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

Use the accompanying Venn diagram, which shows the number of elements in regions I through IV, to answer the questions in Exercises 1–8. How many elements belong to set A?

Use the accompanying Venn diagram, which shows the number of elements in regions I through IV, to answer the questions in Exercises 1–8. How many elements belong to set B?

Use the accompanying Venn diagram, which shows the number of elements in regions I through IV, to answer the questions in Exercises 1–8. How many elements belong to set A but not set B?

Use the accompanying Venn diagram, which shows the number of elements in regions I through IV, to answer the questions in Exercises 1–8. How many elements belong to set B but not set A?

Use the accompanying Venn diagram, which shows the number of elements in regions I through IV, to answer the questions in Exercises 1–8. How many elements belong to set A or set B?

Use the accompanying Venn diagram, which shows the number of elements in regions I through IV, to answer the questions in Exercises 1–8. How many elements belong to set A and set B?

Use the accompanying Venn diagram, which shows the number of elements in regions I through IV, to answer the questions in Exercises 1–8. How many elements belong to neither set A nor set B?

Use the accompanying Venn diagram, which shows the number of elements in regions I through IV, to answer the questions in Exercises 1–8. 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. If n(A) = 21, n(B) = 29, and n(U) = 48, find the number of elements in each of regions I, III, and IV.

Use the accompanying Venn diagram, which shows the number of elements in region II, to answer Exercises 9–10. If n(A) = 23, n(B) = 27, and n(U) = 53, 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. How many elements belong to set B?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set A?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set A but not set C?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set B but not set A?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set A or set C?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set A or set B?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set A and set C?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set A and set B?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set B and set C, but not to set A?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set A and set C, but not to set B?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set B or set C, but not to set A?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. How many elements belong to set A or set C, but not to set B?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. Considering sets A, B, and C, how many elements belong to exactly one of these sets?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. Considering sets A, B, and C, how many elements belong to exactly two of these sets?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. Considering sets A, B, and C, how many elements belong to at least one of these sets?

Use the accompanying Venn diagram, which shows the cardinality of each region, to answer Exercises 11–26. 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 in each of the other seven regions. n(U) = 30, n(A) = 11, n(B) = 8, n(C) = 14, \(n(A \cap B)=3, n(A \cap C)=5, n(B \cap C)=3\) Text Transcription: n(A cap B)=3, n(A cap C)=5, n(B cap C)=3

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 in each of the other seven regions. n(U) = 32, n(A) = 21, n(B) = 15, n(C) = 14, \(n(A \cap B)=6, n(A \cap C)=7, n(B \cap C)=8\) Text Transcription: n(A cap B)=6, n(A cap C)=7, n(B cap C)=8

In 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 to meet. n(U) = 38, n(A) = 26, n(B) = 21, n(C) = 18, \(n(A \cap B)=17, n(A \cap C)=11, n(B \cap C)=8\), \(n(A \cap B \cap C)=7\) Text Transcription: n(A cap B)=17, n(A cap C)=11, n(B cap C)=8 n(A cap B cap C)=7

In 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 to meet. n(U) = 42, n(A) = 26, n(B) = 22, n(C) = 25, \(n(A \cap B)=17, n(A \cap C)=11, n(B \cap C)=9\), \(n(A \cap B \cap C)=5\) Text Transcription: n(A cap B)=17, n(A cap C)=11, n(B cap C)=9 n(A cap B cap C)=5

In 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 to meet. \(\mathrm{n}(\mathrm{U})=40, \mathrm{n}(\mathrm{A})=10, \mathrm{n}(\mathrm{B})=11, \mathrm{n}(\mathrm{C})=12, n(A \cap B)=6, n(A \cap C)=9, n(B \cap C)=7, n(A \cap B \cap C)=2\)

In 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 to meet. n(U) = 25, n(A) = 8, n(B) = 9, n(C) = 10, \(n(A \cap B)=6, n(A \cap C)=9, n(B \cap C)=8\), \(n(A \cap B \cap C)=5\) Text Transcription: n(A cap B)=6, n(A cap C)=9, n(B cap C)=8 n(A cap B cap C)=5

As 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. How many respondents agreed with the statement?

As 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. How many respondents disagreed with the statement?

As 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. How many women agreed with the statement?

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

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

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

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 the ban. 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.

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 behind in 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. 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.

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 a terminally 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 ill person 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.

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), and whether 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. 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 news from 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?

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

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

In Exercises 43–48, construct a Venn diagram and determine the cardinality for each region. Use the completed Venn diagram to answer the questions. A survey of 180 college men was taken to determine participation in various campus activities. Forty-three students were in fraternities, 52 participated in campus sports, and 35 participated in various campus tutorial programs. Thirteen students participated in fraternities and sports, 14 in sports and tutorial programs, and 12 in fraternities 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?

An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 894 drank alcohol regularly, 665 smoked cigarettes, 192 used illegal drugs, 424 drank alcohol regularly and smoked cigarettes, 114 drank alcohol regularly and used illegal 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? 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?

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 better using 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 improved using 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?

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 a circle representing women is needed, rather than two separate circles representing the women surveyed and the men surveyed.

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 by regions II and IV in the Venn diagram shown. What conclusion can you draw?

In Exercises 51–54, determine whether each statement makes sense or does not make sense, and explain your reasoning. 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 whether each statement makes sense. I represented the data for depression using the following Venn diagram:

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 whether each statement makes sense. I improved the Venn diagram in Exercise 52 by adding a third circle for nonsmokers.

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 whether each statement makes sense. 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. In a survey, 110 students were taking mathematics, 90 were taking psychology, and 20 were taking neither. Thus, 220 students were surveyed.

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. If \(A \cap B=\varnothing\), then \(n(A \cup B)=n(A)+n(B)\).

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. 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.

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. \(n\left(A^{\prime}\right)\) can be obtained by subtracting n(A) from n(U).

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

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; 42 take 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 these courses. The applicant was not hired. Explain why

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 circles within 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.