Definition:Relation/Relation as Ordered Pair

Definition
Some sources define a relation between $S$ and $T$ as an ordered pair:
 * $\left({S \times T, P \left({s, t}\right)}\right)$

where:
 * $S \times T$ is the Cartesian product of $S$ and $T$
 * $P \left({s, t}\right)$ is a propositional function on ordered pairs $\left({s, t}\right)$ of $S \times T$.

Note that this approach leaves the domain and codomain inadequately defined.

This situation arises in the case that $S$ or $T$ are empty, whence it follows that $S \times T$ is empty, but $T$ or $S$ are not themselves uniquely determined.