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

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

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

The transitive closure of $\mathcal R$ is denoted $\mathcal R^+$.