# Transitive Closure of Reflexive Symmetric Relation is Equivalence

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## Theorem

Let $S$ be a set.

Let $\mathcal R$ be a symmetric and reflexive relation on $S$.

Then the transitive closure of $\mathcal R$ is an equivalence relation.

## Proof

Let $\sim$ be the transitive closure of $\mathcal R$.

Checking in turn each of the criteria for equivalence:

### Reflexivity

By Transitive Closure of Reflexive Relation is Reflexive:

- $\sim$ is reflexive.

$\Box$

### Symmetry

By Transitive Closure of Symmetric Relation is Symmetric:

- $\sim$ is symmetric.

$\Box$

### Transitivity

By the definition of transitive closure:

- $\sim$ is transitive.

$\Box$

$\sim$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.

$\blacksquare$