Editing Multiset (section)

==Applications==
Multisets have various applications.<ref name="Singh2007" /> They are becoming fundamental in [[combinatorics]].<ref name="Aigner1979">{{cite book|last=Aigner|first=M.|year=1979|title=Combinatorial Theory|publisher=Springer Verlag | location=New York/Berlin}}</ref><ref name="Anderson1987">{{cite book|last=Anderson|first=I.|year=1987|title=Combinatorics of Finite Sets|url=https://archive.org/details/combinatoricsoff0000ande|url-access=registration|publisher=Clarendon Press | location=Oxford|isbn=978-0-19-853367-2 }}</ref><ref name="Stanley1997">{{cite book|last=Stanley|first=Richard P.|author-link=Richard P. Stanley|year=1997|title=Enumerative Combinatorics|volume=1|publisher=Cambridge University Press | url=http://www-math.mit.edu/~rstan/ec/|isbn=0-521-55309-1}}</ref><ref name="Stanley1999">{{cite book | last=Stanley|first=Richard P. |year=1999|title=Enumerative Combinatorics|volume=2|publisher=Cambridge University Press|isbn=0-521-56069-1}}</ref> Multisets have become an important tool in the theory of [[relational database]]s, which often uses the synonym ''bag''.<ref name="GrumbachMilo1996">{{cite journal| last1=Grumbach|first1=S.| last2=Milo|first2=T| year=1996| title=Towards tractable algebras for bags|journal=Journal of Computer and System Sciences| volume=52| issue=3| pages=570–588| doi=10.1006/jcss.1996.0042|doi-access=free}}</ref><ref name="LibkinWong1994">{{cite book| last1=Libkin|first1=L.|author1link = Leonid Libkin| last2=Wong|first2=L.|year=1994|chapter=Some properties of query languages for bags| title=Proceedings of the Workshop on Database Programming Languages|publisher=Springer Verlag|pages=97–114}}</ref><ref name="LibkingWong1995">{{cite journal|last1=Libkin|first1=L.|last2=Wong|first2=L.|year=1995|title=On representation and querying incomplete information in databases with bags|journal=[[Information Processing Letters]]| volume=56| issue=4| pages=209–214|doi=10.1016/0020-0190(95)00154-5}}</ref> For instance, multisets are often used to implement relations in database systems. In particular, a table (without a primary key) works as a multiset, because it can have multiple identical records. Similarly, [[SQL]] operates on multisets and returns identical records. For instance, consider "SELECT name from Student". In the case that there are multiple records with name "Sara" in the student table, all of them are shown. That means the result of an SQL query is a multiset; if the result were instead a set, the repetitive records in the result set would have been eliminated. Another application of multisets is in modeling [[multigraph]]s. In multigraphs there can be multiple edges between any two given [[vertex (graph theory)|vertices]]. As such, the entity that specifies the edges is a multiset, and not a set.

There are also other applications. For instance, [[Richard Rado]] used multisets as a device to investigate the properties of families of sets. He wrote, "The notion of a set takes no account of multiple occurrence of any one of its members, and yet it is just this kind of information that is frequently of importance. We need only think of the set of roots of a polynomial ''f''{{hairsp}}(''x'') or the [[spectrum of an operator|spectrum]] of a [[linear operator]]."<ref name="Blizard1991" />{{rp|pp=328–329}}