I have two symmetric relations on a set. How can I prove that the symmetric difference is irreflexive?

Question

I have this problem.

Let R and S be symmetric relations on a set A. Prove or disprove: $R \oplus S$ is irreflexive.

Now I'm assuming it's not true, because $(x,x)$ can be an element of $R$ without being an element of $S$, but I'm not sure how to prove it. Any tips?

Answer 1

You are on the right track. Recall that a relation on $A$ is irreflexive if it does not contain any of the diagonal elements $(a,a), a \in A$.

Fix $a \in A$ and take $R$ to be the singleton set $\{(a,a)\}$ and $S$ to be the empty set. Then $R$ and $S$ are symmetric relations but their symmetric difference contains a diagonal element, and hence is not irreflexive. Thus the symmetric difference of two symmetric relations is not irreflexive in general. This gives a counterexample and hence disproves the statement that the symmetric difference of symmetric relations is irreflexive.

Note that if we take $R$ and $S$ to be arbitrary symmetric relations that do not contain any diagonal elements (for eg $R=\{(a,b),(b,a)\}=S$), then their symmetric difference would be missing all the diagonal elements, and this makes the symmetric difference irreflexive. Thus, the given implication does hold in some special cases.