Cartesian product

Let $A$ and $B$ be sets . Their Cartesian product is

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

Cartesian products provide a uniform language for describing relations (subsets of $A\times B$) and functions (special relations).

Examples:

• If $A=\{0,1\}$ and $B=\{a,b\}$ then $A\times B=\{(0,a),(0,b),(1,a),(1,b)\}$.
• $\mathbb{R}\times\mathbb{R}$ can be identified with the Euclidean plane $\mathbb{R}^2$.
• If $B=\emptyset$ (see empty set ) then $A\times B=\emptyset$.