Intersection of Equivalences

Theorem
The intersection of two equivalence relations is itself an equivalence relation.

Proof
Let $\RR_1$ and $\RR_2$ be equivalence relations on $S$.

Let $\RR_3 = \RR_1 \cap \RR_2$.

Checking in turn each of the criteria for equivalence:

Reflexive
Equivalence relations are by definition reflexive.

So, by Intersection of Reflexive Relations is Reflexive, so is $\RR_3$.

Symmetric
Equivalence relations are by definition symmetric.

So, by Intersection of Symmetric Relations is Symmetric, so is $\RR_3$.

Transitive
Equivalence relations are by definition transitive.

So, by Intersection of Transitive Relations is Transitive, so is $\RR_3$.

Thus $\RR_3$ is an equivalence.