Cartesian product - Imported Knowlpedia Content

A Cartesian product of sets $A$ and $B$ is the set

A\times B=\{(a,b) : a\in A \text{ and } b\in B\},

where $(a,b)$ is an ordered pair.

Remarks

Cartesian products provide the ambient sets in which relations live: a relation from $A$ to $B$ is a subset of $A\times B$ .

Examples

If $A=\{0,1\}$ and $B=\{a,b\}$ , then $A\times B=\{(0,a),(0,b),(1,a),(1,b)\}$ .
$\mathbb{N}\times\mathbb{N}$ is the set of all ordered pairs of natural numbers.