How to verbalise ':' in a definition?

Question

When I write a definition, e.g.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$

How does one pronounce the colon ':' symbol?

I would naively guess 'Let f be-a-function-mapping from the reals to the reals, but this seems inelegant. Is there another way to read this statement?

Thank you!

Answer 1

You don't :)

This is an instance where mathematical usage corresponds with actual English-writing usage. In English, a colon introduces a list or a new idea (e.g. your tasks today are: wash the horse, replace the gaskets and remember where you left your grandmother) and is not pronounced -- it just indicates that you pause slightly so your listeners know what's coming. Here the colon is introducing what kind of function $f$ is (by naming the domain and range).

I would go with MPW's comment for reading aloud: "$f$ maps the reals to the reals".

If you're talking to people new to functions, you may wish to spell it out per Saucy O'Path's comment: "Let $f$ be a function from ℝ to ℝ".

Answer 2

I would guess it would take the meaning of "such that" or "with the property that"

Let $f : \mathbb{R} \rightarrow \mathbb{R}$

Would therefore be analogous to "The function f with the property that maps elements of the reals to elements of the reals.

Or More General

Let $f : X \rightarrow Y$

"The function f with the property that maps elements of $X$ to elements of $Y$.

This however does seem very cumbersome and I prefer the usage as given by Saucy O'Path