Cartesian product

The Cartesian product of two set $A$ and $B$ , denoted $A \times B$ , is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$ :

A \times B = {(a, b) : a \in A and b \in B}

The $n$ -ary Cartesian product over $n$ sets $X_{1}, \dots, X_{n}$ is the set of $n$ -tuples

X_{1} \times \dots \times X_{n} = {(x_{1}, \dots, x_{n}) : x_{i} \in X_{i} for each i \in {1, \dots, n}}