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

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## 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 $\RR^+$.