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Venn diagram

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   A Venn diagram of sets A, B, and C
   A Venn diagram of sets A, B, and C

   Venn diagrams are illustrations used in the branch of mathematics known
   as set theory. They show all of the possible mathematical or logical
   relationships between sets (groups of things).

Origins

   Stained glass window in the dining hall of Gonville and Caius College,
   Cambridge
   Stained glass window in the dining hall of Gonville and Caius College,
   Cambridge

   The Hull-born British philosopher and mathematician, John Venn
   (1834-1923), introduced the Venn diagram in 1881.

   A stained glass window in Caius College, Cambridge, where Venn studied
   and spent most of his life, commemorates him and represents a Venn
   diagram.

Examples

   The orange circle (set A) might represent, for example, all living
   creatures which are two-legged. The blue circle, (set B) might
   represent living creatures which can fly. The area where the blue and
   orange circles overlap (which is called the intersection) contains all
   living creatures which can fly and which have two legs — for example,
   parrots. (Imagine each separate type of creature as a point somewhere
   in the diagram.)

   Humans and penguins would be in the orange circle, in the part which
   does not overlap with the blue circle. Mosquitos have six legs, and
   fly, so the point for mosquitos would be in the part of the blue circle
   which does not overlap with the orange one. Things which do not have
   two legs and cannot fly (for example, whales and rattlesnakes) would
   all be represented by points outside both circles. Technically, the
   Venn diagram above can be interpreted as "the relationships of set A
   and set B which may have some (but not all) elements in common".

   The combined area of sets A and B is called the union of sets A and B.
   The union in this case contains all things which either have two legs,
   or which fly, or both.

   The area in both A and B, where the two sets overlap, is defined as
   A∩B, that is A intersected with B. The intersection of the two sets is
   not empty, because the circles overlap, i.e. there are creatures that
   are in both the orange and blue circles.

   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).

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 with any number of curves, where each
   successive curve is interleaved with previous curves, starting with the
   3-circle diagram.

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. In 2003, work
   by Griggs, Killian, and Savage showed that this condition is also
   sufficient.

Edwards' Venn diagrams

     Edwards' Venn diagram of three sets
     Edwards' Venn diagram of three sets

                                        Edwards' Venn diagram of four sets
                                        Edwards' Venn diagram of four sets

      Edwards' Venn diagram of five sets
      Edwards' Venn diagram of five sets

                                         Edwards' Venn diagram of six sets
                                         Edwards' Venn diagram of six sets

   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.

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.

   Smith devised similar n-set diagrams using sine curves with equations
   y=sin(2^ix)/2^i, 0≤i≤n-2.

   Charles Lutwidge Dodgson (a.k.a. Lewis Carroll) devised a five set
   diagram.

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.
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