Venn diagrams

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 drawn on a plane to represent sets. Very often, these curves are circles or ellipses.

Similar ideas had been proposed before Venn. Christian Weise in 1712 (''Nucleus Logicoe Wiesianoe'') and Leonhard Euler (''Letters to a German Princess'') in 1768, for instance, came up with similar ideas. The idea was popularised by Venn in ''Symbolic Logic'', Chapter V "Diagrammatic Representation", 1881.

Details

A Venn diagram may also be called a ''set diagram'' or ''logic diagram''. It 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, 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 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''.

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 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.

Example

This example involves two sets, A and B, represented here as colored circles. 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. Living creatures that can fly ''and'' have two legs—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. This overlapping region would only contain those elements (in this example, creatures) that are members of both set A (two-legged creatures) and set B (flying creatures).

Humans and penguins are bipedal, and 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 can fly, but have six, not two, legs, 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 (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 . 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 . 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.

History

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, 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 " 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, 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. Of these schemes one only, viz. that commonly called 'Eulerian 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''.

Venn diagrams are very similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. Margaret Baron has noted that Leibniz (1646–1716) produced similar diagrams before Euler in the 17th century, but much of it was unpublished. She also observes even earlier Euler-like diagrams by Ramon Llull in the 13th Century.

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.

Overview

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 r ''sets''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".

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. 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:
* $A = \$
* $B = \$
* $C = \$

The Euler and the Venn diagram of those sets are:

File:3-set Euler diagram.svg, Euler diagram
File:3-set Venn diagram.svg, Venn diagram

Extensions to higher numbers of 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 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).

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.

Image:Venn4.svg, Venn's construction for four sets (use Gray code to compute, the digit 1 means in the set, and the digit 0 means not in the set)
Image:Venn5.svg, Venn's construction for five sets
Image:Venn6.svg, Venn's construction for six sets
Image:Venn's four ellipse construction.svg, Venn's four-set diagram using ellipses
Image:CirclesN4xb.svg, Non-example: This Euler diagram is a Venn diagram for four sets as it has only 14 regions (and not 2⁴ = 16 regions); there is no region where only the yellow and blue, or only the red and green circles meet.
File:Symmetrical 5-set Venn diagram.svg, Five-set Venn diagram using congruent ellipses in a five-fold rotationally symmetrical arrangement devised by Branko Grünbaum

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