Venn diagram that shows the intersections of the Greek, Latin and Russian alphabets (upper case letters)

Venn diagrams or set diagrams are diagrams that show all possible logical relations between a finite collection of sets (aggregation of things). 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 (see logical connectives).

Overview[link]

• 

  Intersection of two sets:
  Failed to parse (Missing texvc executable; please see math/README to configure.): ~A \cap B

• 

  Union of two sets:
  Failed to parse (Missing texvc executable; please see math/README to configure.): ~A \cup B

• 

  Relative complement
  of A (left) in B (right):
  Failed to parse (Missing texvc executable; please see math/README to configure.): A^c \cap B~=~B \setminus A

• 

  Symmetric difference
  of two sets: Failed to parse (Missing texvc executable; please see math/README to configure.): A~\Delta~B

• 

  Absolute complement
  of A in U: Failed to parse (Missing texvc executable; please see math/README to configure.): A^c~=~U \setminus A

A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis (1918), 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".^[1]

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 another circle may represent the set of all tables. The overlapping area 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; i.e. they are schematic diagrams.

Venn diagrams are similar to Euler diagrams. However, a Venn diagram for n component sets must contain all 2ⁿ 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.^[2]

History[link]

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 propositions by diagrams.^[3] The use of these types of diagrams 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".^[4]

Venn himself did not use the term "Venn diagram" but kept speaking of "Eulerian Circles".^[3] In the opening sentence of his 1880 article Venn declared: "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 nation and object of such devices. Of these schemes one only, viz. that commonly called 'Eulerian circles,' has met with any general acceptance..."^[5] The first to use the term "Venn diagram" was Clarence Irving Lewis in 1918, in his book "A Survey of Symbolic Logic".^[4]

Venn diagrams are very similar to Euler diagrams, which were invented by Leonhard Euler (1708–1783) in the 18th century.^[6] M. E. Baron has noted that 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.^[7]

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 prime.^[8] 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 symmetric Venn diagrams exist if and only if n is a prime number.^[9]

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.^[10]

Example[link]

The following example involves two 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.

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 area of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that are either two-legged or that can fly (or both). The area in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by 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.

Extensions to higher numbers of sets[link]

Venn diagrams typically support 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 is completely symmetric 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,"^{[citation needed]} that represented higher numbers of sets, and he devised a 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.

• 

  Venn's construction for 4 sets

• 

  Venn's construction for 5 sets

• 

  Venn's construction for 6 sets

• 

  Venn's four-set diagram using ellipses

• 

  This symmetrical or 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.

• 

  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^c ∩ C^c ∩ D^c ∩ E^c (or A ∩ ~B ∩ ~C ∩ ~D ∩ ~E), while BCE denotes A^c ∩ B ∩ C ∩ D^c ∩ E (or ~A ∩ B ∩ C ∩ ~D ∩ E).

Edwards' Venn diagrams[link]

• Edwards' Venn diagrams
• 

  three sets

• 

  four sets

• 

  five sets

• 

  six sets

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.

Other diagrams[link]

Edwards' Venn diagrams are topologically equivalent to diagrams devised by Branko Grünbaum, which were based around intersecting polygons with increasing numbers of sides. They are also 2-dimensional representations of hypercubes.

Smith^{[citation needed]} devised similar n-set diagrams using sine curves with the series of equations

Failed to parse (Missing texvc executable; please see math/README to configure.): y_i = \frac {\sin(2^{i }x)}{2 i} \text{ where } 0 \leq i \leq n-2 \text{ and } i \in \mathbb{N}.

Charles Lutwidge Dodgson devised a five-set diagram.

Related concepts[link]

Venn diagram as a truth table

Venn diagrams corresponds to truth tables for the propositions Failed to parse (Missing texvc executable; please see math/README to configure.): x\in A , Failed to parse (Missing texvc executable; please see math/README to configure.): x\in B , etc., in the sense that each region of Venn diagram corresponds to one row of the truth table. ^[11][12]

References[link]

Further reading[link]

• A Survey of Venn Diagrams by F. Ruskey and M. Weston, is an extensive site with much recent research and many beautiful figures.
• Stewart, Ian (2004). "Ch. 4 Cogwheels of the Mind". Another Fine Math You've Got Me Into. Dover Publications. pp. 51–64. ISBN 0-486-43181-9. http://books.google.com.au/books?id=u5GPE97-ZhsC&pg=PA51. 
• Edwards, A.W.F. (2004). Cogwheels of the mind: the story of Venn diagrams. JHU Press. ISBN 978-0-8018-7434-5. http://books.google.com/books?id=7_0Thy4V3JIC. 
• Venn, John (1880). "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings". Dublin Philosophical Magazine and Journal of Science 9 (59): 1–18. 

External links[link]

Wikimedia Commons has media related to: Venn diagrams

• Lewis Carroll's Logic Game — Venn vs. Euler at cut-the-knot
• A Survey of Venn Diagrams
• Area proportional 3-way venn diagram applet
• Generating Venn Diagrams to explore Google Suggest results
• 6-set Venn diagrams made from triangles
• Postscript for 9-set Venn and more.

Lisa Hannigan
Lisa Hannigan in Temple Bar, Dublin
Background information
Birth name	Lisa Margaret Hannigan[1]
Born	(1981-02-12) 12 February 1981 (age 31)
Origin	Kilcloon, County Meath, Republic of Ireland
Genres	Indie folk
Occupations	Musician
Years active	2001–present
Labels	Barp/ATO Records/Play It Again Sam[2]
Associated acts	Damien Rice, Vyvienne Long, Tom Osander, The Frames, Mic Christopher, David Kitt, Herbie Hancock, Paul Noonan, David Geraghty, Bell X1, The Cake Sale, Gary Lightbody,[3] Cathy Davey,[4] Jason Mraz,[5] Mick Flannery.[6][7]
Website	http://www.lisahannigan.ie/

Lisa Margaret Hannigan (born 12 February 1981) is an Irish singer, songwriter, and musician. For several years she was a member of Damien Rice's band before beginning her solo career in 2008. To date she has released two albums: Sea Sew (2008) and Passenger (2011).

Hannigan has been the recipient of various award nominations and critical praise both in Ireland and the USA. She has performed and been associated with Cathy Davey, Mic Christopher, Mick Flannery, David Kitt and members of Snow Patrol, The Frames and Bell X1. She has performed at the Glastonbury Festival both as a solo artist and with Damien Rice.

Early life and education[link]

Hannigan was born in Dublin but grew up in Kilcloon, County Meath, Ireland.^[8] She attended primary school at Scoil Oilibhéir Naofa^[9] in Kilcloon and secondary school at The King's Hospital in Palmerstown.^[10] At the age of eighteen, she enrolled at Trinity College, Dublin, where she studied Art History.^[11]

Career[link]

2001–2007[link]

Hannigan began her six year collaboration with Damien Rice at a concert in the Temple Bar Music Centre in Dublin in early 2001.^[12][13][14] She sang on most of the tracks for Rice's 2002 album O. In 2006, she sang the lead vocal on Rice's song "9 Crimes" and performed at the Electric Picnic music festival in Stradbally, County Laois.^{[12][13][13][15]} As a member of Rice's live performance band, she contributed vocals and on occasion electric guitar, bass and drums.^[12][13]

In 2007, Hannigan left the Damien Rice band to pursue a solo career.^[14][16] “It was upsetting, but worse things have happened. For everybody it was coming to the end of its cycle.”^[15] Hannigan returned to Dublin and began her solo career.^[7][8][16] The singer later said of the split: "I suppose it was coming for a long time. I had become pretty outspoken about my frustrations at the direction the band was going in, and I'm sure he was starting to find me a pain."^[8] Rice said that their incompatible song writing styles were to blame for the inadequacies in his song "Unplayed Piano".^[17] She was comforted by members of The Magic Numbers and has not spoken to Rice since the split.^[8]

2008–2010[link]

Some of Hannigan's live recordings were made available through trading networks radio shows. These recordings included: "Willy" by Joni Mitchell, "Be My Husband" by Nina Simone (from the 1965 album Pastel Blues), "Mercedes Benz" by Janis Joplin and "Love Hurts" by Roy Orbison.^[13] Hannigan also performed live with her own band, called The Daisy Okell Quartet and contributed guest vocals to the recordings of Mic Christopher, The Frames and Herbie Hancock.^[13]

Lisa Hannigan's debut solo album, titled Sea Sew, was rehearsed in a barn in Thomastown and recorded in Dublin before being released in Ireland in September 2008.^[18] The lead single, "Lille", was made available as a free Internet download and other tracks were available for preview on her Myspace page.^[19] The sleeve featured needle-work by Hannigan.^[4][20] Some music critics called the recording one of the best Irish albums of the year.^[21][22]

Sea Sew was played on the American radio station KCRW and received favourable reviews in the Los Angeles Times and The New York Times.^[8][23] The single "Lille", released in August 2008 on Irish and American radio stations.^[7] Hannigan performed at Electric Picnic 2008.^[24] Hannigan supported singer-songwriter Jason Mraz on a 42-date U.S. tour in 2008.^[5]

Also in 2008, she appeared on the charity album Even Better Than the Disco Thing and performed a duet of Mick Flannery's new song "Christmas Past" with Flannery on Tony Fenton's Christmas Special on Today FM.^[6][25] In December 2008, she made her UK solo debut at St Johns Church in London.^[26] Hannigan signed with ATO Records in the U.S., where her album was released in February 2009.^[27]

Sea Sew was nominated for the Choice Music Prize and Best Irish Album at the Meteor Music Awards in January 2009.^[28][29][30]

That same year, Hannigan appeared on the American television shows Jay Leno, The Colbert Report and was well received by their respective hosts.^[8][15][31] In 2009, Hannigan also appeared on the BBC's Later... with Jools Holland, performing her song "I Don't Know".^{[15][32][33][34]} Sea Sew rose in the UK charts following this appearance^{[citation needed]} and she performed at Glastonbury 2009 music festival and went on tour later in the year.^{[26][34][35][36]} She performed at the nomination ceremony and she was greeted by confused journalists wondering "Lisa who?".^[18]

Hannigan performed at Electric Picnic 2009^{[8][37][38][39][40][41]} and Le Chéile.^[42] Later in 2009, she toured the United States with David Gray and performed solo shows in New York, Los Angeles and London.^[43] She then began a tour of Ireland to finish the year.^[43] Hannigan's song "Ocean and Rock" was used in a 2009 Irish video supporting same-sex marriage entitled "Sinéad's Hand".^[44]

In 2009, a broadcast of Other Voices was recorded.^[45]

2011–present[link]

Hannigan recorded her second album, titled Passenger, at Bryn Derwen Studios in North Wales with producer Joe Henry and engineer Ryan Freeland. The album was released in the US and Canada on 20 September 2011, and on 7 October in Ireland and the UK.^[46][47]

Hannigan performed at the Eurosonic Festival in 2012 when Ireland was the "Spotlight Country".^[48]

Reception[link]

Her music has received critical praise, airplay and award nominations both in her native Ireland and the United States. In 2008, her debut album Sea Sew spawned the single "Lille", a Choice Music Prize nomination, two Meteor Music Awards nominations and a Mercury Prize nomination. There has also been praise from fellow musicians and, in recent times, Jools Holland of the United Kingdom and US television hosts Jay Leno and Stephen Colbert.

Hannigan performs using "broken-down, wheezy old instruments", assisting her band members in the transportation of this equipment. Her blog posts are noted for containing recipes for baking cakes.^[15] Herbie Hancock said of her vocals, "there's so much jazz in the notes and phrases that she picks. She was singing the ninths, the elevenths of the chords...I mean some of the things sound like choices that Miles would have made." Possibilities DVD.^[13]

To date,^[when?] Sea Sew has sold 45,000 copies worldwide – with 15,000 of those copies sold in Ireland.

Personal life[link]

As part of an advertisement campaign for Oxfam's Make Trade Fair, Hannigan was drenched in melted chocolate and she participated in the Irish musical collective The Cake Sale with lead vocal on the track Some Surprise, which was played on the US television series Grey's Anatomy.^{[3][12][13][16][49][50]}

Hannigan also contributed to the 2009 charity album, Sparks n' Mind, released in aid of Aware.^[51]

Discography[link]

• Sea Sew (2008)
• Passenger (2011)

Awards[link]

[edit] Hot Press Readers' Poll

Hannigan has four Hot Press Readers' Poll awards. She won Best Debut Album, Best Irish Album, Best Irish Track and Best Female in 2009.^[26]

Choice Music Prize[link]

Hannigan's debut album, Sea Sew, was nominated for the Choice Music Prize in January 2009.^[52] The award was won by Jape for the album Ritual on 4 March 2009.^{[53][54][55][56][57][58][59]}

Hannigan's second album, Passenger, was nominated for the Choice Music Prize in January 2012.^[60]

Meteor Music Awards[link]

Hannigan and her work was nominated in the Best Irish Female and Best Irish Album categories at the Meteor Music Awards in 2009.^[61] She was the only solo artist nominated in more than one category at the awards. She lost to Imelda May and The Script's self-titled album respectively.^[62]

Mercury Prize[link]

Sea Sew was nominated for the Mercury Prize on 21 July 2009.^{[63][64][65][66][67][68][69][70]} It was Hannigan's first nomination.

Hannigan was referred to as 2009's "token folk nominee" in the UK,^[16] with the NME calling her a "token folkie".^[71] One British journalist even claimed she was "truly obscure" and part of the "moribund sensitive singer-songwriter genre".^[16] Ed Power, writing in the Irish Independent, criticised such claims, wondering if Jape (whose album Ritual beat Hannigan to the Choice Music Prize) had come close to receiving a Mercury nomination—"Or, for that matter, how many of the judges had even heard of him".^[16]

See also[link]

Side projects of Lisa Hannigan

References[link]