Irreflexivity occurs where nothing is related to itself. Anti-symmetry provides that whenever 2 elements are related "in both directions" it is because they are equal. These two concepts appear mutually exclusive but it is possible for an irreflexive relation to also be anti-symmetric.
Put another way: why does irreflexivity not preclude anti-symmetry?
These are the definitions I have in my lecture slides that I am basing my question on:
$\forall x \in X : (x, x) \notin R$
Or in plain English "no elements of $X$ satisfy the conditions of $R$" i.e. no elements are related to themselves.
$\forall x, y \in A ((xR y \land yRx) \rightarrow x = y)$
We were told that this is essentially saying that if two elements of $A$ are related in both directions (i.e. $xRy$ and $yRx$), this can only be the case where these two elements are equal.
These concepts appear mutually exclusive: anti-symmetry proposes that the bidirectionality comes from the elements being equal, but irreflexivity says that no element can be related to itself. And yet there are irreflexive and anti-symmetric relations.
I have read through a few of the related posts on this forum but from what I saw, they did not answer this question.