Given two sets *X* and *Y*, the **Cartesian product** (or *direct product*) of the two sets, written as *X* × *Y* is the set of all ordered pairs with the first element of each pair selected from *X* and the second element selected from *Y*.

*X*×*Y*= { (*x*,*y*) |*x*in*X*and*y*in*Y*}

For example, if set X is the 13-element set {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} and set Y is the 4-element set {spades, hearts, diamonds, clubs}, then the Cartesian product of those two sets is the 52-element set { <A, spades>, <K, spades>, ... <2, spades>, <A, hearts>, ... <3, clubs>, <2, clubs> }. Another example is the 2-dimensional plane **R** × **R** where **R** is the set of real numbers. Subsets of the Cartesian product are called binary relations.

The binary Cartesian product can be generalized to the *n*-ary Cartesian product over *n* sets *X*_{1},... ,*X _{n}*:

*X*_{1}× ... ×*X*= { (_{n}*x*_{1},... ,*x*) |_{n}*x*_{1}in*X*_{1}and ... and*x*in_{n}*X*}_{n}

An example of this is the Euclidean 3-space **R** × **R** × **R**, with **R** again the set of real numbers.

The Cartesian product is named after Rene Descartes whose formulation of analytic geometry gave rise to this concept.

See also Mathematics -- Set theory