Cartesian product

In set theory, the Cartesian product of two sets, \(X\) and \(Y\), is denoted by \(X \times Y\), and is equal to the set of ordered pairs whose first coordinate is an element of \(X\) and whose second coordinate is an element of \(Y\). Cartesian product is used to give an alternate characterisation of being infinite - that \(X\) is equinumerous with \(X \times X\). A bijection witnessing this is called a pairing function. However, note that, if \(X\) is an infinite well-ordered set, \(X\) is never order-isomorphic to \(X \times X\), where the order on \(X \times X\) is lexicographical.

Cartesian product is also used to formalize the notions of a relation and a function, originally non-set theoretical concepts, within set theory (without urelements).

It's possible to also define general Cartesian products such as \(X \times Y \times Z\), or even Cartesian products of infinitely many sets.