Talk:Function (mathematics)/Archive 3

message: "A function need not involve numbers. An example of a function that does not use numbers is the function that assigns to each nation its capital. In this case Capital(France) = Paris."
I'm not a mathematician, but I question the validity of this. In the example given, the fucntion, Capital(x) = y does not yield exactly one output per input. Capital(France) = Vichy and Capital(France) = Aachen are arguably valid outputs for the input "France." Generally a nation has only one capital (at a time). However it's misleading to equate the mathematical certainty of a fuction (exactly one output per input) with a historical rule-of-thumb (unless a war has disrupted the normal political process, a nation, during modern times, shall usually have only one capital). Perhaps someone could come up with a better example of a Non-Numeric Function? Thanks! 66.17.118.207 15:11, 26 June 2006 (UTC)
replies: text: Absolute non-ambiguity is impossible.  Even f(x) = x + 5, which seems non-ambiguous, would admit a question whether the addition took place in the natural number system, in the real number system, or in the integers modulo 7.  I suspect that any example of a non-numerical function would suffer from similar problems.  One has to use common sense. Rick Norwood 15:37, 26 June 2006 (UTC)

text: Are you serious? First, this is a non-mathematical example. Second, France has only one capital now. Third, if some country happens to have simultaneous multiple capitals we can simply allow the function to return a set. Fourth, if we really need to accommodate different facts at different times we have tools such as modal logic. Finally, are you serious?! --KSmrqT 23:49, 26 June 2006 (UTC)

text: Well just as with mathmatical functions you could say that the complete function should include the domain (i.e. year = 2006)DougBrown 16:39, 2 November 2006 (UTC)
thread_title: Non-Numeric Functions

message: I suppose that this means that you can write 
:f=(x \mapsto \sin(x)/x)=(\lambda x.\sin(x)/x). 
Can you omit the parentheses and write 
:f=x \mapsto \sin(x)/x=\lambda x.\sin(x)/x ?
Can you write the shift like this
:F=x \mapsto n \mapsto x_{n+1}=\lambda x.\lambda n.x_{n+1}, 
or do you need parentheses: 
:F=x \mapsto (n \mapsto x_{n+1})=\lambda x.(\lambda n.x_{n+1}) 
or even 
:F=(x \mapsto (n \mapsto x_{n+1}))=(\lambda x.(\lambda n.x_{n+1})) ?
Is the function expression used like the function name?  
:(x \mapsto \sin(x)/x)(a)=(\lambda x.\sin(x)/x)(a)=\sin(a)/a.
Bo Jacoby 13:07, 16 August 2006 (UTC)
thread_title: Thank you very much

message: I will try to find a referance to this fact but I was told recently by a tutor of mathmatics that the strict definition of a function is that for each function there must be a corresponding inverse function and therefore each input can only correspond to one output AND each output can only correspond to one input. 
So f(x)=x^2 can only be a function if restricted to x >= 0
I know that this is not the place to discuss mathmatics, however if the above is true I think it should be noted. DougBrown 16:32, 2 November 2006 (UTC)
replies: text: That's an Injective function, f(x)=x^2 is a function without the restriction x >= 0. skip (t / c) 14:00, 2 November 2006 (UTC)

text: Thanks, Do you think that a short mention of Injective Functions would be useful on this page? with a link to the Injective function page? DougBrown 16:32, 2 November 2006 (UTC)
replies: text: It's already mentioned in the vocabulary section, linked from "injections".  It may be helpful to link to the Horizontal line test and/or Vertical line test articles. skip (t / c) 07:19, 3 November 2006 (UTC)
thread_title:  Definition of a function

message: I've just reverted a "clarification" that said a function is a "rule". Though that may sound helpful, we have deliberately avoided that kind of language, for reasons discussed previously. Briefly, while many commonly encountered functions are specified by a rule, that is mathematically impossible for most functions. We have only countably many rules, but uncountably many functions.
The reversion also removed a new leading image and caption. I actually prefer the cubic, since the parabola is a little too special; and the formula for the parabola had a bad line break in the middle. The only problem I see with the cubic is that within the image is a "Fortran"-style text version of the formula using "" for the power. I'd actually prefer to see a more interesting function generated by an elementary formula. We could even include bounds on the range, to make it even more informative. For example,
:\frac{\left( 4 x^3-6 x^2+1 \right) \sqrt{x+1}}{3-x}
has some nice wiggles between −1 and 3⁄2, but we must bound x to avoid problems with the square root. --KSmrqT 16:13, 25 January 2007 (UTC)
thread_title:  Explanation of revert 

message: While the words "function" and "map" are indeed defined synonymously, mathematicians do not use them interchangeably. Look for example at the distinction in differential geometry between "smooth maps" and "smooth functions", or in algebraic geometry between "rational maps" and "rational function": it seems to me that "function" is usually reserved for a map with numerical codomain (depending on context, "numerical" may be "real or complex", or "with values in the ground field", etc.).
Maybe something to that effect should be mentioned in the article. But I thought I'd discuss it here first. (I also note that similar remarks have been made by several contributors in the French and German equivalents of this discussion page.) Mathanor 23:09, 22 February 2007 (UTC)
replies: text: I don't see that any such distinction holds, nor is it helpful. Perhaps "map" is used more often in a topological context, but "function" is used with sets, and the reals have topology. --KSmrqT 00:27, 23 February 2007 (UTC)
replies: text:  The point is not that "map" is used in some contexts and "function" in others, but that several contexts use both terms, with the customary difference I mentioned. Mathanor 09:32, 23 February 2007 (UTC)
thread_title:  Function vs. map 

message: First, I'd like to apologize for not providing the summary of the edit — clicked on the wrong button!
Moreover, I've changed quite a bit more than originally intended, however, instead of apologizing, I would like to explain some of my motives. After reading back-and-forth between the editors, it appeared that the article is "eyeless by virtue of having too many nurses" (a russian proverb). For an encyclopedic article intended for general audience, and even for general mathematical audience, its introduction was practically worthless. On the other hand, Battles of the Bulge were fought over every word in the first sentence. For 99.74% of the reading public, the resulting difference didn't amount to much.
However, I finally came across very well thought out comments from User:Geometry guy and especially the bike shed metaphor,
:: This was pointed out by User:CMummert, not myself, but thank you for the compliment. Geometry guy 22:40, 19 March 2007 (UTC)
replies: text: And good luck to you! I agree with almost all these comments (although the word torsor may be enjoying a renaissance, simply because mathematicians like myself are beginning to get fed up of writing principal homogeneous space or free transitive action in its place!). I also entirely support this edit, not least because it finally supplies this article with a proper lead and hence moves some of the hair-splitting disputes out of the introduction and into the body of the text where they belong. I hope other editors will build on this lead. Geometry guy 22:40, 19 March 2007 (UTC)
thread_title:  Changes to the introduction 

message: KSmrq did a good job, but the work done by Norwood and me about "inverse function vs function inverse" was lost.
Some sentences in the first paragraph of KSmrq's new version contained a very interesting analogy, but were linguistically incorrect. I tried to fix them, moved first part in new section (identity functions) and added a short sentence ("Intuitively, ƒ−1 is a function that undoes and can be undone by ƒ"). KSmrq summarily reverted my edits without explaining why (this is not the first time he behaves this way). Today, I restored and condensed my revision. KSmrq, you teach "Patience... patience... patience..." (see below). You should not forget that politeness favours patience. Paolo.dL 16:35, 2 October 2007 (UTC)
thread_title:  New version by KSmrq