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 $\RR$.