Cartesian product

The Cartesian product of sets $A$ and $B$ is

Cartesian products encode “simultaneous choices” and underlie coordinate descriptions: $\mathbb{R}^2=\mathbb{R}\times\mathbb{R}$, graphs of functions are subsets of $X\times Y$, and relations are subsets of products.

Examples:

• If $A=\{1,2\}$ and $B=\{a,b\}$, then $A\times B=\{(1,a),(1,b),(2,a),(2,b)\}$.
• $\mathbb{R}^3=\mathbb{R}\times\mathbb{R}\times\mathbb{R}$.
• If $A=\varnothing$, then $A\times B=\varnothing$ for any set $B$.