Cartesian product

Given two sets A and B, we are often interested in all ordered pairs of their elements. For example, if A = \{a, b\} and B = \{1, 2\}, the ordered pairs are as follows:

(a, 1) (a, 2) (b, 1) (b, 2)

The set of all such pairs is called the Cartesian product of A and B. The name comes from the French mathematician René Descartes. Formally, Cartesian product can be described like this:

A \times B = \{(a,b) | a \in A, b \in B\}

Descartes saw that the number plane x, y could be represented as a product of two sets of real numbers.

\mathbb{R} \times \mathbb{R} = \{(x, y) | x \in \mathbb{R}, y \in \mathbb{R}\}