NationMaster - Encyclopedia: Venn diagram

Venn diagrams (or set diagrams) are illustrations used in the branch of mathematics known as set theory. Invented in 1881 by John Venn, they show all of the possible mathematical or logical relationships between sets (groups of things). They normally consist of overlapping circles. 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 (intersection) would then represent the set of all wooden tables. Shapes other than circles can be employed (see below), and this is necessary for more than three sets. Image File history File links Venn_diagram_cmyk. ... Image File history File links Venn_diagram_cmyk. ... Illustration by Jessie Willcox Smith. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... John Venn. ... Logic (from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... This article is about the shape and mathematical concept of circle. ... For other uses, see Wood (disambiguation). ...

A simple example

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. Image File history File links Venn-diagram-AB.svgâ€Ž hola File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Venn-diagram-AB.svgâ€Ž hola File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

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 things that either have two legs, or that fly, or both. In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

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 the example, the intersection of the two sets is not empty, because there are points representing creatures that are in both the orange and blue circles. In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

Sometimes a rectangle called the "Universal set" is drawn around the Venn diagram to show the space of all possible things. As mentioned above, a whale would be represented by a point that is not in the union, but is in the Universe (of living creatures, or of all things, depending on how one chose to define the Universe for a particular diagram). In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ...

Extensions to higher numbers of sets

Venn diagrams typically have three sets. Venn was keen to find symmetrical figures…elegant in themselves representing higher numbers of sets and he devised a four-set diagram using ellipses. He also gave a construction for Venn diagrams for any number of sets, where each successive curve delimiting a set is interleaved with previous curves, starting with the 3-circle diagram. For other uses, see Ellipse (disambiguation). ...

Simple symmetric Venn diagrams

D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was prime.^[1] 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.^[2] The triskelion appearing on the Isle of Man flag. ... In mathematics, a prime number (or a prime) is a natural number greater than 1 which has exactly two distinct natural number divisors: 1 and itself. ...

Edwards' Venn diagrams

A. W. F. Edwards gave a construction to higher numbers of sets that features some symmetries. His construction is achieved by projecting the Venn diagram onto a sphere. Three sets can be easily represented by taking three hemispheres at right angles (x≥0, y≥0 and z≥0). A fourth set can be represented by taking a curve similar to the seam on a tennis ball which winds up and down around the equator. The resulting sets can then be projected back to the plane to give cogwheel diagrams with increasing numbers of teeth. These diagrams were devised while designing a stained-glass window in memoriam to Venn. Image File history File links Venn-three. ... Image File history File links Venn-three. ... Image File history File links Edwards-Venn-four. ... Image File history File links Edwards-Venn-four. ... Image File history File links Edwards-Venn-five. ... Image File history File links Edwards-Venn-five. ... Image File history File links Edwards-Venn-six. ... Image File history File links Edwards-Venn-six. ... Professor Anthony William Fairbank Edwards (born 1935) is a British statistician, geneticist and evolutionary biologist. ... For other uses, see Sphere (disambiguation). ... A large Perpendicular style Gothic window of eight lights in Canterbury Cathedral, c. ...

Other diagrams

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. Branko GrÃ¼nbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ... Look up polygon in Wiktionary, the free dictionary. ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ...

Smith devised similar n-set diagrams using sine curves with equations y=sin(2ⁱx)/2ⁱ, 0≤i≤n-2. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Charles Lutwidge Dodgson (a.k.a. Lewis Carroll) devised a five set diagram. The Reverend Charles Lutwidge Dodgson (IPA: ) (27 January 1832 â€“ 14 January 1898), better known by the pen name Lewis Carroll (), was an English author, mathematician, logician, Anglican clergyman and photographer. ...

Classroom use

Venn diagrams are often used by teachers in the classroom as a mechanism to help students compare and contrast two items. Characteristics are listed in each section of the diagram, with shared characteristics listed in the overlapping section.

See also

Boolean algebra is the finitary algebra of two values. ... A simple Carroll diagram. ... Sample flowchart diagram A diagram is a 2D symbolic representation of information according to some visualization technique. ... An Euler diagram does not need to show all possible intersections. ... Graphic organizers are visual representations of knowledge, concepts or ideas. ... Mrs. ... A spider diagram adds existential points to an Euler diagram. ... A bubble map is an organizational tool that enables one to project their thoughts onto paper with a diagram resembling a web. It usually consists of a single large circle in the center, which states the main idea. ... A double bubble map is an organizational tool that allows one to place their thoughts about two things that are both different and similar into an easy-to-read format. ...

Notes

^ D. W. Henderson, "Venn diagrams for more than four classes". American Mathematical Monthly, 70 (1963) 424–426.
^ Ruskey, Frank; Carla D. Savage, and Stan Wagon (December 2006). "The Search for Simple Symmetric Venn Diagrams" (PDF). Notices of the AMS 53 (11): 1304-1311. Retrieved on 2007-04-27.

The American Mathematical Monthly is a mathematical journal published 10 times each year by the Mathematical Association of America since 1894. ... Notices of the AMS, March 2005 issue. ... Year 2007 (MMVII) was a common year starting on Monday of the Gregorian calendar in the 21st century. ... is the 117th day of the year (118th in leap years) in the Gregorian calendar. ...

References

A Survey of Venn Diagrams by F. Ruskey and M. Weston, is an extensive site with much recent research and many beautiful figures.
I. Stewart Another Fine Math You've Got Me Into 1992 ch4
A.W.F. Edwards. Cogwheels of the Mind: the story of Venn diagrams, Johns Hopkins University Press, Baltimore and London, 2004.
- Review of Cogwheels of the Mind
"Venn Diagram Survey: Symmetric Diagrams", The Electronic Journal of Combinatorics, June 2005).
John Venn (1880). "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings". Dublin Philosophical Magazine and Journal of Science 9 (59): 1--18.

External links

cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...

Tools for making Venn Diagrams

A plot of Wikipedia statistics in Ploticus Ploticus is a free, GPL program for producing plots and charts from data. ... ConceptDraw V is a professional cross-platform drawing and diagramming software for quick creation of business diagrams, flowcharts, network diagrams, floor plans, technical drawings, home and office layouts and much more. ... SmartDraw is the first design tool for people who have never designed anything. ... Power point redirects here. ...

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