This document discusses applications of sets and the inclusion-exclusion principle through examples involving surveys and percentages. It contains examples such as a music survey of students' music preferences that is solved using a Venn diagram. The inclusion-exclusion principle is then introduced and used to solve problems involving counting the number of students in bands and choirs. It also provides a formula for inclusion-exclusion that can be used when dealing with percentages.

1. Copyright © Cengage Learning. All rights reserved.
CHAPTER
Sets
2

2. Copyright © Cengage Learning. All rights reserved.
Applications of Sets
Section2.4

3. 3
Surveys: An Application of Sets

4. 4
Surveys: An Application of Sets
Counting problems occur in many areas of applied
mathematics. To solve these counting problems, we often
make use of a Venn diagram and the inclusion-exclusion
principle, which will be presented in this section.
In the next example we consider a complicated survey that
involves three types of music.

5. 5
Example 2 – A Music Survey
A music teacher has surveyed 495 students. The results of
the survey are listed below.
320 students like rap music.
395 students like rock music.
295 students like heavy metal music.
280 students like both rap music and rock music.
190 students like both rap music and heavy metal music.
245 students like both rock music and heavy metal music.
160 students like all three.

6. 6
Example 2 – A Music Survey
How many students
a. like exactly two of the three types of music?
b. like only rock music?
c. like only one of the three types of music?

7. 7
Example 2 – Solution
The Venn diagram below shows three overlapping circles.
Region i represents the set of students who like all three
types of music. Each of the regions v, vi, and vii represent
the students who like only one type of music.

8. 8
Example 2 – Solution
a. The survey shows that 245 students like rock and heavy
metal music, so the numbers we place in regions i and iv
must have a sum of 245.
Since region i has 160 students, we see that region iv
must have 245 – 160 = 85 students. In a similar manner,
we can determine that region ii has 120 students and
region iii has 30 students.
Thus 85 + 120 + 30 = 235 students like exactly two of
the three types of music.
cont’d

9. 9
Example 2 – Solution
b. The sum of the students represented by regions i, ii, iv,
and v must be 395. The number of students in region v
must be the difference between this total and the sum of
the numbers of students in region i, ii, and iv.
Thus the number of student who like only rock music is
395 – (160 + 120 + 85 ) = 30.
See the Venn diagram below.
cont’d

10. 10
Example 2 – Solution
c. Using the same reasoning as in part b, we find that
region vi has 10 students and region vii has 20 students.
To find the number of students who like only one type of
music, find the sum of the numbers of students in
regions v, vi, and vii, which is 30 + 10 + 20 = 60.
Refer the earlier Venn diagram.
cont’d

11. 11
The Inclusion-Exclusion Principle

12. 12
The Inclusion-Exclusion Principle
A music director wishes to take the band and the choir on a
field trip. There are 65 students in the band and 30 students
in the choir. The number of students in both the band and the
choir is 16. How many students should the music director
plan on taking on the field trip?
Using the process developed in the previous example, we
find that the number of students that are in only the band is
65 – 16 = 49. The number of students that are in only the
choir is 30 – 16 = 14.

13. 13
The Inclusion-Exclusion Principle
See the Venn diagram below. Adding the numbers of
students in regions i, ii, and iii gives us a total of
49 + 16 + 14 = 79 students that might go on the field trip.

14. 14
The Inclusion-Exclusion Principle
Although we can use Venn diagrams to solve counting
problems, it is more convenient to make use of the following
technique.
First add the number of students in the band to the number
of students in the choir. Then subtract the number of students
who are in both the band and the choir.
This technique gives us a total of (65 + 30) – 16 = 79
students, the same result as before.

15. 15
The Inclusion-Exclusion Principle
The reason we subtract the 16 students is that we have
counted each of them twice. Note that first we include the
students that are in both the band and the choir twice, and
then we exclude them once.
This procedure leads us to the following result.

16. 16
Example 3 – An Application of the Inclusion-Exclusion Principle
A school finds that 430 of its students are registered in
chemistry, 560 are registered in mathematics, and 225 are
registered in both chemistry and mathematics. How many
students are registered in chemistry or mathematics?
Solution:
and let

17. 17
Using the inclusion-exclusion principle, we see that 765
students are registered in chemistry or mathematics.
Example 3 – Solution cont’d

18. 18
The Inclusion-Exclusion Principle
The inclusion-exclusion formula can be adjusted and
applied to problems that involve percents. In the following
formula we denote “the percent in set A” by the notation
p(A).

19. 19
Example 5 – An Application of the Percent Inclusion-Exclusion Formula
A blood donation organization reports that about
44% of the U.S. population has the
A antigen.
15% of the U.S. population has the
B antigen.
4% of the U.S. population has both
the A and the B antigen.
Use the percent inclusion-exclusion
formula to estimate the percent of
the U.S. population that has the
A antigen or the B antigen.

20. 20
Example 5 – Solution
We are given p(A) = 44%, p(B) = 15%, and p(AB) = 4%.
Substituting in the percent inclusion-exclusion formula gives
Thus about 55% of the U.S. population has the A antigen or
the B antigen.