The text for my 148 class says
A relation is a correspondence between sets.The boldface type (implicitly) means that this is given as a definition. But of course this won’t do: what’s a “correspondence”, after all? To go on to say that “If x and y are two elements in these sets and if a relation exists between x and y, then we say that x corresponds to y or that y depends on x, and we write ” isn’t helpful. In fact, it proves very clearly that our authors don’t believe their own words: if a “relation” is a correspondence between sets, then how the Heck can a “relation” exist between x and y (which are [again implicitly] numbers, not sets)?
There’s more—a lot more—to complain of even in the two sentences I’ve copied out here (and it stays this bad for at least a few paragraphs), but let me leave my analysis at that for a moment.
Actually, of course, a relation is a set of ordered pairs. Our authors know this of course. They’ve even tried to tell us so: an earlier edition had “A relation is a correspondence between two variables, say, x and y, and can be written as a set of ordered pairs (x, y).” … but this was evidently too clear for somebody at Prentice-Hall, so now we’ve got the handwaving.
“Oh, this is hard to understand”, we can imagine somebody saying … “so let’s make it impossible to understand instead.”. Somebody out there prefers imprecise “definitions” to the real ones. I’m convinced that they don’t want students to know how simple the truth can be: math is supposed to be hard for these villains.