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Survey
of Venn diagrams
Spider diagrams combine and extend Venn diagrams and Euler circles to
express constraints on sets and their relationships with other sets. They
arose from the work on constraint diagrams which are diagrams that can
be used in conjunction with the Unified Modelling Language (UML) and the
Object Constraint Language (OCL) which are the Object Management Group’s
(OMG) standard for modeling in object-oriented software development.
An example of a spider diagram is:
A spider is a tree with nodes (called feet) placed in different basic
regions. A spiders denotes the existence of an element belonging to the
set corresponding to the region ‘inhabited’ by the spider. A shaded region
(here shown in blue) indicates that there are no elements belonging to
the corresponding set other than those denoted by spiders. Thus the spider
diagram above represents a number of facts about the sets A, B
and C:
See also Stuart
Kent's "Reasoning with diagrams" page.
Constraint diagrams show relations between sets and their elements.
They use topological properties to express containment, intersection and
disjointedness; they can express relations between sets using the arrow
notation; and universal and existential quantification. The constraint
diagram shown below expresses (among other constraints) an invariant on
a model of a car hire business.
The specification of the car assigned to a reservation must be the
same or better than the specification reserved.
All the contours, whether rectangular or elliptical, represent sets.
The * in the Reservations set represents a wildcard, a universally
quantified element over the region(s) which contain it, in this case the
set Reservations. * is also known as a universal spider.
Other kinds of spider are derived, given and existential.
Spiders denote elements. In general a spider is a set of "blobs" connected
by single lines, where the blobs must all be the same symbol. The symbol
used determines the nature of the spider, so a set of * connected by single
lines represents a universal spider. The union of regions touched by the
feet of a spider is the habitat of that spider. In any semantic interpretation,
a spider must denote an element drawn from its habitat. An arrow shows
the range of a relation when its domain is restricted to the set or element
at its source. The relation is identified by the label on the arrow. Venn/Euler
diagrams then show relationships between all the sets and elements involved
(there are examples of containment and disjointedness in the given example).
Thus to read this diagram, one begins with the spider r, which is
universally quantified over Reservations, then traces through the relations
assigned and spec to identify an element, x, from Car Specifications.
One can also trace from r via reserved to identify another element,
y,
from Car Specifications. The relationship between x and y
is then that x is either contained in the set of Car Specifications
that are better than y, or x is equal to y, which
is indicated by the tie (like a large equals sign) between
x and
y.
Visual
Language Research Bibliography
Reasoning with Diagrams Project, University of Kent site
Spider
diagrams
Constraint diagrams
