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For a standard maximization problem in standard form, with two decision variables x1 and x2, explain why the feasible region is not empty.
A dietitian arranges a special diet using three foods: L, M, and N. Each ounce of food L contains 20 units of calcium, 10 units of iron, 10 units of vitamin A, and 20 units of cholesterol. Each ounce
In Problem solve the given linear programming problem using the table method. Maximize subject to P = 10x₁ + 9x₂ 4x₁ + 5x₂ = 20 x1, x₂ = 0
For a standard maximization problem in standard form, with k decision variables, x1, x2, . . ., xk, explain why the feasible region is not empty.
Many colleges and universities use computer-assisted testing. Suppose that a screening test is to consist of 5 questions, and a computer stores 5 comparable questions for the first test question, 8
Let the universal set U be the set of positive integers less than or equal to 100. Let A be the set of multiples of 3 in U, and let B be the set of multiples of 7 in U.(A) Find n(A ∩ B), n(A ∩
In Problem solve the given linear programming problem using the table method. Maximize subject to = P 15x₁ + 20x₂ x₁ + x₂ ≤ 6 x₁ + 4x₂ = 12 X1, X₂ ≥ 0
In Problem solve the given linear programming problem using the table method. Maximize subject to P= 5x₁ + 20x₂ 5x₁ + x₂ = 15 x₁ + x₂ = 7 x1, x₂ = 0 x2
In Problem solve the given linear programming problem using the table method. Maximize subject to P = 25x₁ + 10x2 2x₁ + 5x₂
In Problem solve the given linear programming problem using the table method. Maximize subject to P = 40x₁ + 50x2 x₁ + 3x₂ 18 5x₁ + 4x2 = 35 X1, X₂0
In Problem solve the given linear programming problem using the table method. Maximize subject to P = x₁ + x₂ 5x₁ + 4x₂ = 240 5x1 + 2x2 = 150 5x1 + x ≤ 120 X1, X₂0
In Problem solve the given linear programming problem using the table method. Maximize subject to P = x₁ + x₂ 5x₁ + 4x₂ = 240 5x12x2 150 5x1 + x < 120 x1, x₂ = 0
Consider the propositions p and q:Express each of the following propositions as an English sentence and determine whether it is true or false.(A) ¬ p (B) ¬ q (C) p ∨ q (D) p ∧ q(E) p → q
Let A, B, and C be three sets. Use a Venn diagram (Fig. 3) to explain the following equation: n(AUBUC) = n(A) + n(B) + n(C) − n(AMB) - n(ANC) - n(BNC) + n(ANBNC)
Solve for x.50 = 34 + 29 - x
From a committee of 12 people,(A) In how many ways can we choose a chairperson, a vice-chairperson, a secretary, and a treasurer, assuming that one person cannot hold more than one position?(B) In
In Problems answer yes or no.Is the set of integers the union of the set of even integers and the set of odd integers?
The LOGIC menu on the TI-84 Plus CE contains the operators “and,” “or,” “xor,” and “not.” The truth table for p ∧ q, for example, can be displayed by entering all combinations of
Which of the following statements are true? (A)o cơ (B) Ø E Ø (C) Ø= {0} (D) Ø C {0}
Find(A) 6! (B) 10!/9!(C) 10!/7!(D) 5!/0!3!(E) 20!/3!17!
(A) List alphabetically by the first letter, all 3-letter license plate codes consisting of 3 different letters chosen from M, A, T, H. Discuss how this list relates to nPr.(B) Reorganize the list
Consider the propositions p and q:Express each of the following propositions as an English sentence and determine whether it is true or false.(A) ¬ p (B) ¬ q (C) p ∨ q (D) p ∧ q (E) p → q
Answer yes or no.Is the set of even integers a subset of the set of odd integers?
According to a survey of business firms in a certain city, 750 firms offer their employees health insurance, 640 offer dental insurance, and 280 offer health insurance and dental insurance. How many
The survey in Example 1 also indicated that 345 firms offer their employees group life insurance, 285 offer long-term disability insurance, and 115 offer group life insurance and long-term disability
Given A = {0, 2, 4, 6}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {2, 6, 0, 4}, indicate whether the following relationships are true (T) or false (F): (A) AC B (D) CC B (B) ACC (E) BA (C) A = C (F) Ø CB
Let G be the set of all numbers such that x2 = 9.(A) Denote G by the rule method.(B) Denote G by the listing method.(C) Indicate whether the following are true or false: 3 ∈ G, 9 ∉ G.
Consider the propositions p and q:Express each of the following propositions as an English sentence and determine whether it is true or false.(A) p → q (B) The converse of p → q (C) The
Students at a university have the option to stream TV shows over the internet or to watch cable TV. A survey of 100 college students produced the following results: In the past 30 days, 65 people
Solve for x.124 = 73 + 87 - x
If a compound proposition contains three variables p, q, and r, then its truth table will have eight rows, one for each of the eight ways of assigning truth values to p, q, and r (TTT, TTF, TFT, TFF,
Given the set {A, B, C}, how many permutations are possible for this set of 3 objects taken 2 at a time? Answer the question(A) Using a tree diagram(B) Using the multiplication principle(C) Using the
In Example 6, find the number of voters in the set (C ∪ M) ∩ M′. Describe this set verbally and with a Venn diagram.Data from Example 6In the 2016 presidential election, an exit poll of 100
Given the set [A, B, C, D], how many permutations are possible for this set of 4 objects taken 2 at a time? Answer the question(A) Using a tree diagram(B) Using the multiplication principle(C) Using
Consider the propositions p and q:Express each of the following propositions as an English sentence and determine whether it is true or false.(A) p → q (B) The converse of p → q (C) The
Answer yes or no.Is the set of rational numbers a subset of the set of integers?
A survey of 100 families found that 35 families subscribe to the video streaming service Webfilms, 60 families subscribe to the video streaming service Nile Prime, and 20 families subscribe to both
Construct the truth table for p ∧ ¬ q.
A company offers its employees health plans from three different companies: R, S, and T. Each company offers two levels of coverage, A and B, with one level requiring additional employee
Solve for x.4x = 23 + 42 – x
Find the number of permutations of 13 objects taken 8 at a time. Compute the answer using a calculator.
Find the number of permutations of 30 objects taken 4 ata time. Compute the answer using a calculator.
List all subsets of the set {a, b, c}.
Construct the truth table for ¬p ∨ q.
Answer yes or no.Is the set of integers the intersection of the set of even integers and the set of odd integers?
A retail store stocks windbreaker jackets in small, medium, large, and extra large. All are available in blue or red. What are the combined choices, and how many combined choices are there?
If R = {1, 2, 3, 4}, S = {1, 3, 5, 7}, T = {2, 4}, and U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, find(A) R ∪ S (B) R ∩ S(C) S ∩ T(D) S′
Construct the truth table for [(p → q) ∧ ¬ q] → ¬ p.
Each question on a multiple-choice test has 5 choices. If there are 5 such questions on a test, how many different responses are possible if only 1 choice is marked for each question?
Solve for x.7x = 51 + 45 – x
From a committee of 10 people,(A) In how many ways can we choose a chairperson, a vice-chairperson, and a secretary, assuming that one person cannot hold more than one position?(B) In how many ways
Construct the truth table for [(p → q) ∧ p] → q.
Construct the truth table for (p → q) ∧ (p ∧ ¬q).
How many 4-letter code words are possible using the first 10 letters of the alphabet under the three different conditions stated in Example 5?Data from Example 5How many 3-letter code words are
Solve for x.3(x + 11) = 65 + x – 14
Find the number of combinations of 13 objects taken 8 at a time. Compute the answer using a calculator.
Find the number of combinations of 30 objects taken 4 at a time. Compute the answer using a calculator.
Let the universal set U be the set of positive integers less than or equal to 100. Let A be the set of multiples of 3 in U, and let B be the set of multiples of 5 in U.(A) Find n(A ∩ B), n(A ∩
Construct the truth table for p ∧ ¬(p ∨ q).
Answer yes or no.If the universal set is the set of integers, is the set of positive integers the complement of the set of negative integers?
How many 3-letter code words are possible using the first 8 letters of the alphabet if(A) No letter can be repeated?(B) Letters can be repeated?(C) Adjacent letters cannot be alike?
Show that [(p → q) → p)] ⇒ (q → p).
Solve for x.12(x + 5) = x + 122 – 29
How many 5-card hands have 3 aces and 2 kings?
Refer to Example 6.(A) Find n(T ⋂ W), n(T ∩ W´), n(W ∩ T´), and n(T´ ∩ W´).(B) Draw a Venn diagram with circles labeled T and W, indicating the numbers of elements in the subsets of part
How many 5-card hands have 3 hearts and 2 spades?
Show that [(p → q) ∧ ¬ q] ⇒ ¬ p.
In the 2016 presidential election, an exit poll of 100 voters produced the results in the table (23 men voted for Clinton, 29 women for Clinton, 26 men for Trump, and 22 women for Trump).Let the
Answer yes or no.If the universal set is the set of integers, is the set of even integers the complement of the set of odd integers?
Show that ¬(p ∧ q) ≡ ¬ p ∨ ¬ q.
Solve Problem two ways:(A) Using a tree diagram, (B) Using the multiplication principle.How many ways can 2 coins turn up—heads, H, or tails, T—if the combined outcome (H, T) is to be
Serial numbers for a product are made using 2 lettersfollowed by 3 numbers. If the letters are taken from the first 8 letters of the alphabetwith no repeats and the numbers are taken from the 10
Repeat Example 7 under the same conditions, except that the serial numbers will now have 3 letters followed by 2 digits (no repeats).Data from Example 7Serial numbers for a product are made using 2
Using a random sample of 100 insurance customers, an insurance company generated the Venn diagram in Figure 10 where A is the set of customers who purchased auto insurance, H is the set of customers
Show that ¬ (p ∨ q) ≡ ¬ p ∧ ¬ q.
Use equivalences from Table 2 to show thatp → q ≡ ¬(¬ q ∧ p).Data from Table 2 Table 2 Some Logical Equivalences (1) (2) (3) -(-p) - P PV q = q V p PAq=qAp PqPV q (p V q) = (p^ q)
In Problem indicate true (T) or false (F).{1, 2} ⊂ {2, 1}
A company has 7 senior and 5 junior officers. It wants to form an ad hoc legislative committee. In how many ways can a 4-officer committee be formed so that it is composed of(A) Any 4 officers?(B) 4
Given the information in Example 8, answer the following questions:(A) How many 4-officer committees with 1 senior officer and 3 junior officers can be formed?(B) How many 4-officer committees with 4
Solve Problem two ways:(A) Using a tree diagram, (B) Using the multiplication principle.A coin is tossed with possible outcomes of heads H, or tails T. Then a single die is tossed with possible
From a standard 52-card deck, how many 3-card hands have all cards from the same suit?
In Problem 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
In Problem 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′) = 40, n(U) =
In Problem use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram.n(A′) = 20, n(B′) = 40, n(A′ ∪ B′) = 50, n(U) =
In Problem 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′) = 35, n(U) =
For U = {1, 2, 3, 4, 5} and A = {2, 3, 4}, find A′.
Refer to Example 7.(A) How many customers purchased homeowner’s insurance?(B) Shade the region A ∪ L in Figure 10. Find n(A ∪ L).(C) Shade the region A′ ∩ H′ ∩ L in Figure 10. Find
From a standard 52-card deck, how many 5-card hands have all cards from the same suit?
In Problem use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram.n(A′) = 110, n(B′) = 220, n(A′ ¨ B′) = 60, n(U) =
In Problem determine whether or not the given sets are disjoint. For the definitions of M, K, and E, refer to the instructions for Problems 37–40.M′ and K′Data from instructions for Problems
In Problem determine whether or not the given sets are disjoint. For the definitions of M, K, and E, refer to the instructions for Problems 37–40.M and E′Data from instructions for Problems
In Example 3, does it make any difference in which order the selection operations are performed? That is, if we select a jacket color first and then select a size, are there as many combined choices
Draw a Venn diagram for sets A, B, and C and shade the region described by A′ ∩ (B ∪ C).
If R = {1, 2, 3, 4} and T = {2, 4, 6}, find(A) {x| x ∈ R or x ∈ T}(B) R ∪ T
How many 3-letter code words are possible using the first 8 letters of the alphabet if no letter can be repeated? If letters can be repeated? If adjacent letters cannot be alike?
From a standard 52-card deck, how many 5-card hands contain 3 cards of one rank and 2 cards of a different rank?
From a standard 52-card deck, how many 5-card hands contain 5 different ranks of cards?
In Problem verify each equivalence using formulas from Table 2.¬(p → q) ≡ p ∧ ¬ q Table 2 Some Logical Equivalences -(-p) = p (1) PVq=qVp (2) P^ q = q ^p (3) P→q = p V q (pv q) = p ^ -q (p
In Problem verify each equivalence using formulas from Table 2.¬(¬ p → ¬ q) ≡ q ∧ ¬ p Table 2 Some Logical Equivalences -(-p) = p (1) PVq=qVp (2) PAq = q ^ p (3) P→q = p V q (pv q) = p ^
Can a conditional proposition be false if its contrapositive is true? Explain.

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