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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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