Cartesian product

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.

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 X₁,... ,X_n:

X₁ × ... × X_n = { (x₁,... ,x_n) | x₁ in X₁ and ... and x_n in X_n }

Indeed, it can be identified to (X₁ × ... × X_n-1) × X_n. It is a set of n-tuples.

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