Definition:Cartesian Product/Class Theory

Definition
Let $A$ and $B$ be classes.

The cartesian product $A \times B$ of $A$ and $B$ is the class of ordered pairs $\tuple {x, y}$ with $x \in A$ and $y \in B$:


 * $A \times B = \set {\tuple {x, y}: x \in A \land y \in B}$

Thus:
 * $\forall p: \paren {p \in A \times B \iff \exists x: \exists y: x \in A \land y \in B \land p = \tuple {x, y} }$

$A \times B$ can be voiced $A$ cross $B$.

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 $\powerset S$ of $S$.

Also see

 * Cartesian Product Exists and is Unique