[Math] What do I not understand about one-to-one functions

Question

Firstly, a definition:

Definition 1: A function $\phi : X \rightarrow Y$ is one-to-one if $\phi(x_1) = \phi(x_2)$ only when $x_1 = x_2$.

Now the question:

Students often misunderstand the concept of a one-to-one function (mapping). You see, a mapping has a direction associated with it, from $A$ to $B$. It seems reasonable to expect a one-to-one mapping that carries one point $A$ 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 1 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 in Definition 1 two-to-two functions instead.

Definition and question ever so slightly modified from A First Course in Abstract Algebra by Fraleigh.

So this is my thoughts: First of all, despite the fact that Definition 1 did not say this, I thought it could be more or less inferred. As in, there is a formal, and intuitive definition of a derivative, but they more or less mean the same thing. But, I guess if this were considered a two-to-two the best argument I could think of is given $x_1 = x_2 \in A$, and $\phi(x_1):=y_1 = \phi(x_2):=y_2 \in B$, then the "diagram", so to speak looks more or less like:

$$ x_1 \rightarrow y_1 \leftrightarrow y_2 \leftarrow x_2 $$

So there is a kind of a two-to-two symmetry, but to me, it is a very weak (couldn't we just cut out the middle anyway?) argument, and don't believe this is what the question is getting at. Can someone explain to me what most students don't understand?

Answer 1

Searched and found this comment by Terrence Tao on a blog at this link:

Too long for a comment.

Terence Tao Says:

John H. Conway once observed that the common terminology “one-to-one” for injections is horribly misleading, as it suggests that the definition of an injection is “something that sends one input to one output” – which is instead the definition of a function, not an injective function. (The problem is that “one-to-one” is intended to be in contrast to “many-to-one”, but is often instead viewed by beginners as a contrast to “one-to-many”.) He proposed instead the more accurate terminology “two-to-two” (sending two distinct inputs to two distinct outputs). This hasn’t caught on, however (not surprising though, given the severe alliteration).