Defining Euler diagrams: simple or what?

Abstract

Euler diagrams are emerging as a powerful tool in a variety of application areas, such as logical reasoning and for representing statistical data. In order to be a reliable means of visualizing information, Euler diagrams must have a firm theoretical underpinning and various formalizations have been given. We illustrate, via pertinent examples, misconceptions and pitfalls that occur as a result of imposing certain well-formedness conditions. In particular, we consider the effect on the interpretation of diagrams, on giving precise definitions and on reasoning systems. First, we highlight some consequences of stipulating that the closed curves must be simple. Secondly, we demonstrate some of the difficulties associated with choosing not to enforce simplicity. We also consider the consequences of defining Euler diagrams inductively and enforcing the connectedness of minimal regions. Choices made when formalizing Euler diagrams can have a profound effect on reasoning systems due to the potential inability to draw a diagram which represents a given collection of set intersections. The issues raised are likely to be interesting to any researcher who is defining diagrams based on closed curves.