Copyright © 2005 Pearson Education, Inc.

2.5

Applications of Sets

Slide 2-2 Copyright © 2005 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?

Slide 2-3 Copyright © 2005 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

Regular Mint

35

Slide 2-4 Copyright © 2005 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.

74 – 35 = 39

62 – 35 = 27

U

Regular Mint

35 both

39 regular only

27 mintonly

Slide 2-5 Copyright © 2005 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.

You can also use the following formula:

N(A or B) = N(A)+N(B)-N(A and B)

N(One or the other or both)=N(regular)+N(mint)-N(both)

= 74+62-35 = 101

Slide 2-6 Copyright © 2005 Pearson Education, Inc.

Solution continued

Take the total number of people in the entire sample (120) and subtract the number who liked one or the other or both (101, from previous step).

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

U

Regular Mint

35 both

62-35=27 Liked mint only

74-35=39 Liked mint only

19 liked neither

Slide 2-7 Copyright © 2005 Pearson Education, Inc.

Next Steps

Read Examples 1-3 Work Problems in text: p. 80: #1-6, all Do Online homework corresponding to this

section Take Online quiz corresponding to Secs. 2.4

and 2.5