Cartesian Product

set-theory

Definition

Cartesian Product

The Cartesian product of two sets $A$ and $B$ is the set of all ordered pairs whose first component belongs to $A$ and whose second component belongs to $B$:

$$A\times B:=\{(a,b)\mid a\in A\wedge b\in B\}.$$

Equivalently, an object $z$ belongs to the product precisely when

$$z\in A\times B⟺\exists a\in A\exists b\in B:z=(a,b).$$

Examples

Example

For $A=\{1,2\}$ and $B=\{x,y\}$,

$$A\times B=\{(1,x),(1,y),(2,x),(2,y)\}.$$

The order of the components matters: $(1,x)\in A\times B$, whereas $(x,1)\in B\times A$.