Let A be a set that contains exactly n elements. Find a formula in terms of n for the number of subsets
> Politics. If 12,457 people voted for a politician in his first election, 15,322 voted for him in his second election, and 9,345 voted for him in the first and second elections, how many people voted for this politician in the first or second election?
> A couple is planning to have 3 children. How many boy–girl combinations are possible? Distinguish between combined outcomes such as (B, B, G), (B, G, B), and (G, B, B). (A) Solve using a tree diagram. (B) Solve using the multiplication principle.
> Refer to the table in Problem 61. (A) How many females are of age 16–19 and earn minimum wage? (B) How many males are of age 16–24 and earn below minimum wage? (C) How many workers are of age 20–24
> A cable company offers its 10,000 customers two special services: high-speed internet and digital phone. If 3,770 customers use high-speed internet, 3,250 use digital phone, and 4,530 do not use either of these services, how many customers use both high-
> A survey of 800 small businesses indicates that 250 own a video conferencing system, 420 own projection equipment, and 180 own a video conferencing system and projection equipment. (A) How many businesses in the survey own either a video conferencing sy
> A manufacturing company in city A wishes to truck its product to 4 different cities: B, C, D, and E. If roads interconnect all 4 cities, how many different route plans can be constructed so that a single truck, starting from A, will visit each city exact
> A corporation plans to fill 2 different positions for vice-president, V1 and V2, from administrative officers in 2 of its manufacturing plants. Plant A has 6 officers and plant B has 8. How many ways can these 2 positions be filled if the V1 position is
> A high school football team with 40 players includes 16 players who played offense last year, 17 who played defense, and 12 who were not on last year’s team. How many players from last year played both offense and defense?
> A class of 30 music students includes 13 who play the piano, 16 who play the guitar, and 5 who play both the piano and the guitar. How many students in the class play neither instrument?
> Refer to the following Venn diagram. Which of the numbers x, y, z, or w must equal 0 if A ∪ B = U?
> If $2,000 is invested at 7% compounded (A) annually (B) quarterly (C) monthly what is the amount after 5 years? How much interest is earned?
> Refer to the following Venn diagram. Which of the numbers x, y, z, or w must equal 0 if A and B are disjoint?
> Explain how three sets, A, B, and C, can be related to each other in order for the following equation to hold true (Venn diagrams may be helpful):
> How many 5-digit ZIP code numbers are possible? How many of these numbers contain no repeated digits?
> A small combination lock has 3 wheels, each labeled with the 10 digits from 0 to 9. How many 3-digit combinations are possible if no digit is repeated? If digits can be repeated? If successive digits must be different?
> Using the English alphabet, how many 5-character case sensitive passwords are possible if each character is a letter or a digit?
> A delicatessen serves meat sandwiches with the following options: 3 kinds of bread, 5 kinds of meat, and lettuce or sprouts. How many different sandwiches are possible, assuming that one item is used out of each category
> Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample (A) If A and B are disjoint, then n(A ∩ B) = n(A) + n(B). (B) If n(A ∪ B) = n(A) + n(B), then A and B are disjoint.
> Use the given information to complete the following table.
> Use the given information to complete the following table.
> Use the given information to complete the following table.
> Use formula (2) for the amount to find each of the indicated quantities. A = $22,135; P = $19,000; t = 39 weeks; r = ?
> Use the given information to complete the following table.
> Use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. n(A′) = 30, n(B′) = 10, n(A′ ∪ B′
> Use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. n(A′) = 70, n(B′) = 170, n(A′ ∩ B′
> Use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. n(A) = 65, n(B) = 150, n(A ∪ B) = 175, n(U) = 200
> Use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. n(A) = 40, n(B) = 60, n(A ∩ B) = 20, n(U) = 100
> The 14 colleges of interest to a high school senior include 6 that are expensive (tuition more than $30,000 per year), 7 that are far from home (more than 200 miles away), and 2 that are both expensive and far from home. (A) If the student decides to se
> How many 4-letter code words can be formed from the letters A, B, C, D, E, F, G if no letter is repeated? If letters can be repeated? If adjacent letters must be different?
> A college offers 2 introductory courses in history, 3 in science, 2 in mathematics, 2 in philosophy, and 1 in English. (A) If a freshman takes one course in each area during her first semester, how many course selections are possible? (B) If a part-tim
> Indicate true (T) or false (F). 1 ∊ {10, 11}
> Indicate true (T) or false (F). {3, 2, 1} ⊂ {1, 2, 3, 4}
> Use the given interest rate i per compounding period to find r, the annual rate. 0.47% per month
> Answer yes or no. (If necessary, review Section A.1). If the universal set is the set of integers, is the set of even integers the complement of the set of odd integers?
> Answer yes or no. (If necessary, review Section A.1). Is the set of integers the union of the set of even integers and the set of odd integers?
> Answer yes or no. (If necessary, review Section A.1). Is the set of rational numbers a subset of the set of integers?
> Use the Venn diagram to indicate which of the eight blood types are included in each set. Rh′ ∩ A
> Use the Venn diagram to indicate which of the eight blood types are included in each set. (A ∪ B ∪ Rh)′
> Use the Venn diagram to indicate which of the eight blood types are included in each set. A ∪ B
> Use the Venn diagram to indicate which of the eight blood types are included in each set. A ∩ B
> Voting coalition. The company’s leaders in Problem 89 decide for or against certain measures as follows: The president has 2 votes and each vice-president has 1 vote. Three favorable votes are needed to pass a measure. List all minimal winning coalitions
> Find the indicated number of elements by referring to the following table of enrollments in a finite mathematics class: Let the universal set U be the set of all 120 students in the class, A the set of students from the College of Arts & Sciences, B
> Find the indicated number of elements by referring to the following table of enrollments in a finite mathematics class: Let the universal set U be the set of all 120 students in the class, A the set of students from the College of Arts & Sciences, B
> Use formula (2) for the amount to find each of the indicated quantities. A = $6,608; r = 24%; t = 3 quarters; P = ?
> Find the indicated number of elements by referring to the following table of enrollments in a finite mathematics class: Let the universal set U be the set of all 120 students in the class, A the set of students from the College of Arts & Sciences, B
> Find the indicated number of elements by referring to the following table of enrollments in a finite mathematics class: Let the universal set U be the set of all 120 students in the class, A the set of students from the College of Arts & Sciences, B
> Find the indicated number of elements by referring to the following table of enrollments in a finite mathematics class: Let the universal set U be the set of all 120 students in the class, A the set of students from the College of Arts & Sciences, B
> Find the indicated number of elements by referring to the following table of enrollments in a finite mathematics class: Let the universal set U be the set of all 120 students in the class, A the set of students from the College of Arts & Sciences, B
> Find the indicated number of elements by referring to the following table of enrollments in a finite mathematics class: Let the universal set U be the set of all 120 students in the class, A the set of students from the College of Arts & Sciences, B
> Discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counter example The empty set is a subset of the empty set.
> Discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counter example If A ⊂ B, then B′ ⊂ A′.
> Discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counter example If A = ∅, then A ∩ B = ∅.
> Discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counter example If A ∩ B = A, then A ⊂ B.
> Use the given interest rate i per compounding period to find r, the annual rate. 3.69% per half-year
> Discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counter example If A ⊂ B, then A ∪ B = A.
> Are the given sets disjoint? Let H, T, P, and E denote the sets in Problems 49, 50, 51, and 52, respectively. E and E
> Are the given sets disjoint? Let H, T, P, and E denote the sets in Problems 49, 50, 51, and 52, respectively. E and P
> Draw a Venn diagram for sets A, B, and C and shade the given region. (A ∩ B)′ ∪ C
> Draw a Venn diagram for sets A, B, and C and shade the given region. (A ∪ B)′
> Draw a Venn diagram for sets A, B, and C and shade the given region. A′ ∩ B′ ∩ C
> Determine whether the given set is finite or infinite. Consider the set N of positive integers to be the universal set. {2, 4, 6, 8, 10, … }
> Determine whether the given set is finite or infinite. Consider the set N of positive integers to be the universal set. {n ∈ N| n < 1000}
> For P, Q, and R in Problem 47, find P ∩(Q ∪ R). Data from Problem 47: For P = {1, 2, 3, 4}, Q = {2, 4, 6}, and R = {3, 4, 5,6 }, find P ∪ (Q ∩ R).
> If R = {1, 3, 4} and T = {2, 4, 6}, find (A) {x |x ∊ R and x ∊ T}
> Use formula (2) for the amount to find each of the indicated quantities. P = $3,000; r = 4.5%; t = 30 days; A = ?
> Refer to the Venn diagram below and find the indicated number of elements. n(A ∩ A′)
> Refer to the Venn diagram below and find the indicated number of elements. n(A′ ∩ B′)
> Refer to the Venn diagram below and find the indicated number of elements. n((A ∩ B)′)
> Refer to the Venn diagram below and find the indicated number of elements. n(A ∩ B′)
> Refer to the Venn diagram below and find the indicated number of elements. n(B′)
> Refer to the Venn diagram below and find the indicated number of elements. n(A ∩ B)
> Refer to the Venn diagram below and find the indicated number of elements. n(A)
> For U = {7, 8, 9, 10, 11} and A = {7, 11}, find A′.
> write the resulting set using the listing method. {x | x is a month starting with M}
> write the resulting set using the listing method. {x | x4 = 16}
> Use the given interest rate i per compounding period to find r, the annual rate. 0.012% per day
> write the resulting set using the listing method. {x | x2 = 36}
> write the resulting set using the listing method. {-3, -1,} ∪ {1, 3}
> write the resulting set using the listing method. {-3, -1} ∩ {1, 3}
> write the resulting set using the listing method. {1, 2, 4} ∩{4, 8, 16}
> write the resulting set using the listing method. {1, 2, 4} ∪ {4, 8, 16}
> Indicate true (T) or false (F). ∅ ⊂ {1, 2, 3}
> Indicate true (T) or false (F). {0, 6} = {6}
> Explain why the product of any two odd integers is odd
> Refer to the footnote for the definitions of divisor, multiple, prime, even, and odd. List the primes between 10 and 20.
> Refer to the footnote for the definitions of divisor, multiple, prime, even, and odd. List the positive multiples of 9 that are less than 50
> Use formula (1) for simple interest to find each of the indicated quantities. I = $96; P = $3,200; r = 4%; t = ?
> Refer to the footnote for the definitions of divisor, multiple, prime, even, and odd. List the positive integers that are divisors of 24.
> Can a conditional proposition be false if its converse is true? Explain.
> If the conditional proposition p is a contradiction, is ¬p a contingency, a tautology, or a contradiction? Explain.
> Let p be the proposition “every politician is honest.” Explain why the statement “every politician is dishonest” is not equivalent to ¬p. Express ¬p as an English sentence without using the word not.
> verify each equivalence using formulas from Table 2. ¬(¬p → ¬q) ( q ( ¬p
> verify each equivalence using formulas from Table 2. ¬p → q ( p ( q
> construct a truth table to verify each equivalence. p → (p ( q) ( p → q
> construct a truth table to verify each equivalence. p ( (p → q) ( p ( q
> Construct a truth table to verify each equivalence. q → (¬p ( q) ( ¬(p ( q)
> Construct a truth table to verify each implication. (p ( ¬p) ( q
> Use the given interest rate i per compounding period to find r, the annual rate. 1.57% per quarter
> Evaluate the expression. If the answer is not an integer, round to four decimal places.
> construct a truth table to verify each implication. p ( q ( p → q
