nowrap|''A'' × ''B''}}, results in a new set which has the following elements:
: ''A'' × ''B'' = {(a,5), (a,6), (b,5), (b,6)}.

Each element of ''A'' is paired with each element of ''B''. 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 {{nowrap|''A'' × ''B''}} is [[infinite set|infinite]] if either ''A'' or ''B'' is infinite and the other set is not the empty set.<ref>Peter S. (1998). A Crash Course in the Mathematics of Infinite Sets. ''St. John's Review, 44''(2), 35–59. Retrieved August 1, 2011, from http://www.mathpath.org/concepts/infinity.htm</ref>

== Cartesian products of several sets ==

=== <var>n</var>-ary Cartesian product ===
The Cartesian product can be generalized to the '''''n''-ary Cartesian product''' over ''n'' sets ''X''<sub>1</sub>, ..., ''X<sub>n</sub>'' as the set

: <math>X_1\times\cdots\times X_n = \{(x_1, \ldots, x_n) \mid x_i \in X_i \ \text{for every} \ i \in \{1, \ldots, n\} \}.</math>

of [[tuple|''n''-tuple]]s. If tuples are defined as [[Tuple#Tuples_as_nested_ordered_pairs|nested ordered pairs]], it can be identified with {{nowrap|(''X''<sub>1</sub> × ... × ''X<sub>n−1</sub>'') × ''X<sub>n</sub>''}}. If a tuple is defined as a function on {{nowrap|{1, 2, ..., ''n''} }} that takes its value at ''i'' to be the ''i''th element of the tuple, then the Cartesian product ''X''<sub>1</sub>×...×''X''<sub>n</sub> is the set of functions

: <math>\{ x:\{1,\ldots,n\}\to X_1\cup\ldots\cup X_n \ | \ x(i)\in X_i \ \text{for every} \ i \in \{1, \ldots, n\} \}.</math>

=== <var>n</var>-ary Cartesian power ===
The '''Cartesian square''' of a set ''X'' is the Cartesian product {{nowrap|1=''X''<sup>2</sup> = ''X'' × ''X''}}.
An example is the 2-dimensional [[plane (mathematics)|plane]] {{nowrap|1='''R'''<sup>2</sup> = '''R''' × '''R'''}} where '''R''' is the set of [[real number]]s: '''R'''<sup>2</sup> is the set of all points {{nowrap|(''x'',''y'')}} where ''x'' and ''y'' are real numbers (see the [[Cartesian coordinate system]]).

The ''' ''n''-ary Cartesian power''' of a set ''X'' can be defined as

: <math> X^n = \underbrace{ X \times X \times \cdots \times X }_{n}= \{ (x_1,\ldots,x_n) \ | \ x_i \in X \ \text{for every} \ i \in \{1, \ldots, n\} \}.</math>

An example of this is {{nowrap|1='''R'''<sup>3</sup> = '''R''' × '''R''' × '''R'''}}, with '''R''' again the set of real numbers, and more generally '''R'''<sup>''n''</sup>.

The ''n''-ary cartesian power of a set ''X'' is [[isomorphism|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]], corresponding to the [[empty function]] with [[codomain]] ''X''.

=== Infinite Cartesian products ===
It is possible to define the Cartesian product of an arbitrary (possibly [[Infinity|infinite]]) [[indexed family]] of sets. If ''I'' is any [[index set]], and <math>\{X_i\}_{i\in I}</math> is a family of sets indexed by ''I'', then the Cartesian product of the sets in ''X'' is defined to be 

: <math>\prod_{i \in I} X_i = \left\{\left. f: I \to \bigcup_{i \in I} X_i\ \right|\ (\forall i)(f(i) \in X_i)\right\},</math> 

that is, the set of all functions defined on the [[index set]] such that the value of the function at a particular index ''i'' is an element of ''X<sub>i</sub>''.  Even if each of the ''X<sub>i</sub>'' is nonempty, the Cartesian product may be empty if the [[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 
: <math>  \pi_{j}: \prod_{i \in I} X_i \to X_{j},</math>
defined by <math>\pi_{j}(f) = f(j)</math> is called the '''''j''th [[Projection (mathematics)|projection map]]'''.

'''Cartesian power''' is a Cartesian product where all the factors ''X<sub>i</sub>'' are the same set ''X''. In this case, 
: <math>\prod_{i \in I} X_i = \prod_{i \in I} X</math> 
is the set of all functions from ''I'' to ''X'', and is frequently denoted ''X<sup>I</sup>''. This case is important in the study of [[cardinal exponentiation]]. An important special case is when the index set is <math>\mathbb{N}</math>, the [[natural numbers]]: this Cartesian product is the set of all infinite sequences with the ''i''th term in its corresponding set ''X<sub>i</sub>''. For example, each element of 
: <math>\prod_{n = 1}^\infty \mathbb R = \mathbb R \times \mathbb R \times \cdots</math>
can be visualized as a [[Euclidean vector|vector]] with countably infinite real number components. This set is frequently denoted <math>\mathbb{R}^\omega</math>, or <math>\mathbb{R}^{\mathbb{N}}</math>.

== Other forms ==

=== Abbreviated form ===
If several sets are being multiplied together, e.g. ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, …, then some authors<ref>Osborne, M., and Rubinstein, A., 1994. ''A Course in Game Theory''. MIT Press.</ref> choose to abbreviate the Cartesian product as simply <big>&times;</big>''X''<sub>''i''</sub>.

=== Cartesian product of functions ===
If ''f'' is a function from ''A'' to ''B'' and ''g'' is a function from ''X'' to ''Y'', their Cartesian product {{nowrap|''f'' × ''g''}} is a function from {{nowrap|''A'' × ''X''}} to {{nowrap|''B'' × ''Y''}} with
: <math>(f\times g)(a, x) = (f(a), g(x)).</math>

This can be extended to [[tuple]]s and infinite collections of functions.
This is different from the standard cartesian product of functions considered as sets.

=== Cylinder ===
Let <math>A</math> be a set and <math>B \subseteq A</math>. Then the ''cylinder'' of <math>B</math> with respect to <math>A</math> is the Cartesian product <math>B \times A</math> of <math>B</math> and <math>A</math>. 

Normally, <math>A</math> is considered to be the universe of the context and is left away. For example, if <math>B</math> is a subset of the natural numbers <math>\mathbb{N}</math>, then the cylinder of <math>B</math> is <math>B \times \mathbb{N}</math>.

== Definitions outside set theory ==

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

[[Exponential object|Exponentiation]] is the [[right adjoint]] of the Cartesian product; thus any category with a Cartesian product (and a [[final object]]) is a [[Cartesian closed category]].

===Graph theory===
In [[graph theory]] the [[Cartesian product of graphs|Cartesian product of two graphs]] ''G'' and ''H'' is the graph denoted by {{nowrap|''G'' × ''H''}} whose [[vertex (graph theory)|vertex]] set is the (ordinary) Cartesian product {{nowrap|''V''(''G'') × ''V''(''H'')}} and such that two vertices (''u'',''v'') and (''u''′,''v''′) are adjacent in {{nowrap|''G'' × ''H''}} if and only if {{nowrap|1=''u'' = ''u''′}} and ''v'' is adjacent with ''v''′ in ''H'', ''or'' {{nowrap|1=''v'' = ''v''′}} and ''u'' is adjacent with ''u''′ in ''G''.  The Cartesian product of graphs is not a [[product (category theory)|product]] in the sense of category theory.  Instead, the categorical product is known as the [[tensor product of graphs]].

==See also==
* [[Binary relation]]
* [[Concatenation#Concatenation of sets of strings|Concatenation of sets of strings]]
* [[Coproduct]]
* [[Empty product]]
* [[Euclidean space]]
* [[Exponential object]]
* [[Finitary relation]]
* [[Join (SQL)#Cross join|Join (SQL) § Cross join]]
* [[Total order#Orders on the Cartesian product of totally ordered sets|Orders on the Cartesian product of totally ordered sets]]
* [[Product (category theory)]]
* [[Product topology]]
* [[Product type]]
* [[Ultraproduct]]

== References ==
{{Reflist}}

==External links==
* [http://www.apronus.com/provenmath/cartesian.htm Cartesian Product at ProvenMath]
* {{springer|title=Direct product|id=p/d032730}}
* [http://education-portal.com/academy/lesson/how-to-find-the-cartesian-product.html How to find the Cartesian Product, Education Portal Academy]

{{Set theory}}

[[Category:Axiom of choice]]
[[Category:Basic concepts in set theory]]
[[Category:Binary operations]]