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

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 $\mathcal R^+$.