Definition:Relation/Abuse of Notation

Notation for Relation
By abuse of notation, the usual technique for denoting a relation $\mathcal R$ on $S \times T$ is:
 * $\mathcal R \subseteq S \times T$

thereby endorsing the approach of defining a relation as a subset of a Cartesian product.

Similarly, it is equally common to denote the expression $s \mathrel {\mathcal R} t$ as:
 * $\tuple {s, t} \in \mathcal R$

While this approach conflates the relation with its truth set, it is sufficiently convenient and widespread as to be endorsed by.

We have not been able to find more mathematically rigorous notations for this that are at the same time not overly unwieldy.