Set of Relations can be Ordered by Inclusion

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Theorem

Let $S \times T$ be the product of two sets.

Let $\mathcal R$ be a set of relations on $S \times T$.


Then $\mathcal R$ can be ordered by inclusion.


Proof

By the definition of relation, any relation $R$ is associated with a subset $R \subseteq S \times T$.

Thus $\mathcal R$ is a subset of the power set $\mathcal P\left({S \times T}\right)$.

The result follows from Subset Relation is Ordering.

$\blacksquare$


Also see