Definition:Transitive Closure (Relation Theory)/Smallest Transitive Superset

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

The transitive closure of $\RR$ is defined as the smallest transitive relation on $S$ which contains $\RR$ as a subset.

The transitive closure of $\RR$ is denoted $\RR^+$.

Also see

 * Equivalence of Definitions of Transitive Closure (Relation Theory)