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Article currently says:
As stated, this excludes the (rather natural) possibility of elements with infinite multiplicity.
I see that this was the very first complaint on this talk page, and references were even given, but nothing seems to have been done about it. This strikes me as rather serious. If you're allowing infinite sets, then I can't see much point in disallowing infinite multiplicities too. -- Trovatore ( talk) 01:54, 19 April 2012 (UTC)
An IP editor and Marc van Leeuwen just went back and forth on some changes w.r.t. the finite versus infinite problem. Right now, the article is entirely about finite multisets (though it doesn't say this in the intro) except for the section "formal definition" which allows infinite multisets (the set A can be infinite, even though the multiplicities are currently restricted to be finite). This is obviously untenable. I suggest that the intro and formal definition be rewritten to make it unambiguously clear that this article is about finite multisets. This will leave a bunch of material (some added by the IP editor then removed, some in the "formal definition" section) homeless; this material should be put into a new section somewhere that discusses the issue of infinite multisets (of both sorts: infinite base sets and infinite multiplicities). Are there any thoughts or objections? -- Joel B. Lewis ( talk) 15:30, 19 June 2012 (UTC)
The article is deficient. Infinite multisets are important even in elementary combinatorics (e.g., see Brualdi, Introductory Combinatorics). Choosing a finite sub(multi)set of given size from an infinite multiset is a basic combinatorial question. Zaslav ( talk) 08:37, 25 August 2023 (UTC)
It is possible to extend the definition of a multiset by allowing multiplicities of individual elements to be infinite cardinals instead of positive integers, but not all properties carry over to this generalization.
Wouldn't a simple application be, say, the roots of ? In set notation it's {−4, +1}, but multisets allow you to show multiplicity of roots, like {−4, +1, +1}. — Preceding unsigned comment added by 68.198.133.72 ( talk) 20:53, 7 July 2014 (UTC)
{1,1,1,3}\{1,1,2}= {1,3} was it? -- Peiffers ( talk) 16:21, 14 July 2014 (UTC)
-- Ernsts ( talk) 19:38, 4 December 2017 (UTC)
For non-mathematics and introductory text, to avoid confusion with set notation in the same text, is usual to adopt square brackets notation: A={1,2} is a set, B=[1,2,2] is a multiset.
See full notation here. -- Krauss ( talk)
Multiplicity and frequency are the same. To model a name/frequency table of statistics, we use the cartesian product Name X Name_frency... As multiplicity, same semantic in with the same product of sets modeling it. -- Krauss ( talk)
It can be easily proven that . I think this formula should be included in the article. Aside from its intrinsic interest, it comes handy when studying dimensions of some subspaces of vector spaces and/or problems related to polynomials.
Example 1: If we consider a polynomial of degree d in n variables along with all its first k-derivatives, this formula computes, from the number of derivatives of order i for each i, the total number of polynomials arising in the system.
Example 2: Since the homogeneous polynomials of degree i in n variables over a field form a subspace of dimension , this formula shows that is isomorphic to , what can also be inferred from the homogeneization process.
What do you think?
Jose Brox ( talk) 10:49, 21 August 2018 (UTC)
A correspondent mathematician questions the whole concept of multisets, basically saying it is poorly defined. He therefore cannot believe Dedekind would have had anything to do with them (Reference 11). Does anybody here happen to know the page number where Dedekind describes/uses multisets in his book?
Failing that, is there some authoritative source which defines this concept, preferably a book on set theory? It seems that discrete maths does not count for my correspondent; it is probably not real maths ;-) KarlFrei ( talk) 16:59, 11 December 2018 (UTC)
![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||
|
|
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Article currently says:
As stated, this excludes the (rather natural) possibility of elements with infinite multiplicity.
I see that this was the very first complaint on this talk page, and references were even given, but nothing seems to have been done about it. This strikes me as rather serious. If you're allowing infinite sets, then I can't see much point in disallowing infinite multiplicities too. -- Trovatore ( talk) 01:54, 19 April 2012 (UTC)
An IP editor and Marc van Leeuwen just went back and forth on some changes w.r.t. the finite versus infinite problem. Right now, the article is entirely about finite multisets (though it doesn't say this in the intro) except for the section "formal definition" which allows infinite multisets (the set A can be infinite, even though the multiplicities are currently restricted to be finite). This is obviously untenable. I suggest that the intro and formal definition be rewritten to make it unambiguously clear that this article is about finite multisets. This will leave a bunch of material (some added by the IP editor then removed, some in the "formal definition" section) homeless; this material should be put into a new section somewhere that discusses the issue of infinite multisets (of both sorts: infinite base sets and infinite multiplicities). Are there any thoughts or objections? -- Joel B. Lewis ( talk) 15:30, 19 June 2012 (UTC)
The article is deficient. Infinite multisets are important even in elementary combinatorics (e.g., see Brualdi, Introductory Combinatorics). Choosing a finite sub(multi)set of given size from an infinite multiset is a basic combinatorial question. Zaslav ( talk) 08:37, 25 August 2023 (UTC)
It is possible to extend the definition of a multiset by allowing multiplicities of individual elements to be infinite cardinals instead of positive integers, but not all properties carry over to this generalization.
Wouldn't a simple application be, say, the roots of ? In set notation it's {−4, +1}, but multisets allow you to show multiplicity of roots, like {−4, +1, +1}. — Preceding unsigned comment added by 68.198.133.72 ( talk) 20:53, 7 July 2014 (UTC)
{1,1,1,3}\{1,1,2}= {1,3} was it? -- Peiffers ( talk) 16:21, 14 July 2014 (UTC)
-- Ernsts ( talk) 19:38, 4 December 2017 (UTC)
For non-mathematics and introductory text, to avoid confusion with set notation in the same text, is usual to adopt square brackets notation: A={1,2} is a set, B=[1,2,2] is a multiset.
See full notation here. -- Krauss ( talk)
Multiplicity and frequency are the same. To model a name/frequency table of statistics, we use the cartesian product Name X Name_frency... As multiplicity, same semantic in with the same product of sets modeling it. -- Krauss ( talk)
It can be easily proven that . I think this formula should be included in the article. Aside from its intrinsic interest, it comes handy when studying dimensions of some subspaces of vector spaces and/or problems related to polynomials.
Example 1: If we consider a polynomial of degree d in n variables along with all its first k-derivatives, this formula computes, from the number of derivatives of order i for each i, the total number of polynomials arising in the system.
Example 2: Since the homogeneous polynomials of degree i in n variables over a field form a subspace of dimension , this formula shows that is isomorphic to , what can also be inferred from the homogeneization process.
What do you think?
Jose Brox ( talk) 10:49, 21 August 2018 (UTC)
A correspondent mathematician questions the whole concept of multisets, basically saying it is poorly defined. He therefore cannot believe Dedekind would have had anything to do with them (Reference 11). Does anybody here happen to know the page number where Dedekind describes/uses multisets in his book?
Failing that, is there some authoritative source which defines this concept, preferably a book on set theory? It seems that discrete maths does not count for my correspondent; it is probably not real maths ;-) KarlFrei ( talk) 16:59, 11 December 2018 (UTC)