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 :  \[\mathrm{dom}(R):=\{x : \exists y, \langle x,y \rangle\in R\},\] \[\mathrm{ran}(R):=\{y : \exists x, \langle x,y \rangle \in R\}.\] 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...