Union of Relations is Relation

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 $T$.

This article is complete as far as it goes, but it could do with expansion. In particular: Binary case You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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 $T$.

$\blacksquare$