Definition:Cartesian Product

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

This is denoted:


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

Another way of defining it is by:


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

More specifically:
 * $\forall p:\left({p \in S \times T \iff \exists x: \exists y: x \in S \land y \in T \land p = \left({x, y}\right)}\right)$

$S \times T$ can be voiced $S$ cross $T$.

Countable Cartesian Product
The same notation can be used to define the (countable) cartesian product of an infinite sequence:

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

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

Some authors use uppercase for the initial, that is: Cartesian product.

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

The notation for the cartesian power of a set $S^n$ should not be confused with the notation used for the conjugate of a set.

Also beware not to confuse the name of the concept itself with that of the power set $\mathcal P \left({S}\right)$ of $S$.

Also see

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