AVenn diagram is a widely useddiagram style that shows the logical relation betweensets, popularized byJohn Venn (1834–1923) in the 1880s. The diagrams are used to teach elementaryset theory, and to illustrate simple set relationships inprobability,logic,statistics,linguistics andcomputer 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 byChristian Weise in 1712 (Nucleus Logicoe Wiesianoe) andLeonhard Euler in 1768 (Letters to a German Princess). The idea was popularised by Venn inSymbolic Logic, Chapter V "Diagrammatic Representation", published in 1881.
A Venn diagram, also called aset diagram orlogic diagram, showsall possible logical relations between a finite collection of different sets. These diagrams depictelements as points in the plane, and sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve labelledS represent elements of the setS, while points outside the boundary represent elements not in the setS. This lends itself to intuitive visualizations; for example, the set of all elements that are members of both setsS andT, denotedS ∩ T and read "the intersection ofS andT", is represented visually by the area of overlap of the regionsS andT.[1]
In Venn diagrams, the curves are overlapped in every possible way, showing all possible relations between the sets. They are thus a special case ofEuler diagrams, 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 anarea-proportional (orscaled)Venn diagram.
Sets of creatures with two legs, and creatures that fly
This example involves two sets of creatures, represented as overlapping circles: one circle that represents all types of creatures that have two legs, and another representing creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that have two legsand can fly—for example, parrots—are then in both sets, so they correspond to points in the region where the two circles overlap. This overlapping region would only contain those elements (in this example, creatures) that are members of both the set of two-legged creatures and set of flying creatures.
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 theirunion, denoted byA ∪ B, 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 theintersection of A and B, denoted byA ∩ B.
Venn diagrams were introduced in 1880 byJohn Venn in a paper entitled "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings"[2] in thePhilosophical Magazine and Journal of Science,[3] about the different ways to representpropositions by diagrams.[4][5][6] The use of these types of diagrams informal logic, according toFrank Ruskey and Mark Weston, predates Venn but are "rightly associated" with him as he "comprehensively surveyed and formalized their usage, and was the first to generalize them".[7]
Diagrams of overlapping circles representing unions and intersections were introduced by Catalan philosopherRamon Llull (c. 1232–1315/1316) in the 13th century, who used them to illustrate combinations of basic principles.[8]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 describingChristian Weise's contributions to logic.[9][8]Euler diagrams, which are similar to Venn diagrams but don't necessarily contain all possible unions and intersections, were first made prominent by mathematicianLeonhard Euler in the 18th century.[note 1][10][11]
Venn did not use the term "Venn diagram" and referred to the concept as "Eulerian Circles".[6] 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 toBoolean logic.[12] 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".[4][5]
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 thesyllogism: 'AllA is someB. NoB is anyC. Hence, noA is anyC.'[12]
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 bookSymbolic Logic (4th edition published in 1896). The term "Venn diagram" was later used byClarence Irving Lewis in 1918, in his bookA Survey of Symbolic Logic.[7][13]
In the 20th century, Venn diagrams were further developed.David Wilson Henderson showed, in 1963, that the existence of ann-Venn diagram withn-foldrotational symmetry implied thatn was aprime number.[14] He also showed that such symmetric Venn diagrams exist whenn is five or seven. In 2002, Peter Hamburger found symmetric Venn diagrams forn = 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 ifn is a prime number.[15]
Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory, as part of thenew math movement in the 1960s. Since then, they have also been adopted in the curriculum of other fields such as reading.[16]
A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis,[13] the "principle of these diagrams is that classes [orsets] 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 initially leaves room for any possible relation of the classes, and the actual or given relation, can then be specified by indicating that some particular region is null or is not-null".[13]: 157
Venn diagrams normally comprise overlappingcircles. 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 allwooden objects, while the other circle may represent the set of all tables. The overlapping region, orintersection, 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 areschematic diagrams generally not drawn to scale.
Venn diagrams are similar to Euler diagrams. However, a Venn diagram forn component sets must contain all 2n hypothetically possible zones, that correspond to some combination of inclusion or exclusion in each of the component sets.[19] 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 representsdairy products and anothercheeses, the Venn diagram contains a zone for cheeses that are not dairy products. Assuming that in the contextcheese 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.[20]
The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets:
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 asimplex and can be visually represented. The 16 intersections correspond to the vertices of atesseract (or the cells of a16-cell, respectively).
For higher numbers of sets, some loss of symmetry in the diagrams is unavoidable. Venn was keen to find "symmetrical figures ... elegant in themselves,"[10] that represented higher numbers of sets, and he devised anelegant four-set diagram usingellipses (see below). He also gave a construction for Venn diagrams forany number of sets, where each successive curve that delimits a set interleaves with previous curves, starting with the three-circle diagram.
Venn's construction for four sets (useGray code to compute, the digit 1 means in the set, and the digit 0 means not in the set)
Venn's construction for five sets
Venn's construction for six sets
Venn's four-set diagram using ellipses
Non-example: ThisEuler diagram isnot a Venn diagram for four sets as it has only 14 regions as opposed to 24 = 16 regions (including the white region); 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-foldrotationally symmetrical arrangement devised byBranko Grünbaum. Labels have been simplified for greater readability; for example,A denotesA ∩Bc ∩Cc ∩Dc ∩Ec, whileBCE denotesAc ∩B ∩C ∩Dc ∩E.
Anthony William Fairbank 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.[21] For example, three sets can be easily represented by taking three hemispheres of the sphere at right angles (x = 0,y = 0 andz = 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. The resulting sets can then be projected back to a plane, to givecogwheel diagrams with increasing numbers of teeth—as shown here. These diagrams were devised while designing astained-glass window in memory of Venn.[21]
Charles Lutwidge Dodgson (also known as Lewis Carroll) devised a five-set diagram known asCarroll'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 usefirst-order logic and set theory to treat categorical statements as statements about sets. Additionally, they propose to treat singular statements as statements aboutset 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.[22]
Venn diagrams correspond totruth tables for the propositions,, etc., in the sense that each region of Venn diagram corresponds to one row of the truth table.[23][24] This type is also known as Johnston diagram. Another way of representing sets is with John F. Randolph'sR-diagrams.
Spherical octahedron – A stereographic projection of a regular octahedron makes a three-set Venn diagram, as three orthogonal great circles, each dividing space into two halves.
^In Euler'sLettres à une princesse d'Allemagne sur divers sujets de physique et de philosophie [Letters to a German Princess on various physical and philosophical subjects] (Saint Petersburg, Russia: l'Academie Impériale des Sciences, 1768), volume 2,pages 95-126. In Venn's article, however, he suggests that the diagrammatic idea predates Euler, and is attributable toChristian Weise or Johann Christian Lange (in Lange's bookNucleus Logicae Weisianae (1712)).
^Leibniz, Gottfried Wilhelm (1903) [ca. 1690]. "De Formae Logicae per linearum ductus". InCouturat, Louis (ed.).Opuscules et fragmentes inedits de Leibniz (in Latin). pp. 292–321.
^Mac Queen, Gailand (October 1967).The Logic Diagram(PDF) (Thesis).McMaster University. Archived fromthe original(PDF) on 2017-04-14. Retrieved2017-04-14. (NB. Has a detailed history of the evolution of logic diagrams including but not limited to the Venn diagram.)
Watkinson, John (1990). "4.10. Hamming distance".Coding for Digital Recording. Stoneham, MA, USA:Focal Press. pp. 94–99, foldout in backsleeve.ISBN978-0-240-51293-8. (NB. The book comes with a 3-page foldout of a seven-bit cylindrical Venn diagram.)
