# Cartesian product

## Mathematical set formed from two given sets / From Wikipedia, the free encyclopedia

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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

- $A\times B=\{(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\}.$
^{[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.