SUMMARY / RELATED TOPICS

A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves circles, each representing a set; the points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends to read visualizations. In Venn diagrams the curves are overlapped in every possible way, showing all possible relations between the sets, they are thus a special case of Euler diagrams, which do not show all relations. Venn diagrams were conceived around 1880 by John Venn, they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, statistics and computer science. A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram.

This example involves A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures; the blue circle, set B, represents the living creatures. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that both can fly and have two legs—for example, parrots—are in both sets, so they correspond to points in the region where the blue and orange circles overlap, it is important to note that this overlapping region would only contain those elements that are members of both set A and are members of set B Humans and penguins are bipedal, so are in the orange circle, but since they cannot fly they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes have six legs, fly, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly would all be represented by points outside both circles.

The combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that can fly; the region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles. Venn diagrams were introduced in 1880 by John Venn in a paper entitled "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings" in the Philosophical Magazine and Journal of Science, about the different ways to represent propositions by diagrams; the use of these types of diagrams in formal logic, according to Frank Ruskey and Mark Weston, is "not an easy history to trace, but it is certain that the diagrams that are popularly associated with Venn, in fact, originated much earlier. They are rightly associated with Venn, because he comprehensively surveyed and formalized their usage, was the first to generalize them".

Venn himself did not use the term "Venn diagram" and referred to his invention as "Eulerian Circles". For example, in the opening sentence of his 1880 article Venn writes, "Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that called'Eulerian circles,' has met with any general acceptance..." Lewis Carroll includes "Venn's Method of Diagrams" as well as "Euler's Method of Diagrams" in an "Appendix, Addressed to Teachers" of his book Symbolic Logic. The term "Venn diagram" was used by Clarence Irving Lewis in 1918, in his book A Survey of Symbolic Logic. Venn diagrams are similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. M. E. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler, but much of it was unpublished.

She observes earlier Euler-like diagrams by Ramon Llull in the 13th Century. In the 20th century, Venn diagrams were further developed. D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number, he showed that such symmetric Venn diagrams exist when n is five or seven. In 2002 Peter Hamburger found symmetric Venn diagrams for n = 11 and in 2003, Griggs and Savage showed that symmetric Venn diagrams exist for all other primes, thus rotationally symmetric Venn diagrams exist. Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since they have been adopted in the curriculum of other fields such as reading. A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis, the "principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram.

That is, the diagram leaves room for any possible relation of