Venn Diagrams
A Venn diagram is a schematic representation used in depicting collections of sets and the interrelationships between those sets. Although Venn diagrams are often employed in literary studies they are more common in mathematical studies involving the validity of deduction and proof. Physically, Venn diagrams are collections of intersecting, simple, two-dimensional polygons. The order of a Venn diagram n is often specified and refers to the number of simple closed curves.
A simple Venn diagram is one in which no two curves intersect in more than a finite number of points and no three or more curves intersect at one common point. This says that the curves defining sets meet at points and not in segments of curves. Below is an example of an order-three Venn diagram:
This particular diagram consists of three intersecting circles that are symmetrically placed and comprise a total of eight regions. A, B, and C are regions that consist of members of a set that are not common to any other set. The regions labeled A ( B, A ( C, and B ( C consist of members that are common to two sets but not the third. The region labeled A ( B ( C consists of members that are common to all three sets. In this particular case, since the circles are symmetrically placed such that the center of each circle is located at the intersection of the other two, the region A ( B ( C is a geometric shape known as a Reuleaux triangle. The last region is the region outside of the three sets and is indicative of the empty set, the set that contains no members. So an order-n Venn diagram is a collection of n sets represented by simple closed curves in a plane that partition the plane into 2n connected regions called subsets (including the region outside of the curves). Each of these subsets represents a unique region formed upon the intersection of the interiors of the closed curves.
The introduction of Venn diagrams is attributed to John Venn, an English mathematician, although there is strong proof that they originated earlier. Aristotle, a Greek scientist, used a diagram called the tree of Porphyry to represent logical concepts as early as 350B.C.. Swiss mathematician Leonhard Euler formulated a geometric system that generated class logic solutions in 1761. This system was replaced by Venn's system in 1880 in a paper published in the Philosophical Magazine and Journal of Science. Venn's formulation of this type of system was a natural evolution of Boolean algebra introduced by George Boole in 1847 and 1854. Boolean algebra represents logical expressions in a mathematical form using a standard set of operators. Venn diagrams were initially based on the relationships between overlapping circles or ellipses. In 1881 Allan Marquand introduced the first logical diagrams based on squares or rectangles. Since that time the study of Venn diagrams and their uses has been expanded to literary uses.
There are several different aspects of particular Venn diagrams that have earned specific terminology. On the subject of extending Venn diagrams they can be reducible, irreducible or extendible. A Venn diagram is called reducible if the removal of one of the curves results in a Venn diagram with n-1 curves. If this is not the case then the diagram is called irreducible. Many elliptical Venn diagrams are irreducible and it was unknown for a long time whether a reducible one existed. An article by Hamburger and Pippert appearing in a 1996 "American Scientist" showed that there was such a Venn diagram. If there is a curve such that when added to an order-n Venn diagram C produces an order-(n + 1) Venn diagram then C is called extendible. The new Venn diagram is said to be an extension of C.
There are other interesting aspects of particular Venn diagrams that are associated with their geometry. A Venn diagram is called convex if the interiors of all of its curves are convex. A Venn diagram is called exposed if each of the curves that compose it touches the outer face, the empty set, at some point of non-intersection. Every convex Venn diagram is also exposed. A Venn diagram is said to have a hidden curve if it has a curve that does not touch the outer empty region. Every simple Venn diagram with five or less curves are exposed.
Three other interesting characteristics of particular Venn diagrams are congruence, symmetry, and monotonicity. A congruent Venn diagram is one that is composed of congruent curves. The first congruent Venn diagrams were constructed by Grünbaum in 1975. Symmetric Venn diagrams are ones that are first congruent and that also have an n-fold rotational symmetry. That is that there is a point about which the diagrams may be rotated by 2i/n, where i = 0, 1,...n - 1, and remain invariant. All symmetric Venn diagrams are exposed. A Venn diagram can be described as monotone if every k-region, for 0 < k < n, is adjacent to both a (k - 1)-region and a (k + 1)-region. All Venn diagrams constructed according to Venn original paper describing such diagrams are monotone.
The last interesting feature of the Venn diagram that will be discussed here is another graph that is associated with the Venn diagram. This graph is called the Venn graph and is another two-dimensional graph that is the planar dual of the Venn diagram. The vertices of the Venn graph are the connected open regions of the Venn diagram. A line connects two vertices if they share a common boundary. Below is shown the Venn graph for another three-order Venn diagram.
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