Talk:Multivalued function

Improving content

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The documentation is out of date.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:

1. a multivalued fonction is single valued and uniquely determined.
2. a multivalued fonction is single valued but nonuniquely determined.

Multivalued functions are actually single valued! Paolo.dL 21:29, 27 September 2007 (UTC)

I dunno, why don't we keep the notion of "valued" (instead of throwing it out as a contradiction), and use it to mean one of two isomorphic objects:

1. a multivalued function is singly selected aka uniquely determined, but with set-valued output (if viewed as a set-valued output of possibilities).
2. a multivalued function is not singly selected aka nonuniquely determined, but with individual-element-valued output (if viewed as a non-deterministic relation).

:>

--RProgrammer (talk) 16:44, 14 March 2014 (UTC)

Notation for multifunctions

I have seen $[\![$ and $]\!]$ used to donate multivalued functions, as in:

$[\![\,\log z\,]\!]:=\{\,\log |z|+i\theta :\theta \in [\![\,\arg z\,]\!]\,\},$

$[\![\,\arg z\,]\!]:=\{\,\theta \in \mathbb {R} :z=|z|e^{i\theta }\,\}$

and

$[\![\,z^{\alpha }\,]\!]:=\{\,e^{\alpha (\log |z|+i\theta )}:\theta \in [\![\,\arg z\,]\!]\,\}.$

(Priestley, H. A. (2006). Introduction to Complex Analysis, Second Edition, Oxford University Press. Chapters 7 and 9.)

129.67.19.252 02:18, 26 October 2007 (UTC)

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)

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 $x$ is contained $n$ times in the domain, the multifunction must have exactly $n$ 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)

Total has been removed and Heterogeneous relation included as it is standard terminology. — Rgdboer (talk) 22:49, 18 June 2018 (UTC)

Splitting proposal

I propose that section Set-valued analysis be split into a separate page called Set-valued analysis since multivalued functions is just a particular topic of set-valued analysis. Saung Tadashi (talk) 14:07, 22 January 2019 (UTC)

Conditional support: This article is about two different topics: The lead and the "Example section" are about multi-valued functions as they are considered in complex analysis. The remainder of the article is about set-valued functions. Although a multi-valued functions (complex analysis) may be viewed as a set-valued function (with discrete sets as values), the methods and the properties that are studied are completely different. This justifies splitting the article. Thus I support such a split. However, "set-valued analysis" is not a common terminology (at least for people that are not specialist of this subject). Set-valued function is a title that is much clearer for everybody, and includes the analysis with such functions. For the moment, it redirects here. Thus, I suggest to transform it in an article, which, at the beginning would contain the section Set-valued analysis and most of what follows. Both resulting articles must have a disambiguating hatnote linking to the other, and deserve to be largely expanded. IMO, you can be WP:BOLD and proceed. D.Lazard (talk) 17:11, 22 January 2019 (UTC)

Hi @D.Lazard, thanks for your feedback. I finally found some time to work on the splitting of the articles and followed your suggestion.

As you are a highly experienced mathematician and Wikipedian editor, I'd greatly appreciate if you could review these last edits.

I also was thinking in splitting the "Multivalued function" in two new pages: one called "Multivalued function (Complex analysis)" with the major part of this article, and another one simply called "Multivalued function", which would contain an elementary set-theoretical description and would have links pointing to the `"Multivalued function (Complex analysis)" and the new article "Set-valued function". Do you think it makes sense? Gratefully, Saung Tadashi (talk) 21:38, 1 January 2023 (UTC)

The split is fine, but improvements are needed. In particular both leads need to be completely rewritten and both article need to be largely expanded.

Set-valued function is not presently a long article, and its lead is much too short. So, it is reasonable to include in this article what you intend to include in your suggested version for Multivalued function. In any case, the lead must contain the fact that multivalued functions of analysis are set valued functions which satisfy the further condition that choosing a value at a point defines a function in a neighbourhood of this point.

On the other hand, the lead of Multivalued function is too long, and most of it should be moved in a section "Motivation". Also it should be explained in the lead that multivalued functions are set-valued functions with continuity properties that allow considering them locally as ordinary functions. More important, the article is presently restricted to complex analysis although it is commonly used in all analysis, in particular in the context of the implicit function theorem and solutions of partial differential equations. D.Lazard (talk) 11:33, 3 January 2023 (UTC)

Real square root example

For real numbers, the radix sign ${\sqrt {x}}$ usually only denotes the non-negative root (see Square root); it is precisely defined like that to avoid multivaluedness. Using it as an example is likely to increase confusion. (The complex square root is different, of course.) RealSkeime (talk) 08:42, 4 March 2021 (UTC)

There was only one use of use the radix sign for a number that is not real and nonnegative. I have fixed this. D.Lazard (talk) 10:01, 4 March 2021 (UTC)

Link to German Page seems wrong

The link to the German page "Mengenwertige Abbildung" seems wrong. The German "Mengenwertige Abbildung" should rather be linked to "Set-valued function". A suitable German page to link from here ("multivalued faction") should rather be "Multifunktion" or "Korrespondenz_(Mathematik)" (see also first item in the discussion). 82.83.165.210 (talk) 12:35, 2 January 2023 (UTC)

Confusing

I don't understand:"Write f(x) for the set of those y ∈ Y with (x,y) ∈ Γf. If f is an ordinary function, it is a multivalued function by taking its graph ... They are called single-valued functions to distinguish them."

What is an ordinary function? It should be explained or referenced.

"Write f(x) for the set of those y ∈ Y with (x,y) ∈ Γf." So f(x) is a set. For example srqt(4) = {2,-2}. Then "If f is an ordinary function, it is a multivalued function by taking its graph" but it is already a multivalued function. No need to take it's graph. But if you were to take it's graph as suggested this would give for example (4,{2,-2}) as an element of Гf. But this would make Гf no longer a subset of X x Y.

"it is a multivalued function" and "They are called single-valued functions" seems contradictory. BartYgor (talk) 12:38, 27 December 2023 (UTC)

Proposed merger

Since all functions are univalent relations, the title of this article is self-contradictory. The article should be merged into Relation (mathematics). Rgdboer (talk) 01:23, 10 March 2024 (UTC)

No, no, no: Most mathematical texts that use "Multivaued function" do not talk of relations, and do not contain the word "relation". So such a merge would confuse many readers, and would contradict the main usage. Relations are not the alpha and omega of calculus and mathematical analysis. As an example, the principal value is fundamental for multivalued functions and cannot easily be defined for relations. D.Lazard (talk) 10:02, 10 March 2024 (UTC)

This argument is circular. The article you linked to says that the "principal value" is applied to what you call "Multi-Valued Functions". --Felix Tritschler (talk) 19:42, 14 April 2024 (UTC)

I'm also against a merge for the following reasons:

• Even though "all functions are univalent relations", I'm not sure we could reasonably argue a move from Function to Relation (mathematics), so why do it here?
• Multivalued function states that it is "about multivalued functions as they are considered in mathematical analysis" and this is a very specific context with its own rich theory. (Perhaps it needs a title change to make this clear?)
• Relation (mathematics) is largely about generic relations within a single set. It's very much from a set theoretic standpoint and there's very little overlap between the existing text of the two articles.
• Binary relation generalises to heterogeneous relations (on sets X and Y); but this again is at a fairly low, generic set theoretic level, with little overlap here.
• I agree that "multi-valued function" sounds self-contradictory; but that's probably more about how language works, where terms get ingrained even though our understanding changes. We live with terms like imaginary number, however reluctantly. NeilOnWiki (talk) 19:43, 13 May 2024 (UTC)

I oppose the merger too even though, mathematically, there is no difference between relation and a multivalued function. The principal issue, in addition to what said above, is that the two articles, this one and the relation, have quite a different style so the merger wouldn’t be easy. If either article is short, the merger might work. But since the two articles are sufficiently well-developed already, the merger would probably not work. As the consensus seems clear, I have gone ahead and removed the merger tag. —- Taku (talk) 17:06, 17 July 2024 (UTC)