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| {{Redirect|Cartesian square|Cartesian squares in category theory|Cartesian square (category theory)}}
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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>]]
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| In [[mathematics]], a '''Cartesian product''' is a [[mathematical operation]] which returns a [[set (mathematics)|set]] (or '''product set''') from multiple sets. That is, for sets ''A'' and ''B'', the Cartesian product {{nowrap|''A'' × ''B''}} is the set of all ordered pairs {{nowrap|(a, b)}} where {{nowrap|a ∈ ''A''}} and {{nowrap|b ∈ ''B''}}.
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| The simplest case of a Cartesian product is the '''Cartesian square''', which returns a set from two sets. 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)}}.
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| A Cartesian product of ''n'' sets can be represented by an array of ''n'' dimensions, where each element is an ''n''-[[tuple]]. An ordered pair is a 2-tuple.
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| The Cartesian product is named after [[René Descartes]],<ref>cartesian. (2009). In Merriam-Webster Online Dictionary. Retrieved December 1, 2009, from http://www.merriam-webster.com/dictionary/cartesian</ref> whose formulation of [[analytic geometry]] gave rise to the concept.
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| == Examples ==
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| === A deck of cards ===
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| 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|{♠, ♥, ♦, ♣} }} form a 4-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.
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| {{nowrap|''Ranks'' × ''Suits''}} returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}.
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| {{nowrap|''Suits'' × ''Ranks''}} returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}.
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| === A two-dimensional coordinate system ===
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| An example in [[analytic geometry]] is the [[Cartesian plane]]. The Cartesian plane is the result of the Cartesian product of two sets ''X'' and ''Y'', which refer to points on the x-axis and points on the y-axis, respectively. This Cartesian product can be denoted as {{nowrap|''X'' × ''Y''}}. This produces the set of all possible [[ordered pair]]s whose first component is a member of ''X'' and whose second component is a member of ''Y'' (e.g., the whole of the x–y plane). Alternatively, the Cartesian product can be denoted as {{nowrap|''Y'' × ''X''}}, in which case the first component of the order pair is a member of ''Y'' and the second component of the ordered pair is a member of ''X''. The Cartesian product is consequently not [[commutative]].
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| :<math>X\times Y = \{\,(x,y)\mid x\in X \ \and \ y\in Y\,\}.</math> <ref>Warner, S: ''Modern Algebra'', page 6. Dover Press, 1990.</ref>
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| :<math>Y\times X = \{\,(y,x)\mid y\in Y \ \and \ x\in X\,\}.</math>
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| :<math>X\times Y \neq Y\times X</math>
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| == Most common implementation (set theory) ==
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| {{main|Implementation of mathematics in set theory}}
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| 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>. Note that, under this definition, <math>X\times Y \subseteq \mathcal{P}(\mathcal{P}(X \cup Y))</math>, where <math>\mathcal{P}</math> represents the [[power set]]. 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.
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| === Non-commutativity and non-associativity ===
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| Let ''A'', ''B'', ''C'', and ''D'' be sets.
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| The Cartesian product {{nowrap|''A'' × ''B''}} is not [[commutative]],
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| :<math>A \times B \neq B \times A,</math>
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| because the [[ordered pair]]s are reversed except if at least one of the following conditions is satisfied:<ref name="cnx"/>
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| * ''A'' is equal to ''B'', or
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| * ''A'' or ''B'' is the [[empty set]].
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| For example: | |
| :''A'' = {1,2}; ''B'' = {3,4}
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| ::''A'' × ''B'' = {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)}
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| ::''B'' × ''A'' = {3,4} × {1,2} = {(3,1), (3,2), (4,1), (4,2)}
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| :''A'' = ''B'' = {1,2}
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| ::''A'' × ''B'' = ''B'' × ''A'' = {1,2} × {1,2} = {(1,1), (1,2), (2,1), (2,2)}
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| :''A'' = {1,2}; ''B'' = ∅
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| ::''A'' × ''B'' = {1,2} × ∅ = ∅
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| ::''B'' × ''A'' = ∅ × {1,2} = ∅
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| Strictly speaking, the Cartesian product is not [[associative]] (unless one of the above conditions occurs).
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| :<math>(A\times B)\times C \neq A \times (B \times C)</math>
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| === Intersections, unions, and subsets ===
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| The Cartesian product behaves nicely with respect to [[Intersection (set theory)|intersections]].
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| :<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>
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| Notice that in most cases the above statement is not true if we replace intersection with [[Union (set theory)|union]].
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| :<math>(A \cup B) \times (C \cup D) \neq (A \times C) \cup (B \times D)</math>
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| Here are some rules demonstrating distributivity with other operators:<ref name="cnx">Singh, S. (2009, August 27). ''Cartesian product''. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/</ref>
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| :<math>A \times (B \cap C) = (A \times B) \cap (A \times C),</math>
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| :<math>A \times (B \cup C) = (A \times B) \cup (A \times C),</math>
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| :<math>A \times (B \setminus C) = (A \times B) \setminus (A \times C),</math>
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| :<math>(A \times B)^c = (A^c \times B^c) \cup (A^c \times B) \cup (A \times B^c).</math><ref name="planetmath"/>
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| Other properties related with [[subset]]s are:
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| :<math>\text{if } A \subseteq B \text{ then } A \times C \subseteq B \times C,</math>
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| :<math>\text{if both } A,B \neq \emptyset \text{ then } A \times B \subseteq C \times D \iff A \subseteq C \and B \subseteq D.</math><ref>Cartesian Product of Subsets. (2011, February 15). ''ProofWiki''. Retrieved 05:06, August 1, 2011 from http://www.proofwiki.org/w/index.php?title=Cartesian_Product_of_Subsets&oldid=45868</ref>
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| === Cardinality ===
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| 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)}.
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| Each element of ''A'' is paired with each element of ''B''. Each pair makes up one element of the output set.
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| The number of values in each pair is equal to the number of sets whose cartesian product is being taken; 2 in this case.
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| The cardinality of the output set is equal to the product of the cardinalities of all the input sets. That is,
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| :|''A'' × ''B''| = |''A''| · |''B''|.
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| Similarly
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| :|''A'' × ''B'' × ''C''| = |''A''| · |''B''| · |''C''|
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| and so on. | |
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| The cardinality of {{nowrap|''A'' × ''B''}} is [[infinity]] if either ''A'' or ''B'' has infinite elements 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> | |
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| == Cartesian square and Cartesian power ==
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| The '''Cartesian square''' (or '''binary Cartesian product''') of a set ''X'' is the Cartesian product {{nowrap|1=''X''<sup>2</sup> = ''X'' × ''X''}}.
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| 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 – all points {{nowrap|(''x'',''y'')}} where ''x'' and ''y'' are real numbers (see the [[Cartesian coordinate system]]).
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| == Higher powers of a set ==
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| The '''cartesian power''' of a [[Set (mathematics)|set]] ''X'' can be defined as:
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| : <math> X^n = \underbrace{ X \times X \times \cdots \times X }_{n}= \{ (x_1,\ldots,x_n) \ | \ x_i \in X \ \text{for all} \ 1 \le i \le n \}.</math>
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| 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>.
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| == Generalized powers from different sets ==
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| === <var>n</var>-ary product ===
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| The Cartesian product can be generalized to the '''''n''-ary Cartesian product''' over ''n'' sets ''X''<sub>1</sub>, ..., ''X<sub>n</sub>'':
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| :<math>X_1\times\cdots\times X_n = \{(x_1, \ldots, x_n) : x_i \in X_i \}.</math>
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| It is a set of [[tuple|''n''-tuple]]s. If tuples are defined as [[Tuple#Tuples_as_nested_ordered_pairs|nested ordered pairs]], it can be identified to {{nowrap|(''X''<sub>1</sub> × ... × ''X<sub>n−1</sub>'') × ''X<sub>n</sub>''}}.
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| The ''n''-ary cartesian power of a set ''X'' is [[isomorphism|isomorphic]] to the space of [[function (mathematics)|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''.
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| === Infinite products ===
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| 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>\left\{X_i\,|\,i\in I\right\}</math> is a collection of sets indexed by ''I'', then the Cartesian product of the sets in ''X'' is defined to be
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| :<math>\prod_{i \in I} X_i = \left\{ f : I \to \bigcup_{i \in I} X_i\ \Big|\ (\forall i)(f(i) \in X_i)\right\},</math>
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| 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.
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| For each ''j'' in ''I'', the function | |
| :<math> \pi_{j} : \prod_{i \in I} X_i \to X_{j},</math>
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| defined by <math>\pi_{j}(f) = f(j)</math> is called the '''''j''th [[Projection (mathematics)|projection map]]'''.
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| An important 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
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| :<math>\prod_{n = 1}^\infty \mathbb R = \mathbb R \times \mathbb R \times \cdots</math>
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| 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>.
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| The special case '''Cartesian exponentiation''' occurs when all the factors ''X<sub>i</sub>'' involved in the product are the same set ''X''. In this case,
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| :<math>\prod_{i \in I} X_i = \prod_{i \in I} X</math>
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| 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]].
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| The definition of finite Cartesian products can be seen as a special case of the definition for infinite products. In this interpretation, an ''n''-tuple can be viewed as a function on {{nowrap|{1, 2, ..., ''n''} }} that takes its value at ''i'' to be the ''i''th element of the tuple (in some settings, this is taken as the very definition of an ''n''-tuple).
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| Nothing in the definition of an infinite Cartesian product implies that the Cartesian product of nonempty sets must itself be nonempty. This assertion is equivalent to the [[axiom of choice]].
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| == Other forms ==
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| === Abbreviated form ===
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| 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>×</big>''X''<sub>''i''</sub>.
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| === Cartesian product of functions ===
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| 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
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| :<math>(f\times g)(a, b) = (f(a), g(b)).</math>
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| This can be extended to [[tuple]]s and infinite collections of functions.
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| Note that this is different from the standard cartesian product of functions considered as sets.
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| == Definitions outside of Set theory ==
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| ===Category theory===
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| 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]].
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| [[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]].
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| ===Graph theory===
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| 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]].
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| ==See also==
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| * [[Exponential object]]
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| * [[Binary relation]]
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| * [[Coproduct]]
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| * [[Empty product]]
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| * [[Product (category theory)]]
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| * [[Product topology]]
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| * [[Finitary relation]]
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| * [[Ultraproduct]]
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| * [[Product type]]
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| * [[Euclidean space]]
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| * [[total order#Orders on the Cartesian product of totally ordered sets|orders on '''R'''<sup>''n''</sup>]]
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| == References ==
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| {{reflist}}
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| ==External links==
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| * [http://www.apronus.com/provenmath/cartesian.htm Cartesian Product at ProvenMath]
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| * {{springer|title=Direct product|id=p/d032730}}
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| * [http://education-portal.com/academy/lesson/how-to-find-the-cartesian-product.html How to find the Cartesian Product, Education Portal Academy]
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| {{Set theory}}
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| [[Category:Basic concepts in set theory]]
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| [[Category:Binary operations]]
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| [[Category:Axiom of choice]]
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