Definition:Relation/Notation

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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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

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