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 X1,... ,Xn:
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