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A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram uses simple closed curves on a plane to represent sets. The curves are often circles or ellipses.
Similar ideas had been proposed before Venn such as by Christian Weise in 1712 ( Nucleus Logicoe Wiesianoe) and Leonhard Euler in 1768 ( Letters to a German Princess). The idea was popularised by Venn in Symbolic Logic, Chapter V "Diagrammatic Representation", published in 1881.
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 , which do not necessarily 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, logic, statistics, linguistics, 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.
Humans and penguins are bipedal, and so are in the "has two legs" circle, but since they cannot fly, they appear in the part of the that circle that does not overlap with the "can fly" circle. Mosquitoes can fly, but have six, not two, legs, so the point for mosquitoes is in the part of the "can fly" circle that does not overlap with the "has two legs" circle. Creatures that are neither two-legged nor able to fly (for example, whales and spiders) would all be represented by points outside both circles.
The combined region of the two sets is called their union, denoted by , where A is the "has two legs" circle and B the "can fly" circle. The union in this case contains all living creatures that either are two-legged or can fly (or both). The region included in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by .
Diagrams of overlapping circles representing unions and intersections were introduced by Catalan philosopher Ramon Llull (c. 1232–1315/1316) in the 13th century, who used them to illustrate combinations of basic principles. Gottfried Wilhelm Leibniz (1646–1716) produced similar diagrams in the 17th century (though much of this work was unpublished), as did Johann Christian Lange in a work from 1712 describing Christian Weise's contributions to logic. , which are similar to Venn diagrams but don't necessarily contain all possible unions and intersections, were first made prominent by mathematician Leonhard Euler in the 18th century.
Venn did not use the term "Venn diagram" and referred to the concept as "Eulerian Circles". He became acquainted with Euler diagrams in 1862 and wrote that Venn diagrams did not occur to him "till much later", while attempting to adapt Euler diagrams to Boolean logic. In the opening sentence of his 1880 article Venn wrote that Euler diagrams were the only diagrammatic representation of logic to gain "any general acceptance".
Venn viewed his diagrams as a pedagogical tool, analogous to verification of physical concepts through experiment. As an example of their applications, he noted that a three-set diagram could show the syllogism: 'All A is some B. No B is any C. Hence, no A is any C.'
Charles L. Dodgson (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 (4th edition published in 1896). The term "Venn diagram" was later used by Clarence Irving Lewis in 1918, in his book A Survey of Symbolic Logic.
In the 20th century, Venn diagrams were further developed. David Wilson 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 also 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, Killian, and Savage showed that symmetric Venn diagrams exist for all other primes. These combined results show that rotationally symmetric Venn diagrams exist, if and only if n is a prime number.
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 then, they have also been adopted in the curriculum of other fields such as reading.
Venn diagrams normally comprise overlapping . The interior of the circle symbolically represents the elements of the set, while the exterior represents elements that are not members of the set. For instance, in a two-set Venn diagram, one circle may represent the group of all objects, while the other circle may represent the set of all tables. The overlapping region, or intersection, would then represent the set of all wooden tables. Shapes other than circles can be employed as shown below by Venn's own higher set diagrams. Venn diagrams do not generally contain information on the relative or absolute sizes (cardinality) of sets. That is, they are schematic diagrams generally not drawn to scale.
Venn diagrams are similar to Euler diagrams. However, a Venn diagram for n component sets must contain all 2 n hypothetically possible zones, that correspond to some combination of inclusion or exclusion in each of the component sets. Euler diagrams contain only the actually possible zones in a given context. In Venn diagrams, a shaded zone may represent an empty zone, whereas in an Euler diagram, the corresponding zone is missing from the diagram. For example, if one set represents dairy products and another cheeses, the Venn diagram contains a zone for cheeses that are not dairy products. Assuming that in the context cheese means some type of dairy product, the Euler diagram has the cheese zone entirely contained within the dairy-product zone—there is no zone for (non-existent) non-dairy cheese. This means that as the number of contours increases, Euler diagrams are typically less visually complex than the equivalent Venn diagram, particularly if the number of non-empty intersections is small.
The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets:
The Euler and the Venn diagram of those sets are:
For higher numbers of sets, some loss of symmetry in the diagrams is unavoidable. Venn was keen to find "symmetrical figures ... elegant in themselves," that represented higher numbers of sets, and he devised an elegant four-set diagram using (see below). He also gave a construction for Venn diagrams for any number of sets, where each successive curve that delimits a set interleaves with previous curves, starting with the three-circle diagram.
Henry John Stephen Smith devised similar n-set diagrams using sine curves with the series of equations
Charles Lutwidge Dodgson (also known as Lewis Carroll) devised a five-set diagram known as Carroll's square. Joaquin and Boyles, on the other hand, proposed supplemental rules for the standard Venn diagram, in order to account for certain problem cases. For instance, regarding the issue of representing singular statements, they suggest to consider the Venn diagram circle as a representation of a set of things, and use first-order logic and set theory to treat categorical statements as statements about sets. Additionally, they propose to treat singular statements as statements about set membership. So, for example, to represent the statement "a is F" in this retooled Venn diagram, a small letter "a" may be placed inside the circle that represents the set F.
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