Cartesian product

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

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

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.