In

_{1}, ..., ''X_{n}'' as the set
: $X\_1\backslash times\backslash cdots\backslash times\; X\_n\; =\; \backslash $
of tuple, ''n''-tuples. If tuples are defined as Tuple#Tuples_as_nested_ordered_pairs, 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 Cartesian coordinate system).
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 R^{''n''}.
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''.

_{i}''. Even if each of the ''X_{i}'' 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
: $\backslash pi\_:\; \backslash prod\_\; X\_i\; \backslash to\; X\_,$
defined by $\backslash pi\_(f)\; =\; f(j)$ is called the ''j''th Projection (mathematics), projection map.
Cartesian power is a Cartesian product where all the factors ''X_{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 natural numbers: this Cartesian product is the set of all infinite sequences with the ''i''th term in its corresponding set ''X_{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 Euclidean vector, vector with countably infinite real number components. This set is frequently denoted $\backslash mathbb^\backslash omega$, or $\backslash mathbb^$.

_{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

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{{Set theory Axiom of choice Operations on sets

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 has no generally ...

, specifically 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 type of object c ...

, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, 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. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.
The Cartesian product is named after René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French-born philosopher, mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Anc ...

, 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 product.
Examples

A deck of cards

An illustrative example is the standard 52-card deck. The Playing cards#Anglo-American, 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 inanalytic 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 ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French-born philosopher, mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Anc ...

assigned to each point in the plane a pair of real numbers, 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 , 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 theory, set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, Ordered pair#Kuratowski's definition, 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 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 is not commutative, : $A\; \backslash times\; B\; \backslash neq\; B\; \backslash times\; A,$ because the ordered pairs 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. 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 (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 Intersection (set theory), 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 (set theory), 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: : $(A\; \backslash times\; C)\; \backslash cup\; (B\; \backslash times\; D)\; =\; [(A\; \backslash setminus\; B)\; \backslash times\; C]\; \backslash cup\; [(A\; \backslash cap\; B)\; \backslash times\; (C\; \backslash cup\; D)]\; \backslash cup\; [(B\; \backslash setminus\; A)\; \backslash times\; D]$ For the set difference, we also have the following identity: : $(A\; \backslash times\; C)\; \backslash setminus\; (B\; \backslash times\; D)\; =\; [A\; \backslash times\; (C\; \backslash setminus\; D)]\; \backslash cup\; [(A\; \backslash setminus\; B)\; \backslash times\; C]$ 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/ : $\backslash begin\; A\; \backslash times\; (B\; \backslash cap\; C)\; \&=\; (A\; \backslash times\; B)\; \backslash cap\; (A\; \backslash times\; C),\; \backslash \backslash \; A\; \backslash times\; (B\; \backslash cup\; C)\; \&=\; (A\; \backslash times\; B)\; \backslash cup\; (A\; \backslash times\; C),\; \backslash \backslash \; A\; \backslash times\; (B\; \backslash setminus\; C)\; \&=\; (A\; \backslash times\; B)\; \backslash setminus\; (A\; \backslash times\; C),\; \backslash end$ : $(A\; \backslash times\; B)^\backslash complement\; =\; \backslash left(A^\backslash complement\; \backslash times\; B^\backslash complement\backslash right)\; \backslash cup\; \backslash left(A^\backslash complement\; \backslash times\; B\backslash right)\; \backslash cup\; \backslash left(A\; \backslash times\; B^\backslash complement\backslash right)\backslash !,$ where $A^\backslash complement$ denotes the absolute complement of ''A''. Other properties related with subsets 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

The cardinality 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 set, infinite 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

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 $\backslash \_$ 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 such that the value of the function at a particular index ''i'' is an element of ''XOther 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 to tuples 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 the Universe (mathematics), universe 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 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 , whose vertex (graph theory), vertex 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 (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 * Cross product * Direct product of groups * 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 * Axiom of power set#Consequences, Axiom of power set (to prove the existence of the Cartesian product) * Product (category theory) * Product topology * Product type * UltraproductReferences

External links

Cartesian Product at ProvenMath

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{{Set theory Axiom of choice Operations on sets