Equivalence of Definitions of Transitive Closure (Relation Theory)/Intersection is Smallest/Proof 1
Jump to navigation Jump to search
Note also that by the definition of endorelation, $\mathcal R \subseteq S \times S$.
Next, note that the Intersection of Transitive Relations is Transitive.
Thus $\mathcal R^+$ is the smallest transitive relation on $S$ containing $\mathcal R$.