To quote Halmos:

If $ R$ is an equivalence relation in $ X$ , and if $ x$ is in $ X$ , the equivalence class of $ x$ with respect to $ R$ is the set of all elements $ y$ in $ X$ for which $ x R y$ . Examples: if $ R$ is equality in $ X$ , then each equivalence class is a singleton; if $ R = X \times X$ , then the set $ X$ itself is the only equivalence class.

~P. R. Halmos, *Naive Set Theory* (p. 28)

The first one, I think I understand. Each equivalence class is a singleton because each element $ x$ in $ X$ is only equal to itself.

The second is confusing me further the more I think about it, perhaps because of the wording. If $ R = X \times X$ , do I still consider it to be ‘in’ $ X$ or is it ‘in’ the result of $ X \times X$ ? If it’s the former, how is that any difference than ‘equality in $ X$ ,’ which should yield singletons? If it’s the latter, then surely we’re now dealing with a series of ordered pairs that did not exist in the set $ X$ beforehand, precluding it from being the equivalence class.

Or, is it that it’s neither of these, and the set $ X$ used here is being treated like the $ x$ we are seeking equivalence classes for in his initial definition? This latter definition seems to be the only way I can get my head around how $ X$ itself ends up being the equivalence class, but also seems like I’m missing something vital in making that assumption.