# [Om] range & image in fns1

Arjeh Cohen amc at win.tue.nl
Wed Nov 25 08:52:34 CET 2009

```Dear Bruce,

I think you are correct in asking

> Shouldn't the FMP be:
>    image(f) \subseteq range(f)
>

that is, the answer is yes. It seesm liek very few people have
actually checked what was written. As for the definition of "range",
more clarity is given in fns3, which I (think I) wrote a couple of years ago.
I see several typos that have never been repaired, like "aplied" and "stes".

Greetings, Arjeh Cohen

On Tue, Nov 24, 2009 at 01:32:30PM -0500, Bruce Miller wrote:
> Hi all;
>    In trying to answer a student's question
> about the difference between range and image
> (& implicitly codomain) ... and rather stumbling...
>
> Math question:
> My naive understanding was that the image is
> _exactly_ the set of values produced by a function
> over some domain (or subset of the domain)
> --- no more, no less.
> And, that range is allowed to be some larger set
> than the image if it is more convenient for some
> purposes (eg continuous, convex, whatever) ...
>
> Is that correct, and are there any formal conditions
> on how a range is defined w.r.t an image?
> Or is it simply expanded to get whatever desired properties?
>
> OpenMath question:
>   The description, CMP & FMP for the symbol range at
>      http://www.openmath.org/cd/fns1.xhtml#range
> are contradictory.  The description ("range ... merely
> required to contain the image") is consistent
> with my understanding (such that it is).
>
> However, the CMP says that the range is a subset of
> the image, while the FMP goes further saying that
> the range is a proper subset (to the extent the
> implied "proper" is significant).
>
> Shouldn't the FMP be:
>    image(f) \subseteq range(f)
>
>
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```