Set Theory and Beyond

So, a binary relation is a set of ordered pairs, any such set. A binary relation on a set $X$ is a binary relation which is a subset of $X\times X$. A binary relation between sets $X$ and $Y$ is a binary relation which is a subset of $X\times Y$. Now, I want to say what is a function: a function will be a special kind of a binary relation. Let us fix a binary relation $R$. This just means that $R$ is a set of ordered pairs. Two natural sets assigned to $R$ are its domain  and its range :  $$\mathrm{dom}(R):=\{x:\mathrm{\exists }y,⟨x,y⟩\in R\},$$ $$\mathrm{ran}(R):=\{y:\mathrm{\exists }x,⟨x,y⟩\in R\}.$$ Of course, it's not immediate that these are sets, but there is no doubt that they are classes. In order for a class to be a set, it suffices for it to be contained in a set. More precisely, if a class $C$ is contained in a set $X$, then $C$ is equal to the intersection $X\cap C$. The intersection of a set and a class is a set by Axiom of...

 [I continue writing in the form of a dialogue between Master (M) and Apprentice (A).] M: As I promised last time we met, I'll explain to you today what is an ordering. A: I've been waiting so eagerly for this! Please, go on! M: Let's look at some set $X$. You now know what a binary relation on $X$ is. Let $<$ be a binary relation on $X$. By the way, since $<\subseteq X\times X$, we can have that $(x,y)\in <$ for some elements $x$ and $y$ of $X$. However, we usually write "$x<y$" instead of "$(x,y)\in <$". A: Oh, yes, that's a standard mathematical notation. Now I understand what it formally means. M: Good. Okay, so $<$ is a binary relation on $X$, but in order for it to be an ordering, it needs to have some additional properties. In fact, I'll first tell you what is a strict ordering. Binary relation $<$ will be a strict ordering just in case it is irreflexive and transitive. Being irreflexive mea...