9.78. Let $R_1$ and $R_2$ be equivalence relations on a nonempty set A. Prove or disprove the following:
If $R_1$ ∩ $R_2$ is symmetric, then so are $R_1$ and $R_2$.
The statement is false. Let A = {1, 2, 3} and suppose that $R_1$ = {(1, 2),(2, 1),(2, 3)} and $R_2$ = {(1, 2),(2, 1),(3, 2)}.Thus, neither $R_1$ nor $R_2$ is symmetric; however, $R_1$ ∩ $R_2$ = {(1, 2),(2, 1)} is symmetric.
I have a question concerning the definition of $R_1$ in the proof given by the textbook. $R_1$ includes (2,3) as an element, and since $R_1$ is an equivalence relation, shouldn't it include (3,2) as well? In other words, it seems to me like we cannot define a non-symmetric relation $R_1$ in the first place, since the question says "Let $R_1$ and $R_2$ be equivalence relations".
Re-read the question and proof multiple times, but just can't see what I'm missing.