Talk:Multivalued function
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As best I can tell, the relation discussed here is what economists call a correspondence. I've put a cross-reference in here, and added a mention of multivalued functions in the correspondence article. As far as I can tell these are two names for the same thing, used in different areas of math. Isomorphic 22:25, 29 July 2006 (UTC)
Perhaps the following part of History should be moved to Applications: [BeginQuote] In physics, multivalued functions play an increasingly important role (...) They are the origin of gauge field structures in many branches of physics. [EndQuote] Megaloxantha (talk) 14:36, 3 December 2008 (UTC)
Correspondence, multiset - even mathematicians like various terms for the same thing (to assert the context). In case of natural languages it is fair and has it's own name synonymy (sorry for math sarcasm). We only shall ensure that none of the contexts (e.g. correspondence in economy) is omitted. Megaloxantha
Misnomer?
What this "misnomer" is supposed to mean? I do not know the formal mathematical definition of such a term. Usually the functions are assumed single-valued but the general definition of a function relates elements from one set to the elements of another (or same) one. Even the ordinary sqrt(x) is having two values (not to mention sin-1(z))! It's just for convinience that usually only one of the values is deliverately chosen. Or the implicit functions are also a "misnomer". -- Goldie (tell me) 22:19, 24 August 2006 (UTC)
- 'Implicit function' is a misnomer, in the large. A careful statement of the implicit function theorem will only give a local existence theorem. Charles Matthews 14:03, 12 October 2006 (UTC)
A graphical, interactive example of a multi-valued function
Go to [1] to see an example of a multi-valued function. This came from a class titled Complex Analysis. This demonstrates how a function can be analytic in a region, but not in the entire complex plane. The input is shown in black, and the three possible outputs are shown in red, green, and blue. As long as you don’t go around or through one of the "bad" points (shown in pink) you can view this as three ordinary functions.
For additional examples see [2].
The documentation is out of date. If you want to download TCL, you will need to go to [3].
Output: single multiset
The square root of 4 is the multiset {+2,−2). The square root of zero is the multiset {0,0}, because zero is a double root of the equation x2=0. Using the concept of a multiset, the term 'multivalued function' ceases to be a misnomer. Any comments? Bo Jacoby 16:33, 14 December 2006 (UTC)
Output: single multiset or single value
As far as I know, some authors accept that the codomain of a multivalued function is a set of sets or multisets, but many others interpret multivalued functions as functions which return a single (arbitrarily selected) value. For instance, many define the indefinite integral of f as one of the infinitely many antiderivatives of f. This is obviously convenient. Consider this question:
- Does the square root of x return (1) all the numbers the square of which is x, or (2) any number the square of which is x?
In other words, is its output a multiset with two elements or a single (not uniquely determined, and arbitrarily selected) number? In other words, does the algorithm imply the process of "collecting all possible solutions" or the process of "arbitrary selection of only one solution"? I guess that some mathematicians will defend the second option.
It is quite intersting to notice that the second option implies what follows:
- the square root is regarded as a multivalued function but, paradoxically, it has a single-valued output, and
- its codomain is simply R (rather than a set of multi-sub-sets of R).
Note that, in both cases, the square root returns a single value (either a single multiset or a single number). THis example can be generalized to all multivalued functions.
Conclusion. It seems that we have only two options for defining a multivalued function:
- a multivalued fonction is single valued and uniquely determined.
- a multivalued fonction is single valued but nonuniquely determined.
Multivalued functions are actually single valued! Paolo.dL 21:29, 27 September 2007 (UTC)
Notation for multifunctions
I have seen and used to donate multivalued functions, as in:
![{\displaystyle ]\!]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c6cef2c94369cb280cd2122fdaba4ca6a4f5f9)
![{\displaystyle [\![\,\log z\,]\!]:=\{\,\log |z|+i\theta :\theta \in [\![\,\arg z\,]\!]\,\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b52bb4b961f8ec39ef0e6e7d840e0de4086a45ec)
![{\displaystyle [\![\,\arg z\,]\!]:=\{\,\theta \in \mathbb {R} :z=|z|e^{i\theta }\,\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef8d8f7240d07d5530947c0c121a9d0499e81f14)
and
![{\displaystyle [\![\,z^{\alpha }\,]\!]:=\{\,e^{\alpha (\log |z|+i\theta )}:\theta \in [\![\,\arg z\,]\!]\,\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1353cc2702b9d90c7994431e3aa9f96bd078c8c)
(Priestley, H. A. (2006). Introduction to Complex Analysis, Second Edition, Oxford University Press. Chapters 7 and 9.)
Definition
Is there something amiss with this definition: "a multivalued function ... is a total relation; i.e. every input is associated with one or more outputs"? Suppose we have a function where every input is associated with only one output. Since "one" qualifies as "one or more", such a function would be multivalued according to this definition, wouldn't it? But I thought the idea was that a multivalued function musButt have more than one output associated with some inputs. I don't understand what this has to do with a total relation. Dependent Variable (talk) 13:02, 30 July 2009 (UTC)
- You wrote "associated with only one output", and of course you cannot say that "only one" qualifies as "only one or more".
- Anyway and besides, there should be no such thing as a multi-valued function. By definition, a function is single valued, and the value can be a tuple or a set, or whatever which can be considered as a single object. So i.m.o. this entire article is bunk and should be removed as such. But someone's MMV :-) DVdm (talk) 15:03, 30 July 2009 (UTC)
- I don't know what you mean. "Only one" is one way that the condition "one" could be fulfilled, and so it's also one way that the condition "one or more" could be fulfilled. The possiblility of it being taken to mean "only one" is accepted as real enough to warrent the clarification that the definition also covers cases where it's more than one. But shouldn't it be saying that it only refers to cases where at least one input is associated with more than one output?
- My point is just that a "multivalued function" (depracated, ill-advised misnomer though it may be) seems to refer explicitly to the case where at least one input is associated with more than one output. Otherwise it would be a function in the usual sense of the word function, a single-valued function. As it stands, the definition suggests that a single-valued function is a special case of a multi-valued function. By the way, is there a better name for the relation called a "multi-valued function", a name which respects the convention that a function must have no more than one output associated with any input? Dependent Variable (talk) 13:55, 1 August 2009 (UTC)
- You wrote: "Suppose we have a function where every input is associated with only one output".
- Then you wrote: "Since "one" qualifies as "one or more", such a function would be multivalued according to this definition."
- In my understanding of the English language, the usage of "only one" does not allow for an interpretation as "one or more", and therefore, still in my understanding of the English language, such a function would not be multivalued according to this definition.
- DVdm (talk) 14:08, 1 August 2009 (UTC)
- I would have thought this depends on the context and could go either way. If I say complex logarithm is a multivalued function then it obviously means that it can have more than one output value per input value. If I talk about a set of multivalued functions then I'd almost certainly want to include any normal single valued functions. So I'd go with that it included normal functions. Otherwise it's like having a set of natural numbers but excluding one. I think the article is fine as it is at that point, but of course as always if a citation can be found saying otherwise that would override what I consider as common sense. Dmcq (talk) 16:21, 1 August 2009 (UTC)
You're right that "only one" excludes the possibility of "more than one", but the article doesn't say "only one". It says "one or more". My point is that "one or more" includes the possibility of "only one", so - according to this definition - all single-valued functions would belong to the set of multi-valued functions. Of course, if that's the intended meaning, then okay, although it might be worth noting that a multi-valued function is often defined in a different way, in contrast to single-valued function, e.g. at Wolfram Mathworld: "A multivalued function, also known as a multiple-valued function (Knopp 1996, part 1 p. 103), is a "function" that assumes two or more distinct values in its range for at least one point in its ___domain." Similarly Borowsky & Borwein: "Set-valued function, multi-valued function, multifunction, carrier or point-to-set mapping, n. a mapping that associates a number of different elements of the second set with the same element of the first set ..." (Collins Dictionary of Mathematics, 1989) Dependent Variable (talk) 10:27, 2 August 2009 (UTC)
- Knopp doesn't say that exactly. He distinguishes between single-valued and multiple-valued functions saying multiple valued ones are the ones that aren't single valued and includes them both as functions, though he puts brackets in when referring to our functions as in "(single-valued) function". By the way the contents of the book I date to a 1945 translation of an edition of a 1913 book, it doesn't date to 1996 like the reference to the Dover reprint might indicate. As to the second definition a number includes 1 and even 0 quite often so it is as imprecise as the current definition in the article. I'm happy for you to change the definition if you want though and stick in a citation. I like citations on articles. Dmcq (talk) 11:30, 2 August 2009 (UTC)
- In this case, "a number" must mean "more than one". It would make no sense to talk about a single element different from itself, so logically it can't mean one. And, while "zero different elements" is grammatical, and zero is a number, the English expression "a number of different elements" really excludes that possibility. The only natural interpretation is that Borowsky & Borwein really do mean "more than one". And if Knopp says no more than that the terms multi-valued and single-valued are mutally exclusive, he's still implied a definition that differs from that of the Wikipedia article. But I'm new to this subject, so I feel I'd better leave any edits to people more knowledgable than me. Dependent Variable (talk) 16:47, 2 August 2009 (UTC)
- Well you can always work through the Wikipedia:Introduction but why not just dive in? It isn't as though this article is a featured article or numero uno in the popularity stakes so it's a good place to start. You searched for some references to check your facts, have written reasonably on the talk page, and signed yourself. That's way more than enough qualification. One thing I'd add is putting comments into the edit summary is a good idea. Dmcq (talk) 17:05, 2 August 2009 (UTC)
- In this case, "a number" must mean "more than one". It would make no sense to talk about a single element different from itself, so logically it can't mean one. And, while "zero different elements" is grammatical, and zero is a number, the English expression "a number of different elements" really excludes that possibility. The only natural interpretation is that Borowsky & Borwein really do mean "more than one". And if Knopp says no more than that the terms multi-valued and single-valued are mutally exclusive, he's still implied a definition that differs from that of the Wikipedia article. But I'm new to this subject, so I feel I'd better leave any edits to people more knowledgable than me. Dependent Variable (talk) 16:47, 2 August 2009 (UTC)
Set-Valued functions and multivalued functions are the same thing? I think not.
Set-valued functions are strictly functions, while multivalued functions strictly are not. —Preceding unsigned comment added by 67.194.132.91 (talk) 05:51, 25 March 2010 (UTC)
- That depends on how you define a multivalued function. If it is defined the same as a set-valued function then they are the same. Dmcq (talk) 15:24, 25 March 2010 (UTC)
- For example, consider functions with multiset-values. Also, a multiset-valued relation need not be a function. Kiefer.Wolfowitz (talk) 16:29, 25 March 2010 (UTC)
Multifunction with Multiset Domains?
Is there a notion of multifunctions with multiset domains, i.e. if an object is contained times in the ___domain, the multifunction must have exactly values for it? -- 132.231.1.56 (talk) 12:52, 20 September 2010 (UTC)
Total relation?
In the first sentence 'In mathematics, a multivalued function (shortly: multifunction, other names: set-valued function, set-valued map, multi-valued map, multimap, correspondence, carrier) is a total relation' I do not get the connection between multivalued function and total relation. In which sense exactly is it supposed to be a total relation? O.mangold (talk) 12:10, 23 December 2010 (UTC)
- I think they meant left-total but that seems overkill in terminology. I'll stick it in instead. Dmcq (talk) 12:43, 23 December 2010 (UTC)