View source for Venn diagram

{{Technical|introduction|date=September 2019}}
[[File:Venn diagram gr la ru.svg|thumb|'''Venn diagram''' showing the uppercase [[glyph]]s shared by the [[Greek alphabet|Greek]], [[Latin alphabets|Latin]], and [[Russian Cyrillic]] alphabets]]
{{Probability fundamentals}}
{{anchor|Primary|Simple|Cylindrical|Metric|2|3}}A '''Venn diagram''' (also called '''primary diagram''', '''set diagram''' or '''logic diagram''') is a [[diagram]] that shows ''all'' possible [[logic]]al relations between a finite collection of different [[set (mathematics)|sets]].  These diagrams depict [[element (mathematics)|element]]s as points in the plane, and [[set (mathematics)|set]]s 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 to easily read visualizations; for example, the set of all elements that are members of both sets ''S'' and ''T'', ''S''&nbsp;∩&nbsp;''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 diagram|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==
[[File:Венов дијаграм.svg|thumb|left|Sets A (creatures with two legs) and B (creatures that can fly)]]
This example involves two [[Set (mathematics)|sets]], A and B, represented here as coloured circles. The orange circle, set A, represents all 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 both 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. It is important to note that this overlapping region would only contain those elements (in this example creatures) that are members of both set A (two-legged creatures) and are also members of set B (flying creatures.)

Humans and penguins are bipedal, and so are then 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, and 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 (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 (set theory)|union]]'' of A and B, denoted by {{nowrap|A ∪ B}}. The union in this case contains all living creatures that are either two-legged or that can fly (or both).

The region in both A and B, where the two sets overlap, is called the ''[[intersection (set theory)|intersection]]'' of A and B, denoted by {{nowrap|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.

{{anchor|Symmetric|n}}

== History ==
[[File:Venn-stainedglass-gonville-caius.jpg|thumb|right|150px|[[Stained-glass]] window with Venn diagram in [[Gonville and Caius College, Cambridge]]]]

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 [[proposition]]s by diagrams.<ref name="Venn1880_1"/><ref name="Venn1880_2"/><ref name="Sandifer2003"/> The use of these types of [[diagram]]s 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".<ref name="Ruskey2005"/>

Venn himself did not use the term "Venn diagram" and referred to his invention as "[[Eulerian Circle]]s".<ref name="Sandifer2003"/> 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..."<ref name="Venn1880_1"/><ref name="Venn1880_2"/> Lewis Carroll (Charles 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''.<ref name="Ruskey2005"/><ref name="Lewis1918"/>

Venn diagrams are very similar to [[Euler diagram]]s, which were invented by [[Leonhard Euler]] in the 18th century.<ref group="note" name="NB_1"/><ref name="Venn1881"/><ref name="Gailand_1967"/> M.&nbsp;E.&nbsp;Baron has noted that [[Gottfried Wilhelm Leibniz|Leibniz]] (1646–1716) in the 17th century produced similar diagrams before Euler, but much of it was unpublished.<ref name="Leibniz_1690"/> She also observes even earlier Euler-like diagrams by [[Ramon Llull]] in the 13th Century.<ref name="Baron_1969"/>

In the 20th century, Venn diagrams were further developed. D.&nbsp;W.&nbsp;Henderson showed in 1963 that the existence of an ''n''-Venn diagram with ''n''-fold [[rotational symmetry]] implied that ''n'' was a [[prime number]].<ref name="Henderson_1963"/> 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. Thus rotationally symmetric Venn diagrams exist if and only if ''n'' is a prime number.<ref name="Ruskey_2006"/>

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.<ref name="Stategies"/>

==Overview==
{{See also|Set (mathematics)#Basic operations}}
{{Gallery|lines=3
|File:Venn0001.svg|[[Intersection (set theory)|Intersection]] of two sets <math>~A \cap B</math>
|File:Venn0111.svg|[[Union (set theory)|Union]] of two sets <math>~A \cup B</math>
|File:Venn0110.svg|[[Symmetric difference]] of two sets <math>A~\triangle~B</math>
|File:Venn0010.svg|[[Complement (set theory)#Relative complement|Relative complement]] of ''A'' (left) in ''B'' (right) <math>A^c \cap B~=~B \setminus A</math>
|File:Venn1010.svg|[[Complement (set theory)#Absolute complement|Absolute complement]] of A in U  <math>A^c~=~U \setminus A</math>
}}
A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis,<ref name="Lewis1918"/> the "principle of these diagrams is that classes [or ''[[set (mathematics)|set]]s''] 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".<ref name="Lewis1918"/>{{rp|157}}

Venn diagrams normally comprise overlapping [[circle]]s. The interior of the circle symbolically represents the [[element (mathematics)|element]]s 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 [[wood]]en objects, while another circle may represent the set of all tables. The overlapping region or ''[[intersection (set theory)|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; i.e. they are [[schematic]] diagrams.

Venn diagrams are similar to [[Euler diagram]]s. However, a Venn diagram for ''n'' component sets must contain all 2<sup>''n''</sup> 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.<ref name="Kent_2004"/>

The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets:

* <math>A = \{1,\, 2,\, 5\}</math>
* <math>B = \{1,\, 6\}</math>
* <math>C = \{4,\, 7\}</math>

The Venn and the Euler diagram of those sets are:

<gallery widths="300">
File:3-set Euler diagram.svg|Euler diagram
File:3-set Venn diagram.svg|Venn diagram
</gallery>

=={{anchor|Elegant|4|5|6|7|8|9|10|11}}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).

{|class="wikitable" style="text-align:center; width: 100%;"
| style="vertical-align:top;"|[[File:Venn 1000 0000 0000 0000.png|150px]]
| style="vertical-align:top;"|[[File:Venn 0110 1000 1000 0000.png|150px]]<br />
[[File:Venn 0100 0000 0000 0000.png|37px]][[File:Venn 0010 0000 0000 0000.png|37px]][[File:Venn 0000 1000 0000 0000.png|37px]][[File:Venn 0000 0000 1000 0000.png|37px]]
| style="vertical-align:top;"|[[File:Venn 0001 0110 0110 1000.png|150px]]<br />
[[File:Venn 0001 0000 0000 0000.png|24px]][[File:Venn 0000 0100 0000 0000.png|24px]][[File:Venn 0000 0010 0000 0000.png|24px]][[File:Venn 0000 0000 0100 0000.png|24px]][[File:Venn 0000 0000 0010 0000.png|24px]][[File:Venn 0000 0000 0000 1000.png|24px]]
| style="vertical-align:top;"|[[File:Venn 0000 0001 0001 0110.png|150px]]<br />
[[File:Venn 0000 0001 0000 0000.png|37px]][[File:Venn 0000 0000 0001 0000.png|37px]][[File:Venn 0000 0000 0000 0100.png|37px]][[File:Venn 0000 0000 0000 0010.png|37px]]
| style="vertical-align:top;"|[[File:Venn 0000 0000 0000 0001.png|150px]]
|}

For higher numbers of sets, some loss of symmetry in the diagrams is unavoidable. Venn was keen to find "symmetrical figures...elegant in themselves,"<ref name="Venn1881"/> that represented higher numbers of sets, and he devised an ''elegant'' four-set diagram using [[ellipse]]s (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.

<gallery widths="200px"><!---perrow=3-->
Image:Venn4.svg|Venn's construction for four sets
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 '''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.
File:Symmetrical 5-set Venn diagram.svg|Five-set Venn diagram using congruent ellipses in a five-fold [[rotational symmetry|rotationally symmetrical]] arrangement devised by [[Branko Grünbaum]]. Labels have been simplified for greater readability; for example, '''A''' denotes {{nowrap|'''A''' ∩ '''B'''<sup>c</sup> ∩ '''C'''<sup>c</sup> ∩ '''D'''<sup>c</sup> ∩ '''E'''<sup>c</sup>}}, while '''BCE''' denotes {{nowrap|'''A'''<sup>c</sup> ∩ '''B''' ∩ '''C''' ∩ '''D'''<sup>c</sup> ∩ '''E'''}}.
File:6-set_Venn_diagram.svg|Six-set Venn diagram made of only triangles [http://upload.wikimedia.org/wikipedia/commons/5/56/6-set_Venn_diagram_SMIL.svg (interactive&nbsp;version)]
</gallery>

==={{anchor|Edwards-Venn|Adelaide|Hamilton|Massey|Victoria|Palmerston North|Manawatu}}Edwards–Venn diagrams===
<gallery widths="150px">
Image:Venn-three.svg| Three sets
Image:Edwards-Venn-four.svg| Four sets
Image:Edwards-Venn-five.svg| Five sets
Image:Edwards-Venn-six.svg| Six sets
</gallery>
[[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.<ref name="Edwards_2004"/> For example, three sets can be easily represented by taking three hemispheres of the sphere at right angles (''x''&nbsp;=&nbsp;0, ''y''&nbsp;=&nbsp;0 and ''z''&nbsp;=&nbsp;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 give ''cogwheel'' diagrams with increasing numbers of teeth, as shown here. These diagrams were devised while designing a [[stained-glass]] window in memory of Venn.<ref name="Edwards_2004"/>

===Other diagrams===
Edwards–Venn diagrams are [[topological equivalence|topologically equivalent]] to diagrams devised by [[Branko Grünbaum]], which were based around intersecting [[polygon]]s with increasing numbers of sides. They are also two-dimensional representations of [[hypercube]]s.

[[Henry John Stephen Smith]] devised similar ''n''-set diagrams using [[sine]] curves<ref name="Edwards_2004"/> with the series of equations
: <math>y_i = \frac{\sin\left(2^i x\right)}{2i} \text{ where } 0 \leq i \leq n-2 \text{ and } i \in \mathbb{N}. </math>

[[Charles Lutwidge Dodgson]] (aka [[Lewis Carroll]]) devised a five-set diagram known as [[Carroll's square (diagram)|Carroll's square]]. Joaquin and Boyles, on the other hand, proposed supplemental rules for the standard Venn diagram in able 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.<ref>Joaquin, J.J. and Boyles, R.J.M. (2017). [https://www.pdcnet.org/teachphil/content/teachphil_2017_0040_0002_0161_0180 Teaching Syllogistic Logic via a Retooled Venn Diagrammatical Technique]. ''Teaching Philosophy'', Volume 40(2), pp. 161-180.</ref>

=={{anchor|Johnston}}Related concepts==
[[File:Venn3tab.svg|thumb|Venn diagram as a truth table]]
Venn diagrams correspond to [[truth table]]s for the propositions <math>x\in A</math>, <math>x\in B</math>, etc., in the sense that each region of Venn diagram corresponds to one row of the truth table.<ref name="Grimaldi_2004"/><ref name="Johnson_2001"/> This type is also known as Johnston diagram. Another way of representing sets is with John F. Randolph's [[R-diagram]]s.

==See also==
* [[Existential graph]] (by [[Charles Sanders Peirce]])
* [[Logical connectives]]
* [[Octahedron#Spherical tiling|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.
* [[Marquand diagram]] (and as further derivation [[Veitch chart]] and [[Karnaugh map]])
* [[Vesica piscis]]
* [[Triquetra]]

==Notes==
{{reflist|group="note"|refs=
<ref group="note" name="NB_1">In Euler's ''Lettres 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, [https://books.google.com/books?id=gxsAAAAAQAAJ&pg=PA95#v=onepage&q&f=false pages 95-126.]  In Venn's article, however, he suggests that the diagrammatic idea predates Euler, and is attributable to [[Christian Weise]] or Johann Christian Lange (in Lange's book ''Nucleus Logicae Weisianae'' (1712)).</ref>
}}

==References==
{{Reflist|refs=
<ref name="Lewis1918">{{cite book |author-link=Clarence Irving Lewis |author-first=Clarence Irving |author-last=Lewis |date=1918 |url=https://archive.org/details/asurveyofsymboli00lewiuoft |title=A Survey of Symbolic Logic |location=Berkeley |publisher=[[University of California Press]]}}</ref>
<ref name="Sandifer2003">{{cite web |author-first=Ed |author-last=Sandifer |date=2003 |title=How Euler Did It |url=http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2003%20Venn%20Diagrams.pdf |publisher=[[The Mathematical Association of America]] (MAA) |work=MAA Online |access-date=2009-10-26}}</ref>
<ref name="Ruskey2005">{{cite journal |author-last1=Ruskey |author-first1=Frank |author-link1=Frank Ruskey |author-last2=Weston |author-first2=Mark |title=A Survey of Venn Diagrams |journal=[[The Electronic Journal of Combinatorics]] |date=2005-06-18 |url=http://www.combinatorics.org/files/Surveys/ds5/VennEJC.html}}</ref>
<ref name="Gailand_1967">{{cite book |title=The Logic Diagram |author-first=Gailand |author-last=Mac Queen |date=October 1967 |type=Thesis |publisher=[[McMaster University]] |url=https://macsphere.mcmaster.ca/bitstream/11375/10794/1/fulltext.pdf |access-date=2017-04-14 |url-status=dead |archive-url=https://web.archive.org/web/20170414163921/https://macsphere.mcmaster.ca/bitstream/11375/10794/1/fulltext.pdf |archive-date=2017-04-14}} (NB. Has a detailed history of the evolution of logic diagrams including but not limited to the Venn diagram.)</ref>
<ref name="Venn1880_1">{{cite journal |author-last=Venn |author-first=John |author-link=John Venn |title=I. On the Diagrammatic and Mechanical Representation of Propositions and Reasonings |journal=[[The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science]] |volume=10 |issue=59 |date=July 1880 |series=5 |doi=10.1080/14786448008626877 |pages=1–18 |url=https://www.cis.upenn.edu/~bhusnur4/cit592_fall2014/venn%20diagrams.pdf |url-status=live |archive-url=https://web.archive.org/web/20170516204620/https://www.cis.upenn.edu/~bhusnur4/cit592_fall2014/venn%20diagrams.pdf |archive-date=2017-05-16}} [http://www.tandfonline.com/doi/abs/10.1080/14786448008626877] [https://books.google.com/books?id=k68vAQAAIAAJ&pg=PA1]</ref>
<ref name="Venn1880_2">{{cite journal |author-last=Venn |author-first=John |author-link=John Venn |date=1880 |url=https://archive.org/stream/proceedingsofcam4188083camb#page/47/mode/1up |title=On the employment of geometrical diagrams for the sensible representations of logical propositions |journal=[[Proceedings of the Cambridge Philosophical Society]] |volume=4 |pages=47–59}}</ref>
<ref name="Baron_1969">{{cite journal |author-last=Baron |author-first=Margaret E. |title=A Note on The Historical Development of Logic Diagrams |journal=[[The Mathematical Gazette]] |volume=53 |issue=384 |pages=113–125 |date=May 1969 |jstor=3614533 |doi=10.2307/3614533}}</ref>
<ref name="Henderson_1963">{{cite journal |author-last=Henderson |author-first=D. W. |title=Venn diagrams for more than four classes |journal=[[American Mathematical Monthly]] |volume=70 |issue=4 |pages=424–6 |date=April 1963 |jstor=2311865 |doi=10.2307/2311865}}</ref>
<ref name="Ruskey_2006">{{cite journal |author-last1=Ruskey |author-first1=Frank |author-link1=Frank Ruskey |author-first2=Carla D. |author-last2=Savage |author-link2=Carla Savage |author-first3=Stan |author-last3=Wagon |author-link3=Stan Wagon |date=December 2006 |title=The Search for Simple Symmetric Venn Diagrams |journal=[[Notices of the AMS]] |volume=53 |issue=11 |pages=1304–11 |url=http://www.ams.org/notices/200611/fea-wagon.pdf }}</ref>
<ref name="Venn1881">{{cite book |author-first=John |author-last=Venn |author-link=John Venn |title=Symbolic logic |date=1881 |publisher=[[Macmillan (publisher)|Macmillan]] |page=108 |url=https://books.google.com/books?id=nisCAAAAQAAJ&pg=PA108 |access-date=2013-04-09}}</ref>
<ref name="Edwards_2004">{{citation |title=Cogwheels of the Mind: The Story of Venn Diagrams |author-first=Anthony William Fairbank |author-last=Edwards |author-link=Anthony William Fairbank Edwards<!-- |contribution=Foreword |contributor-first=Ian |contributor-last=Stewart --> |publisher=[[Johns Hopkins University Press]] |date=2004 |isbn=978-0-8018-7434-5 |location=Baltimore, Maryland, USA |page=65 |url=https://books.google.com/books?id=7_0Thy4V3JIC&pg=PA65}}.</ref>
<ref name="Grimaldi_2004">{{cite book |author-last=Grimaldi |author-first=Ralph P. |author-link=Ralph Grimaldi |title=Discrete and combinatorial mathematics |publisher=[[Addison-Wesley]] |location=Boston |date=2004 |page=143 |isbn=978-0-201-72634-3}}</ref>
<ref name="Johnson_2001">{{cite book |author-last=Johnson |author-first=D. L. |title=Elements of logic via numbers and sets |publisher=[[Springer-Verlag]] |location=Berlin, Germany |date=2001 |series=Springer Undergraduate Mathematics Series |page=62 |isbn=978-3-540-76123-5 |chapter=3.3 Laws |chapter-url=https://books.google.com/books?id=8KtRMofBKc0C&lpg=PP1&pg=PA62}}</ref> 
<ref name="Kent_2004">{{cite web |title=Euler Diagrams 2004: Brighton, UK: September 22–23 |url=http://www.cs.kent.ac.uk/events/conf/2004/euler/eulerdiagrams.html |date=2004 |publisher=Reasoning with Diagrams project, University of Kent |access-date=2008-08-13}}</ref>
<ref name="Stategies">{{Cite web |url=http://www.readingquest.org/strat/venn.html |title=Strategies for Reading Comprehension Venn Diagrams |access-date=2009-06-20 |archive-url=https://web.archive.org/web/20090429093334/http://readingquest.org/strat/venn.html |archive-date=2009-04-29 |url-status=dead }}</ref>
<ref name="Leibniz_1690">{{cite book |author-first=Gottfried Wilhelm |author-last=Leibniz |author-link=Gottfried Wilhelm Leibniz |chapter=De Formae Logicae per linearum ductus |orig-year=ca. 1690 |date=1903 |editor-first=Louis |editor-last=Couturat |editor-link=Louis Couturat |title=Opuscules et fragmentes inedits de Leibniz |language=Latin |pages=292–321}}</ref>
}}

==Further reading==
* {{cite web |author-first1=Ebadollah S. |author-last1=Mahmoodian |author-link1=Ebadollah S. Mahmoodian |author-first2=M. |author-last2=Rezaie |author-first3=F. |author-last3=Vatan  |title=Generalization of Venn Diagram |work=Eighteenth Annual Iranian Mathematics Conference |location=Tehran and Isfahan, Iran |date=March 1987 |url=http://sharif.ir/~emahmood/papers/Generalized-Venn-Diagram1987.pdf |access-date=2017-05-01 |url-status=live |archive-url=https://web.archive.org/web/20170501202223/http://sharif.ir/~emahmood/papers/Generalized-Venn-Diagram1987.pdf |archive-date=2017-05-01}}
* {{cite journal |title=Venn diagrams for many sets |author-first=Anthony William Fairbank |author-last=Edwards |author-link=Anthony William Fairbank Edwards |journal=[[New Scientist]] |volume=121 |number=1646 |date=1989-01-07 |pages=51–56}}
* {{cite book |title=Coding for Digital Recording |chapter=4.10. Hamming distance |author-first=John |author-last=Watkinson |publisher=[[Focal Press]] |location=Stoneham, MA, USA |date=1990 |isbn=978-0-240-51293-8 |pages=94–99, foldout in backsleeve}} (NB. The book comes with a 3-page foldout of a seven-bit cylindrical Venn diagram.)
* {{cite book |author-first=Ian |author-last=Stewart |author-link=Ian Stewart (mathematician) |chapter=Chapter 4. Cogwheels of the Mind |chapter-url=https://books.google.com/books?id=u5GPE97-ZhsC&pg=PA51 |title=Another Fine Math You've Got Me Into |publisher=[[Dover Publications, Inc.]] ([[W. H. Freeman]]) |edition=reprint of 1st |location=Mineola, New York, USA |year=June 2003 |orig-year=1992 |isbn=978-0-486-43181-9 |pages=51–64}}
* {{cite book |author-last=Glassner |author-first=Andrew |author-link=Andrew Glassner |title=Morphs, Mallards, and Montages: Computer-Aided Imagination |date=2004 |publisher=[[A. K. Peters]] |location=Wellesley, MA, USA |isbn=978-1568812311 |pages=161–184 |chapter=Venn and Now}}
* {{anchor|Newroz}}{{Cite web |author-first1=Khalegh |author-last1=Mamakani |author-first2=Frank |author-last2=Ruskey |author-link2=Frank Ruskey |date=2012-07-27 |title=A New Rose: The First Simple Symmetric 11-Venn Diagram |arxiv=1207.6452 |url=http://webhome.cs.uvic.ca/~ruskey/Publications/Venn11/Venn11.html |access-date=2017-05-01 |url-status=live |archive-url=https://web.archive.org/web/20170501204303/http://webhome.cs.uvic.ca/~ruskey/Publications/Venn11/Venn11.html |archive-date=2017-05-01 |bibcode=2012arXiv1207.6452M |volume=1207 |pages=6452{{inconsistent citations}}}}

==External links==
{{Commons category|Venn diagrams}}
* {{springer|title=Venn diagram|id=p/v096550}}
* {{MathWorld|title=Venn Diagram|id=VennDiagram}}
* [http://www.cut-the-knot.org/LewisCarroll/dunham.shtml Lewis Carroll's Logic Game – Venn vs. Euler] at [[Cut-the-knot]]
* [http://www.combinatorics.org/Surveys/ds5/VennTriangleEJC.html Six sets Venn diagrams made from triangles]
*[http://moebio.com/research/sevensets/ Interactive seven sets Venn diagram]

{{Mathematical logic}}
{{Set theory}}

[[Category:Graphical concepts in set theory]]
[[Category:Diagrams]]
[[Category:Statistical charts and diagrams]]