Definition:Cartesian Product

Definition
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 $T$.

Some call it the cartesian product set, others just the product set.

Another way of defining it is by:


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

It is also known as the cross product of two sets, but this can be confused with other usages of this term.

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

Axiomatic Set Theory
The concept of the cartesian product is shown in Kuratowski Formalization of Ordered Pair to be constructible from the Zermelo-Fraenkel axioms.

Also see

 * Cartesian products of algebraic structures:
 * External Direct Product
 * Internal Direct Product
 * (External) Group Direct Product
 * Internal Group Direct Product