Venn Diagram

Elaine Sutton
• Monday, 26 July, 2021
• 8 min read

Diagram that shows all possible logical relations between a collection of sets These diagrams depict elements as points in the plane, and sets as regions inside closed curves.

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A Venn diagram consists of multiple overlapping closed curves, usually 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 itself to intuitive visualizations; for example, the set of all elements that are members of both sets S and T, denoted S T and read “the intersection of S and T “, is represented visually by the area of overlap of the regions S and T. The orange circle, set A, represents all types of living creatures that are two-legged.

The blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram.

Creatures that are not two-legged and cannot fly (for example, whales and spiders) 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 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 A B.

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In this 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's diagrams were introduced in 1880 by John Venn in a paper entitled “On the Diagrammatic and Mechanical Representation of Propositions and Reasoning” 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 Runway 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, however, because he comprehensively surveyed and formalized their usage, and was the first to generalize them”.

Venn himself did not use the term Venn diagram and referred to his invention as Sumerian 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, even those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices.

That commonly called 'Sumerian circles,' has met with any general acceptance...” Lewis Carroll (Charles L. Dodgson) 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.

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.

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These combined results show that rotationally symmetric Venn diagrams exist, if and only if n is a prime number. Venn's diagrams and Euler diagrams were incorporated as part of instruction in set theory, as part of the new math movement in the 1960s.

In 2020, sexologist Dr. Lindsey Doe began a trend of using the word cunt to refer to the intersection of A and B in a Venn diagram. This was an attempt to add this usage of the word to a dictionary by 2023.

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.

Venn diagrams normally comprise overlapping circles. 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 wooden 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.

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Shapes other than circles can be employed as shown below by Venn's own higher set diagrams. Venn's 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. 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. For example, if one set represents dairy products and another cheeses, the Venn diagram contains a zone for cheeses that are not dairy products.

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. Venn diagrams typically represent two or three sets, but there are forms that allow for higher numbers.

Shown below, four intersecting spheres form the highest order Venn diagram that has the symmetry of a simplex and can be visually represented. The 16 intersections correspond to the vertices of a tesseract (or the cells of a 16-cell, respectively).

compounds elements mixtures
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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 ellipses (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. Non-example: This Euler diagram is not a Venn diagram for four sets as it has only 13 regions (excluding the outside); there is no region where only the yellow and blue, or only the red and green circles meet.

Five-set Venn diagram using congruent ellipses in a five-fold rotationally symmetrical arrangement devised by Brando Grandam. Anthony William Fairbanks Edwards constructed a series of Venn diagrams for higher numbers of sets by segmenting the surface of a sphere, which became known as Edwards– Venn diagrams.

For example, three sets can be easily represented by taking three hemispheres of the sphere at right angles (x = 0, y = 0 and z = 0). A fourth set can be added to the representation, by taking a curve similar to the seam on a tennis ball, which winds up and down around the equator, and so on.

Joaquin and Boyle's, 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 considering 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.

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Venn diagram as a truth table Venn diagrams correspond to truth tables for the propositions XNA{\display style x\in A}, job{\display style x\in B}, etc., in the sense that each region of Venn diagram corresponds to one row of the truth table. Another way of representing sets is with John F. Randolph's R-diagrams.

^ In Euler's Letters à one princess d'Charlemagne SUR divers sets DE physique ET DE philosophies (Saint Petersburg, Russia: l'Academic Imperial DES Sciences, 1768), volume 2, pages 95-126. In Venn's article, however, he suggests that the diagrammatic idea predates Euler, and is attributable to Christian Was or Johann Christian Large (in Large's book Nucleus Logical Variance (1712)).

^ a b c “Comprehensive List of Set Theory Symbols”. On the Diagrammatic and Mechanical Representation of Propositions and Reasoning” (PDF).

The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. “On the employment of geometrical diagrams for the sensible representations of logical propositions”.

“A Note on The Historical Development of Logic Diagrams”. ^ Runway, Frank ; Savage, Carla D. ; Wagon, Stan (December 2006).

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Joaquin, Jeremiah Oven; Boyle's, Robert James M. (June 2017). “Teaching Syllogistic Logic via a Retooled Venn Diagrammatic Technique”.

Mineral, New York, USA: Dover Publications, Inc. (W. H. Freeman). “A New Rose: The First Simple Symmetric 11- Venn Diagram ".

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