Set of Images of Reflexive Relation is Cover of Set

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Theorem

Let $\RR \subseteq S \times S$ be a reflexive relation in $S$.

Let $\II = \set{\map \RR x : x \in S}$ be the set of images under $\RR$.

Then:

$\II$ is a cover of $S$

Proof

By definition of reflexive relation:

$\forall x \in S : \tuple{x, x} \in \RR$

By definition of image:

$\forall x \in S : x \in \map \RR x$

Hence, $\II$ is a cover of $S$ by definition.

$\blacksquare$