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?
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!