If I didn't miscalculate, the following is a counterexample:

A = {0 1 2}

R1 = {<0 0> <0 1> <1 0> <1 1> <2 2>}

R2 = {<0 0> <1 1> <1 2> <2 1> <2 2>}

I found this by first trying to prove that R1oR2 is an equivalence relation. Reflexivity was okay, but symmetry didn't seem to follow, and as I tried to prove symmetry the above example kind of suggested itself from the variables I had working in the failed attempt at proving symmetry.