A counterexample for an equation between arbitrary relations

Question

It seems that the following equation does not hold always for the arbitrary relations R1,R1 and R3:

$R1.(R2\cap R3)=(R1.R2)\cap(R1.R3)$

Instead, the right axiom is the following:

$R1.(R2\cap R3) \subseteq (R1.R2)\cap(R1.R3)$

I wrote dot for relation composition, and $\cap$ sign for relation intersection.

Can anybody please think of any counterexample for the first equation?

Thanks

Answer 1

Take for a simplest example $$R_1=\{(a,x),(a,y)\},\\ R_2=\{(x,z)\},\quad R_3=\{(y,z)\}\,.$$ Note also that we have equality with union instead of intersection.