If $R,S$ are reflexive relations, so are $R \oplus S$ and $R \setminus S$?

Question

Suppose $R$ and $S$ are reflexive relations on a set $A$. Prove or disprove each of these statements.

a) $R\oplus S$ is reflexive.

b) $R\setminus S$ is reflexive.

I think both of a) and b) are false, but I'm having trouble with coming up with counterexamples.

Answer 1

Hint: What does it mean to say $(x,x) \in R \oplus S$, respectively $(x,x) \in R \setminus S$? Can both $R$ and $S$ be reflexive if this is the case?

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The above amounts to a proof by contradiction. But we can avoid this; for example, by the following argument:

Let $x \in A$. We know that $(x,x)\in R$ and $(x,x) \in S$. Therefore, by definition of $R \oplus S$ (resp. $R \setminus S$), $(x,x)\notin R \oplus S$. Hence $R \oplus S$ is not reflexive.

Answer 2

Assuming $A \neq \emptyset$ consider $R=S= \{ (a,a) : a \in A \}$.