I'm reading some text about binary relations and the author is about to introduce the concept of reflexivity. Before doing this he gives an example of a relation that is not reflexive.
Similarly, the relation “$(x, y) = (y, x)$” over ordered pairs sometimes relates ordered pairs to themselves (for example, $(0, 0) = (0, 0)$), but sometimes does not (for example, $(1, 0) \neq (0, 1)$)
I can't understand how this relations can be a binary one because we always have only one tuple. And how can we even check if two tuples relate to each other?. I mean let's say we want to check if $(1, 0)$ and $(0, 1)$ relate to each other by this relation. For me it seems that only the first tuple will participate in the relation and the other one is just ignored.
I understand relation "less than" ($x < y$) over natural numbers. For example, we have two numbers 1 and 3 and we see that they relate to each other, because 1 < 3. But that relation over pairs is very weird to me. I'd be grateful if someone could explain how to deal with it and how I can treat similarly like the "less than" relation.
I'm sorry if my question is too vague or something, but I hope it's clear what I'm worrying about here.