Reflexive Closure is Closure Operator/Proof 1
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Theorem
Let $S$ be a set.
Let $R$ be the set of all endorelations on $S$.
Then the reflexive closure operator on $R$ is a closure operator.
Proof
Let $\QQ$ be the set of reflexive relations on $S$.
By Intersection of Reflexive Relations is Reflexive, the intersection of any subset of $\QQ$ is in $Q$.
By the definition of reflexive closure as the intersection of reflexive supersets:
- The reflexive closure of a relation $\RR$ on $S$ is the intersection of elements of $\QQ$ that contain $S$.
From Closure Operator from Closed Sets we conclude that reflexive closure is a closure operator.
$\blacksquare$