Equivalence of Definitions of Transitive Closure (Relation Theory)

Proof
Let $\RR$ be a relation on a set $S$.

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

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

First we have:

Intersection of Transitive Supersets is Smallest Transitive Superset
=== The Finite Chain Definition is Equivalent to the Union of Compositions Definition ===

Also see

 * Recursive Construction of Transitive Closure