Application of venn diagrams1/4/2024 ![]() Further conditions for being well-formed are defined slightly differently by different authors (e.g. ![]() In a well-formed Euler diagram, zones correspond to minimal regions. The reason for distinguishing minimal regions and zones is that zones are the smallest set-theoretically meaningful areas in a diagram whereas minimal regions are the smallest visible areas in a diagram. In other words, E( L) is a subset of the powerset of L that corresponds to the zones of an Euler diagram. For a set L of curve labels, the notation E( L) is used in this paper for a set of zones. Zones are maximal regions that are within a set of curves and outwith the remaining curves. Minimal regions are the smallest areas in a diagram which are surrounded by lines and not divided further. The following terminology applies to Venn and Euler diagrams in this paper: Venn and Euler diagrams consist of closed curves which have labels. A possibly provocative conclusion of this paper is that although many people may find Euler diagrams “intuitive” as a representation of sets, from a structural viewpoint Hasse diagrams are potentially more suitable for visualising set theory than Venn and Euler diagrams. Although most of the individual mathematical aspects presented in this paper are not new, we believe that the compilation and elaboration of details with respect to the examples presented in this paper is new. ![]() Sections 5, 6, 7 discuss different aspects of the relationship between Euler and Hasse diagrams. Section 4 covers Venn diagrams and their (well-known) relationship to Boolean lattices. Sections 2 and 3 of this paper provide introductions to Venn, Euler and Hasse diagrams and FCA. Many questions about the relationship between well-formed Euler diagrams and lattices still remain open. Because each field has a slightly different focus, it is conceivable that a combination might provide further interesting results. We suspect that many researchers from either field are not aware of all of the connections. It discusses the application of some lattice-theoretical properties to Euler diagrams. This paper provides an introduction to both fields and basic translations between Venn/Euler and Hasse diagrams. The intention of this paper is to elaborate the basic connections between both fields. ( 2004) show, however, that novice users can be instructed to use Hasse diagrams fairly effectively.Īs far as we know, the relationship between FCA and Euler diagrams has so far not been investigated in any great depth Footnote 1. If restricted to specific tasks, Eklund et al. Priss ( 2017) discusses misconceptions that students initially have about Hasse diagrams of concept lattices in general. Hasse diagrams may be less intuitive at first sight and require some training. Venn and Euler diagrams are well-established as a visualisation of sets that is used, for example, in schools when students are first introduced to set theory. The version of lattice theory used in this paper is called Formal Concept Analysis (FCA) and has been developed since the 1980s as an applied mathematical theory of knowledge representation (Ganter and Wille 1999). The research about Venn and Euler diagrams provides, for example, applications and algorithms which could be of interest for Hasse diagrams as well. Lattice theory has produced a large body of knowledge which could potentially be beneficial for research about well-formed Euler diagrams. While a translation between lattices and Venn diagrams is straightforward, the connection between well-formed Euler diagrams and lattices is not trivial. ![]() ![]() It is therefore of interest to compare Euler and Hasse diagrams both with respect to what can be observed from the diagrams but also with respect to underlying theoretical constructs. Sets and their intersections can be visualised with Venn and Euler diagrams but also using mathematical lattice theory and a certain type of diagram ( Hasse diagram) that is commonly used with lattices. ![]()
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