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Chapter 2: Problem 57
Find each of the following sets. \(A \cup \varnothing\)
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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?
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?
In Exercises 13-24, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\\B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \((A \cup B \cup C)^{\prime}\)
Use the formula for the cardinal number of the union of two sets to solve Exercises 93-96. Set \(A\) contains 30 elements, set \(B\) contains 18 elements, and 5 elements are common to sets \(A\) and \(B\). How many elements are in \(A \cup B\) ?
The group should define three sets, each of which categorizes \(U\), the set of students in the group, in different ways. Examples include the set of students with blonde hair, the set of students no more than 23 years old, and the set of students whose major is undecided. Once you have defined the sets, construct a Venn diagram with three intersecting sets and eight regions. Each student should determine to which region he or she belongs. Illustrate the sets by writing each first name in the appropriate region.
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