How to prove or disprove that a union of two anti-reflexive relations is also anti-reflexive

Question

Let $R$ and $S$ be binary relations on a nonempty set $A$. Then $S \cup R$ is also a binary relation on $A$. Suppose $S$ and $R$ are anti-reflexive relations on $A$. How can we prove or disprove that $S \cup R$ is also anti-reflexive?

Answer 1

A relation $T$ on $X$ is anti-reflexive iff for all $x \in X$, $(x,x)$ is not in $T$.

Now, given that $S$ and $R$ are both antireflexive, and if $x \in X$ is arbitrary, $(x,x) \notin S$ and also $(x,x) \notin R$ and so $(x,x) \notin S \cup R$. Hence $S \cup R$ is anti-reflexive.