In
mathematics, specifically
set theory, the **Cartesian product** of two
sets *A* and *B*, denoted *A* × *B*, is the set of all
ordered pairs (*a*, *b*) where *a* is in *A* and *b* is in *B*.^{
[1]} In terms of
set-builder notation, that is

^{ [2]}^{ [3]}

A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product *rows* × *columns* is taken, the cells of the table contain ordered pairs of the form (row value, column value).^{
[4]}

One can similarly define the Cartesian product of *n* sets, also known as an ** n-fold Cartesian product**, which can be represented by an

The Cartesian product is named after
René Descartes,^{
[5]} whose formulation of
analytic geometry gave rise to the concept, which is further generalized in terms of
direct product.

An illustrative example is the standard 52-card deck. The 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 {♠, ♥, ♦, ♣} 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.

*Ranks* × *Suits* returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), …, (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}.

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

These two sets are distinct, even disjoint, but there is a natural bijection between them, under which (3, ♣) corresponds to (♣, 3) and so on.

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 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 ℝ×ℝ, with ℝ denoting the real numbers) is thus assigned to the set of all points in the plane.^{[
citation needed]}

A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, Kuratowski's definition, is . Under this definition, is an element of , and is a subset of that set, where represents the power set operator. 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 two-set Cartesian product is necessarily prior to most other definitions.

Let *A*, *B*, *C*, and *D* be sets.

The Cartesian product *A* × *B* is not
commutative,

^{ [4]}

because the
ordered pairs are reversed unless at least one of the following conditions is satisfied:^{
[6]}

*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).

If for example *A* = {1}, then (*A* × *A*) × *A* = {((1, 1), 1)} ≠ {(1, (1, 1))} = *A* × (*A* × *A*).

The Cartesian product satisfies the following property with respect to intersections (see middle picture).

In most cases, the above statement is not true if we replace intersection with union (see rightmost picture).

In fact, we have that:

For the set difference, we also have the following identity:

Here are some rules demonstrating distributivity with other operators (see leftmost picture):^{
[6]}

where denotes the
absolute complement of *A*.

Other properties related with subsets are:

^{ [7]}

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

*A*×*B*= {(a,5), (a,6), (b,5), (b,6)}.

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*|.^{ [4]}

In this case, |*A* × *B*| = 4

Similarly

- |
*A*×*B*×*C*| = |*A*| · |*B*| · |*C*|

and so on.

The set *A* × *B* is
infinite if either *A* or *B* is infinite, and the other set is not the empty set.^{
[8]}

The Cartesian product can be generalized to the ** n-ary Cartesian product** over

of
*n*-tuples. If tuples are defined as
nested ordered pairs, it can be identified with (*X*_{1} × ⋯ × *X*_{n−1}) × *X _{n}*. If a tuple is defined as a function on {1, 2, …,

The **Cartesian square** of a set *X* is the Cartesian product *X*^{2} = *X* × *X*.
An example is the 2-dimensional
plane **R**^{2} = **R** × **R** where **R** is the set of
real numbers:^{
[1]} **R**^{2} is the set of all points (*x*,*y*) where *x* and *y* are real numbers (see the
Cartesian coordinate system).

The ** n-ary Cartesian power** of a set

An example of this is **R**^{3} = **R** × **R** × **R**, with **R** again the set of real numbers,^{
[1]} and more generally **R**^{n}.

The *n*-ary Cartesian power of a set *X* is
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*.

It is possible to define the Cartesian product of an arbitrary (possibly
infinite)
indexed family of sets. If *I* is any
index set, and is a family of sets indexed by *I*, then the Cartesian product of the sets in is defined to be

that is, the set of all functions defined on the
index set *I* such that the value of the function at a particular index *i* is an element of *X _{i}*. Even if each of the

For each *j* in *I*, the function

defined by is called the ** jth
projection map**.

**Cartesian power** is a Cartesian product where all the factors *X _{i}* are the same set

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 , the
natural numbers: this Cartesian product is the set of all infinite sequences with the

can be visualized as a vector with countably infinite real number components. This set is frequently denoted , or .

If several sets are being multiplied together (e.g., *X*_{1}, *X*_{2}, *X*_{3}, …), then some authors^{
[9]} choose to abbreviate the Cartesian product as simply ×*X*_{i}.

If *f* is a function from *X* to *A* and *g* is a function from *Y* to *B*, then their Cartesian product *f* × *g* is a function from *X* × *Y* to *A* × *B* with

This can be extended to tuples and infinite collections of functions. This is different from the standard Cartesian product of functions considered as sets.

Let be a set and . Then the *cylinder* of with respect to is the Cartesian product of and .

Normally, is considered to be the universe of the context and is left away. For example, if is a subset of the natural numbers , then the cylinder of is .

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.

In
graph theory, the
Cartesian product of two graphs *G* and *H* is the graph denoted by *G* × *H*, whose
vertex set is the (ordinary) Cartesian product *V*(*G*) × *V*(*H*) and such that two vertices (*u*,*v*) and (*u*′,*v*′) are adjacent in *G* × *H*, if and only if *u* = *u*′ and *v* is adjacent with *v*′ in *H*, *or* *v* = *v*′ 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.

- Binary relation
- Concatenation of sets of strings
- Coproduct
- Cross product
- Direct product of groups
- Empty product
- Euclidean space
- Exponential object
- Finitary relation
- Join (SQL) § Cross join
- Orders on the Cartesian product of totally ordered sets
- Axiom of power set (to prove the existence of the Cartesian product)
- Product (category theory)
- Product topology
- Product type
- Ultraproduct

- ^
^{a}^{b}^{c}Weisstein, Eric W. "Cartesian Product".*mathworld.wolfram.com*. Retrieved September 5, 2020. **^**Warner, S. (1990).*Modern Algebra*. Dover Publications. p. 6.**^**Nykamp, Duane. "Cartesian product definition".*Math Insight*. Retrieved September 5, 2020.- ^
^{a}^{b}^{c}"Cartesian Product".*web.mnstate.edu*. Archived from the original on July 18, 2020. Retrieved September 5, 2020. **^**"Cartesian".*Merriam-Webster.com*. 2009. Retrieved December 1, 2009.- ^
^{a}^{b}Singh, S. (August 27, 2009).*Cartesian product*. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/ **^**Cartesian Product of Subsets. (February 15, 2011).*ProofWiki*. Retrieved 05:06, August 1, 2011 from https://proofwiki.org/?title=Cartesian_Product_of_Subsets&oldid=45868**^**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**^**Osborne, M., and Rubinstein, A., 1994.*A Course in Game Theory*. MIT Press.