Cartesian product

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

${\displaystyle A\times B=\{(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\}.}$^[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 n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.

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.