TOPICS
Search

Venn Diagram


VennDiagram

A schematic diagram used in logic theory to depict collections of sets and represent their relationships.

The Venn diagrams on two and three sets are illustrated above. The order-two diagram (left) consists of two intersecting circles, producing a total of four regions, A, B, A intersection B, and emptyset (the empty set, represented by none of the regions occupied). Here, A intersection B denotes the intersection of sets A and B.

The order-three diagram (right) consists of three symmetrically placed mutually intersecting circles comprising a total of eight regions. The regions labeled A, B, and C consist of members which are only in one set and no others, the three regions labelled A intersection B, A intersection C, and B intersection C consist of members which are in two sets but not the third, the region A intersection B intersection C consists of members which are simultaneously in all three, and no regions occupied represents emptyset.

In general, an order-n Venn diagram is a collection of n simple closed curves in the plane such that

1. The curves partition the plane into 2^n connected regions, and

2. Each subset S of {1,2,...,n} corresponds to a unique region formed by the intersection of the interiors of the curves in S (Ruskey).

Since there are (n; k) (the binomial coefficient) ways to pick k members from a total of n, the number of regions in an order n Venn diagram is

 N=sum_(k=0)^n(n; k)=2^n,

(where the region outside the diagram is included in the count).

The region of intersection of the three circles A intersection B intersection C in the order three Venn diagram in the special case of the center of each being located at the intersection of the other two is a geometric shape known as a Reuleaux triangle.

Venn diagrams

The left figure at left above shows an n=5 Venn diagram due to Branko Grünbaum, while the attractive 7-fold rosette illustrated in the middle figure is an n=7 Venn diagram called "Victoria" by Ruskey. The right figure shows a recently constructed symmetric Venn diagram on n=11 due to Ruskey, Carla Savage, and Stan Wagon.

In Season 4 episode "Power" of the television crime drama NUMB3RS, mathematical genius Charles Eppes constructs a Venn diagram to determine suspects who match a particular description and have a history of violence.


See also

Circle, Flower of Life, Haruki's Theorem, Intersection, Lens, Magic Circles, Reuleaux Triangle, Seed of Life

Explore with Wolfram|Alpha

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 255-256, 1989.Grünbaum, B. "On Venn Diagrams and the Counting of Regions." College Math. J. 15, 433-435, 1984.Grünbaum, B. "Venn Diagrams and Independent Families of Sets." Math. Mag. 48, 12-23, 1975.Henderson, D. W. "Venn Diagrams for More Than Four Classes." Amer. Math. Monthly 70, 424-426, 1963.Ogilvy, C. S. "Solution to Problem E 1154." Amer. Math. Monthly 62, 584-585, 1955.Ruskey, F. "A Survey of Venn Diagrams." Electronic J. Combinatorics Dynamical Survey DS5, June 18, 2005. http://www.combinatorics.org/Surveys/#DS5.Ruskey, F. "Venn Diagrams." http://www.theory.csc.uvic.ca/~cos/inf/comb/SubsetInfo.html#Venn.Ruskey, F.; Savage, C. D., and Wagon, S. "The Search for Simple Symmetric Venn Diagrams." Not. Amer. Math. Soc. 53, 1304-1311, 2006.Venn, J. "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings." Dublin Philos. Mag. J. Sci. 9, 1-18, 1880.

Referenced on Wolfram|Alpha

Venn Diagram

Cite this as:

Weisstein, Eric W. "Venn Diagram." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VennDiagram.html

Subject classifications