Reflexive Closure is Inflationary

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Theorem

Let $S$ be a set.

Let $R$ denote the set of all endorelations on $S$.

Then the reflexive closure operator is an inflationary mapping on $R$.


Proof

Let $\RR \in R$.

The reflexive closure $\RR^=$ of $\RR$ is defined as:

$\RR^= := \RR \cup \Delta_S$

From Set is Subset of Union:

$\RR \subseteq \RR^=$

Hence the reflexive closure operator is an inflationary mapping.

$\blacksquare$