Venn Diagrams and Applications 

Course:  For finite mathematics students majoring in Business, Management, or Economics /Social & Life sciences

Created by Kameswari Tekumalla for Synergy Project (Feb 2002)

Uses websites about Venn diagrams produced by A.J.Ronald, Campbell, and NY State Exam Prep. Center.

Background: In the mid-nineteenth century, John Venn (1834-1923), a Fellow of Cambridge University, devised a scheme for visualizing logical relationships. His single contribution to the field of Mathematics made him immortal. That contribution is the Venn diagram. Venn wrote Logic of Chance in 1866 which his colleague Keynes described as  “strikingly original and considerably influenced the development of the theory of Statistics. “  It is a technique for analyzing visually and solving many problems and logical relationships. A Venn diagram is simply a field within which circular areas represent groups of items sharing common properties. The Venn diagram is made up of two or more overlapping circles and a rectangle is used to represent the Universal Set (Universal Set includes all objects being analyzed). Venn Diagrams are often used in mathematics to show relationships between sets.Consider a Universal set with two subsets A and B. The union of A and B is everything which is in either A or B.  We may represent this as AÈB. The intersection of two sets is that which is in both sets, as represented as AÇB. The complement of a set A is everything that is not in A; it is represented by A' or Ac. To know more about them, visit the website. Venn Diagrams . .  A set is a list of objects in no particular order; they could be numbers, letters,  or even words. A Venn diagram is a way of representing sets visually. To learn more, visit the site Sets and Venn diagrams. Venn Diagrams are useful to solve many problems that involve critical thinking. To see examples visit the website Practice Problems . If you like to create  a Venn diagram on web, Click here.


Using the data from a survey of critical care hospital patients, complete the Venn diagram and answer the questions below.

80 patients were surveyed with the following results;

40 patients had Diabetes (D)

30 Patients had Heart Disease (H)

30 Patients had Cancer (C)

15 Patients had Diabetes and Heart Disease

14 Patients had Diabetes and Cancer

18 Patients had Heart Disease and Cancer

10 Patients had all three

  a) How many patients had exactly two   of the ailments?        _________


  b) How many patients suffered from none of the three ailments? ________


   c) How many patients suffered from Diabetes    only?  ______


   d) How many patients had at least two of the   ailments? ______