Union of Relations is Relation

Theorem
Let $S$ and $T$ be sets

Let $\mathscr F$ be a family of relations from $S$ to $T$.

Let $\displaystyle \mathcal R = \bigcup \mathscr F$, the union of all the elements of $\mathscr F$.

Then $\mathcal R$ is a relation from $S$ to $T$.

Proof
By the definition of a relation from $S$ to $T$, each element of $\mathscr F$ is a subset of $S \times T$.

By Union Smallest:
 * $\mathcal R \subseteq S \times T$

Therefore, by the definition of a relation from $S$ to $T$, $\mathcal R$ is a relation from $S$ to $T$.