C h a n g e t e x t s i ze S h o w / H i d e TO C · 8 / 2 / 2 0 1 7 T h i n ki n g M a t h e m a t...

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2 Set Theory > 2.5 Survey Problems

2.5 Survey Problems

What am I Supposed to Learn?After you have read this section, you should be able to:

1 Use Venn diagrams to visualize a survey's results.

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

MEXICO'S 2006 ELECTION, ITS closest-ever race for president, exposed an emerging trend in Latin America: a sharpening divide between the rich and the poor.Although Mexico's economy expanded by 4.4% in 2004 and 3% in 2005, at the end of 2005, almost half of the country's 107 million people still lived in poverty. (Source:Newsweek, July 17, 2006) Figure 2.25 shows that Mexicans tend to see societal injustice, rather than personal laziness, as the primary cause of poverty.

dFIGURE 2.25 Percentages for each country may not total 100% because less frequently identified primary causes of poverty were omitted from the graph. Source: Ronald Inglehart et al., World Values Surveys and European Values Surveys

Suppose a survey is taken that asks randomly selected adults in the United States and Mexico the following question:

Do you agree or disagree that the primary cause of poverty is societal injustice?

In this section, you will see how sets and Venn diagrams are used to tabulate information collected in such a survey. In survey problems, it is helpful to remember thatand means intersection, or means union, and not means complement. Furthermore, but means the same thing as and. Thus, but means intersection.

Visualizing the Results of a Survey

1 Use Venn diagrams to visualize a survey's results.

In Section 2.1, we defined the cardinal number of set A, denoted by as the number of elements in set A. Venn diagrams are helpful in determining a set'scardinality.

Example 1 Using a Venn Diagram to Visualize the Results of a SurveyWe return to the campus survey in which students were asked two questions:

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Would you be willing to donate blood?

Would you be willing to help serve a free breakfast to blood donors?

Set A represents the set of students willing to donate blood. Set B represents the set of students willing to help serve breakfast to donors. The survey results aresummarized in Figure 2.26 below. Use the diagram to answer the following questions:

FIGURE 2.26 Results of a survey

a. How many students are willing to donate blood?

b. How many students are willing to help serve a free breakfast to blood donors?

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2 Set Theory > 2.5 Survey Problems

c. How many students are willing to donate blood and serve breakfast?

d. How many students are willing to donate blood or serve breakfast?

e. How many students are willing to donate blood but not serve breakfast?

f. How many students are willing to serve breakfast but not donate blood?

g. How many students are neither willing to donate blood nor serve breakfast?

h. How many students were surveyed?

SOLUTION

a. The number of students willing to donate blood can be determined by adding the numbers in regions I and II. Thus, There are 490students willing to donate blood.

b. The number of students willing to help serve a free breakfast to blood donors can be determined by adding the numbers in regions II and III. Thus, There are 340 students willing to help serve breakfast.

c. The number of students willing to donate blood and serve breakfast appears in region II, the region representing the intersection of the two sets. Thus, There are 120 students willing to donate blood and serve breakfast.

d. The number of students willing to donate blood or serve breakfast is found by adding the numbers in regions I, II, and III, representing the union of the two sets.We see that Therefore, 710 students in the survey are willing to donate blood or serve breakfast.

e. The region representing students who are willing to donate blood but not serve breakfast, is region I. We see that 370 of the students surveyed arewilling to donate blood but not serve breakfast.

f. Region III represents students willing to serve breakfast but not donate blood: We see that 220 students surveyed are willing to help serve breakfast butnot donate blood.

g. Students who are neither willing to donate blood nor serve breakfast, fall within the universal set, but outside circles A and B. These students fall inregion IV, where the Venn diagram indicates that there are 290 elements. There are 290 students in the survey who are neither willing to donate blood nor servebreakfast.

h. We can find the number of students surveyed by adding the numbers in regions I, II, III, and IV. Thus, There were1000 students surveyed.

Check Point 1In a survey on musical tastes, respondents were asked: Do you listen to classical music? Do you listen to jazz? The survey results are summarized in Figure 2.27.Use the diagram to answer the following questions.

FIGURE 2.27

a. How many respondents listened to classical music?

b. How many respondents listened to jazz?

c. How many respondents listened to both classical music and jazz?

d. How many respondents listened to classical music or jazz?

e. How many respondents listened to classical music but not jazz?

f. How many respondents listened to jazz but not classical music?

g. How many respondents listened to neither classical music nor jazz?

h. How many people were surveyed?



n (A) = 370 + 120 = 490.

n (B) = 120 + 220 = 340.

n (A ∩ B) = 120.

n (A ∪ B) = 370 + 120 + 220 = 710.

A ∩ B',

B ∩ A'.

A' ∩ B',

n (U) = 370 + 120 + 220 + 290 = 1000.







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2 Set Theory > 2.5 Survey Problems > Solving Survey Problems

Solving Survey Problems

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

Venn diagrams are used to solve problems involving surveys. Here are the steps needed to solve survey problems:

Solving Survey Problems1. Use the survey's description to define sets and draw a Venn diagram.

2. Use the survey's results to determine the cardinality for each region in the Venn diagram. Start with the intersection of the sets, the innermost region, andwork outward.

3. Use the completed Venn diagram to answer the problem's questions.

Example 2 Surveying People's AttitudesA survey is taken that asks 2000 randomly selected U.S. and Mexican adults the following question:

Do you agree or disagree that the primary cause of poverty is societal injustice?

The results of the survey showed that

1060 people agreed with the statement.

400 Americans agreed with the statement.

Source: World Values Surveys

If half the adults surveyed were Americans,

a. How many Mexicans agreed with the statement?

b. How many Mexicans disagreed with the statement?

SOLUTION

Step 1 Define sets and draw a Venn diagram. The Venn diagram in Figure 2.28 shows two sets. Set U.S. is the set of Americans surveyed. Set A (labeled“Agree”) is the set of people surveyed who agreed with the statement. By representing the Americans surveyed with circle U.S., we do not need a separate circle forthe Mexicans. The group of people outside circle U.S. must be the set of Mexicans. Similarly, by visualizing the set of people who agreed with the statement ascircle A, we do not need a separate circle for those who disagreed. The group of people outside circle A (Agree) must be the set of people disagreeing with thestatement.

FIGURE 2.28

Step 2 Determine the cardinality for each region in the Venn diagram, starting with the innermost region and working outward. We are given the followingcardinalities:

There were 2000 people surveyed:

Half the people surveyed were Americans:

The number of people who agreed with the statement was 1060:

There were 400 Americans who agreed with the statement:

Now let's use these numbers to determine the cardinality of each region, starting with region II, moving outward to regions I and III, and ending with region IV.



n (U) = 2000.

n (U. S. ) = 1000.

n (A) = 1060.

n (U. S. ∩A) = 400.









Start with region II.

Move out to region I.

d

Move out to region III.

d

End with IV, the outer region.

d

Step 3 Use the completed Venn diagram to answer the problem's questions. The completed Venn diagram that illustrates the survey's results is shown inFigure 2.29.










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dFIGURE 2.29

a. The Mexicans who agreed with the statement are those members of the set of people who agreed who are not Americans, shown in region III. This means that660 Mexicans agreed that societal injustice is the primary cause of poverty.

b. The Mexicans who disagreed with the statement can be found outside the circles of people who agreed and people who are Americans. This corresponds toregion IV, whose cardinality is 340. Thus, 340 Mexicans disagreed that societal injustice is the primary cause of poverty.

Check Point 2In a Gallup poll, 2000 U.S. adults were selected at random and asked to agree or disagree with the following statement:

Job opportunities for women are not equal to those for men.

The results of the survey showed that

1190 people agreed with the statement.

700 women agreed with the statement.

Source: The People's Almanac

If half the people surveyed were women,

a. How many men agreed with the statement?

b. How many men disagreed with the statement?










When tabulating survey results, more than two circles within a Venn diagram are often needed. For example, consider a Time/CNN poll that sought to determine howAmericans felt about reserving a certain number of college scholarships exclusively for minorities and women. Respondents were asked the following question:

Do you agree or disagree with the following statement: Colleges should reserve a certain number of scholarships exclusively for minorities and women? Source: TimeAlmanac

Suppose that we want the respondents to the poll to be identified by gender (man or woman), ethnicity (African American or other), and whether or not they agreed withthe statement. A Venn diagram into which the results of the survey can be tabulated is shown in Figure 2.30.

FIGURE 2.30

Based on our work in Example 2, we only used one circle in the Venn diagram to indicate the gender of the respondent. We used M for men, so the set of womenrespondents, consists of the regions outside circle M. Similarly, we used B for the set of African American respondents, so the regions outside circle B account forall other ethnicities. Finally, we used A for the set of respondents who agreed with the statement. Those who disagreed lie outside circle A.

In the next example, we create a Venn diagram with three intersecting sets to illustrate a survey's results. In our final example, we use this Venn diagram to answerquestions about the survey.

Example 3 Constructing a Venn Diagram for a SurveySixty people were contacted and responded to a movie survey. The following information was obtained:

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

b. 13 people liked comedies and dramas.

c. 10 people liked comedies and science fiction.

d. 11 people liked dramas and science fiction.

e. 26 people liked comedies.

f. 21 people liked dramas.

g. 25 people liked science fiction.

Use a Venn diagram to illustrate the survey's results.

SOLUTION

The set of people surveyed is a universal set with 60 elements containing three subsets:

We draw these sets in Figure 2.31. Now let's use the numbers in (a) through (g), as well as the fact that 60 people were surveyed, which we call condition (h), todetermine the cardinality of each region in the Venn diagram.



M',

C

D

S

=

=

=

the set of those who like comedies

the set of those who like dramas

the set of those who like science fiction.








0

FIGURE 2.31










d

With a cardinality in each region, we have completed the Venn diagram that illustrates the survey's results.

Great Question!Can you briefly summarize the procedure used to construct a Venn diagram for a survey?

After entering the cardinality of the innermost region, work outward and use subtraction to obtain subsequent cardinalities. The phrase

SURVEY SUBTRACT is a helpful reminder of these repeated subtractions.

Check Point 3A survey of 250 memorabilia collectors showed the following results: 108 collected baseball cards. 92 collected comic books. 62 collected stamps. 29 collectedbaseball cards and comic books. 5 collected baseball cards and stamps. 2 collected comic books and stamps. 2 collected all three types of memorabilia. Use a Venndiagram to illustrate the survey's results.

Example 4 Using a Survey's Venn DiagramThe Venn diagram in Figure 2.32 shows the results of the movie survey in Example 3. How many of those surveyed liked










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FIGURE 2.32

a. comedies, but neither dramas 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?

SOLUTION

a. Those surveyed who liked comedies, but neither dramas nor science fiction, are represented in region I. There are 9 people in this category.


▶C ∩ ( ∩ )D′ S ′








2 Set Theory > 2.5 Survey Problems > Concept and Vocabulary Check

b. Those surveyed who liked dramas and science fiction, but not comedies, are represented in region VI. There are 5 people in this category.

c. We are interested in those surveyed who liked dramas or science fiction, but not comedies:

FIGURE 2.32 (repeated)

The intersection of the regions in the voice balloons consists of the common regions shown in red, III, VI, and VII. There are elements in theseregions. There are 18 people who liked dramas or science fiction, but not comedies.

d. Those surveyed who liked exactly one movie style are represented in regions I, III, and VII. There are elements in these regions. Thus, 22people liked exactly one movie style.

e. Those surveyed who liked at least two movie styles are people who liked two or more types of movies. People who liked two movie styles are represented inregions II, IV, and VI. Those who liked three movie styles are represented in region V. Thus, we add the number of elements in regions II, IV, V, and VI:

Thus, 22 people liked at least two movie styles.

f. Those surveyed who liked none of the movie styles are represented in region VIII. There are 16 people in this category.

Check Point 4Use the Venn diagram you constructed in Check Point 3 to determine how many of those surveyed collected

a. comic books, but neither baseball cards nor stamps.

b. baseball cards and stamps, but not comic books.

c. baseball cards or stamps, but not comic books.

d. exactly two types of memorabilia.

e. at least one type of memorabilia.

f. none of the types of memorabilia.

Concept and Vocabulary CheckFill in each blank so that the resulting statement is true.

1. In survey problems, the word _________________ means “intersection.”

2. In survey problems, the word _________________ means “union.”

3. In survey problems, the word _________________ means “complement.”

4. In order to construct a Venn diagram for a survey, first enter the cardinality of the ___________________ region. Then use ___________________ to obtainsubsequent cardinalities.

The Venn diagram below shows the results of a GfK Roper Public Affairs and Media survey of 1967 American adults taken on October 9, 2012. Use the Venn diagram todetermine whether each statement in Exercises 5–8 is true or false. If the statement is false, make the necessary change(s) to produce a true statement.



▶(D ∩ S) ∩C ′

Dramas or science fiction, but not comedies

▼ ▼ ▼

(D ∪ S) ∩ .C ′

▲ ▲

Regions II, III, IV, V, VI, VII Regions III, VI, VII, VIII

3 + 5 + 10 = 18

9 + 3 + 10 = 22

7 + 4 + 6 + 5 = 22.








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Source: GfK Roper Public Affairs and Media Survey

5. 64% of those surveyed liked cats. _________________

6. 29% of those surveyed liked dogs but not cats. _________________

7. 60% of those surveyed liked cats or dogs. _________________

8. 7% of those surveyed liked neither cats nor dogs. _________________









2 Set Theory > 2.5 Survey Problems > Exercise Set 2.5

Exercise Set 2.5Practice ExercisesUse the accompanying Venn diagram, which shows the number of elements in regions I through IV, to answer the questions in Exercises 1–8.

1. How many elements belong to set


3. How many elements belong to set A but not set

4. How many elements belong to set B but not set

5. How many elements belong to set A or set

6. How many elements belong to set A and set

7. How many elements belong to neither set A nor set

8. How many elements are there in the universal set?

Use the accompanying Venn diagram, which shows the number of elements in region II, to answer Exercises 9–10.

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

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

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26.



13. How many elements belong to set A but not set

14. How many elements belong to set B but not set





19. How many elements belong to set B and set C, but not to set



A?

B?

B?

A?

B?

B?

B?

n (A) = 21,n (B) = 29, n (U) = 48,

n (A) = 23,n (B) = 27, n (U) = 53,

B?

A?

C?

A?

C?

B?

C?

B?

A?

B?








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20. How many elements belong to set A and set C, but not to set

21. How many elements belong to set B or set C, but not to set

22. How many elements belong to set A or set C, but not to set

23. Considering sets A, B, and C, how many elements belong to exactly one of these sets?

24. Considering sets A, B, and C, how many elements belong to exactly two of these sets?

25. Considering sets A, B, and C, how many elements belong to at least one of these sets?

26. Considering sets A, B, and C, how many elements belong to at least two of these sets?

The accompanying Venn diagram shows the number of elements in region V. In Exercises 27–28, use the given cardinalities to determine the number of elements ineach of the other seven regions.

27.

28.

Practice PlusIn Exercises 29–32, use the Venn diagram and the given conditions to determine the number of elements in each region, or explain why the conditions are impossible tomeet.

29.


B?

A?

B?

n (U) = 30,n (A) = 11,n (B) = 8,n (C) = 14,

n (A ∩ B) = 3,n (A ∩ C) = 5,n (B ∩ C) = 3

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

n (A ∩ B) = 6,n (A ∩ C) = 7,n (B ∩ C) = 8

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









(In Exercises 30–32, continue to refer to the Venn diagram at the bottom of the previous page.)

30.

31.

32.

Application ExercisesAs discussed in the text on page 100, a poll asked respondents if they agreed with the statement

Colleges should reserve a certain number of scholarships exclusively for minorities and women.

Hypothetical results of the poll are tabulated in the Venn diagram. Use these cardinalities to solve Exercises 33–38.

33. How many respondents agreed with the statement?

34. How many respondents disagreed with the statement?

35. How many women agreed with the statement?

36. How many people who are not African American agreed with the statement?

37. How many women who are not African American disagreed with the statement?

38. How many men who are not African American disagreed with the statement?

39. A pollster conducting a telephone poll of a city's residents asked two questions:

1. Do you currently smoke cigarettes?

2. Regardless of your answer to question 1, would you support a ban on smoking in all city parks?

a. Construct a Venn diagram that allows the respondents to the poll to be identified by whether or not they smoke cigarettes and whether or not they support theban.

b. Write the letter b in every region of the diagram that represents smokers polled who support the ban.

c. Write the letter c in every region of the diagram that represents nonsmokers polled who support the ban.

d. Write the letter d in every region of the diagram that represents nonsmokers polled who do not support the ban.

40. A pollster conducting a telephone poll at a college campus asked students two questions:

1. Do you binge drink three or more times per month?

2. Regardless of your answer to question 1, are you frequently behind in your school work?

a. Construct a Venn diagram that allows the respondents to the poll to be identified by whether or not they binge drink and whether or not they frequently fall behindin school work.

b. Write the letter b in every region of the diagram that represents binge drinkers who are frequently behind in school work.

c. Write the letter c in every region of the diagram that represents students polled who do not binge drink but who are frequently behind in school work.



n (U) = 42, n (A) = 26, n (B) = 22, n (C) = 25,

n (A ∩ B) = 17, n (A ∩ C) = 11, n (B ∩ C) = 9,

n (A ∩ B ∩ C) = 5

n (U) = 40, n (A) = 10, n (B) = 11, n (C) = 12,

n (A ∩ B) = 6, n (A ∩ C) = 9, n (B ∩ C) = 7,

n (A ∩ B ∩ C) = 2

n (U) = 25, n (A) = 8, n (B) = 9, n (C) = 10,

n (A ∩ B) = 6, n (A ∩ C) = 9, n (B ∩ C) = 8,

n (A ∩ B ∩ C) = 5









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d. Write the letter d in every region of the diagram that represents students polled who do not binge drink and who do not frequently fall behind in their school work.

41. A pollster conducting a telephone poll asked three questions:

1. Are you religious?

2. Have you spent time with a person during his or her last days of a terminal illness?

3. Should assisted suicide be an option for terminally ill people?

a. Construct a Venn diagram with three circles that can assist the pollster in tabulating the responses to the three questions.

b. Write the letter b in every region of the diagram that represents all religious persons polled who are not in favor of assisted suicide for the terminally ill.

c. Write the letter c in every region of the diagram that represents the people polled who do not consider themselves religious, who have not spent time with aterminally ill person during his or her last days, and who are in favor of assisted suicide for the terminally ill.

d. Write the letter d in every region of the diagram that represents the people polled who consider themselves religious, who have not spent time with a terminally illperson during his or her last days, and who are not in favor of assisted suicide for the terminally ill.

e. Write the letter e in a region of the Venn diagram other than those in parts (b)–(d) and then describe who in the poll is represented by this region.

42. A poll asks respondents the following question:

Do you agree or disagree with this statement: In order to address the trend in diminishing male enrollment, colleges should begin special efforts to recruit men?

a. Construct a Venn diagram with three circles that allows the respondents to be identified by gender (man or woman), education level (college or no college), andwhether or not they agreed with the statement.

b. Write the letter b in every region of the diagram that represents men with a college education who agreed with the statement.

c. Write the letter c in every region of the diagram that represents women who disagreed with the statement.










d. Write the letter d in every region of the diagram that represents women without a college education who agreed with the statement.

e. Write the letter e in a region of the Venn diagram other than those in parts (b)–(d) and then describe who in the poll is represented by this region.

In Exercises 43–48, construct a Venn diagram and determine the cardinality for each region. Use the completed Venn diagram to answer the questions.

43. A survey of 75 college students was taken to determine where they got the news about what's going on in the world. Of those surveyed, 29 students got the newsfrom newspapers, 43 from television, and 7 from both newspapers and television.

Of those surveyed,

a. How many got the news from only newspapers?

b. How many got the news from only television?

c. How many got the news from newspapers or television?

d. How many did not get the news from either newspapers or television?

44. A survey of 120 college students was taken at registration. Of those surveyed, 75 students registered for a math course, 65 for an English course, and 40 for bothmath and English.

Of those surveyed,

a. How many registered only for a math course?

b. How many registered only for an English course?

c. How many registered for a math course or an English course?

d. How many did not register for either a math course or an English course?

45. A survey of 80 college students was taken to determine the musical styles they listened to. Forty-two students listened to rock, 34 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?

46. A survey of 180 college men was taken to determine participation in various campus activities. Forty-three students were in fraternities, 52 participated in campussports, and 35 participated in various campus tutorial programs. Thirteen students participated in fraternities and sports, 14 in sports and tutorial programs, and 12 infraternities and tutorial programs. Five students participated in all three activities.

Of those surveyed,

a. How many participated in only campus sports?

b. How many participated in fraternities and sports, but not tutorial programs?

c. How many participated in fraternities or sports, but not tutorial programs?

d. How many participated in exactly one of these activities?

e. How many participated in at least two of these activities?

f. How many did not participate in any of the three activities?

47. An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 894drank alcohol regularly, 665 smoked cigarettes, 192 used illegal drugs, 424 drank alcohol regularly and smoked cigarettes, 114 drank alcohol regularly and usedillegal drugs, 119 smoked cigarettes and used illegal drugs, 97 engaged in all three behaviors, and 309 engaged in none of these behaviors.

Source: Jamie Langille, University of Nevada Las Vegas

a. How many students were surveyed?

Of those surveyed,

b. How many drank alcohol regularly or smoked cigarettes?

c. How many used illegal drugs only?

d. How many drank alcohol regularly and smoked cigarettes, but did not use illegal drugs?










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e. How many drank alcohol regularly or used illegal drugs, but did not smoke cigarettes?

f. How many engaged in exactly two of these behaviors?

g. How many engaged in at least one of these behaviors?

48. In the August 2005 issue of Consumer Reports, readers suffering from depression reported that alternative treatments were less effective than prescription drugs.Suppose that 550 readers felt better taking prescription drugs, 220 felt better through meditation, and 45 felt better taking St. John's wort. Furthermore, 95 felt betterusing prescription drugs and meditation, 17 felt better using prescription drugs and St. John's wort, 35 felt better using meditation and St. John's wort, 15 improvedusing all three treatments, and 150 improved using none of these treatments. (Hypothetical results are partly based on percentages given in Consumer Reports.)

a. How many readers suffering from depression were included in the report?

Of those included in the report,

b. How many felt better using prescription drugs or meditation?

c. How many felt better using St. John's wort only?

d. How many improved using prescription drugs and meditation, but not St. John's wort?

e. How many improved using prescription drugs or St. John's wort, but not meditation?

f. How many improved using exactly two of these treatments?

g. How many improved using at least one of these treatments?

Writing in Mathematics49. Suppose that you are drawing a Venn diagram to sort and tabulate the results of a survey. If results are being tabulated along gender lines, explain why only acircle representing women is needed, rather than two separate circles representing the women surveyed and the men surveyed.










50. Suppose that you decide to use two sets, M and W, to sort and tabulate the responses for men and women in a survey. Describe the set of people represented byregions II and IV in the Venn diagram shown. What conclusion can you draw?

Critical Thinking ExercisesMake Sense? In Exercises 51–54, determine whether each statement makes sense or does not make sense, and explain your reasoning.

51. A survey problem must present the information in exactly the same order in which I determine cardinalities from innermost to outermost region.

Exercises 52–54 are based on the graph that shows the percentage of smokers and nonsmokers suffering from various ailments. Use the graph to determine whethereach statement makes sense.

dSource: MARS OTC/DTC

52. I represented the data for depression using the following Venn diagram:

53. I improved the Venn diagram in Exercise 52 by adding a third circle for nonsmokers.

54. I used a single Venn diagram to represent all the data displayed by the bar graph.

In Exercises 55–58, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

55. In a survey, 110 students were taking mathematics, 90 were taking psychology, and 20 were taking neither. Thus, 220 students were surveyed.

56. If then

57. When filling in cardinalities for regions in a two-set Venn diagram, the innermost region, the intersection of the two sets, should be the last region to be filled in.

58. can be obtained by subtracting from

59. In a survey of 150 students, 90 were taking mathematics and 30 were taking psychology.

a. What is the least number of students who could have been taking both courses?

b. What is the greatest number of students who could have been taking both courses?



A ∩ B = Ø, n (A ∪ B) = n (A) + n (B) .

n (A') n (A) n (U) .








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c. What is the greatest number of students who could have been taking neither course?

60. A person applying for the position of college registrar submitted the following report to the college president on 90 students: 31 take math; 28 take chemistry; 42take psychology; 9 take math and chemistry; 10 take chemistry and psychology; 6 take math and psychology; 4 take all three subjects; and 20 take none of thesecourses. The applicant was not hired. Explain why.

Group Exercise61. This group activity is intended to provide practice in the use of Venn diagrams to sort responses to a survey. The group will determine the topic of the survey.Although you will not actually conduct the survey, it might be helpful to imagine carrying out the survey using the students on your campus.

a. In your group, decide on a topic for the survey.

b. Devise three questions that the pollster will ask to the people who are interviewed.

c. Construct a Venn diagram that will assist the pollster in sorting the answers to the three questions. The Venn diagram should contain three intersecting circleswithin a universal set and eight regions.

d. Describe what each of the regions in the Venn diagram represents in terms of the questions in your poll.







C h a n g e t e x t s i ze S h o w / H i d e TO C · 8 / 2 / 2 0 1 7 T h i n ki n g M a t h e m a t...

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