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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
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)
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The documentation is out of date.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].
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)
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:
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:
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:
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:
:>
-- RProgrammer ( talk) 16:44, 14 March 2014 (UTC)
I have seen and used to donate multivalued functions, as in:
and
(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)
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'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)
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)
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)
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 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)
For real numbers, the radix sign 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)
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)
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)
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)
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Daily pageviews of this article
A graph should have been displayed here but
graphs are temporarily disabled. Until they are enabled again, visit the interactive graph at
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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
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)
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].
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)
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:
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:
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:
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:
:>
-- RProgrammer ( talk) 16:44, 14 March 2014 (UTC)
I have seen and used to donate multivalued functions, as in:
and
(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)
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'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)
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)
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)
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 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)
For real numbers, the radix sign 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)
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)
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)
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)