# Equivalence of Definitions of Transitive Closure (Relation Theory)/Intersection is Smallest/Proof 1

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## Theorem

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

Then the intersection of all transitive relations on $S$ that contain $\RR$ is the smallest transitive relation on $S$ that contains $\RR$.

## Proof

Let $\RR^+$ be the intersection of all transitive relations containing $\RR$.

By Trivial Relation is Equivalence:

- the trivial relation $S \times S$ is transitive.

By the definition of endorelation:

- $\RR \subseteq S \times S$.

Next we have that the Intersection of Transitive Relations is Transitive.

Thus $\RR^+$ is the smallest transitive relation on $S$ containing $\RR$.