Formal Definition

\[ A_1 \times A_2 \times ... \times A_n = \{(a_1, a_2, ..., a_n) | a_i \in A_i \textrm{ for all } i \in \{1, 2, ..., n\}\} \]

Examples

Let $A$ and $B$ be sets as such:

\[ A = \{ 1, 2, 3 \}\\ B = \{ 4, 5, 6\} \]

We can define

\[ A \times B = \{(a, b) | a \in A \textrm{ and } b \in B\} \]
(1, 4)(2, 4)(3, 4)
(1, 5)(2, 5)(3, 5)
(1, 6)(2, 6)(3, 6)

Another Example

\[ A = \{a, b\}\\ B = \{1, 2, 3\}\\ \\ \begin{aligned} A \times B = \{&\\ &(a, 1),\\ &(a, 2),\\ &(a, 3),\\ &(b, 1),\\ &(b, 2),\\ &(b, 3)\\ \} \end{aligned} \]

Final Example!

\[ \begin{aligned} \mathbb{N} \times \{1, 2, 3\} = \{&\\ &(0, 1),\\ &(0, 2),\\ &(0, 3),\\ &(1, 1),\\ &(2, 2),\\ &...\\ \} \end{aligned} \]