Venn Diagrams for Multisets

We introduce Venn diagrams for multisets and show how they simplify the analysis of multisets. Venn diagrams are very useful in proofs involving multisets and multiset orders, especially considering the complications introduced by the multiplicity of elements in multisets. We compare the Venn diagrams for multisets with the corresponding ones for sets. Thus, we present two types of Venn diagrams for multisets, a simple one that looks like a diagram for sets, but with areas that are not necessarily disjoint, and a complex one (compared to sets), but with certain delimited disjoint areas. We determine the number of non-composite areas (disjoint or not) in a Venn diagram for multisets, 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 multisets can also be used for databases, knowledge representation systems, in artificial intelligence, Semantic Web.