This is an old revision of this page, as edited by Charles Matthews (talk | contribs) at 19:37, 4 February 2004 (Comment on partial functions etc.). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.
Revision as of 19:37, 4 February 2004 by Charles Matthews (talk | contribs) (Comment on partial functions etc.)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)There should probably be a distinction between the range and codomain. I say this because I can't remember the exact definition of each, and I would like to be able to look it up on wikipedia!
If you consider a function f from S to T, where S, T are non-empty, as a rule that assigns to each element of S (the domain) an unique element in T (the codomain), then:
The subset of T defined as {all y in T, such that y=f(x) for some x in S} is the range of f.
Since you have defined function differently (not in terms of sets), it is hard to know exactly where to put this.
Actually, I wrote the above, but I did not write the definition on the main page. For that I would have used a more precise formulation, e.g., something out Singer and Thorpe's ugrad topology text. I let the main page more or less stand as the level of math here (on wikipedia) seems to be about 2d year level, and this is just fine for that. DMD
IMO, there should be a variety of "levels" of discourse on Misplaced Pages. The function article, for example, right now, is totally useless for nonmathematicians. It also, no doubt, does not reflect the latest research about functions. Eventually, we're going to want to include information to enlighten and please everyone.
Personally, I think basic explanations for a lay audience are crucial to an encyclopedia, and the math section is sorely lacking those. Is it because you're incapable of producing such explanations, mathematicians? Or is it that you just don't wanna? --LMS---- The definition of a function has not changed since I first studied Math. I am not aware of a state of the art definition. And yes, the definition could be more understandable to the lay man. More important the second definition of a function as a set of ordered pairs is wrong. No function can contain the ordered pairs (0,1) and (0,-1) because a function would assign to the first element "0" a unique element from the codomain, not both "1" and "-1."
You misrepresent what it says: a function can be considered as a set of ordered pairs, but not every set of ordered pairs defines a function. GWO
I stand corrected. Apologies.
However, to define a function in a really understandable way would require something like tables and even diagrams. I think the Math people are literally crippled by the limitations of the Wiki software. I have been trying to figure out how to explain what Calculus is about for months--but the inability to use diagrams always stops me in my tracks. The Mapping entry describes the same basic concept. Is that page so un-understandable??? AnonymousCoward
Part of the problem is that mathematical notions such as functions are just plain hard to understand. For example, the statement found on the function page "Continuous Functions are those whose domains and ranges are sets of real numbers and which satisfy additional constraints." is just wrong. For example, finite dimensional linear operators are perfectly reasonable continuous functions taking elements from one linear space (domain) to another (codomain, which may be the same space). Explaining this requires that reader understand a slightly more abstract and precise definition of function than so far given, and understanding the structure of a finite dimensional linear space. And we have a further conundrum, because the "range" of a linear operator has a precise technical meaning, which is not the same as codomain.
But I just cannot see how to explain such things in "layman's terms".
Taking a larger view, I suspect that part of the problem with math on wiki is the same as with math in general. It's hard. I can scream through a sci-fi or detective paperback at something like 100 pages/hour. Getting through a page of math at the level I am capable of comprehending (4th year/1st year grad) is something like 2-4 hours per page, on average. Sometimes more: one particular paper out of (the journal) Solid Mechanics Archives has occupied the better part of 6 weekends so far, and I have only gotten through 4 of 6 sections.
Math is hard. It's hard for almost all mathematicians too. Can wikipedia offer something for both the lay reader and the amateur mathematician? DMD
Well, my point, which I evidently didn't make clear enough, is that basic concepts like "function" and "set" and a lot of other basic concepts can receive a relatively unrigorous introduction. See Set Theory for an excellent example. Anyone who has read any basic text in math or logic knows that it can be done and to great effect. Now, I'm not saying (what would indeed be absurd) that a "simple" version of everything in math category should or can be given. Obviously, it can't, nor should we try to give one. (Of course, in any case, we should always strive for maximum clarity, and not just conceptual clarity but clarity to all sorts of adequately-prepared readers.) I just would like to see, associated with very basic concepts and theorems, some attempt at giving the sorts of explanations that one ordinarily gives to beginners, in order to help them get a fix on those concepts. If you don't think that can be done, it must have been too long since you were a beginner yourself. :-)
By the way, A.C., if you want, you can always e-mail images to jasonr at bomis dot com, and he can upload them to the server and tell you where they live. It's as simple as that... --LMS
It would like to suggest to merge this entry with the one for "Mapping". The two are already indicated as synonymous anyway. I have prepared a version in Suggestion. If I don't hear any big objections I will make the neccessary changes. (My own feeling is that there should be something added about n-ary functions with more than 1 arguement.)
Another suggestion I would like to make is move the whole thing to "Mathematical function". The current beginning "In Mathematics" already suggests something like that and currently there are not too many references to "function" so I could easiliy fix that.
So, waddayasay Wikipedians? -- Jan Hidders
I approve, since I already pointed out that functions and mappings are used interchangeably most of the time today. I only worry that as simple as you make it, someone will claim the material is too "specialist." For that reason, I would include a large variety of examples. Functions, limits and continuity need all need precise, yet understandable explanations. That what I goddasay..RoseParks
Eventually it should be titled "Function (Mathematics)", but we don't have parens yet (Usemod 0.92). For now I don't see any problem with an article titled "function". --LDC
I totally agree, Jan. I also agree with LDC's comment (function is an important concept in Ancient philosophy, philosophy of biology, and ethics as well). Examples are always helpful, too... The article for now could live on function or mapping, and then what I'd love to see (though I can't produce myself) is an explanation, on the page where the article doesn't live, a brief explanation of any differences in meaning between "function" and "mapping." --LMS
Ok. Thank you for your replies. I will put it under 'function' and make a small remark in 'mapping' on the difference (of which I am not really too sure). I'll think about some more examples, but I'm a bit timepressed. Maybe somebody else has some ideas. -- JanHidders
The following is imported from "Mathematical Function/Talk". I redirected Mathematical Function to Function. --AxelBoldt
How do people think about merging this article with Function and then redirecting there? This article's definition isn't even very mathematical. --AxelBoldt
I think they should be merged. The author of Mathematical Function probably felt that function is still too formal, and that is probably true. So perhaps we should merge this article with the part of Function that gives some examples. But maybe other people would like to start the article with an informal introduction, and only after that introduce the formal definition. So perhaps the following would be best:
- (initial abstract description) A function is a means of associating with every element in a certain set an unique element in another set.
- (informal discussion) some examples, use in mathematics, et cetera (this would contain the stuff from Mathematical Function
- (formal definition) the formal definition in set theory that function now starts with
- (terminology) one-to-one, injective, et cetera
- (see also:)
-- JanHidders
I think the difference between a mapping and a function is that a mapping X->Y can send one element of X to more than one element of Y.
Eg. Let f: R -> R be defined by f(x) = x
Then f is a function, but f is only a mapping. -- Tarquin
I have seen "multi-valued function" for that kind of thing, but not mapping. --AxelBoldt
...so that a multi-valued function cannot be called a function? I don't like that... isn't there another term for that? --Seb
good point, I've seen & used "multi-valued function". On reflection, I think usage of function / mapping / transformation varied among my lecturers at uni, and some sought to make a useful distinction. -- Tarquin
It's a shame we have to wait until a consistent vocabulary becomes well-established among mathematicians before we can clear things up... --Seb
What about "binary relation"? --Seb
Yes, functions are binary relations, but special ones. Maybe we can use "set-valued function" instead of "multi-valued function". This seems to be a logical and self-explanatory term. AxelBoldt Yes, functions are binary relations, but special ones. Maybe we can use "set-valued function" instead of "multi-valued function". It seems to be a logical and self-explanatory name. AxelBoldt
what is the difference between the codomain and range of a function?
- The function f : R -> R with f(x) = x has codomain R and range the non-negative reals. Range is the set of all outputs that the function produces; codomain is the set that all those outputs have to be a member of. If the range and the codomain are the same, then the function is onto, or surjective. AxelBoldt 04:38 Nov 26, 2002 (UTC)
I have just uploaded three images: mathmap.png, an example on functions; notMap1.png, an example on not a function (but a multivalued function); and notMap2.png, an example on not a function (but a partial function). I will soon put the images in this page. Wshun
After rewriting, the article is shorter and the paragraph below doesn't seem to fit. It is because the difference between the set-valued definition and the usual explicit formula definition is hidden by the "magic" word relation. However, this is a good paragraph and someone may like to put it back:
This definition of function provides mathematicians with a number of useful advantages over the "explicit formula" definition:
- Clearly, if we already have an explicit formula for f(x), we can construct the set of pairs f; so nothing is lost by this definition.
- The definition allows us to consider functions in the abstract which, as a practical matter, could never be evaluated. In particular, functions which would require an infinite number of operations to evaluate can be considered as a set which has been given as a whole, rather than calculated.
- In many cases, there is no explicit formula which amounts to more than a "table" of arguments and values; the above definition accommodates these types of functions naturally.
- We can talk about sets of functions without specifying them exactly; for example, given the domain R of real numbers, and the codomain N of natural numbers, we can talk about the set of all functions of the form f: R → N, or some subset of these functions which satisfy other criteria. Mathematicians are often interested in such generalizations.
- In some cases, it may be unclear whether or not some binary relation f is "truly" a function; the definition provides a way of proving the existence of a well-defined function without actually specifying an explicit formula.
Would it make sense to replace this with a decent disambiguation page? Right now we start with a history paragraph about the mathematical concept, then the computer science paragraph mentions concepts that haven't even been defined, then the sociology paragraph uses a completely different notion of function, then the math starts. The notion of function in biology should also be explained somewhere, and the etymology. AxelBoldt 02:48 14 Jun 2003 (UTC)
There obviously is a case for separating out the biology and sociology. But I think putting the computer science elsewhere would actually diminish the content.
(Later)
Well, now the computer science wording on domain and codomain is gone. Really, I think this is unhelpful.
You are right. I reintroduced it at the end of the paragraph on domain and codomain. I realized that the correspondence sets<->datatypes makes a perfect source of examples and motivation for the concept of "partial function" and the distinction between "codomain" and "range", but I don't know how to write that coherently and, more important, briefly. -- Miguel
Well, maybe a solution is to have a function (mathematics) page and a function (computing) page. Or maybe there is a way to talk about function-as-relation, and function-as-expression; that could be put on a function page that was really disambiguating and fixing ideas. Currently a function is a "machine" and then that idea is quickly dropped. Perhaps I do favour a 'function (mathematics)' page for the bulk of the current material.
Much of the content that would go into 'function (programming)' or 'function (computing)' is covered in 'subprogram'. but in a way that IMHO doesn't do justice to the importance of functions and makes too many explicit references to specific computer languages. There seems to be a tension between the way computer scientists and mathematicians want pages to look here on Misplaced Pages. -- Miguel
Well, I'm new round here, but I'm aware of that sort of thing. It's one reason I added something today to the Nicolas Bourbaki page. The subprogram page isn't great. The 'neutral point of view' might regard 'function (mathematics)' as a cop-out; so in a sense it is better to have the old debate about this out in the open somewhere.
I have moved the section on explicit function and implicit function to how to specify a function
The two sections below works for real or complex functions, not general functions:
Elementary Functions
Elementary functions are "basic" or "simple" functions, these are the most commonly used types of functions. These functions involve addition, division, exponents, logarithms, multiplication, polynomials, radicals, rationals, subtraction, and trigonometric expressions. Example of non-elementary functions are Bessel functions and gamma functions.
Calculus
The derivative of a function, at some point, is a measure of the rate at which that function is changing (as an argument undergoes change).
--Wshun
Well, Im obviously not getting along with the mathematicians here; but, shouldn't there be some kind of link here to derivative and some kind of mention of calculus etc? Pizza Puzzle
Yeah, that's the difficulty. Functions in daily uses-physics, engineering, high schools-are those kind of functions that derivatives are well-defined. But functions in pure mathematical senses (as the piecewise function in the article) have nothing to do with derivatives. A good introduction paragraph could settle the issue, but all versions so far just focus on what a function is theoretically. Wshun
A rewritten of introduction. Settle some questions about calculus and the strange definitions. Wshun
Put the "Elementary function" in "List of functions", with a statement saying that this term has no fixed meaning. Wshun
For such kind of functions, one can talk about limits and derivatives, both are measurements of the change of output values with respect to the change of input values. They are the basics of calculus. <--- Good Pizza Puzzle
This article is improving in a quick pace. Cheers for all Wikipedians!!! Wshun
MyRedDice moved several paragraphs here that I had written at the Misplaced Pages:Reference desk. I don't think they're really suitable for this article. Had I been writing for function, I would have written something different. It would have been more complete and less informal. I also don't think it is particularly relevant to the 'functions' page; the new section doesn't seem to fit with the other material on function. I like what I wrote as a partial answer to a reference desk question, but not as part of an encyclopedia article on functions.
I've moved the 'enumeration' section back to the reference desk. Dominus
- I thought it was a good start and should remain here. Sure, it could be improved further, but so could everything. Personally, I found its informal tone rather refreshing. :) Not that bothered, though. Martin
single and multi-valued functions
I came to this page looking for an answer to a specific question. Forty years ago, when I learned what little math I know, a function could be either single or multi-valued, and example of the latter being y = x^0.5. (I can cite textbooks to support this claim).
Now, the definition has changed and functions must be single-valued, that is in y = f(x) there must be one and only one y to every x.
I have spoken to a couple of mathematicians at my local university and neither were aware of this change of definition, because 40 years is too long ago for most people, but not (unfortunately) for me.
So my question is, how where when and why did the definition change? In my subject, economics, it is very hard to get a definition generally accepted and even harder to get one changed!
g.t.renshaw@warwick.ac.uk
- Nothing's changed. There hasn't been any change in definition. A function has always been and is now single-valued. However, the term "multi-valued function" is used to describe relations whose "output" is not unique (i.e. a multi-valued function from X to Y is really a single-valued function from X to the power set of Y.) Let me be very clear: the term "multi-valued function" is meant to be taken as a WHOLE, the "multi-valued" part is NOT an adjective describing a particular type of function. Revolver 05:28, 4 Feb 2004 (UTC)
Actually, I think the whole discussion of multivalued and partial functions is unnecessary on this page, not simply because the vast majority of people reading the article don't need to know about these things, but because the terminology is almost bound to confuse people. There are separate articles for each of these topics. Revolver 18:54, 4 Feb 2004 (UTC)
The 'change' can be tracked in Hardy's Pure Mathematics, as I recall. As of post-Bourbaki days, functions are strictly a certain kind of relation, and single-valued is the defining property.
I do however disagree with lumping multi-valued and partial as confusing. I think the multi-valued stuff is just (mildly) confusing language for talking about relations. But partial functions occur every day on the pages of mathematics books - caused by trying to divide by zero. And they occur very day in computer programs that loop. It is more confusing, and less tolerable in my view, to have to define the domain of every function on the real line before giving a formula, which may be as simple as a rational function.
Charles Matthews 19:37, 4 Feb 2004 (UTC)