Union of Relations is Relation
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Theorem
Let $S$ and $T$ be sets.
Let $\FF$ be a family of relations from $S$ to $T$.
Let $\ds \RR = \bigcup \FF$, the union of all the elements of $\FF$.
Then $\RR$ is a relation from $S$ to $T$.
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Proof
By the definition of a relation from $S$ to $T$, each element of $\FF$ is a subset of $S \times T$.
By Union of Subsets is Subset: Set of Sets:
- $\RR \subseteq S \times T$
Therefore, by the definition of a relation from $S$ to $T$, $\RR$ is a relation from $S$ to $T$.
$\blacksquare$