Set of Relations can be Ordered by Subset Relation

Theorem

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

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

Then $\RR$ can be ordered by the subset relation.

Proof

Let $R$ be a relation on $S \times T$.

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

Thus $\RR$ is a subset of the power set $\powerset {S \times T}$.

The result follows from Subset Relation is Ordering.

$\blacksquare$

Also see

• Set of Mappings can be Ordered by Subset Relation