Transitive Closure of Reflexive Symmetric Relation is Equivalence
Let $S$ be a set.
Let $\sim$ be the transitive closure of $\mathcal R$.
Checking in turn each of the criteria for equivalence:
- $\sim$ is reflexive.
- $\sim$ is symmetric.
By the definition of transitive closure:
- $\sim$ is transitive.
Hence by definition it is an equivalence relation.