Definition:Cartesian Product

The cartesian product (or Cartesian product) of two sets $$S$$ and $$T$$ is the set of ordered pairs $$\left({x, y}\right)$$ with $$x \in S$$ and $$y \in T$$.

This is denoted:

$$S \times T = \left\{{\left({x, y}\right) : x \in S \land y \in T}\right\}$$

Some authors call this the direct product of $$S$$ and $$T$$.

In a cartesian product $$S \times T$$, the sets $$S$$ and $$T$$ are called the factors of $$S \times T$$.

Another way of defining it is by:

$$\left({x, y}\right) \in S \times T \iff x \in S, y \in T$$