Is this always true about symmetric relations?

Question

Statement : If $R_1$ and $R_2$ are both symmetric, then $R_1 \cap R_2$ is symmetric.

Is this always true? As far as I could understand this with an example, suppose I have a set $A = \{1,2,3\}$

And $R_1$ and $R_2$ be symmetric relations

where,

$R_1$ = $\{(1,2),(2,1)\}$ and $R_2$ = $\{(1,3),(3,1)\}$, then the statement turns out to be false. Am I right?

Answer 1

A relation, say $R_1$ or $R_2$, is symmetric if for any ordered pair $(x,y)$ in that relation, then (what?).

Take any point $(x,y)$ in $R_1\cap R_2$.   When given that $R_1, R_2$ are symmetric, can you show that (what?)?

In the case where the intersection is empty, this is vacuously true.