Discrete Mathematics and Its Applications 7th edition Kenneth H. Rosen - Solutions

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In a drilling operation, four factors (A, B, C, and D), each at three levels, are thought to be of importance in influencing the volume of crude oil pumped. Using an L9 orthogonal array, the factors A, B, C, and D are assigned to columns 1, 2, 3, and 4, respectively. The response variable showing
With the assignment of factors A, B, C, and D to columns 1, 2, 3, and 4, respectively, of an L9 orthogonal array, the output is as follows for another replication of the nine experiments:Determine the main effects. Plot the average response curves. What are the optimal settings of the design
In a food processing plant, four design parameters, A, B, C, and D, each at three levels, have been identified as having an effect on the moisture content in packaged meat. Three noise factors, E, F, and G, each at two levels, are also to be investigated in the experiment. In the inner array, an
The design factors are A, B, C, and D, each at three levels. These are assigned to an orthogonal array (inner array) with the factors A, B, C, and D assigned to columns 1, 2, 3, and 4, respectively. Suppose that in addition to the three main effects of the noise factors E, F, and G, it is felt
What is the difference between qualitative and quantitative variables? Give examples of each in the transportation industry. Which of these two classes permit interpolation of the response variable?
In a textile processing plant the quality of the output fabric is believed to be influenced by four factors (A, B, C, and D), each of which can be controlled at three levels. The fabric is classified into three categories: acceptable, second-class, or reject. An L9 orthogonal array is selected for
Four factors (A, B, C, and D), each at three levels, are controlled in an experiment using an L9 orthogonal array. The output quality is classified as acceptable or unacceptable; unacceptable includes both the second class and reject classes. Consider the data shown in Table 12-47. Combine the
What is the difference between a fixed effects model and a random effects model? Give some examples in the logistics area.
Explain the difference between the completely randomized design and the randomized block design. Discuss in the context of a gasoline refining process. Under what conditions would you prefer to use each design?
Distinguish between a randomized block design and a Latin square design. What are the advantages and disadvantages of a Latin square design?
Explain why it does not make sense to test for the main effects in a factorial experiment if the interaction effects are significant.
Which of these sentences are propositions? What are the truth values of those that are propositions? a) Boston is the capital of Massachusetts. b) Miami is the capital of Florida. c) 2 + 3 = 5. d) 5 + 7 = 10. e) x + 2 = 11. f) Answer this question.
For each of these sentences, determine whether an inclusive or, or an exclusive or, is intended. Explain your answer. a) Coffee or tea comes with dinner. b) A password must have at least three digits or be at least eight characters long. c) The prerequisite for the course is a course in number
For each of these sentences, state what the sentence means if the logical connective or is an inclusive or (that is, a disjunction) versus an exclusive or. Which of these meanings of or do you think is intended? a) To take discrete mathematics, you must have taken calculus or a course in computer
Write each of these statements in the form "if p, then q" in English. [Hint: Refer to the list of common ways to express conditional statements.] a) It snows whenever the wind blows from the northeast. b) The apple trees will bloom if it stays warm for a week. c) That the Pistons win the
Write each of these propositions in the form "p if and only if q" in English. a) If it is hot outside you buy an ice cream cone, and if you buy an ice cream cone it is hot outside. b) For you to win the contest it is necessary and sufficient that you have the only winning ticket. c) You get
State the converse, contrapositive, and inverse of each of these conditional statements. a) If it snows today, I will ski tomorrow. b) I come to class whenever there is going to be a quiz. c) A positive integer is a prime only if it has no divisors other than 1 and itself.
How many rows appear in a truth table for each of these compound propositions? a) p → ¬p b) (p ∨ ¬r) ∧ (q ∨ ¬s) c) q ∨ p ∨ ¬s ∨ ¬r ∨ ¬t ∨ u d) (p ∧ r ∧ t) ↔ (q ∧ t)
Construct a truth table for each of these compound propositions. a) p ∧ ¬p b) p ∨¬p c) (p ∨ ¬q) → q d) (p ∨ q) → (p ∧ q) e) (p → q) ↔ (¬q → ¬p) f) (p → q) → (q → p)
Construct a truth table for each of these compound propositions. a) (p ∨ q) → (p ⊕ q) b) (p ⊕ q) → (p ∧ q) c) (p ∨ q) ⊕ (p ∧ q) d) (p ↔ q) ⊕ (¬p ↔ q) e) (p ↔ q) ⊕ (¬p ↔¬r) f) (p ⊕ q) → (p ⊕¬q)
Construct a truth table for each of these compound propositions. a) p →¬q b) ¬p ↔ q c) (p → q) ∨ (¬p → q) d) (p → q) ∧ (¬p → q) e) (p ↔ q) ∨ (¬p ↔ q) f) (¬p ↔¬q) ↔ (p ↔ q)
Construct a truth table for each of these compound propositions. a) p → (¬q ∨ r) b) ¬p → (q → r) c) (p → q) ∨ (¬p → r) d) (p → q) ∧ (¬p → r) e) (p ↔ q) ∨ (¬q ↔ r) f) (¬p ↔¬q) ↔ (q ↔ r)
What is the negation of each of these propositions? a) Mei has an MP3 player. b) There is no pollution in New Jersey. c) 2 + 1 = 3. d) The summer in Maine is hot and sunny.
Construct a truth table for (p ↔ q) ↔ (r ↔ s).
Explain, without using a truth table, why (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r) is true when at least one of p, q, and r is true and at least one is false, but is false when all three variables have the same truth value.
Find the bitwise OR, bitwise AND, and bitwise XOR of each of these pairs of bit strings. a) 101 1110, 010 0001 b) 1111 0000, 1010 1010 c) 00 0111 0001, 10 0100 1000 d) 11 1111 1111, 00 0000 0000
The truth value of the negation of a proposition in fuzzy logic is 1 minus the truth value of the proposition. What are the truth values of the statements "Fred is not happy" and "John is not happy?"
The truth value of the disjunction of two propositions in fuzzy logic is the maximum of the truth values of the two propositions. What are the truth values of the statements "Fred is happy, or John is happy" and "Fred is not happy, or John is not happy?"
The nth statement in a list of 100 statements is "Exactly n of the statements in this list are false." a) What conclusions can you draw from these statements? b) Answer part (a) if the nth statement is "At least n of the statements in this list are false." c) Answer part (b) assuming that the list
What is the negation of each of these propositions? a) Steve has more than 100 GB free disk space on his laptop. b) Zach blocks e-mails and texts from Jennifer. c) 7 ∙ 11 ∙ 13 = 999. d) Diane rode her bicycle 100 miles on Sunday.
Suppose that during the most recent fiscal year, the annual revenue of Acme Computer was 138 billion dollars and its net profit was 8 billion dollars, the annual revenue of Nadir Software was 87 billion dollars and its net profit was 5 billion dollars, and the annual revenue of Quixote Media was
Let p and q be the propositions "Swimming at the New Jersey shore is allowed" and "Sharks have been spotted near the shore," respectively. Express each of these compound propositions as an English sentence. a) ¬q b) p ∧ q c) ¬p ∨ q d) p →¬q e) ¬q → p f) ¬p →¬q g) p ↔¬q h) ¬p ∧
Let p and q be the propositions p: It is below freezing. q: It is snowing. Write these propositions using p and q and logical connectives (including negations). a) It is below freezing and snowing. b) It is below freezing but not snowing. c) It is not below freezing and it is not snowing. d) It is
Let p and q be the propositions p :You drive over 65 miles per hour. q :You get a speeding ticket. Write these propositions using p and q and logical connectives (including negations). a) You do not drive over 65 miles per hour. b) You drive over 65 miles per hour, but you do not get a speeding
Let p, q, and r be the propositionsp: Grizzly bears have been seen in the area.q: Hiking is safe on the trail.r: Berries are ripe along the trail.Write these propositions using p, q, and r and logical connectives (including negations).a) Berries are ripe along the trail, but grizzly bears have not
Determine whether each of these conditional statements is true or false. a) If 1 + 1 = 2, then 2 + 2 = 5. b) If 1 + 1 = 3, then 2 + 2 = 4. c) If 1 + 1 = 3, then 2 + 2 = 5. d) If monkeys can fly, then 1 + 1 = 3.
You can graduate only if you have completed the requirements of your major and you do not owe money to the university and you do not have an overdue library book. Express your answer in terms of g: "You can graduate," m: "You owe money to the university," r: "You have completed the requirements of
A says "I am the knight," B says "I am the knave," and C says "B is the knight."
Are these system specifications consistent? "The router can send packets to the edge system only if it supports the new address space. For the router to support the new address space it is necessary that the latest software release be installed. The router can send packets to the edge system if the
A says "I am the knight," B says "A is telling the truth," and C says "I am the spy."
A says "I am the knight," B says "I am the knight," and C says "I am the knight."
A says "I am not the spy," B says "I am not the spy," and C says "I am not the spy." Exercises 32-38 are puzzles that can be solved by translating statements into logical expressions and reasoning from these expressions using truth tables.
Steve would like to determine the relative salaries of three coworkers using two facts. First, he knows that if Fred is not the highest paid of the three, then Janice is. Second, he knows that if Janice is not the lowest paid, then Maggie is paid the most. Is it possible to determine the relative
A detective has interviewed four witnesses to a crime. From the stories of the witnesses the detective has concluded that if the butler is telling the truth then so is the cook; the cook and the gardener cannot both be telling the truth; the gardener and the handyman are not both lying; and if the
Suppose there are signs on the doors to two rooms. The sign on the first door reads "In this room there is a lady, and in the other one there is a tiger"; and the sign on the second door reads "In one of these rooms, there is a lady, and in one of them there is a tiger." Suppose that you know that
Freedonia has fifty senators. Each senator is either honest or corrupt. Suppose you knowthat at least one of the Freedonian senators is honest and that, given any two Freedonian senators, at least one is corrupt. Based on these facts, can you determine how many Freedonian senators are honest and
Find the output of each of these combinatorial circuits.(a)(b)
Express these system specifications using the propositions p "The message is scanned for viruses" and q "The message was sent from an unknown system" together with logical connectives (including negations).a) "The message is scanned for viruses whenever the message was sent from an unknown
Construct a combinatorial circuit using inverters, OR gates, and AND gates that produces the output ((¬p ∨ ¬r)∧ ¬q) ∨ (¬p ∧ (q ∨ r)) from input bits p, q, and r.
You cannot edit a protected Wikipedia entry unless you are an administrator. Express your answer in terms of e: "You can edit a protected Wikipedia entry" and a: "You are an administrator."
Are these system specifications consistent? "The system is in multiuser state if and only if it is operating normally. If the system is operating normally, the kernel is functioning. The kernel is not functioning or the system is in interrupt mode. If the system is not in multiuser state, then it
What Boolean search would you use to look for Web pages about beaches in New Jersey? What if you wanted to findWeb pages about beaches on the isle of Jersey (in the English Channel)?
When three professors are seated in a restaurant, the hostess asks them: "Does everyone want coffee?" The first professor says: "I do not know." The second professor then says: "I do not know." Finally, the third professor says: "No, not everyonewants coffee." The hostess comes back and gives
A says "At least one of us is a knave" and B says nothing.
A says "I am a knave or B is a knight" and B says nothing.
A says "We are both knaves" and B says nothing. Exercises 24-31 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals bySmullyan [Sm78]) who can either lie or tell the truth. You encounter
Use truth tables to verify these equivalences.a) p ∧ T ≡ pb) p ∨ F ≡ pc) p ∧ F ≡Fd) p ∨ T ≡ Te) p ∨ p ≡ pf) p ∧ p ≡ p
Show that ¬p ↔ q and p ↔¬q are logically equivalent.
Show that ¬(p ↔ q) and ¬p ↔ q are logically equivalent.
Show that (p → r) ∧ (q → r) and (p ∨ q) → r are logically equivalent.
Show that (p → r) ∨ (q → r) and (p ∧ q) → r are logically equivalent.
Show that (p → q) ∧ (q → r) → (p → r) is a tautology.
Show that (p → q) → r and p → (q → r) are not logically equivalent.
Show that (p → q) → (r → s) and (p → r) → (q → s) are not logically equivalent.
Find the dual of each of these compound propositions.a) p ∧¬q ∧¬rb) (p ∧ q ∧ r) ∨ sc) (p ∨ F) ∧ (q ∨ T)
Show that (s∗)∗ = s when s is a compound proposition.
Why are the duals of two equivalent compound propositions also equivalent, where these compound propositions contain only the operators ∧, ∨, and ¬?
Use truth tables to verify the commutative laws a) p ∨ q ≡ q ∨ p. b) p ∧ q ≡ q ∧ p.
Find a compound proposition involving the propositional variables p, q, and r that is true when exactly two of p, q, and r are true and is false otherwise.
Show that ¬, ∧, and ∨ form a functionally complete collection of logical operators.
Show that ¬ and ∨ form a functionally complete collection of logical operators.The following exercises involve the logical operators NAND and NOR. The proposition p NAND q is true when either p or q, or both, are false; and it is false when both p and q are true. The proposition p NOR q is true
Show that p | q is logically equivalent to ¬(p ∧ q).
Show that p ↓ q is logically equivalent to ¬(p ∨ q).
Find a compound proposition logically equivalent to p → q using only the logical operator ↓.
How many different truth tables of compound propositions are there that involve the propositional variables p and q?
The following sentence is taken from the specification of a telephone system: "If the directory database is opened, then the monitor is put in a closed state, if the system is not in its initial state." This specification is hard to understand because it involves two conditional statements. Find an
How many of the disjunctions p ∨¬q ∨ s, ¬p ∨ ¬r ∨ s, ¬p ∨¬r ∨¬s, ¬p ∨ q ∨¬s, q ∨ r ∨¬s, q ∨¬r ∨¬s, ¬p ∨¬q ∨¬s, p ∨ r ∨ s, and p ∨ r ∨¬s can be made simultaneously true by an assignment of truth values to p, q, r, and s?
Determine whether each of these compound propositions is satisfiable.a) (p ∨¬q) ∧ (¬p ∨ q) ∧ (¬p ∨¬q)b) (p → q) ∧ (p →¬q) ∧ (¬p → q) ∧ (¬p →¬q)c) (p ↔ q) ∧ (¬p ↔ q)
Use a truth table to verify the distributive law p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r).
Show how the solution of a given 4 × 4 Sudoku puzzle can be found by solving a satisfiability problem.
Explain the steps in the construction of the compound proposition given in the text that asserts that every column of a 9 × 9 Sudoku puzzle contains every number.
Use De Morgan's laws to find the negation of each of the following statements. a) Jan is rich and happy. b) Carlos will bicycle or run tomorrow. c) Mei walks or takes the bus to class. d) Ibrahim is smart and hard working.
Show that each of these conditional statements is a tautology by using truth tables. a) (p ∧ q) → p b) p → (p ∨ q) c) ¬p → (p → q) d) (p ∧ q) → (p → q) e) ¬(p → q) → p f) ¬(p → q)→¬q
Show that each conditional statement in Exercise 9 is a tautology without using truth tables.In Figure 9a) (p ∧ q) → pb) p → (p ∨ q)c) ¬p → (p → q)d) (p ∧ q) → (p → q)e) ¬(p → q) → pf) ¬(p → q)→¬q
Use truth tables to verify the absorption laws.a) p ∨ (p ∧ q) ≡ pb) p ∧ (p ∨ q) ≡ p
Determine whether (¬q ∧ (p → q)) →¬p is a tautology. Each of Exercises 16-28 asks you to show that two compound propositions are logically equivalent. To do this, either show that both sides are true, or that both sides are false, for exactly the same combinations of truth values of the
Show that ¬(p ↔ q) and p ↔¬q are logically equivalent.
Let Q(x, y) denote the statement "x is the capital of y." What are these truth values? a) Q(Denver, Colorado) b) Q(Detroit, Michigan) c) Q(Massachusetts, Boston) d) Q(NewYork, NewYork)
Establish these logical equivalences, where x does not occur as a free variable in A. Assume that the domain is nonempty. a) (∀xP(x)) ∧ A ≡ ∀x(P(x) ∧ A) b) (∃xP(x)) ∧ A ≡ ∃x(P(x) ∧ A)
Show that ∃xP(x) ∧ ∃xQ(x) and ∃x(P(x) ∧ Q(x)) are not logically equivalent.
What are the truth values of these statements? a) ∃!xP(x) → ∃xP(x) b) ∀xP(x) → ∃!xP(x) c) ∃!x¬P(x)→¬∀xP(x)
Given the Prolog facts in Example 28, what would Prolog return given these queries? a) ?instructor(chan,math273) b) ?instructor(patel,cs301) c) ?enrolled(X,cs301) d) ?enrolled(kiko,Y) e) ?teaches(grossman,Y)
Suppose that Prolog facts are used to define the predicates mother(M, Y) and father(F,X), which represent that M is the mother of Y and F is the father of X, respectively. Give a Prolog rule to define the predicate sibling(X, Y), which represents that X and Y are siblings (that is, have the same
Let P(x) be the statement "x spends more than five hours every weekday in class," where the domain for x consists of all students. Express each of these quantifications in English. a) ∃x P(x) b) ∀x P(x) c) ∃x ¬P(x) d) ∀x ¬P(x)
Translate these statements into English, where C(x) is "x is a comedian" and F(x) is "x is funny" and the domain consists of all people. a) ∀x(C(x) → F(x)) b) ∀x(C(x) ∧ F(x)) c) ∃x(C(x) → F(x)) d) ∃x(C(x) ∧ F(x))
Determine the truth value of each of these statements if the domain consists of all integers. a) ∀n(n + 1 > n) b) ∃n(2n = 3n) c) ∃n(n = −n) d) ∀n(3n ≤ 4n)
Determine the truth value of each of these statements if the domain for all variables consists of all integers. a) ∀n(n2 ≥ 0) b) ∃n(n2 = 2) c) ∀n(n2 ≥ n) d) ∃n(n2 < 0)
Suppose that the domain of the propositional function P(x) consists of the integers 0, 1, 2, 3, and 4. Write out each of these propositions using disjunctions, conjunctions, and negations.a) ∃x P(x) b) ∀x P(x) c) ∃x - P(x)d) ∀x - P(x)e) -∃x P(x) f) -∀x P(x)
For each of these statements find a domain for which the statement is true and a domain for which the statement is false. a) Everyone is studying discrete mathematics. b) Everyone is older than 21 years. c) Every two people have the same mother. d) No two different people have the same grandmother.
Suppose that the domain of Q(x, y, z) consists of triples x, y, z, where x = 0, 1, or 2, y = 0 or 1, and z = 0 or 1. Write out these propositions using disjunctions and conjunctions.a) ∀y Q(0, y, 0) b) ∃x Q(x, 1, 1)c) ∃z - Q(0, 0, z) d) ∃x - Q(x, 0, 1)

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Discrete Mathematics and Its Applications 7th edition Kenneth H. Rosen - Solutions

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