Venn Diagrams for Multisets

We introduce Venn diagrams for multisets and showhow they simplify the analysis of multisets. Venn diagrams arevery useful in proofs involving multisets and multiset orders, especially considering the complications introduced by the multiplicity of elements in multisets. We compare the Venn diagramsfor multisets with the corresponding ones for sets. Thus, wepresent two types of Venn diagrams for multisets, a simple onethat looks like a diagram for sets, but with areas that are notnecessarily disjoint, and a complex one (compared to sets), butwith certain delimited disjoint areas. We determine the numberof non-composite areas (disjoint or not) in a Venn diagram formultisets, for which we give two sequences of integers. We compare several properties of Venn diagrams for sets and multisets, like symmetry and Hamiltonicity. Venn diagrams for multisetscan also be used for databases, knowledge representation systems, in artificial intelligence, Semantic Web.