Editing Venn diagram

{{Short description|Diagram that shows all possible logical relations between a collection of sets}}
{{Use list-defined references|date=January 2022}}
{{Use dmy dates|date=May 2020|cs1-dates=y}}
[[File:Venn diagram gr la ru.svg|class=skin-invert-image|thumb|Venn diagram showing the uppercase [[glyph]]s shared by the [[Greek alphabet|Greek]] (upper left), [[Latin alphabets|Latin]] (upper right), and [[Russian alphabet|Russian Cyrillic]] (bottom) alphabets]]
{{Probability fundamentals}}

A '''Venn diagram''' is a widely used [[diagram]] style that shows the logical relation between [[set (mathematics)|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.

== Details ==
{{anchor|Primary|Simple|Cylindrical|Metric|2|3}}A Venn diagram, also called a ''set diagram'' or ''logic diagram'', shows ''all'' possible logical relations between a finite collection of different sets.  These diagrams depict [[element (mathematics)|element]]s 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''&nbsp;∩&nbsp;''T'' and read "the intersection of ''S'' and ''T''", is represented visually by the area of overlap of the regions ''S'' and ''T''.<ref name="Peil_2020"/>

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]]s, 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:Venn diagram of legs and flying.svg|class=skin-invert-image|thumb|left|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 legs ''and'' 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 their ''[[union (set theory)|union]]'', denoted by {{nowrap|A ∪ 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 the ''[[intersection (set theory)|intersection]]'' of A and B, denoted by {{nowrap|A ∩ B}}.

== {{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"<ref name="Venn_2014"/> in the ''Philosophical Magazine and Journal of Science'',<ref name="PM"/> 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 diagrams in [[formal logic]], according to [[Frank 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".<ref name="Ruskey2005"/>

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.<ref name="Baron_1969" /> [[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.<ref name="Leibniz_1690" /><ref name="Baron_1969" /> [[Euler diagram]]s, 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.<ref group="note" name="NB_1"/><ref name="Venn1881"/><ref name="Gailand_1967"/>

Venn did not use the term "Venn diagram" and referred to the concept as "Eulerian Circles".<ref name="Sandifer2003"/> 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]].<ref name="Maths Today">{{cite magazine|title=The Venn Behind the Diagram|last=Verburgt|first=Lukas M.|volume=59|issue=2|date=April 2023|pages=53–55|magazine=Mathematics Today|publisher=[[Institute of Mathematics and its Applications]]}}</ref> 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".<ref name="Venn1880_1"/><ref name="Venn1880_2"/>

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''.'<ref name="Maths Today"/>

[[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''.<ref name="Ruskey2005"/><ref name="Lewis1918"/>

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]].<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. These combined results show that 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="Strategies"/>

==Popular culture==
Venn diagrams have been commonly used in [[meme]]s.<ref>{{cite news |last1=Leo |first1=Alex |title=Jesus, Karaoke, And Serial Killers: The Funniest Venn Diagrams The Web Has To Offer |url=https://www.huffpost.com/entry/funniest-venn-diagrams-th_n_347552 |access-date=2 October 2024 |work=Huffpost |date=March 18, 2010}}</ref> At least one politician has been mocked for misusing Venn diagrams.<ref>{{cite news |last1=Moran |first1=Lee |title=Scott Walker Gets Mercilessly Mocked By Twitter Users Over Venn Diagram Fail |url=https://www.huffpost.com/entry/scott-walker-venn-diagram-meme_n_5c14e6d5e4b05d7e5d8258cc |access-date=2 October 2024 |work=HuffPost |date=December 15, 2018}}</ref>

==Overview==
{{See also|Set (mathematics)#Basic operations}}
<div class=skin-invert-image>{{Gallery |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>
}}</div>
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 ''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".<ref name="Lewis1918"/>{{rp|157}}

Venn diagrams normally comprise overlapping [[circle]]s. 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 [[wood]]en 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<sup>''n''</sup> hypothetically possible zones, that correspond to some combination of inclusion or exclusion in each of the component sets.<ref name="Weisstein_2020"/> 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 Euler and the Venn 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:4 spheres, cell 00, solid.png|180px]]
| style="vertical-align:top;"| [[File:4 spheres, weight 1, solid.png|180px]]<br>
[[File:4 spheres, cell 01, solid.png|45px]][[File:4 spheres, cell 02, solid.png|45px]][[File:4 spheres, cell 04, solid.png|45px]][[File:4 spheres, cell 08, solid.png|45px]]
| style="vertical-align:top;"| [[File:4 spheres, weight 2, solid.png|180px]]<br>
[[File:4 spheres, cell 03, solid.png|30px]][[File:4 spheres, cell 05, solid.png|30px]][[File:4 spheres, cell 06, solid.png|30px]][[File:4 spheres, cell 09, solid.png|30px]][[File:4 spheres, cell 10, solid.png|30px]][[File:4 spheres, cell 12, solid.png|30px]]
| style="vertical-align:top;"| [[File:4 spheres, weight 3, solid.png|180px]]<br>
[[File:4 spheres, cell 07, solid.png|45px]][[File:4 spheres, cell 11, solid.png|45px]][[File:4 spheres, cell 13, solid.png|45px]][[File:4 spheres, cell 14, solid.png|45px]]
| style="vertical-align:top;"| [[File:4 spheres, cell 15, solid.png|180px]]
|}

For higher numbers of sets, some loss of symmetry in the diagrams is unavoidable. Venn was keen to find "symmetrical figures&nbsp;... 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 (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 {{em|not}} a Venn diagram for four sets as it has only 14 regions as opposed to 2<sup>4</sup> = 16 regions (including the white region); 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" class="skin-invert-image">
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 display="block">y_i = \frac{\sin\left(2^i x\right)}{2^i} \text{ where } 0 \leq i \leq n-1 \text{ and } i \in \mathbb{N}. </math>

[[Charles Lutwidge Dodgson]] (also known as 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 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.<ref name="Joaquin_2017"/>

=={{anchor|Johnston}}Related concepts==
[[File:Venn3tab.svg|class=skin-invert-image|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 connective]]
* [[Information diagram]]
* [[Marquand diagram]] (and as further derivation [[Veitch chart]] and [[Karnaugh map]])
* [[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.
* [[Stanhope Demonstrator]]
* [[Three circles model]]
* [[Triquetra]]
* [[Vesica piscis]]
* [[UpSet plot]]

==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 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.|author-link=Margaret Baron |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|s2cid=125364002 }}</ref>
<ref name="Henderson_1963">{{cite journal |author-last=Henderson |author-first=David Wilson |author-link=David Wilson Henderson |title=Venn diagrams for more than four classes |journal=[[American Mathematical Monthly]] |volume=70 |issue=4 |pages=424–426 |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=1304–1311 |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=[https://archive.org/details/symboliclogic00venngoog/page/n150 108] |url=https://archive.org/details/symboliclogic00venngoog |access-date=2013-04-09}}</ref>
<ref name="Edwards_2004">{{cite book |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=David L. |title=Elements of logic via numbers and sets |publisher=[[Springer-Verlag]] |location=Berlin, Germany |date=2001 |series=Springer Undergraduate Mathematics Series |page=[https://archive.org/details/elementsoflogicv0000john/page/62 62] |isbn=978-3-540-76123-5 |chapter=3.3 Laws |chapter-url=https://books.google.com/books?id=8KtRMofBKc0C&pg=PA62 |url=https://archive.org/details/elementsoflogicv0000john/page/62 }}</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="Strategies">{{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-date=ca. 1690 |date=1903 |editor-first=Louis |editor-last=Couturat |editor-link=Louis Couturat |title=Opuscules et fragmentes inedits de Leibniz |language=la |pages=292–321}}</ref>
<ref name="Joaquin_2017">{{cite journal |author-last1=Joaquin |author-first1=Jeremiah Joven |author-last2=Boyles |author-first2=Robert James M. |date=June 2017 |title=Teaching Syllogistic Logic via a Retooled Venn Diagrammatical Technique |journal=[[Teaching Philosophy]] |volume=40 |issue=2 |doi=10.5840/teachphil201771767 |pages=161–180 |url=https://www.pdcnet.org/teachphil/content/teachphil_2017_0040_0002_0161_0180 |access-date=2020-05-12 |url-status=live |archive-url=https://web.archive.org/web/20181121120401/https://www.pdcnet.org/teachphil/content/teachphil_2017_0040_0002_0161_0180 |archive-date=2018-11-21}}</ref>
<ref name="Peil_2020">{{cite web |title=Intersection of Sets |url=http://web.mnstate.edu/peil/MDEV102/U1/S3/Intersection4.htm |access-date=2020-09-05 |website=web.mnstate.edu |archive-date=2020-08-04  |archive-url=https://web.archive.org/web/20200804163657/http://web.mnstate.edu/peil/MDEV102/U1/S3/Intersection4.htm |url-status=dead }}</ref>
<ref name="Venn_2014">{{cite web |author-last=Venn |author-first=John |title=On the Diagrammatic and Mechanical Representation of Propositions and Reasonings |url=https://www.cis.upenn.edu/~bhusnur4/cit592_fall2014/venn%20diagrams.pdf |website=Penn Engineering}}</ref>
<ref name="PM">{{cite news |title=The Philosophical Magazine |language=en |work=[[Taylor & Francis]] |url=https://www.tandfonline.com/toc/tphm18/current |access-date=2021-08-06}}</ref>
<ref name="Weisstein_2020">{{cite web |author-last=Weisstein |author-first=Eric W. |title=Venn Diagram |url=https://mathworld.wolfram.com/VennDiagram.html |access-date=2020-09-05 |website=mathworld.wolfram.com |language=en}}</ref>
}}

==Further reading==
{{refbegin}}
* {{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=dead |archive-url=https://web.archive.org/web/20170501202223/http://sharif.ir/~emahmood/papers/Generalized-Venn-Diagram1987.pdf |archive-date=1 May 2017 }}
* {{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 |date=June 2003 |orig-date=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}}
{{refend}}

==External links==
{{Commons category|Venn diagrams}}
* {{springer|title=Venn diagram|id=p/v096550}}
* [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]
* [https://www.usgs.gov/software/vbvenn-visual-basic-venn-diagram-software-page VBVenn, a free open source program for calculating and graphing quantitative two-circle Venn diagrams]
* [https://interactivenn.net/index.html InteractiVenn, a web-based tool for visualizing Venn diagrams]
* [https://www.deepvenn.com/ DeepVenn, a tool for creating area-proportional Venn Diagrams]

{{Mathematical logic}}
{{Set theory}}
{{Authority control}}

{{Diagrams in logic}}

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