Set Theory and Beyond

[I continue writing in the form of a dialogue between Master (M) and Apprentice (A).] A: O Noble Teacher, last time we talked, you said that a unary relation is just an arbitrary set. But you didn't explain why would you introduce another word just to mean "set"! M: O My Dear Apprentice, that is because when we say $R$ is a unary relation, we usually have context in mind and we usually say more. For example, we might be thinking about some set $X$ and then we might say that $R$ is a unary relation on set $X$ . In this way, we want to say more then just "$R$ is a set". Formally, we mean that $R$ is a subset of $X$, but intuitively, we separate elements $x$ of $X$ into those for which the relation $R(x)$ holds and those for which the relation $R(x)$ does not hold. Here, "$R(x)$ holds" just means "$x\in R$" and "$R(x)$ does not hold" means $x\in R-X$. A: Okay, that makes more sense. But what about binary r...