Equivalence of Definitions of Reflexive Transitive Closure

Theorem
Let $\mathcal R$ be a relation on a set $S$.

Then the reflexive transitive closure, $\mathcal R^*$, of $\mathcal R$ by any of the following definitions is the same:

Proof
The result follows from:


 * Transitive Closure of Reflexive Relation is Reflexive


 * Reflexive Closure of Transitive Relation is Transitive


 * Composition of Compatible Closure Operators