2

If it is rather silly, how would you express such a function? For example (to be specific), if $\phi : [a,b] \rightarrow \mathbb{R}$ is injective, how would you express the fact that the same function with the co-domain restricted to the range of $\phi$ is bijective?

EDIT: I am assuming I haven't previously defined what the function $\phi$ is in any way.

ANSWER FROM COMMENTS: It is not silly to write $\phi : [a,b] \rightarrow \phi([a,b])$.

RayaneCTX
  • 147
  • 1
    It's a standard fact that an injective function that is restricted to its image becomes bijective. Mostly any undergraduate math major will know this immediately. – Nicholas Roberts Feb 09 '21 at 17:54
  • 4
    No, it is not silly to write $\phi:[a,b]\to\phi([a,b])$ is bijective. – Shubham Johri Feb 09 '21 at 17:55
  • @NicholasRoberts How would you write down this function? – RayaneCTX Feb 09 '21 at 17:55
  • 3
    Stating the fact in English as you've done in the question is a good way to express it. – Karl Feb 09 '21 at 17:56
  • 2
    The way you have is fine, but more commonly I have seen $\phi:[a,b]\to\operatorname{Im}(\phi)$. But as I said, making this distinction is not really needed because mostly everyone will know that $\phi$ will be bijective when you restrict the range to the image. – Nicholas Roberts Feb 09 '21 at 17:57

1 Answers1

3

It doesn't quite sit so well with me as it seems to with people in the comments, but then I'm usually a bit of a stickler for notation. It strikes me as an abuse of notation.

A function $\phi$ ought be defined with a fixed, known domain and codomain (before you can even define where in the codomain it sends a point $x$ in the domain). In order to parse $\phi([a, b])$, I need to already know where $\phi$ sends the points in its domain, which means I already have some idea of the codomain (these points have to be mapping to somewhere, right?). This original codomain should technically be specified.

If I were doing it properly, I'd prefer to talk about $\phi : [a, b] \to \Bbb{R}$ (or whatever other codomain) first, then define some $\hat{\phi} : [a, b] \to \phi([a, b])$, or something like that.

Now, that said, being an abuse of notation doesn't mean that it's silly, or even that it's at all bad. We use abuses of notation all the time. So long as it is clear what you're saying (and I'd say it is pretty clear), then most people will be fine with it. In fact, it can often be preferable to abuse notation than to, say, define all your functions twice and leave hats over all of them!

Theo Bendit
  • 50,900