Cartesian product of sets

The Cartesian product of sets $A$ and $B$ is the set $$ A \times B = \{ (x, y) | x \in A \text{ and } y \in B \} $$

$A\times B$ consists of ordered pairs $(x,y)$ where $x\in A$ and $y\in B$.

For example, let $A=\{1,2\}$ and $B=\{a,b,c\}$. Hence, $A\times B=\{(1,a),(2,a),(1,b),(2,b),(1,c),(2,c)\}$.

Generic Format

The Cartesian product of sets $A_1,A_2,...,A_k$ is the set $$ A_1 \times A_2 \times ... \times A_k = \{ (x_1, x_2, ..., x_k) | x_i \in A \text{ for } i = 1, 2, ..., k \} $$

$A_1\times A_2\times ...\times A_k$ consists of those $k\text{-tuples }(x_1,x_2,...,x_k)$ where $x_1\in A_1,x_2\in A_2,..$