91Ó°ÊÓ

In Exercises \(1-4\), describe a universal set \(U\) that includes all elements in the given sets. Answers may vary. \(A=\\{\) Bach, Mozart, Beethoven \(\\}\) \(B=\\{\) Brahms, Schubert \(\\}\)

Based on more than 1500 ballots sent to film notables, the American Film Institute rated the top U.S. movies. The Institute selected Citizen Kane (1941), The Godfather (1972), Casablanca (1942), Raging Bull (1980), Singin' in the Rain (1952), and Gone with the Wind (1939) as the top six films. Suppose that you have all six films on DVD and decide to view some, all, or none of these films. How many viewing options do you have?

In Exercises 15-32, express each set using the roster method. The set of natural numbers less than 4

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 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?

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?

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

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 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?

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.