Union of Relations is Relation

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Theorem

Let $S$ and $T$ be sets.

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

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


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

Proof

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

By Union of Subsets is Subset: Set of Sets:

$\mathcal R \subseteq S \times T$

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

$\blacksquare$