Positive semantics of projections in Venn-Euler diagrams
Gil, Joseph (Yossi), HOWSE, JOHN and Tulchinsky, Elena (2002) Positive semantics of projections in Venn-Euler diagrams Journal of Visual Languages and Computing, 13 (2). pp. 197-227. ISSN 1045-926X
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Official URL: http://www.sciencedirect.com/science/article/pii/S...
Venn diagrams and Euler circles have long been used as a means of expressing relationships among sets using visual metaphors such as ‘disjointness’ and ‘containment’ of topological contours. Although the notation is effective in delivering a clear visual representation of set theoretical relationships, it does not scale well. The topology of Venn diagrams of four or more contours is so cluttered that it becomes impractical to use it to visualize set relationships. In this work, we study ‘projection contours’, a new means for presenting sets intersections, which is designed to reduce the clutter in such diagrams. Informally, a projected contour is a contour which describes a set of elements limited to a certain context. The challenge in introducing this notation is in producing precise and consistent semantics for the general case, including a diagram comprising several, possibly interacting projections, which might even be of the same base set. The semantics investigated here assigns a ‘positive’ meaning to a projection, i.e. based on the list of contours with which it interacts, where contours disjoint to it do not change its semantics. This semantics is produced by a novel Gaussian-like elimination process for solving set equations. In dealing with multiple projections of the same base set, we introduce yet another extension to Venn–Euler diagrams in which the same set can be described by multiple contours. This extension is of independent interest as a powerful means for reducing the clutter in Venn–Euler diagrams.
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