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?

the intersection in your example is empty and thus symmetric — user251257, Aug 26 '15 at 03:56
Try a direct argument to prove the claim. A very very direct argument. — Ittay Weiss, Aug 26 '15 at 03:57

Graham Kemp · Answer 1 · 2015-08-26T04:28:10.040

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.

edited Aug 26 '15 at 04:28

answered Aug 26 '15 at 04:21

Graham Kemp

1 Answers1