Corresponding regions in Euler diagrams

Abstract

Euler diagrams use topological properties to represent set-theoretical concepts and thus are `intuitive' to some people. When reasoning with Euler diagrams, it is essential to have a notion of correspondence among the regions in different diagrams. At the semantic level, two regions correspond when they represent the same set. However, we wish to construct a purely syntactic definition of corresponding regions, so that reasoning can take place entirely at the diagrammatic level. This task is interesting in Euler diagrams because some regions of one diagram may be missing from another. We construct the correspondence relation from `zones' or minimal regions, introducing the concept of `zonal regions' for the case in which labels may differ between diagrams. We show that the relation is an equivalence relation and that it is a generalization of the counterpart relations introduced by Shin and Hammer.