I've seen an example that goes as follows:
Let $R$ be an irreflexive, anti-symmetric, and transitive relation on a set $A$.
$A = \{1,2,3\}$
$R = \{(1,1)(1,2),(2,3)(1,3)\} $
How can the relation $R$ include $(1,1)$? Wouldn't that just make it neither reflexive or irreflexive.
I thought for relations to hold they were required to hold for all $x \in A$. Therefore it isn't irreflexive anymore because the condition $(x,x) \notin R$ is broken by $(1,1)$.
As noted in the comments, you are correct. For $R$ a relation on $A$ to be irreflexive, we would require that $\forall a \in A$, $(a,a) \not \in A$. However, that is broken by $a=1$, so $R$ as given is not irreflexive.
Mostly just posting this to get this out of the unanswered queue. Posting as Community Wiki in particular since I have nothing further to add.