Chapter 6: Section 6-2Applications of Venn Diagrams

D. S. MalikCreighton University, Omaha, NE

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 1 / 24

Suppose that the universal set, U, under consideration is finite.

Suppose that U, A, and B are sets such that

U = {1, 2, 3, 4, 5, 6, 7, 8},

A = {1, 3, 7} and B = {2, 5, 6, 8}.Then

n(A) = 3,n(B) = 4,A∩ B = ∅, n(A∩ B) = 0,A∪ B = {1, 2, 3, 5, 6, 7, 8}, and n(A∪ B) = 7.

n(A∪ B) = 7 = 3+ 4− 0 = n(A) + n(B)− n(A∩ B).

A′ = {2, 4, 5, 6, 8}.

n(A′) = 5 = 8− 3 = n(U)− n(A).

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 2 / 24

Example

Let U = Set of English alphabet. Let A = {a, e, i , o, u} andB = {a, b, e, g , i , k,m, u, y , z}. Then

A∪ B = {a, b, e, g , i , k,m, o, u, y , z} and A∩ B = {a, e, i , u}.

Thus,

n(A∪ B) = 11, n(A) = 5, n(B) = 10, and n(A∩ B) = 4.

Note that

n(A∪ B) = 11 = 5+ 10− 4 = n(A) + n(B)− n(A∩ B).

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 3 / 24

TheoremLet A and B be subsets of a finite set U.(i) n(∅) = 0(ii) n(U) = n(A) + n(A′), n(A) = n(U)− n(A′), andn(A′) = n(U)− n(A).(iii) If A∩ B = ∅, then

n(A∪ B) = n(A) + n(B).

(iv)n(A∪ B) = n(A) + n(B)− n(A∩ B).

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 4 / 24

Example

Let A and B be subsets of U such that n(A) = 48, n(B) = 27, andn(A∩ B) = 15. Then

n(A∪ B) = n(A) + n(B)− n(A∩ B) = 48+ 27− 15 = 60.

Example

Let A and B be subsets of U such that n(U) = 55, n(A′) = 32,n(B) = 31, and n(A∩ B) = 9. Then

n(A) = n(U)− n(A′) = 55− 32 = 23.

Thus,

n(A∪ B) = n(A) + n(B)− n(A∩ B) = 23+ 31− 9 = 47.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 5 / 24

Example

Let A and B be subsets of U such that n(A∪ B) = 74, n(A) = 58,n(B) = 47. Let us find n(A∩ B).Suppose that n(A∩ B) = x . Then

n(A∪ B) = n(A) + n(B)− n(A∩ B)⇒ 74 = 58+ 47− x⇒ 74 = 105− x⇒ 74− 105 = −x⇒ −31 = −x⇒ x = 31.

Hence,n(A∩ B) = 31.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 6 / 24

Example

Let U be a universal set with subsets A and B such that n(U) = 12,n(A) = 7, n(B) = 8, and n(A∩ B) = 5. Then

U

A B

5

(a)

A B

U

A B

5

A B

2

2 3

A ­ B B ­ A

(b)

From this figure,

n(A∪ B) = n(A− B) + n(B − A) + n(A∩ B) = 2+ 3+ 5 = 10.

Also note that n(U) = 12 and n(A∪ B) = 10. So the number of elementsthat are not in A∪ B is 12− 10 = 2. That is,

n((A∪ B)′) = 2.D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 7 / 24

ExampleIn a survey of 150 people, the following information was gathered.(i) 110 people use the Internet to pay bills,(ii) 70 people use regular mail to pay bills, and(iii) 40 people use both the Internet and regular mail to pay bills.We determine the number of people who use either the Internet or regularmail to pay their bills.Let U denote the set of people surveyed, I denote the set of people whouse the Internet to pay bills, and M denote the set of people who useregular mail to pay bills. Then,

n(I ) = 110, n(M) = 70.

Because 40 people use both the Internet and regular mail to pay bills,

n(I ∩M) = 40.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 8 / 24

Then,n(I ) = 110, n(M) = 70.

Because 40 people use both the Internet and regular mail to pay bills,

n(I ∩M) = 40.

Thus,I ­ M

U

I M

40

(a)

U

I M

40

10

70 30

M ­ I

(b)

I MI M

From this figure, we have

n(I ∪M) = n(I −M) + n(M − I ) + n(I ∩M) = 70+ 30+ 40 = 140.

This implies that the number of people who use the Internet or regularmail to pay their bills is 140.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 9 / 24

Venn Diagram and Three Subsets

Let A, B, and C be subsets of a universal set U. In general, the Venndiagram containing three sets is:

U

AB

C

Reg 1

Reg 2

Reg 3 Reg 4

Reg 5Reg 6

Reg 7Reg 8

A    B    C

From this region, it is clear that the region is divided into eight regions.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 10 / 24

RemarkAs in the case of a Venn diagram of two sets, in general, when writing theelements of various regions, we start from the inside out.First we write the number of elements of Reg 1, which is the number ofelements in A∩ B ∩ C . If the number of elements in A∩ B ∩ C is notgiven, then we can assume that the number of elements in A∩ B ∩ C is,say x .Next, we write the elements in Reg 2, Reg 3, and Reg 4, taking intoaccount the number of elements in Reg 1. For example, suppose thatn(A∩ B) = 10 and n(A∩ B ∩ C ) = 4. Then

n(Reg 1) = 4

andn(Reg 2) = 10− 4 = 6.

Finally, we write the number of elements in Reg 5, Reg 6, and Reg 7,taking into account the number of elements in Reg 1 to Reg 4.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 11 / 24

Example

Let A, B, and C be subsets of a universal set U such that n(U) = 110,n(A) = 38, n(B) = 23, n(C ) = 50, n(A∩ B) = 14, n(B ∩ C ) = 10,n(A∩ C ) = 12, and n(A∩ B ∩ C ) = 6. Then

U

A

34

B

C

6

8

6 4

18 5

Thus,n(A∪ B ∪ C ) = 6+ 8+ 6+ 4+ 18+ 5+ 34 = 81

Now n(U) = 110 and n(A∪ B ∪ C ) = 81. So

n((A∪ B ∪ C )′) = n(U)− n(A∪ B ∪ C ) = 110− 81 = 29.D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 12 / 24

ExampleIn a survey of 100 students, at a local university, the following informationis collected. 55 students play basketball, 39 play soccer, and 47 play golf.22 play basketball and soccer, 15 play soccer and golf, and 18 playbasketball and golf. 8 students play all three sports. LetB = set of students who play basketball,S = set of students who play soccer, andG = set of students who play golf.Then from the given information, we have

n(B) = 55, n(S) = 39, n(G ) = 47,n(B ∩ S) = 22, n(S ∩ G ) = 15, n(B ∩ G ) = 18,n(B ∩ S ∩ G ) = 8.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 13 / 24

Then from the given information, we have

n(B) = 55, n(S) = 39, n(G ) = 47,n(B ∩ S) = 22, n(S ∩ G ) = 15, n(B ∩ G ) = 18,n(B ∩ S ∩ G ) = 8.

Thus, we have the Venn diagram shown in the following figure:

U

B S

G

Reg 18

Reg 214

Reg 310

Reg 47

Reg 523 Reg 6

10

Reg 722

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 14 / 24

Example1 The number of students who play basketball and soccer, but not golfis the number of players in Reg 2. Thus, the number of students whoplay basketball and soccer, but not golf is 14.

2 The number of students who play basketball and golf, but not socceris 10.

3 The number of students who play soccer and golf, but not basketballis 7.

4 The number of students who play basketball, but neither soccer norgolf is the number of players in Reg 5. Thus, the number of studentswho play basketball, but neither soccer nor golf is 23. This also meansthat the number of students who play only basketball is 23.

5 The number of students who play soccer, but neither basketball norgolf is 10. This also means that the number of students who playonly soccer is 10.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 15 / 24

Example6. The number of students who play golf, but neither basketball norsoccer is 22. This also means that the number of students who playonly golf is 22.

7. The number of students who play at least one of the sports isn(B ∪ S ∪ G ). So we add the numbers in Reg 1 to Reg 7. Thus,

n(B ∪ S ∪ G ) = 8+ 14+ 10+ 7+ 23+ 10+ 22 = 94.

Thus, the number of students who play at least one of the sports is94.

8. The number of students who do not play any of the three sports isthe number of students in Reg 8. Now,

n(Reg 8) = 100− 94 = 6.

Hence, there are 6 students who do not play any of the three sports.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 16 / 24

Exercise: During July, a car dealer sold 100 cars. Of these, 83 cars hadeither leather seats or DVD players, 48 cars had leatherseats, and 56 cars had DVD players.

1 How many cars sold had both leather seats and a DVDplayer?

2 How many cars sold had leather seats but not DVDplayers?

3 How many cars sold had neither leather seats nor DVDplayers?

Solution: Let L denote the set of cars that had leather seats and Ddenote the set of cars that had DVD players. Then,

n(L∪D) = 83, n(L) = 48, and n(D) = 56.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 17 / 24

Solution: Let L denote the set of cars that had leather seats and Ddenote the set of cars that had DVD players. Then,

n(L∪D) = 83, n(L) = 48, and n(D) = 56.

Suppose that n(L∩D) = x .Then

n(L−D) = n(L)− n(L∩D) = 48− x ,

andn(D − L) = n(D)− n(L∩D) = 56− x .

We thus have the following figure:

U

L D

48 ­ x x 56 ­ x

L ­ D L D D ­ L

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 18 / 24

Solution:

U

L D

48 ­ x x 56 ­ x

L ­ D L D D ­ L

From this figure,

n(L∪D) = 48− x + x + 56− x⇒ 83 = 104− x⇒ x = 104− 83⇒ x = 21.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 19 / 24

Solution:U

L D

27 21 35

1 The number of cars with both leather seats and a DVDplayer is

n(L∩D) = x = 21.2 The number of cars with leather seats but not DVDplayers is

n(L−D) = 48− x = 48− 21 = 27.3 The number of cars with neither leather seats nor DVDplayers is

n((L∪D)′) = n(U)− n(L∪D) = 100− 83 = 17.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 20 / 24

Exercise: A campus survey of Internet use by 200 students revealedthe following information.110 students use the Internet to download music files.; 90students use the Internet to download video files; 60 studentsuse the Internet to download research papers to write termpapers; 55 students use the Internet to download music andvideo files; 37 students use the Internet to download musicfiles and research papers; 25 students use the Internet todownload video files and research papers; and 160 studentsuse the Internet to download music files or video files orresearch papers.

1 How many students use the Internet for all threeactivities?

2 How many students use the Internet for activities otherthan either of these three activities?How many students use the Internet to download musicor video files but not research papers?

3 How many students use the Internet to download onlymusic files?

4 How many students use the Internet to download onlyresearch papers?

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 21 / 24

Solution: Let M be the set of students who use the Internet todownload music files, V be set of students who use theInternet to download music files, and R be the set ofstudents who use the Internet to download research papers.Then,

n(M) = 110, n(V ) = 90, n(R) = 60,n(M ∩ V ) = 55, n(M ∩ R) = 37, n(V ∩ R) = 25,n(M ∪ V ∪ R) = 160.

Also n(U) = 200. Let us suppose that n(M ∩ V ∩ R) = x .Then

U

MV

R

18 + x 55 ­ x 10 + x

x37 ­ x 25 ­ x

­2 + x

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 22 / 24

Solution: Because n(M ∪ V ∪ R) = 160, we have

18+ x + 55− x + 10+ x + x + 37− x + 25− x − 2+ x = 160⇒ x + 143 = 160⇒ x = 17.

Hence, n(M ∩ V ∩ R) = 17. So 17 students use the Internetfor all three activities.

U

MV

R

35 38 27

1720 8

15

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 23 / 24

The number of students who use the Internet for activities other thaneither of these three activities is:

n((M ∪ V ∪ R)′) = n(U)− n(M ∪ V ∪ R) = 200− 160 = 40.

From the Venn diagram, the number of students who use the Internetto download music or video files but not research papers is

35+ 38+ 27 = 100.

From the Venn diagram, 35 students use the Internet to downloadonly music files.

From the Venn diagram, 15 students use the Internet to downloadonly research papers.

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-2 Applications of Venn Diagrams 24 / 24