Proof of every asymmetric relation is irreflexive

Question

I came across a question as follows: Show that every asymmetric relation over a set $A$ is irreflexive.

The solution instructs one to use the relation < and suppose that it is asymmetric but not irreflexive, so that one derives a contradiction.

Since the relation is not irreflexive, there is an $a\in A$ with $a < a$. By asymmetry, not $a < a$ - contradiction"

But my understanding is that being non-irreflexive doesn't necessary mean it is reflexive, which seems to have been assumed so here. If this assumption is indeed false, wouldn't this proof fail?

Thank you so much!

Answer 1

You're correct, not being irreflexive does not mean that the relation is reflexive.

But it is not assumed to be reflexive. Remember that reflexive means for all $a\in A$ it holds that $a<a$. Here we only assume that for some $a\in A$ it holds that $a<a$, which is exactly what it means not to be irreflexive.

Answer 2

Definition. A relation that is irreflexive, is a binary relation on a set where no element is related to itself; in other words: $$\forall a\in A; \lnot (a<a)$$
Let < not irreflexive. So $$\lnot (\forall a\in A; \lnot (a<a))$$ which means $$\exists a\in A,a < a.$$ And this is what you want: "the relation is not irreflexive, there is an $a∈A$ with $a < a$"