Intersection of Equivalences

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

Proof
Let $$\mathcal{R}_1$$ and $$\mathcal{R}_2$$ be equivalences on $$S$$, and let $$\mathcal{R}_3 = \mathcal{R}_1 \cap \mathcal{R}_2$$.

Checking in turn each of the criteria for equivalence:

Reflexive

 * $$\mathcal{R}_3$$ is reflexive:

$$ $$ $$ $$

Symmetric

 * $$\mathcal{R}_3$$ is symmetric:

$$ $$ $$ $$ $$

Transitive

 * $$\mathcal{R}_3$$ is transitive:

$$ $$ $$ $$ $$ $$

Thus $$\mathcal{R}_3$$ is an equivalence.