Venn diagram
A Venn diagram uses simple closed curves drawn on a plane to represent sets.
Similar ideas had been proposed before Venn such as by Christian Weise in 1712 (Nucleus Logicoe Wiesianoe) and Leonhard Euler (Letters to a German Princess) in 1768.
The idea was popularised by Venn in Symbolic Logic, Chapter V "Diagrammatic Representation", published in 1881.
These diagrams depict elements as points in the plane, and sets as regions inside closed curves.
A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set.
Living creatures that have two legs and can fly—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap.
The combined region of the two sets is called their union, denoted by A ∪ B, where A is the orange circle and B the blue.
The region included in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. Venn diagrams were introduced in 1880 by John Venn in a paper entitled "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings"[2] in the Philosophical Magazine and Journal of Science,[3] about the different ways to represent propositions by diagrams.
[7] Diagrams of overlapping circles representing unions and intersections were introduced by Catalan philosopher Ramon Llull (c. 1232–1315/1316) in the 13th century, who used them to illustrate combinations of basic principles.
[12] In the opening sentence of his 1880 article Venn wrote that Euler diagrams were the only diagrammatic representation of logic to gain "any general acceptance".
[4][5] Venn viewed his diagrams as a pedagogical tool, analogous to verification of physical concepts through experiment.
The term "Venn diagram" was later used by Clarence Irving Lewis in 1918, in his book A Survey of Symbolic Logic.
David Wilson Henderson showed, in 1963, that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number.
These combined results show that rotationally symmetric Venn diagrams exist, if and only if n is a prime number.
[18] A Venn diagram is constructed with a collection of simple closed curves drawn in a plane.
Venn diagrams do not generally contain information on the relative or absolute sizes (cardinality) of sets.
Shown below, four intersecting spheres form the highest order Venn diagram that has the symmetry of a simplex and can be visually represented.
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 here.
Henry John Stephen Smith devised similar n-set diagrams using sine curves[21] with the series of equations
Joaquin and Boyles, on the other hand, proposed supplemental rules for the standard Venn diagram, in order to account for certain problem cases.
