In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets. That is, for sets A and B, the Cartesian product is the set of all where and . Products can be specified using setbuilder notation, e.g.
 $A\backslash times\; B\; =\; \backslash \{\backslash ,(a,b)\backslash mid\; a\backslash in\; A\; \backslash \; \backslash mbox\{\; and\; \}\; \backslash \; b\backslash in\; B\backslash ,\backslash \}.$
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 .
More generally, a Cartesian product of n sets, also known as an nfold Cartesian product, can be represented by an array of n dimensions, where each element is an ntuple. An ordered pair is a 2tuple or couple.
The Cartesian product is named after René Descartes, whose formulation of analytic geometry 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 52card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13element set. The card suits form a fourelement set. The Cartesian product of these sets returns a 52element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
returns a set of the form {(A, ♠), (A, ), (A, ), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, ), (2, ), (2, ♣)}.
returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}.
Both sets are distinct, even disjoint.
A twodimensional coordinate system
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
, called its coordinates. Usually, such a pair's first and second components are called its
x and
y coordinates, respectively; cf. 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 principles follows from a definition of
ordered pair. The most common definition of ordered pairs, the Kuratowski definition, is
$(x,\; y)\; =\; \backslash \{\backslash \{x\backslash \},\backslash \{x,\; y\backslash \}\backslash \}$. Note that, under this definition,
$X\backslash times\; Y\; \backslash subseteq\; \backslash mathcal\{P\}(\backslash mathcal\{P\}(X\; \backslash cup\; Y))$, where
$\backslash mathcal\{P\}$ represents the
power set. Therefore, the existence of the Cartesian product of any two sets in
ZFC follows from the axioms of pairing, union, power set, and specification. Since functions are usually defined as a special case of relations, and relations are usually defined as subsets of the Cartesian product, the definition of the twoset Cartesian product is necessarily prior to most other definitions.
Noncommutativity and nonassociativity
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
are reversed unless at least one of the following conditions is satisfied:
[
]
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).
 $(A\backslash times\; B)\backslash times\; C\; \backslash neq\; A\; \backslash times\; (B\; \backslash times\; C)$
If for example A = {1}, then ( A × A) × A = { ((1,1),1) } ≠ { (1,(1,1)) } = A × ( A × A).
Intersections, unions, and subsets
Example sets
={ y∈ℝ:1≤ y≤4},
={ x∈ℝ:2≤ x≤5}, and ={ x∈ℝ:4≤ x≤7}, demonstrating
A×( B∩ C) = ()∩(),
A×( B∪ C) = ()∪(), and
A×( B\ C) = ()\().]] 
Example sets
={ x∈ℝ:2≤ x≤5}, ={ x∈ℝ:3≤ x≤7},
={ y∈ℝ:1≤ y≤3}, ={ y∈ℝ:2≤ y≤4}, demonstrating
= .]]

The Cartesian product behaves nicely with respect to intersections, cf. left 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, cf. middle 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:
 $(A\; \backslash times\; C)\; \backslash cup\; (B\; \backslash times\; D)\; =\; (A\; \backslash cup\; (A\; \backslash cup\; (B$
For the set difference we also have the following identity:
 $(A\; \backslash times\; C)\; \backslash setminus\; (B\; \backslash times\; D)\; =\; A\; \backslash cup\; (A$
Here are some rules demonstrating distributivity with other operators (cf. right picture):[Singh, S. (August 27, 2009). Cartesian product. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/]
 $A\; \backslash times\; (B\; \backslash cap\; C)\; =\; (A\; \backslash times\; B)\; \backslash cap\; (A\; \backslash times\; C),$
 $A\; \backslash times\; (B\; \backslash cup\; C)\; =\; (A\; \backslash times\; B)\; \backslash cup\; (A\; \backslash times\; C),$
 $A\; \backslash times\; (B\; \backslash setminus\; C)\; =\; (A\; \backslash times\; B)\; \backslash setminus\; (A\; \backslash times\; C),$
 $(A\; \backslash times\; B)^\backslash complement\; =\; (A^\backslash complement\; \backslash times\; B^\backslash complement)\; \backslash cup\; (A^\backslash complement\; \backslash times\; B)\; \backslash cup\; (A\; \backslash times\; B^\backslash complement),$
where $A^\backslash complement$ denotes the absolute complement of A.
Other properties related with are:
 $\backslash text\{if\; \}\; A\; \backslash subseteq\; B\; \backslash text\{\; then\; \}\; A\; \backslash times\; C\; \backslash subseteq\; B\; \backslash times\; C,$
 $\backslash text\{if\; both\; \}\; A,B\; \backslash neq\; \backslash emptyset\; \backslash text\{\; then\; \}\; A\; \backslash times\; B\; \backslash subseteq\; C\; \backslash times\; D\; \backslash iff\; A\; \backslash subseteq\; C\; \backslash and\; B\; \backslash subseteq\; D.$
[Cartesian Product of Subsets. (February 15, 2011). ProofWiki
/ref>]
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 = {(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.
Similarly
  A × B × C =  A ·  B ·  C
and so on.
The set is infinite set if either A or B is infinite and the other set is not the empty set.[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]
nary product
Cartesian power
The Cartesian square (or binary Cartesian product) of a set X is the Cartesian product .
An example is the 2dimensional plane where R is the set of : R^{2} is the set of all points where x and y are real numbers (see the Cartesian coordinate system).
The cartesian power of a set X can be defined as:
 $X^n\; =\; \backslash underbrace\{\; X\; \backslash times\; X\; \backslash times\; \backslash cdots\; \backslash times\; X\; \}\_\{n\}=\; \backslash \{\; (x\_1,\backslash ldots,x\_n)\; \backslash \; \; \backslash \; x\_i\; \backslash in\; X\; \backslash text\{\; for\; all\; \}\; i=1,\backslash ldots,n\; \backslash \}.$
An example of this is , with R again the set of real numbers, and more generally R^{ n}.
The nary cartesian power of a set X is isomorphism to the space of functions from an nelement set to X. As a special case, the 0ary cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.
Finite nary product
The Cartesian product can be generalized to the nary Cartesian product over n sets X_{1}, ..., X_{n}:
 $X\_1\backslash times\backslash cdots\backslash times\; X\_n\; =\; \backslash \{(x\_1,\; \backslash ldots,\; x\_n):\; x\_i\; \backslash in\; X\_i\; \backslash \}.$
It is a set of tuple. If tuples are defined as nested ordered pairs, it can be identified to .
Infinite products
It is possible to define the Cartesian product of an arbitrary (possibly Infinity) indexed family of sets. If I is any index set, and $\backslash \{X\_i\backslash \}\_\{i\backslash in\; I\}$ is a family of sets indexed by I, then the Cartesian product of the sets in X is defined to be
 $\backslash prod\_\{i\; \backslash in\; I\}\; X\_i\; =\; \backslash left\backslash \{\backslash left.\; f:\; I\; \backslash to\; \backslash bigcup\_\{i\; \backslash in\; I\}\; X\_i\backslash \; \backslash right\backslash \; (\backslash forall\; i)(f(i)\; \backslash in\; X\_i)\backslash right\backslash \},$
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 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
 $\backslash pi\_\{j\}:\; \backslash prod\_\{i\; \backslash in\; I\}\; X\_i\; \backslash to\; X\_\{j\},$
defined by $\backslash pi\_\{j\}(f)\; =\; f(j)$ is called the jth projection map.
An important case is when the index set is $\backslash mathbb\{N\}$, the natural numbers: this Cartesian product is the set of all infinite sequences with the ith term in its corresponding set X_{i}. For example, each element of
 $\backslash prod\_\{n\; =\; 1\}^\backslash infty\; \backslash mathbb\; R\; =\; \backslash mathbb\; R\; \backslash times\; \backslash mathbb\; R\; \backslash times\; \backslash cdots$
can be visualized as a Euclidean vector with countably infinite real number components. This set is frequently denoted $\backslash mathbb\{R\}^\backslash omega$, or $\backslash mathbb\{R\}^\{\backslash mathbb\{N\}\}$.
The special case Cartesian exponentiation occurs when all the factors X_{i} involved in the product are the same set X. In this case,
 $\backslash prod\_\{i\; \backslash in\; I\}\; X\_i\; =\; \backslash prod\_\{i\; \backslash in\; I\}\; X$
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.
The definition of finite Cartesian products can be seen as a special case of the definition for infinite products. In this interpretation, an ntuple can be viewed as a function on that takes its value at i to be the ith element of the tuple (in some settings, this is taken as the very definition of an ntuple).
Other forms
Abbreviated form
If several sets are being multiplied together, e.g. X_{1}, X_{2}, X_{3}, …, then some authors[Osborne, 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, their Cartesian product is a function from to with
 $(f\backslash times\; g)(a,\; x)\; =\; (f(a),\; g(x)).$
This can be extended to and infinite collections of functions.
Note that 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 of the context and is left away. For example, if $B$ is a subset of the natural numbers $\backslash mathbb\{N\}$, then the cylinder of $B$ is $B\; \backslash times\; \backslash mathbb\{N\}$.
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 of mathematical structures. This is distinct from, although related to, the notion of a Cartesian square in category theory, which is a generalization of the fiber product.
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 two graphs G and H is the graph denoted by whose 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 in the sense of category theory. Instead, the categorical product is known as the tensor product of graphs.
See also
External links