# Definition:Transitive Closure (Relation Theory)/Intersection of Transitive Supersets

## Definition

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

The **transitive closure** of $\mathcal R$ is defined as the intersection of all transitive relations on $S$ which contain $\mathcal R$.

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