Definition:Relation/Relation as Subset of Cartesian Product

Definition
Most treatments of set theory and relation theory define a relation on $S \times T$ to refer to just the truth set itself:
 * $\mathcal R \subseteq S \times T$

where:
 * $S \times T$ is the Cartesian product of $S$ and $T$.

Thus under this treatment, $\mathcal R$ is a set of ordered pairs, the first coordinate from $S$ and the second coordinate from $T$.

This approach leaves the precise nature of $S$ and $T$ undefined.

While this definition is usually perfectly adequate, on the full formal definition is preferred.

There may be many places on where this simplified approach is taken. However, there exists an ongoing maintenance task to address this.

Also defined as
As a sideline, it is noted that some sources define a relation $\mathcal R$ as a set of ordered pairs, with no initial reference to the domain or image of $\mathcal R$.

The domain and image of $\mathcal R$ are then defined as the sets:
 * $\operatorname{Dom} \mathcal R = \left\{ {x: \exists y: \left({x, y}\right) \in \mathcal R}\right\}$
 * $\operatorname{Im} \mathcal R = \left\{ {y: \exists x: \left({x, y}\right) \in \mathcal R}\right\}$

Using this approach, the cartesian product $S \times T$ of two sets $S$ and $T$ is defined as the relation consisting of all the ordered pairs $\left({x, y}\right)$ where $x \in S$ and $y \in T$, rather than defining the cartesian product first and the relation as being a subset of it.