SECTION 1.6Survey Problems

OBJECTIVES Use Venn diagrams to visualize a survey’s

results.

Use survey results to complete Venn diagrams and answer questions about the survey.

EXAMPLE 1: Determine which numbered regions make up

the indicated set. A

I IIIII

IV

U

EXAMPLE 2: Determine which numbered regions make up

the indicated set. A – B

I IIIII

IV

U

EXAMPLE 3: Determine which numbered regions make up the

indicated set. B’ ∩ C ∩ A’

I III

VII

II

VIII

U

IVV

VI

EXAMPLE 4: If n(A) = 21, n(B) = 29, and n(U) = 48, find

the number of elements in each regions I, III, and IV.

I IIIII 7

IV

U

EXAMPLE 5: If n(A) = 23, n(B) = 27, and n(U) = 53, find

the number of elements in each regions I, III, and IV.

I IIIII 8

IV

U

EXAMPLE 6: The Venn diagram show the cardinality of each

region. How many elements belong to set B?

5 6

10

1

8

U

27

3

EXAMPLE 7: How many elements belong to set A?

5 6

10

1

8

U

27

3

EXAMPLE 8: How many elements belong to set A but not set

C?

5 6

10

1

8

U

27

3

EXAMPLE 9: How many elements belong to set B but not set

A?

5 6

10

1

8

U

27

3

EXAMPLE 10: How many elements belong to set A or set C?

5 6

10

1

8

U

27

3

EXAMPLE 11: How many elements belong to set A and set C,

but not to set B?

5 6

10

1

8

U

27

3

EXAMPLE 12: Considering sets A, B, and C, how many elements

belong to exactly one of these sets?

5 6

10

1

8

U

27

3

SECTION 1.6 ASSIGNMENTClasswork:

TB pg. 55/1 – 18Must write problems and show ALL work to receive credit for the given assignment.

EXAMPLE 13: The accompanying Venn diagram shows the

number of elements in region V. Use the given cardinalities to determine the number of elements in each of the other seven regions. n(U) = 30, n(A) = 11, n(B) = 8, n(C) = 14, n(A∩B)

= 3, n(A∩C) = 5, n(B∩C) = 3

U

2

EXAMPLE 14: n(U) = 32, n(A) = 21, n(B) = 15, n(C) = 14,

n(A∩B) = 6, n(A∩C) = 7, n(B∩C) = 8

U

2

EXAMPLE 15: n(U) = 38, n(A) = 26, n(B) = 21, n(C) = 18,

n(A∩B) =17, n(A∩C) = 11, n(B∩C) = 8, n(A∩B∩C) = 7

U

EXAMPLE 16: TB pg. 55/23

U

EXAMPLE 17: TB pg. 55/25

EXAMPLE 18: TB pg. 56/31

U

EXAMPLE 19: A survey of 80 college students was taken to determine

the musical styles they listened to. Forty-two students listened to rock, 34 listened to classical, and 27 to jazz. Twelve students listened to rock and jazz, 14 to rock and classical, and 10 to classical and jazz. Seven students listened to all three musical styles. Of those surveyed,

a. How many listened to only rock music?b. How many listened to classical and jazz, but not rock?c. How many listened to classical or jazz, but not rock?d. How many listened to music in exactly one of the musical

styles?e. How many listened to music in at least two of the musical

styles?f. How many did not listen to any of the musical styles?

EXAMPLE 19:

23 17

12

U7

735

6

Rock Classical

Jazz

EXAMPLE 19: Of those surveyed,

a. How many listened to only rock music? 23b. How many listened to classical and jazz, but not

rock? 3c. How many listened to classical or jazz, but not

rock? 32d. How many listened to music in exactly one of

the musical styles? 52e. How many listened to music in at least two of

the musical styles? 22f. How many did not listen to any of the musical

styles? 6

SECTION 1.6 ASSIGNMENTClasswork:

TB pg. 55/24 – 34 EVENMust write problems and show ALL work to receive credit for the given assignment.