Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 5 - Slide 1

Section 5

Applications of Sets

Chapter 2 Section 5 - Slide 2Copyright © 2009 Pearson Education, Inc.

Example: Toothpaste Taste Test

A drug company is considering manufacturing a new toothpaste. They are considering two flavors, regular and mint.

In a sample of 120 people, it was found that 74 liked the regular, 62 liked the mint, and 35 liked both types.

How many liked only the regular flavor? How many liked either one or the other or

both? How many people did not like either flavor?

Chapter 2 Section 5 - Slide 3Copyright © 2009 Pearson Education, Inc.

Solution

Begin by setting up a Venn diagram with sets A (regular flavor) and B (mint flavor). Since some people liked both flavors, the sets will overlap and the number who liked both with be placed in region II.

35 people liked both flavors.

U

A(Regular) B(Mint)

35

II

Chapter 2 Section 5 - Slide 4Copyright © 2009 Pearson Education, Inc.

Solution (continued)

Next, region I will refer to those who liked only the regular and region III will refer to those who liked only the mint.

In order to get the number of people in each region, find the difference between all the people who liked each toothpaste and those who liked both.

I: 74 – 35 = 39

III: 62 – 35 = 27

U

A(Regular) B(Mint)

39 regular only

27 mintonly

III

II

I

both35

Chapter 2 Section 5 - Slide 5Copyright © 2009 Pearson Education, Inc.

Solution (continued)

“One or the other or both” represents the UNION of the two sets.

Therefore, 39 + 27 + 35 = 101 people who liked one or the other or both.

Chapter 2 Section 5 - Slide 6Copyright © 2009 Pearson Education, Inc.

Solution (continued)

Take the total number of people in the entire sample and subtract the number who liked one or the other or both.

120-101 = 19 people did not like either flavor.

U

A(Regular) B(Mint)

35 both

62-35=27 Liked mint only

74-35=39 Liked regular only

19 liked neither

I I I

I I

I

Chapter 2 Section 5 - Slide 7Copyright © 2009 Pearson Education, Inc.

Example: Ice Cream Flavors

The Ice Cream Shoppe surveyed its customers about their preferences of three ice cream flavors. The results of the survey showed the following number of people liked the respective flavor(s):

vanilla 170

vanilla & chocolate 100

chocolate 135

vanilla & strawberry 74

strawberry 103

chocolate & strawberry 52

all three 35

none of the three 7

Chapter 2 Section 5 - Slide 8Copyright © 2009 Pearson Education, Inc.

Using a Venn diagram, determine how many people completed the survey.

Chapter 2 Section 5 - Slide 9Copyright © 2009 Pearson Education, Inc.

Solution

31

65

35

39

18

17

+12

217