Definition talk:Mapping
I'd dispute:
- $f: x \mapsto y$, which emphasizes the interpretation of a mapping as "a thing which associates elements to other elements" rather than "a thing that turns one thing into another"
as the "arrow" notation does rather give the "goes to" impression, whereas the "equal" notation emphasises the "isness" of the relationship between $x$ and $y$.
So from that point of view I would suggest that $f: x \mapsto y$ specifically emphasizes the interpretation of a mapping as "a thing that turns one thing into another" rather than "a thing which associates elements to other elements".
But ultimately it's empty wordage which doesn't actually convey any extra meaning, so I'd vote for not including that statement in there at all.
What does anyone else think? --prime mover 10:40, 11 March 2012 (EDT)
- After deleting my response several times while disagreeing with myself, I think it's best to remove the statement entirely. It confuses rather than enlightens. --Lord_Farin 11:00, 11 March 2012 (EDT)
- Okay, I'm okay with deleting it. Has anyone else seen the other notations below it in common use? It might be a good idea to indicate that $\mapsto$ is common and the others are not. --GFauxPas 11:19, 11 March 2012 (EDT)
The second is very common, particularly in functional analysis; I even use it on this site as it is so convenient to omit brackets. The others I hadn't even seen before. --Lord_Farin 11:23, 11 March 2012 (EDT)
- Well I suppose I've seen them in particular examples even if I haven't seen them generically: $\sin x, x!, x^n$ --GFauxPas 11:37, 11 March 2012 (EDT)
- $f: x \mapsto y$ is common, but I never got used to it myself. $fx=y$ is just $f(x)=y$ without the brackets and doesn't offer much - IMO $f(x)=y$ is so much better because then there's no ambiguity as to exactly what the argument of $f$ actually is. The others are particularly obscure, but they're out there so I documented them.
- In short, $f(x)=y$ is the standard to which I've been working (it's unambiguous and gets the job done) while $f: x \mapsto y$ is also standard. Recommend we add a note to the effect that these are the notations which will be found on this site, I suppose. --prime mover 11:58, 11 March 2012 (EDT)
Implied Domain
I found the following in Chapter $9$: Functions in the book Precalculus (second edition) by Fred Safier:
which could come in handy if the page is expanded with points about domain and codomain of functions being unspecified.
--Ascii 09:15, 24 August 2019 (EDT)
- Already covered somewhere. Can't remember where. --prime mover (talk) 16:53, 24 August 2019 (EDT)
- Ahah! here we go ... Definition:Real Function/Domain
- Feel free to expand on this, if you have an idea of how to present it. --prime mover (talk) 18:20, 24 August 2019 (EDT)
- Incidentally, it's only the domain which needs to be handled like this -- the range (that is, the image) is defined completely by the domain, and the codomain is arbitrary (but usually taken to be $\R$). But for the fact that it has to be a superset of the image, it is of far less importance than the "implied domain". --prime mover (talk) 18:24, 24 August 2019 (EDT)