Research thesis
Positive Semantics of Projections in VennEuler Diagrams
Table of contents:
Abstract
Venn diagrams and Euler circles have long
been used to express relationship among sets using visual metaphors such
as disjointness and containment of topological contours. Although the notation
is effective in delivering a clear visual modeling of set theoretical relationship,
it does not scale well. The topology of Venn diagrams of four contours
or more is so cluttered that it becomes impractical to use it for visualization
of 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 projection 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. Another approach, calling in literature a negative
semantics, assigns to a projection the semantics based on the contours
interacting with the projection as well on the contours disjoint to the
projection.
Positive semantics for a projection is produced by a novel Gaussianlike
elimination process for solving set equations.
In dealing with multiple projections of the same base set, we introduce
yet another extension to VennEuler 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 VennEuler diagrams.
Seminar lecture slides

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Final thesis paper
This work was presented at the International Conference on the Theory and
Application of Diagrams (DAIGRAMS
2000) in September 2000. The final paper, submitted to the Senate of
the Technion, is availabled
All VennEuler diagrams for the thesis were prepared using visual editor,
named CDEditor,
developped as a part of Yan
Sorkin thesis.