This pretty much covers the conversation that you two have been having. (There have been a few other functions offered, but let's stick to this one for now.)
@stribor40
Plato has covered all three possible cases to show when f(a) = f(b) then a = b. The first two examples show that f(a) = f(b) then a = b. So far so good. The third is where things get dicey, but note that if $\displaystyle a \geq 0$ and b < 0 then there is a contradiction...f(a) is even whereas f(b) is odd. This obviously cannot happen as an even number can never be equal to an odd number. This means that there is no "overlap" of the two pieces. My (first) question to you is "Why is the third case necessary?"
My second question is why aren't you seeing this? It sounds very much like you are either looking for a full solution (which we generally do not give) or you aren't putting effort into this. Spend some time with the above quotes and see if things make more sense. You are posting in the Discrete Math forum! Heck, I haven't taken Discrete and I'm following his reasoning so you should be well beyond many of the difficulties I'm seeing you have here. Plato has been very patient about all this. Play him back by working some more at it.
-Dan
I know the argument on counting. But I remember very well in my first years of learning counting the teacher asked how many pencils we have on the table. None. That's 0. It was and it still should be as counting as three pencils on the table. I haven't lived the stone age years, but I find the pencil example as a translation in time transformation to a stone age hunter returning to his group and answering he didn't get any boar that day. That's counting isn't it? Incidentally, the set of natural numbers is postulated infinite. Obvioulsy nobody I am aware of counted that infinite number. Therefore I conclude what is defined natural numbers set is not countable. it is just an abstraction in the mind as Plato (the Greeck guy!) defined it long after the stone age.