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 $S \times T$.

Such sources then define the graph of the relation as:
 * $\mathcal R = \left\{{\left({s, t}\right) \in S \times T: P \left({s, t}\right)}\right\}$

that is, the set of all $\left({s, t}\right)$ in $S \times T$ for which $P \left({s, t}\right)$ holds.

Hence the graph of a relation is simply what is defined on this page as a relation.

Whether there are any advantages to this form of treatment is debatable. In general, will not use this somewhat more elaborate terminology.

Also 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.