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.

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.

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.

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.

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 for any number of sets, where each successive curve delimiting a set 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.^[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]

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.

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(2ⁱx)/2ⁱ, 0≤i≤n-2.

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.

Contents

A simple example

Extensions to higher numbers of sets

Simple symmetric Venn diagrams

Edwards' Venn diagrams

Other diagrams

Classroom use

See also

Notes

References

External links

Tools for making Venn Diagrams