Equivalence of Definitions of Transitive Closure (Relation Theory)

Theorem
The following definitions of the transitive closure of a relation are all equivalent:

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

First note that by Smallest Element is Unique, once it has been shown that some relation, $\mathcal Q$, is the smallest transitive superset of $\mathcal R$, it is the only such.

Thus we need only prove that each of the other definitions lead to relations with this property.

Also see

 * Recursive Construction of Transitive Closure