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== Improving content ==
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Is there a notion of multifunctions with multiset domains, i.e. if an object <math>x</math> is contained <math>n</math> times in the ___domain, the multifunction must have exactly <math>n</math> values for it? -- [[Special:Contributions/132.231.1.56|132.231.1.56]] ([[User talk:132.231.1.56|talk]]) 12:52, 20 September 2010 (UTC)
:"Multiset" by itself is a rather distracting piece of terminology, since a set simply cannot have two different elements that are the same element.
:If one wishes to encode "a multiset that contains the same object several times" as a set, it can be considered as a function (a certain kind of set of Kuratowski-pairs). The ___domain <math> \mathrm{dom}(f) </math> of the multiset-encoding function <math> f </math> contains a different element for "each of the copies of the same object in the multiset", and the range <math> \mathrm{ran}(f) </math> of the function contains "each object that is in the multiset".
:For example, assume that a, b and c are different objects and that one wishes to encode "a multiset that has five copies of a, three copies of b, and a single copy of c" as a set. Define a function <math> f </math> with <math>\mathrm{dom}(f) = \{1,2,3,4,5,6,7,8,9\} </math> so that <math> f(1)=f(2)=f(3)=f(4)=f(5)=a </math>, <math> f(6)=f(7)=f(8)=b </math> and <math> f(9)=c </math>. This function <math> f </math> fulfills the wish (in particular it is a set), and it does a little bit more: It gives five different external labels <math> 1,2,3,4,5 </math> for different copies of the same object <math> a </math>, three different external labels <math> 6,7,8 </math> for different copies of the same object <math> b </math>, and the external label <math> 9 </math> for the object <math> c </math>.
:Of course one can deem those external labels ugly, but then one can come up with another function that encodes "the multiset" with prettier labels. However, if "a multiset" happens to have several copies of the same object, it is quite reasonable that one should be able to attach different external labels to different copies of that object some way or another. [[User:Lapasotka|Lapasotka]] ([[User talk:Lapasotka|talk]]) 05:43, 14 June 2025 (UTC)
== Total relation? ==
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:I think they meant left-total but that seems overkill in terminology. I'll stick it in instead. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 12:43, 23 December 2010 (UTC)
::Total has been removed and [[Heterogeneous relation]] included as it is standard terminology. — [[User:Rgdboer|Rgdboer]] ([[User talk: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. [[User:Saung Tadashi|Saung Tadashi]] ([[User talk: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 function]]s. 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). {{noredirect|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. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 17:11, 22 January 2019 (UTC)
::Hi @[[User:D.Lazard|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, [[User:Saung Tadashi|Saung Tadashi]] ([[User talk: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 (mathematics)|analysis]], in particular in the context of the [[implicit function theorem]] and solutions of [[partial differential equation]]s. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 11:33, 3 January 2023 (UTC)
== Real square root example ==
For real numbers, the radix sign <math>\sqrt{x}</math> 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.) [[User:RealSkeime|RealSkeime]] ([[User talk: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. [[User:D.Lazard|D.Lazard]] ([[User talk: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). [[Special:Contributions/82.83.165.210|82.83.165.210]] ([[User talk: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. [[User:BartYgor|BartYgor]] ([[User talk:BartYgor|talk]]) 12:38, 27 December 2023 (UTC)
== Proposed merger ==
Since all functions are [[univalent relation]]s, the title of this article is self-contradictory. The article should be merged into [[Relation (mathematics)]]. [[User:Rgdboer|Rgdboer]] ([[User talk: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. [[User:D.Lazard|D.Lazard]] ([[User talk: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". --[[User:Felix Tritschler|Felix Tritschler]] ([[User talk: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 relation]]s", 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 {{mvar|''X''}} and {{mvar|''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. [[User:NeilOnWiki|NeilOnWiki]] ([[User talk: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. —- [[User:TakuyaMurata|Taku]] ([[User talk:TakuyaMurata|talk]]) 17:06, 17 July 2024 (UTC)
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