Equivalence relations on intersections

Question

Let $R_1$ and $R_2$ be equivalence relations on a set $S$. Their intersection is on the relations considered as sets of ordered pairs. Identify the equivalence classes of the relation $R_1 \cap R_2$.

I don't know how to go about showing this with proof, any help would be appreciated!

Answer 1

Maybe this will help:

This is a visual representation of two equivalence relations. Do you think you can translate what is going on here into the language of math?

Answer 2

If $E$ denotes an equivalence relation then the following statements are equivalent:

• $\langle x,y\rangle\in E$
• $y\in[x]_E$

Here $[x]_E$ denotes the equivalence class represented by $x$.

By definition of intersection:$$\langle x,y\rangle\in R_1\cap R_2\iff\langle x,y\rangle\in R_1\wedge\langle x,y\rangle\in R_2$$ Apparantly this statement can be translated into:$$y\in[x]_{R_1\cap R_2}\iff y\in[x]_{R_1}\wedge y\in[x]_{R_2}$$

Based on this find a relation between $[x]_{R_1\cap R_2}$, $[x]_{R_1}$ and $[x]_{R_2}$.