Talk:Multivalued function

message: 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)
thread_title:  Improving content 

message: 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)
replies: text: '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)
thread_title:  Misnomer? 

message: 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 x<Sup>2</sup>=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)
replies: text: >
--RProgrammer (talk) 16:44, 14 March 2014 (UTC)
thread_title:  Output: single multiset 

message: 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)
replies: text: >
--RProgrammer (talk) 16:44, 14 March 2014 (UTC)
thread_title:  Output: single multiset or single value 

message: 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)
thread_title:  Notation for multifunctions 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Set-Valued functions and multivalued functions are the same thing?  I think not. 

message: 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)
thread_title:  Multifunction with Multiset Domains? 

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

message: 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)
replies: text: 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)
thread_title:  Real square root example 

message: 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)
thread_title:  Link to German Page seems wrong 

message: 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)
thread_title:  Confusing