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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 ?

Mark
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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.

Walt van Amstel
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    I admire the patience and clarity of this answer. – B. Pasternak Apr 16 '16 at 17:24
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    @rt6 What about the (somewhat trivial case) where $X = \emptyset$? Then $R = \emptyset$ is a relation on $X$ which satisfies both properties, trivially. – Peter Apr 16 '16 at 17:27
  • @Pétur: Please see my edit. Thank you for fleshing out the answer – Walt van Amstel Apr 16 '16 at 17:36
  • @rt6 what you said is perfect and is what i thought but then i found this link and it says " That is, a relation on a set may be both reflexive and irreflexive or it may be neither."

    And here also link see problem 5 which says "Problem 5. Can a relation be both reflexive and irreflexive? A) Yes B) No Solution. A)"

     – Mark Apr 16 '16 at 17:36
  • @Mark : Yes for your 1st link. The empty set is a trivial example. But one might consider it foolish to order a set with no elements :P But it is indeed an example of what you wanted. – Walt van Amstel Apr 16 '16 at 17:41
  • Ok now it makes a bit more sense after edit so does set which is X=∅ and R=∅ have all properties? e.g symmetric, transitive and so on.. – Mark Apr 16 '16 at 17:41
  • See the Wikipedia example: ${\langle x, y\rangle\in\mathbb{N}^2:xy\bmod2=0}$. – Pockets Apr 16 '16 at 17:42
  • The empty set contains all the pink elephants in the universe along with any other possible condition you can put on a set, vacuously - since it has no elements – Walt van Amstel Apr 16 '16 at 17:45