Venn Diagrams

Venn diagrams are a method to help solve problems in market research, in science, in social science, etc. where often overlapping information is collected and needs to be sorted out.

John Venn (1834 - 1923) for more information about Venn, check out: Who was John Venn? http://sue.csc.uvic.ca/~cos/venn/VennJohnEJC.html

 

Examples

1. From a survey of 100 college students, a marketing research company found that 75 students owned stereos, 45 owned cars, and 35 owned cars and stereos. a) How many students owned either a car or a stereo? b) How many students did not own either a car or a stereo?  METHOD: a) Start with a Venn Diagram and label the different categories: b) Fill in the number of students who own both cars and stereos, which would be in the intersection of the two sets: c) Fill in the remaining numbers for the two sets. In this case, since a total of 45 students own cars, and 35 have already been listed, then 45 - 35 = 10 students own cars only. Similarly, since 75 students own stereos and 35 have already been listed, then 75 - 35 = 40 students who own stereos only: d) Finally, interpret and answer the questions: How many students owned either a car or a stereo? The question asks either ... or which is union of the sets. From the diagram, the number of elements in A = 10 + 35 and the number of elements in B which are NOT in A are 40. So the union would be 10+35+40 = 85 How many students did not own either a car or a stereo? The question asks for the number not in either A nor B (namely, the complement of A B or (A B)' ). Since there are 100 students in the universe, then the complement is found by subtracting those who own either a car or stereo from the total number of students surveyed or 100 - 85 = 15.

2. Suppose n(U) = 150, n(A) = 37, and n(B) = 84.  a) If n( A U B) = 100, find n(a B) and draw a Venn diagram illustrating the composition of U. b) How many elements belong to A only?  METHOD: a) Start with a Venn diagram and label the categories: b) Since the number of elements in the union = 100, add the number of elements in A to the numbers of elements in B: 37 + 84 = 121. But 121 is larger than 100, which means that 121 - 100 = 21 must be in both sets or in the intersection! And the first question (If n( A U B) = 100, find n(a B) and draw a Venn diagram illustrating the composition of U.) is answered! c) Using the information about the intersection, the other numbers can then be filled in: And the second question (How many elements belong to A only?) is answered since the total number of elements in A is 37. Then the remaining elements in A will be 37 - 21 = 16!

 

 

Week 2