In mathematics, 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.
For example, if set X is the 13element set {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} and set Y is the 4element set {spades, hearts, diamonds, clubs}, then the Cartesian product of those two sets is the 52element set { <A, spades>, <K, spades>, ... <2, spades>, <A, hearts>, ... <3, clubs>, <2, clubs> }. Another example is the 2dimensional 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 nary Cartesian product over n sets X_{1},... ,X_{n}:
Indeed, it can be identified to (X_{1} × ... × X_{n1}) × X_{n}. It is a set of ntuples.
An example of this is the Euclidean 3space 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
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