Students often misunderstand the concept of a one-to-one function (mapping). I think I know the reason. You see, a mapping ∅ : A → B has a direction associated with it, from A to B. It seems reasonable to expect a one-to-one mapping simply to be a mapping that carries one point of A into one point of B, in the direction indicated by the arrow. But of course, every mapping of A into B does this, and Definition 0.12 did not say that at all. With this unfortunate situation in mind, make as good a pedagogical case as you can for calling the functions described in Definition 0.12 two-to-two functions instead.