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What would you call a finite collection of unordered objects that are not necessarily distinct?



The Next CEO of Stack OverflowDefinition of correspondenceName for variations of elements from several setsComparison of two sets of 4-tuples using combinatoricsComparison of two collections of 4-tuples using combinatorics - more complicated versionPermutation of numbers from multiple sets [May contain duplicate numbers among other sets], resulting in Non-Duplicate SetPredicting the number of unique elements in the Cartesian product of a set with itselfIs there symbol to denote a combination and permutation?Name for a set in which some of the elements are also contained in other set elements?What would you call this?Is there a name for the set of “unique” combinations of the powerset of $2^n$ modulo permutation?










9












$begingroup$


I Just want to know the name for this if there is one because I don't think it satisifies any of the formal definitions for sets, n-tuples, sequences, combinations, permutations, or any other enumerated objects I can think of.



For convenience, I will henceforth use the term $mathbf set^*$ with an asterisk to refer to what I described in the title.



As a quick example, let $mathbfA$ and $mathbfB $ be $mathbf set^*$'s where $$mathbfA = 3,3,4,11,4,8$$
$$mathbfB = 4,3,4,8,11,3$$



Then $mathbfA $ and $mathbf B $ are equal.










share|cite|improve this question









$endgroup$
















    9












    $begingroup$


    I Just want to know the name for this if there is one because I don't think it satisifies any of the formal definitions for sets, n-tuples, sequences, combinations, permutations, or any other enumerated objects I can think of.



    For convenience, I will henceforth use the term $mathbf set^*$ with an asterisk to refer to what I described in the title.



    As a quick example, let $mathbfA$ and $mathbfB $ be $mathbf set^*$'s where $$mathbfA = 3,3,4,11,4,8$$
    $$mathbfB = 4,3,4,8,11,3$$



    Then $mathbfA $ and $mathbf B $ are equal.










    share|cite|improve this question









    $endgroup$














      9












      9








      9


      2



      $begingroup$


      I Just want to know the name for this if there is one because I don't think it satisifies any of the formal definitions for sets, n-tuples, sequences, combinations, permutations, or any other enumerated objects I can think of.



      For convenience, I will henceforth use the term $mathbf set^*$ with an asterisk to refer to what I described in the title.



      As a quick example, let $mathbfA$ and $mathbfB $ be $mathbf set^*$'s where $$mathbfA = 3,3,4,11,4,8$$
      $$mathbfB = 4,3,4,8,11,3$$



      Then $mathbfA $ and $mathbf B $ are equal.










      share|cite|improve this question









      $endgroup$




      I Just want to know the name for this if there is one because I don't think it satisifies any of the formal definitions for sets, n-tuples, sequences, combinations, permutations, or any other enumerated objects I can think of.



      For convenience, I will henceforth use the term $mathbf set^*$ with an asterisk to refer to what I described in the title.



      As a quick example, let $mathbfA$ and $mathbfB $ be $mathbf set^*$'s where $$mathbfA = 3,3,4,11,4,8$$
      $$mathbfB = 4,3,4,8,11,3$$



      Then $mathbfA $ and $mathbf B $ are equal.







      combinatorics elementary-set-theory notation permutations definition






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      share|cite|improve this question










      asked Mar 25 at 11:20









      Nicholas CousarNicholas Cousar

      374212




      374212




















          4 Answers
          4






          active

          oldest

          votes


















          35












          $begingroup$

          If you're looking for something like a set which may have repeated elements, standard terms are multiset or bag. See multiset on wikipedia.






          share|cite|improve this answer









          $endgroup$




















            7












            $begingroup$

            The common term is multiset. For a formal definition, you can for instance define the set of multisets of size $n$ of a given set $A$ as $A^n/mathfrakS_n$ where $mathfrakS_n$ acts by permutation of the factors; or if you don't want to be bothered by size you can define it as a map $f: Ato mathbbN$ where $f(a)$ is supposed to represent the number of times $a$ appears in the multiset.



            These are two interesting models for different situations, and there are probably more.






            share|cite|improve this answer









            $endgroup$




















              4












              $begingroup$

              In this context you can identify what you call a $mathbf set^*$ with a function that has a finite domain and has $mathbb N=1,2,3cdots$ as codomain.



              $A$ and $B$ in your question can both be identified with function: $$langle3,2rangle,langle4,2rangle,langle8,1rangle,langle11,1rangle$$Domain of the function in this case is the set $3,4,8,11$.






              share|cite|improve this answer









              $endgroup$




















                1












                $begingroup$

                If two objects can be distinguished by the number of times an element appears in them, that is called "multiplicity". So the more mathy version of "finite collection of unordered objects that are not necessarily distinct" would be "unordered finite collection with multiplicity" or "finite collection with multiplicity but not order".



                The single-word term for unordered collections with multiplicity is "multi-set", but I don't think there's any single-word term for finite multi-sets. Googling "math collection multiplicity no order" returns http://mathworld.wolfram.com/Set.html and https://en.wikipedia.org/wiki/Multiplicity_(mathematics) , both of which mention multisets.



                Another term that is used in the context of eigenvalues is "spectrum": the multiplicity of the eigenvalues is important, but there is no canonical order (other than the normal order of the real numbers, but that doesn't apply if they are complex). When you diagonalize or take the Jordan canonical form of a matrix, it matters how many times each eigenvalue appears, but putting the eigenvalues in a different order results in the same matrix, up to similarity.






                share|cite|improve this answer









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                  4 Answers
                  4






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                  active

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                  active

                  oldest

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                  35












                  $begingroup$

                  If you're looking for something like a set which may have repeated elements, standard terms are multiset or bag. See multiset on wikipedia.






                  share|cite|improve this answer









                  $endgroup$

















                    35












                    $begingroup$

                    If you're looking for something like a set which may have repeated elements, standard terms are multiset or bag. See multiset on wikipedia.






                    share|cite|improve this answer









                    $endgroup$















                      35












                      35








                      35





                      $begingroup$

                      If you're looking for something like a set which may have repeated elements, standard terms are multiset or bag. See multiset on wikipedia.






                      share|cite|improve this answer









                      $endgroup$



                      If you're looking for something like a set which may have repeated elements, standard terms are multiset or bag. See multiset on wikipedia.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Mar 25 at 11:24









                      Especially LimeEspecially Lime

                      22.7k23059




                      22.7k23059





















                          7












                          $begingroup$

                          The common term is multiset. For a formal definition, you can for instance define the set of multisets of size $n$ of a given set $A$ as $A^n/mathfrakS_n$ where $mathfrakS_n$ acts by permutation of the factors; or if you don't want to be bothered by size you can define it as a map $f: Ato mathbbN$ where $f(a)$ is supposed to represent the number of times $a$ appears in the multiset.



                          These are two interesting models for different situations, and there are probably more.






                          share|cite|improve this answer









                          $endgroup$

















                            7












                            $begingroup$

                            The common term is multiset. For a formal definition, you can for instance define the set of multisets of size $n$ of a given set $A$ as $A^n/mathfrakS_n$ where $mathfrakS_n$ acts by permutation of the factors; or if you don't want to be bothered by size you can define it as a map $f: Ato mathbbN$ where $f(a)$ is supposed to represent the number of times $a$ appears in the multiset.



                            These are two interesting models for different situations, and there are probably more.






                            share|cite|improve this answer









                            $endgroup$















                              7












                              7








                              7





                              $begingroup$

                              The common term is multiset. For a formal definition, you can for instance define the set of multisets of size $n$ of a given set $A$ as $A^n/mathfrakS_n$ where $mathfrakS_n$ acts by permutation of the factors; or if you don't want to be bothered by size you can define it as a map $f: Ato mathbbN$ where $f(a)$ is supposed to represent the number of times $a$ appears in the multiset.



                              These are two interesting models for different situations, and there are probably more.






                              share|cite|improve this answer









                              $endgroup$



                              The common term is multiset. For a formal definition, you can for instance define the set of multisets of size $n$ of a given set $A$ as $A^n/mathfrakS_n$ where $mathfrakS_n$ acts by permutation of the factors; or if you don't want to be bothered by size you can define it as a map $f: Ato mathbbN$ where $f(a)$ is supposed to represent the number of times $a$ appears in the multiset.



                              These are two interesting models for different situations, and there are probably more.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered Mar 25 at 11:27









                              MaxMax

                              15.8k11143




                              15.8k11143





















                                  4












                                  $begingroup$

                                  In this context you can identify what you call a $mathbf set^*$ with a function that has a finite domain and has $mathbb N=1,2,3cdots$ as codomain.



                                  $A$ and $B$ in your question can both be identified with function: $$langle3,2rangle,langle4,2rangle,langle8,1rangle,langle11,1rangle$$Domain of the function in this case is the set $3,4,8,11$.






                                  share|cite|improve this answer









                                  $endgroup$

















                                    4












                                    $begingroup$

                                    In this context you can identify what you call a $mathbf set^*$ with a function that has a finite domain and has $mathbb N=1,2,3cdots$ as codomain.



                                    $A$ and $B$ in your question can both be identified with function: $$langle3,2rangle,langle4,2rangle,langle8,1rangle,langle11,1rangle$$Domain of the function in this case is the set $3,4,8,11$.






                                    share|cite|improve this answer









                                    $endgroup$















                                      4












                                      4








                                      4





                                      $begingroup$

                                      In this context you can identify what you call a $mathbf set^*$ with a function that has a finite domain and has $mathbb N=1,2,3cdots$ as codomain.



                                      $A$ and $B$ in your question can both be identified with function: $$langle3,2rangle,langle4,2rangle,langle8,1rangle,langle11,1rangle$$Domain of the function in this case is the set $3,4,8,11$.






                                      share|cite|improve this answer









                                      $endgroup$



                                      In this context you can identify what you call a $mathbf set^*$ with a function that has a finite domain and has $mathbb N=1,2,3cdots$ as codomain.



                                      $A$ and $B$ in your question can both be identified with function: $$langle3,2rangle,langle4,2rangle,langle8,1rangle,langle11,1rangle$$Domain of the function in this case is the set $3,4,8,11$.







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered Mar 25 at 11:33









                                      drhabdrhab

                                      104k545136




                                      104k545136





















                                          1












                                          $begingroup$

                                          If two objects can be distinguished by the number of times an element appears in them, that is called "multiplicity". So the more mathy version of "finite collection of unordered objects that are not necessarily distinct" would be "unordered finite collection with multiplicity" or "finite collection with multiplicity but not order".



                                          The single-word term for unordered collections with multiplicity is "multi-set", but I don't think there's any single-word term for finite multi-sets. Googling "math collection multiplicity no order" returns http://mathworld.wolfram.com/Set.html and https://en.wikipedia.org/wiki/Multiplicity_(mathematics) , both of which mention multisets.



                                          Another term that is used in the context of eigenvalues is "spectrum": the multiplicity of the eigenvalues is important, but there is no canonical order (other than the normal order of the real numbers, but that doesn't apply if they are complex). When you diagonalize or take the Jordan canonical form of a matrix, it matters how many times each eigenvalue appears, but putting the eigenvalues in a different order results in the same matrix, up to similarity.






                                          share|cite|improve this answer









                                          $endgroup$

















                                            1












                                            $begingroup$

                                            If two objects can be distinguished by the number of times an element appears in them, that is called "multiplicity". So the more mathy version of "finite collection of unordered objects that are not necessarily distinct" would be "unordered finite collection with multiplicity" or "finite collection with multiplicity but not order".



                                            The single-word term for unordered collections with multiplicity is "multi-set", but I don't think there's any single-word term for finite multi-sets. Googling "math collection multiplicity no order" returns http://mathworld.wolfram.com/Set.html and https://en.wikipedia.org/wiki/Multiplicity_(mathematics) , both of which mention multisets.



                                            Another term that is used in the context of eigenvalues is "spectrum": the multiplicity of the eigenvalues is important, but there is no canonical order (other than the normal order of the real numbers, but that doesn't apply if they are complex). When you diagonalize or take the Jordan canonical form of a matrix, it matters how many times each eigenvalue appears, but putting the eigenvalues in a different order results in the same matrix, up to similarity.






                                            share|cite|improve this answer









                                            $endgroup$















                                              1












                                              1








                                              1





                                              $begingroup$

                                              If two objects can be distinguished by the number of times an element appears in them, that is called "multiplicity". So the more mathy version of "finite collection of unordered objects that are not necessarily distinct" would be "unordered finite collection with multiplicity" or "finite collection with multiplicity but not order".



                                              The single-word term for unordered collections with multiplicity is "multi-set", but I don't think there's any single-word term for finite multi-sets. Googling "math collection multiplicity no order" returns http://mathworld.wolfram.com/Set.html and https://en.wikipedia.org/wiki/Multiplicity_(mathematics) , both of which mention multisets.



                                              Another term that is used in the context of eigenvalues is "spectrum": the multiplicity of the eigenvalues is important, but there is no canonical order (other than the normal order of the real numbers, but that doesn't apply if they are complex). When you diagonalize or take the Jordan canonical form of a matrix, it matters how many times each eigenvalue appears, but putting the eigenvalues in a different order results in the same matrix, up to similarity.






                                              share|cite|improve this answer









                                              $endgroup$



                                              If two objects can be distinguished by the number of times an element appears in them, that is called "multiplicity". So the more mathy version of "finite collection of unordered objects that are not necessarily distinct" would be "unordered finite collection with multiplicity" or "finite collection with multiplicity but not order".



                                              The single-word term for unordered collections with multiplicity is "multi-set", but I don't think there's any single-word term for finite multi-sets. Googling "math collection multiplicity no order" returns http://mathworld.wolfram.com/Set.html and https://en.wikipedia.org/wiki/Multiplicity_(mathematics) , both of which mention multisets.



                                              Another term that is used in the context of eigenvalues is "spectrum": the multiplicity of the eigenvalues is important, but there is no canonical order (other than the normal order of the real numbers, but that doesn't apply if they are complex). When you diagonalize or take the Jordan canonical form of a matrix, it matters how many times each eigenvalue appears, but putting the eigenvalues in a different order results in the same matrix, up to similarity.







                                              share|cite|improve this answer












                                              share|cite|improve this answer



                                              share|cite|improve this answer










                                              answered Mar 25 at 15:50









                                              AcccumulationAcccumulation

                                              7,2052619




                                              7,2052619



























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