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2012 Pearson Prentice Hall. All rights reserved.

CHAPTER 2

Set Theory

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2012 Pearson Prentice Hall. All rights reserved. 2

2.5

Survey Problems

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Objectives

1. Use Venn diagrams to visualize a surveysresults.

2. Use survey results to complete Venndiagrams and answer questions about the

survey.

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Example 1: Visualizing the Results of a

Survey

The results of the survey are summarized in this figure.

a. How many students are willing

to donate blood?Represented by regions I and II.

Thus, n(A) = 370 + 120 = 490.

b. How many are willing to donateblood but not serve breakfast?

Region 1 representsAB' = 370

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Example 1: Visualizing the Results of a

Survey

The results of the survey aresummarized in this figure.

c. How many werent willing todo either?

A' B' is region IV; those

areas outside the circles = 290.

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Solving Survey Problems

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Example 2: Teens and Cellphones

A survey was taken that asked 150 randomly

selected teens ages 16 and 17 what they do withtheir cellphones. Results of the survey showed that

30 teens used their cellphones to take pictures and

play games, 100 teens used their cellphones to take

pictures and 70 teens used their cellphones to play

games.

a. Use a Venn diagram to illustrate the surveys results.

b. How many of those surveyed used their cellphonesto take pictures or play games?

c. How many of those surveyed did not use their

cellphones to take pictures or play games?7

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Example 2 continued

Solution:Step 1 Define the sets and draw a Venn diagram.

SetPis the set of teens who use their

cellphones to take pictures.

Set G is the set of teens who use their

cellphones to play games.

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Example 2 continued

We are given the following cardinalities:

There were 150 teens surveyed: n(U) = 150.

30 teens take pictures and play games:

100 teens take pictures: n(P) = 100.

70 teens play games: n(G) = 70.

Step 2 Determine the cardinality for each region in theVenn diagram, starting with the innermost region.

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30.n P G

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Example 2 continued

Starting with region II and moving outwards to

regions I and III, and ending with region IV.

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Example 2 continued

Starting with region II and moving outwards to

regions I and III, and ending with region IV.

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Example 2 continued

b. Those surveyed who used their cellphones to takepictures but not play games are shown in region I.This means that 70 teens used their cellphones to take

pictures but not play games.

c. Those surveyed who did not use their cellphones totake pictures or play games are shown in region IV.This means that 10 teens did not use their cellphones

to take pictures or play games.

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Example 3: Constructing a Venn Diagram

for a SurveySixty people were contacted and responded to a

movie survey. The following information wasobtained.

a. 6 people liked comedies, drams, and science fiction

b. 13 people liked comedies and dramasc. 10 people liked comedies and science fiction

d. 11 people liked dramas and science fiction

e. 26 people liked comediesf. 21 people liked dramas

g. 25 people liked science fiction

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Example 3 continued

Solution:The set of people surveyed is a universal set with 60

elements containing three subsets:

C= those who like comedies

D = those who like dramasS= those who like science fiction

Use the given numbers and the fact that 60 people were

surveyed to determine the cardinality of each region.

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Example 3 continued

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Example 3 continued

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Example 4: Using a Surveys Venn

DiagramUse the Venn diagram from Example 3:

How many of those surveyed liked

a. comedies, but neither drams nor science fiction?

b. dramas and science fiction, but not comedies?

c. dramas or science fiction, but not comedies?

d. exactly one movie style?

e. at least two movie styles?

f. none of the movie styles?

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Example 4: continued


a. comedies, but neither drams nor science fiction?

Region I. There are 9 people in this category.

b. dramas and science fiction, but not comedies?

Region IV. There are 5 people in this category.

c. dramas or science fiction, but not comedies?

There are 3 + 5 + 10 = 18 people.

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Example 4: continued


d. exactly one movie style?

Regions I, III, and VII.

There are 9 + 3 + 10 = 22 people in this category.

e. at least two movie styles?

Regions II, IV, V, and VI.

There are 7 + 4 + 6 + 5 + 2 = 22 people in this

category.f. none of the movie styles?

Region VIII

There are 16 people in this category. 19

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