I'm reading Category Theory for the Sciences by Spivak available online at MIT. And I can't understand this proof.
In particular I have two questions.
in other words ..., l is element of X x X.
If l is equivalence relation and equivalence relation is a subset of X x X shouldn't it be 'l is subset of X x X'? Is that a typo or what?
Next thing is this part.
It is clearly reflexive, because R is
It is not so clear to me though. Because all that is said about R is that is a subset of X x X, which also mean that it is a some relation. But it could be non reflexive, right? Like for example 'is not equal' relation. It would be non reflexive, transitive and symmetric, right?
And even if R is reflexive, how does it proof that S is?
You’ve found two errors in the text. For your first question: yes, it should read $\ell\subseteq X\times X$. That error is probably a typo.
The other error is more substantial. There is nothing in the hypotheses of the lemma that ensures that $R$ is reflexive: for that we’d need to know that $R\supseteq\Delta_X=\{\langle x,x\rangle:x\in X\}$. The simplest way to fix the argument is to replace $R$ by $R'=R\cup\Delta_X$. Any equivalence relation that contains $R$ must contain $R'$ and vice versa, so $L_{R'}=L_R$. $R'$ is reflexive, and the rest of the argument goes through as given.