I didn't know that a relation could be both reflexive and irreflexive. However, now I do, I cannot think of an example.
So what is an example of a relation on a set that is both reflexive and irreflexive ?
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$\begingroup$If you have an irreflexive relation $S$ on a set $X\neq\emptyset$ then $(x,x)\not\in S\ \forall x\in X $
If you have an reflexive relation $T$ on a set $X\neq\emptyset$ then $(x,x)\in T\ \forall x\in X $
We can't have two properties being applied to the same (non-trivial) set that simultaneously qualify $(x,x)$ being and not being in the relation.
This property is only satisfied in the case where $X=\emptyset$ - since it holds vacuously true that $(x,x)$ are elements and not elements of the empty relation $R=\emptyset$ $\forall x \in \emptyset$
$\textbf{Addendum:}$
One possibility I didn't mention is the possibility of a relation being $\textit{neither}$ reflexive $\textit{nor}$ irreflexive.
Consider a set $X=\{a,b,c\}$ and the relation $R=\{(a,b),(b,c)(a,c), (b,a),(c,b),(c,a),(a,a)\}$
It's easy to see that relation is transitive and symmetric but is neither reflexive nor irreflexive, one of the double pairs is included so it's not irreflexive, but not all of them - so it's not reflexive.
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