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| [[File:Venn diagram gr la ru.svg|thumb|'''Venn diagram''' showing which uppercase letter [[glyph]]s are shared by the [[Greek alphabet|Greek]], [[ISO basic Latin alphabet|Latin]] and [[Russian alphabet|Russian]] alphabets]]
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| {{Probability fundamentals}}
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| A '''Venn diagram''' or '''set diagram''' is a [[diagram]] that shows all possible [[logic]]al relations between a finite collection of [[Set (mathematics)|sets]]. 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]].
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| == Example ==
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| [[Image:venn-diagram-AB.svg|thumb|left|Sets A (creatures with two legs) and B (creatures that can fly)]]
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| 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 area where the blue and orange circles overlap. That area contains all such and only such living creatures.
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| 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.
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| The combined area 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).
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| The area 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.
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| == History ==
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| Venn diagrams were introduced in 1880 by [[John Venn]] (1834–1923) 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=Sandifer2003/> The use of these types of [[diagram]]s in [[formal logic]], according to Ruskey and M. 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/>
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| Venn himself did not use the term "Venn diagram" and referred to his invention as "Eulerian Circles."<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/> The first to use the term "Venn diagram" was [[Clarence Irving Lewis]] in 1918, in his book "A Survey of Symbolic Logic".<ref name=Ruskey2005/>
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| Venn diagrams are very similar to [[Euler diagram]]s, which were invented by [[Leonhard Euler]] (1708–1783) in the 18th century.<ref group=note>In Euler's ''Letters to a German Princess.'' In Venn's article, however, he suggests that the diagrammatic idea predates Euler, and is attributable to C. Weise or J. C. Lange.</ref> M. E. 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. She also observes even earlier Euler-like diagrams by [[Ramon Lull]] in the 13th Century.<ref>{{cite journal |author=Baron, M.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>
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| In the 20th century, Venn diagrams were further developed. D.W. 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>{{cite journal |author=Henderson, 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> He also showed that such symmetric Venn diagrams exist when ''n'' is 5 or 7. 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>{{cite journal |last=Ruskey |first=Frank |first2=Carla D. |last2=Savage |authorlink2=Carla Savage|first3=Stan |last3=Wagon |authorlink3=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 | format = PDF}}</ref>
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| 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 by other curriculum fields such as reading.<ref>[http://www.readingquest.org/strat/venn.html Strategies for Reading Comprehension Venn Diagrams]</ref>
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| == Overview ==
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| {{See also|Set (mathematics)#Basic operations}}
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| {{Gallery|lines=3
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| |File:Venn0001.svg|[[Intersection (set theory)|Intersection]] of two sets<br/><math>~A \cap B</math>
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| |File:Venn0111.svg|[[Union (set theory)|Union]] of two sets<br /><math>~A \cup B</math>
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| |File:Venn0110.svg|[[Symmetric difference]] of two sets<math>A~\Delta~B</math>
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| |File:Venn0010.svg|[[Complement (set theory)#Relative complement|Relative complement]] of ''A'' (left) in ''B'' (right)<br /><math>A^c \cap B~=~B \setminus A</math>
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| |File:Venn1010.svg|[[Complement (set theory)#Absolute complement|Absolute complement]] of A in U<br/> <math>A^c~=~U \setminus A</math>
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| }}
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| 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}}
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| 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 area 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.
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| 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>{{cite web|title=Euler Diagrams 2004: Brighton, UK: September 22–23|url=http://www.cs.kent.ac.uk/events/conf/2004/euler/eulerdiagrams.html|year=2004|publisher=Reasoning with Diagrams project, University of Kent|accessdate=13 August 2008}}</ref>
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| ==Extensions to higher numbers of sets==
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| 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).
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| {|class="wikitable" style="text-align:center; width: 100%;"
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| | style="vertical-align:top;"|[[File:Venn 1000 0000 0000 0000.png|150px]]
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| | style="vertical-align:top;"|[[File:Venn 0110 1000 1000 0000.png|150px]]<br />
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| [[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]]
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| | style="vertical-align:top;"|[[File:Venn 0001 0110 0110 1000.png|150px]]<br />
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| [[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]]
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| | style="vertical-align:top;"|[[File:Venn 0000 0001 0001 0110.png|150px]]<br />
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| [[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]]
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| | style="vertical-align:top;"|[[File:Venn 0000 0000 0000 0001.png|150px]]
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| |}
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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,"<ref name="Venn1881">{{cite book|author=Jo Venn|title=Symbolic logic|url=http://books.google.com/books?id=nisCAAAAQAAJ&pg=PA108|accessdate=9 April 2013|year=1881|publisher=Macmillan|page=108}}</ref> that represented higher numbers of sets, and he devised a 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.
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| <gallery widths=200px><!---perrow=3-->
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| Image:Venn4.svg|Venn's construction for 4 sets
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| Image:Venn5.svg|Venn's construction for 5 sets
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| Image:Venn6.svg|Venn's construction for 6 sets
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| Image:Venn's four ellipse construction.svg|Venn's four-set diagram using ellipses
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| Image:CirclesN4xb.svg|'''Counter-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 pink and green circles meet.
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| File:Symmetrical 5-set Venn diagram.svg|Five-set Venn diagram using congruent ellipses in a radially symmetrical arrangement devised by [[Branko Grünbaum]]. Labels have been simplified for greater readability; for example, '''A''' denotes '''A''' ∩ '''B'''<sup>c</sup> ∩ '''C'''<sup>c</sup> ∩ '''D'''<sup>c</sup> ∩ '''E'''<sup>c</sup>, while '''BCE''' denotes '''A'''<sup>c</sup> ∩ '''B''' ∩ '''C''' ∩ '''D'''<sup>c</sup> ∩ '''E'''.
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| File:6-set_Venn_diagram.svg|Six-set Venn diagram made of only triangles.
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| </gallery>
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| ===Edwards' Venn diagrams===
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| <gallery widths=150px>
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| Image:Venn-three.svg| Three sets
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| Image:Edwards-Venn-four.svg| Four sets
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| Image:Edwards-Venn-five.svg| Five sets
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| Image:Edwards-Venn-six.svg| Six sets
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| </gallery>
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| [[A. W. F. Edwards]] constructed a series of Venn diagrams for higher numbers of sets by segmenting the surface of a sphere. 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. The resulting sets can then be projected back to a plane to give ''cogwheel'' diagrams with increasing numbers of teeth, as shown on the right. These diagrams were devised while designing a [[stained-glass]] window in memory of Venn.
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| ===Other diagrams===
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| 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 2-dimensional representations of [[hypercube]]s.
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| Smith{{Citation needed|date=January 2012}} devised similar ''n''-set diagrams using [[sine]] curves with the series of equations
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| : <math>y_i = \frac {\sin(2^{i }x)}{2 i} \text{ where } 0 \leq i \leq n-2 \text{ and } i \in \mathbb{N}. </math>
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| [[Lewis Carroll|Charles Lutwidge Dodgson]] devised a five-set diagram.
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| ==Related concepts==
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| [[File:Venn3tab.svg|thumb|Venn diagram as a truth table]]
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| 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>{{cite book | last=Grimaldi | first=Ralph P. | authorlink=Ralph Grimaldi | title=Discrete and combinatorial mathematics | publisher=[[Addison-Wesley]]| location=Boston | year=2004 | page=143 | isbn=0-201-72634-3}}</ref><ref>{{cite book | last=Johnson | first=D. L. | title=Elements of logic via numbers and sets | publisher=[[Springer-Verlag]]| location=Berlin | year=2001 | series= Springer Undergraduate Mathematics Series | page=62 | isbn=3-540-76123-3 |chapter=3.3 Laws |chapterurl=http://books.google.com.au/books?id=8KtRMofBKc0C&lpg=PP1&pg=PA62 }}</ref>
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| Another way of representing sets is with [[Randolph diagram|R-Diagram]]s.
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| ==See also==
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| *[[Logical connectives]]
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| ==Notes==
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| {{reflist|group=note}}
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| ==References==
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| {{Reflist|refs=
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| <ref name=Lewis1918>{{cite book|authorlink=Clarence Irving Lewis|first=Clarence Irving|last=Lewis|year=1918|url= http://www.archive.org/details/asurveyofsymboli00lewiuoft|title=A Survey of Symbolic Logic|location=Berkeley|publisher=University of California Press}}</ref> | |
| <ref name=Sandifer2003>{{cite web|first=Ed|last=Sandifer|year=2003|title=How Euler Did It|url=http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2003%20Venn%20Diagrams.pdf|format=pdf|publisher=The Mathematical Association of America: MAA Online|accessdate=26 October 2009}}</ref> | |
| <ref name=Ruskey2005>{{cite journal |author=Ruskey, F.; Weston, M. |title=Venn Diagram Survey |journal=The electronic journal of combinatorics |date=June 2005 |url=http://www.combinatorics.org/Surveys/ds5/VennJohnEJC.html}}</ref>
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| <ref name=Venn1880>{{cite journal |author=Venn, J. |title=On the Diagrammatic and Mechanical Representation of Propositions and Reasonings |journal=Philosophical Magazine and Journal of Science |volume=10 |issue=59 |date=July 1880 |series=5}}</ref>
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| }}
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| ==Further reading ==
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| * [http://www.combinatorics.org/Surveys/ds5/VennEJC.html A Survey of Venn Diagrams] by F. Ruskey and M. Weston, is an extensive site with much recent research and many beautiful figures.
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| *{{cite book |first=Ian |last=Stewart |authorlink=Ian Stewart (mathematician) |chapter=Ch. 4 Cogwheels of the Mind |chapterurl=http://books.google.com.au/books?id=u5GPE97-ZhsC&pg=PA51 |title=Another Fine Math You've Got Me Into |publisher=Dover Publications |year=2004 |isbn=0-486-43181-9 |pages=51–64}}
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| *{{cite book |first=A.W.F. |last=Edwards |title=Cogwheels of the mind: the story of Venn diagrams |url=http://books.google.com/books?id=7_0Thy4V3JIC |year=2004 |publisher=[[Johns Hopkins University Press|JHU Press]] |isbn=978-0-8018-7434-5}}
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| * {{cite journal
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| | first = John |last=Venn
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| | title = On the Diagrammatic and Mechanical Representation of Propositions and Reasonings
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| | journal = Dublin Philosophical Magazine and Journal of Science
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| | volume = 9
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| | issue = 59
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| | pages = 1–18
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| | year = 1880
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| | doi = 10.1080/14786448008626877
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| }}
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| *{{Cite web |first=Khalegh |last=Ruskey |first2=Frank |last2=Ruskey |author2-link=Frank Ruskey |date=27 July 2012 |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 |bibcode=2012arXiv1207.6452M |volume=1207 |pages=6452 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| == External links ==
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| {{Commons category|Venn diagrams}}
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| * {{springer|title=Venn diagram|id=p/v096550}}
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| * {{MathWorld |title=Venn Diagram |id=VennDiagram }}
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| * [http://www.eulerdiagrams.com/inductivecircles.html Free software for generating Venn and Euler diagrams using circles]
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| * [http://www.cut-the-knot.org/LewisCarroll/dunham.shtml Lewis Carroll's Logic Game – Venn vs. Euler] at [[cut-the-knot]]
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| * [http://www.combinatorics.org/Surveys/ds5/VennEJC.html A Survey of Venn Diagrams]
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| * [http://www.cs.kent.ac.uk/people/staff/pjr/EulerVennCircles/EulerVennApplet.html Area proportional 3-way venn diagram applet]
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| * [http://www.technomancy.org/google-suggest-venn/ Generating Venn Diagrams to explore Google Suggest results]
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| * [http://moebio.com/research/sevensets seven sets interactive Venn diagram displaying color combinations]
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| * [http://www.combinatorics.org/Surveys/ds5/VennTriangleEJC.html six sets Venn diagrams made from triangles]
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| * [http://qandr.org/quentin/software/venn Postscript for 9-set Venn] and more
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| * [http://webdmamrl.er.usgs.gov/g1/FHWA/VBVenn/default.htm VBVenn – A Visual Basic program for calculating and graphing quantitative two-circle Venn diagrams]
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| {{logic}}
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| {{Set theory}}
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| [[Category:Graphical concepts in set theory]]
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| [[Category:Diagrams]]
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| [[Category:Statistical charts and diagrams]]
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| {{Link GA|es}}
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If you are planning to visit the United States, or have family and friends right here, store at Sears to get all the pieces you need—together with items for yourself or others. Buying online within the U.S. at Sears couldn't be easier. You'll be able to pay for your order in a retailer and even use a global bank card. Once you place an order, you (or someone you know) can choose it up at a Sears store, have it shipped to over one hundred international locations—no matter is most convenient for you. In case you are looking to do some worldwide on-line buying, be sure you visit /international
Tread – The tread area of the tire is the half that comes into contact with the street floor. Tread depth (the quantity of tread remaining on the tire ) is a crucial factor. Tread depth is measured in millimeters, and there should be a minimum of 50% of the tread remaining. In addition, inspect the tread for any uncommon put on patterns, comparable to lower tread within the middle, cupping on the edges or significant edge wear. These could cause vibrations and even scale back the lifespan of your new tires
This is among the most important facet in selling your automotive. Very often automobile house owners have a good quality car on the market , and there're patrons trying to find helpful auto deals. Many homeowners end up shedding good patrons just because individuals do not know a car is offered on the market Promoting your car for sale in native newspapers and auto magazines and fetch in lots of patrons, however you find yourself incurring a price for placing up the advert. Another option is to contact an auto dealer who may help you search for good consumers. Concerning the Writer.
The 2009 Subaru WRX for sale is the right choice for the driving force searching for 4-door sport and enjoyable with out paying a premium. The WRX isn't precisely low cost, but most comparables are at a a lot higher value level. The 2010 mannequin that adopted was only a minor revision on that 2009 success, and the upcoming 2011 model seems to be extra of the identical. If the 2008 WRX was the last one you check-drove, you owe it to your car-loving self to offer it another probability. In regards to the Writer
Harry W. Millis, an impartial tire-trade analyst in Cleveland, said the tires would be only in monitoring tires inside trucking firms' warehouses and for guarantee functions. "The query now is how much information there can be on the chip and how the data can be learn," he said. "However for now, evidently the potential is enormous. It could possibly be an amazing boon to tire managers in controlling value." The brand new tires developed by Goodyear even have robust potential for the passenger-car tire market, some industry analysts suggest. By enabling car homeowners to do a greater job of monitoring the inflation stage of the tire and other efficiency info, a automotive proprietor can lengthen the lifetime of a passenger automotive tire by 10 % as nicely.
These are all straightforward strategies that may increase lifespan of a car, specifically within the case of pre-owned cars. Each automobile proprietor must be conscious that a car is dependable as long as it's looked after. Your car can run for a number of years with none problems, for those who incorporate these simple tips into your long run car maintenance technique. The Mustang is a seasoned veteran since its existence began the 1960's by sticking with their great types which were loved by automobile lovers through your complete years. The actual inexpensive pricing plus nice performance it gives made the car profitable. The brand new fashions of the Kia Mustang will carry on make its point out on Ford's product gross sales this coming season. Concerning the Author
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