DISCRETE MATHEMATICS LOOSELEAF
8th Edition
ISBN: 9781264309689
Author: ROSEN
Publisher: MCG
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Chapter 1.8, Problem 37E
To determine
To prove:
Between every two rational numbers there is an irrational number.
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Chapter 1 Solutions
DISCRETE MATHEMATICS LOOSELEAF
Ch. 1.1 - Which of these sentences are propositions? What...Ch. 1.1 - Which of these are propositions? What are the...Ch. 1.1 - What is the negation of each of these...Ch. 1.1 - What is the negation of each of these...Ch. 1.1 - What is the negation of each of these...Ch. 1.1 - What is the negation of each of these...Ch. 1.1 - What is the negation of each of these...Ch. 1.1 - Suppose that Smartphone A has 256 MB RAM and 32 GB...Ch. 1.1 - Suppose that during the most recent fiscal year,...Ch. 1.1 - Letpandqbe the propositions p:I bought a lottery...
Ch. 1.1 - Le p and q be the prepositions “Swimming at the...Ch. 1.1 - Le p and q be the prepositions “The election is...Ch. 1.1 - Let p and q be the propositions p: It is below...Ch. 1.1 - Let p, q, and r be the propositions p: You have...Ch. 1.1 - Let p and q be the propositions p: You drive over...Ch. 1.1 - Let p, q, and r be the propositions p: You get an...Ch. 1.1 - Letp,q, andrbe the propositions p:Grizzly bears...Ch. 1.1 - Determine whether these biconditionals are true or...Ch. 1.1 - Determine whether each of these conditional...Ch. 1.1 - Determine whether each of these conditional...Ch. 1.1 - For each of these sentences, determine whether an...Ch. 1.1 - For each of these sentences, determine whether an...Ch. 1.1 - For each of these sentences, state what the...Ch. 1.1 - Write each of these statements in the form "ifp,...Ch. 1.1 - Write each of these statements in the form "ifp,...Ch. 1.1 - Write each of these statements in the form "ifp,...Ch. 1.1 - Write each of these propositions in the form "pif...Ch. 1.1 - Write each of these propositions in the form "pif...Ch. 1.1 - State the converse, contrapositive, and inverse of...Ch. 1.1 - State the converse, contrapositive, and inverse of...Ch. 1.1 - How many rows appear in a truth table for each of...Ch. 1.1 - How many rows appear in a truth table for each of...Ch. 1.1 - Construct a truth table for each of these compound...Ch. 1.1 - Construct a truth table for each of these compound...Ch. 1.1 - Construct a truth table for each of these compound...Ch. 1.1 - Construct a truth table for each of these compound...Ch. 1.1 - Construct a truth table for each of these compound...Ch. 1.1 - Construct a truth table for each of these compound...Ch. 1.1 - Construct a truth table for each of these compound...Ch. 1.1 - Construct a truth table for((pq)r)s.Ch. 1.1 - Construct a truth table for(pq)(rs).Ch. 1.1 - Explain, without using a truth table,...Ch. 1.1 - Explain, without using a truth table,...Ch. 1.1 - Ifp1,p2, ...,pnarenpropositions, explain why...Ch. 1.1 - Use Exercise 44 to construct a compound...Ch. 1.1 - What is the value ofxafter each of these...Ch. 1.1 - Find the bitwiseOR, bitwiseAND, and bitwiseXORof...Ch. 1.1 - Evaluate each of these expressions....Ch. 1.1 - Fuzzy logicis used in artificial intelligence. In...Ch. 1.1 - Fuzzy logicis used in artificial intelligence. In...Ch. 1.1 - Fuzzy logicis used in artificial intelligence. In...Ch. 1.1 - Is the assertion “This statement is false” a...Ch. 1.1 - The nth statement in a list of 100 statements is...Ch. 1.1 - An ancient Sicilian legend says that the barber in...Ch. 1.2 - In Exercise 1-6, translate the given statement...Ch. 1.2 - In Exercise 1-6, translate the given statement...Ch. 1.2 - In Exercise 1-6, translate the given statement...Ch. 1.2 - In Exercise 1-6, translate the given statement...Ch. 1.2 - In Exercise 1-6, translate the given statement...Ch. 1.2 - In Exercise 1-6, translate the given statement...Ch. 1.2 - Prob. 7ECh. 1.2 - Express these system specifications using the...Ch. 1.2 - Are these system specifications consistent? "The...Ch. 1.2 - Are these system specifications consistent?...Ch. 1.2 - Are these system specifications consistent? "The...Ch. 1.2 - Are these system specifications consistent? “If...Ch. 1.2 - What Boolean search would you use to look for Web...Ch. 1.2 - What Boolean search would you use to look for Web...Ch. 1.2 - What Google search would you use to look for Web...Ch. 1.2 - What Google search would you use to look for men’s...Ch. 1.2 - Suppose that inExample7, the inscriptions on...Ch. 1.2 - Suppose that inExample 7there are treasures in two...Ch. 1.2 - Each inhabitant of a remote village always tells...Ch. 1.2 - An explorer is captured by a group of cannibals....Ch. 1.2 - When three professors are seated in a restaurant,...Ch. 1.2 - When planning a party you want to know whom to...Ch. 1.2 - Exercises 23-27 relate to inhabitants of the...Ch. 1.2 - Exercises 23-27 relate to inhabitants of the...Ch. 1.2 - Exercises 23-27 relate to inhabitants of the...Ch. 1.2 - Exercises 23-27 relate to inhabitants of the...Ch. 1.2 - Exercises 23-27 relate to inhabitants of the...Ch. 1.2 - Exercises 28-35 relate to inhabitants of an island...Ch. 1.2 - Exercises 28-35 relate to inhabitants of an island...Ch. 1.2 - Exercises 28-35 relate to inhabitants of an island...Ch. 1.2 - Exercises 28-35 relate to inhabitants of an island...Ch. 1.2 - Exercises 28-35 relate to inhabitants of an island...Ch. 1.2 - Exercises 28-35 relate to inhabitants of an island...Ch. 1.2 - Exercises 28-35 relate to inhabitants of an island...Ch. 1.2 - Prob. 35ECh. 1.2 - Exercises 36-42 are puzzles that can be solved by...Ch. 1.2 - Exercises 36-42 are puzzles that can be solved by...Ch. 1.2 - Exercises 36-42 are puzzles that can be solved by...Ch. 1.2 - Exercises 36-42 are puzzles that can be solved by...Ch. 1.2 - Exercises 36-42 are puzzles that can be solved by...Ch. 1.2 - Exercises 36-42 are puzzles that can be solved by...Ch. 1.2 - Exercises 36-42 are puzzles that can be solved by...Ch. 1.2 - Freedonia has 50 senators. Each senator is either...Ch. 1.2 - Find the output of each of these combinatorial...Ch. 1.2 - Find the output of each of these combinatorial...Ch. 1.2 - Construct a combinatorial circuit using inverters,...Ch. 1.2 - Construct a combinatorial circuit using inverters,...Ch. 1.3 - Use truth tables to verify these equivalences....Ch. 1.3 - Show that(p)andpare logically equivalent.Ch. 1.3 - Use truth tables to verify the commutative laws...Ch. 1.3 - Use truth tables to verify the associative laws...Ch. 1.3 - Use a truth table to verify the distributive law...Ch. 1.3 - Use a truth table to verify the first De Morgan...Ch. 1.3 - Use De Morgan's laws to find the negation of each...Ch. 1.3 - Use De Morgan's laws to find the negation of each...Ch. 1.3 - For each of these compound propositions, use the...Ch. 1.3 - For each of these compound propositions, use the...Ch. 1.3 - Show that each of these conditional statements is...Ch. 1.3 - Show that each of these conditional statements is...Ch. 1.3 - Show that each conditional statement in Exercise...Ch. 1.3 - Show that each conditional statement in Exercise...Ch. 1.3 - Show that each conditional statement in Exercise...Ch. 1.3 - Show that each conditional statement in Exercise...Ch. 1.3 - Use truth tables to verify the absorption laws....Ch. 1.3 - Determine whether(p(pq))qis a tautology.Ch. 1.3 - Determine whether(q(pq))qis a tautology.Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Each of Exercises 20-32 asks you to show that two...Ch. 1.3 - Show that(pq)(qr)(pr)is a tautology.Ch. 1.3 - Show that(pq)(pr)(pr)is a tautology.Ch. 1.3 - Show that(pq)randp(qr)are not logically...Ch. 1.3 - Show that(pq)rand(pr)(qr)are not logically...Ch. 1.3 - Show that(pq)(rs)and(pr)(qs)are not logically...Ch. 1.3 - Find the dual of each of these compound...Ch. 1.3 - Find the dual of each of these compound...Ch. 1.3 - Prob. 40ECh. 1.3 - Show that(s*)*=s, wheresis a compound proposition?Ch. 1.3 - Show that the logical equivalences inTable 6,...Ch. 1.3 - Why are the duals of two equivalent compound...Ch. 1.3 - Find a compound proposition involving the...Ch. 1.3 - Find a compound proposition involving the...Ch. 1.3 - Suppose that a truth table innpropositional...Ch. 1.3 - A collection of logical operators is...Ch. 1.3 - A collection of logical operators is...Ch. 1.3 - A collection of logical operators is...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.3 - We now present a group of exercises that involve...Ch. 1.4 - LetP(x) denote the statement "x4. " What are these...Ch. 1.4 - LetP(x) be the statement "The wordxcontains the...Ch. 1.4 - LetQ(x,y) denote the statement "xis the capital...Ch. 1.4 - State the value ofxafter the statement ifP(x)...Ch. 1.4 - LetP(x) be the statement “xspends more than five...Ch. 1.4 - LetN(x) be the statement “x has visited North...Ch. 1.4 - Translate these statements into English, whereC(x)...Ch. 1.4 - Translate these statements into English, whereR(x)...Ch. 1.4 - LetP(x) be the statement "xcan speak Russian" and...Ch. 1.4 - LetC(x) be the statement “xhas a cat,” letD(x) be...Ch. 1.4 - LetP(x) be the statement "x=x2." If the domain...Ch. 1.4 - LetQ(x) be the statementx+12x.” If the domain...Ch. 1.4 - Determine the truth value of each of these...Ch. 1.4 - Determine the truth value of each of these...Ch. 1.4 - Determine the truth value of each of these...Ch. 1.4 - Determine the truth value of each of these...Ch. 1.4 - Suppose that the domain of the propositionalP(x)...Ch. 1.4 - Suppose that the domain of the propositionalP(x)...Ch. 1.4 - Suppose that the domain of the propositionalP(x)...Ch. 1.4 - Suppose that the domain of the propositionalP(x)...Ch. 1.4 - For each of these statements find a domain for...Ch. 1.4 - For each of these statements find a domain for...Ch. 1.4 - Translate in two ways each of these statements...Ch. 1.4 - Translate in two ways using predicates,...Ch. 1.4 - Translate each of these statements into logical...Ch. 1.4 - Translate each of these statements into logical...Ch. 1.4 - Translate each of these statements into logical...Ch. 1.4 - Translate each of these statements into logical...Ch. 1.4 - Express each of these statements using logical...Ch. 1.4 - Suppose the domain of the propositional...Ch. 1.4 - Suppose that the domain ofQ(x,y,z)consists of...Ch. 1.4 - Express each of these statements using...Ch. 1.4 - Express each of these statements using...Ch. 1.4 - Express the negation of these propositions using...Ch. 1.4 - Express the negation of each of these statements...Ch. 1.4 - Express the negation of each of these statements...Ch. 1.4 - Find a counter example, if possible, to these...Ch. 1.4 - Find a counterexample, if possible, to these...Ch. 1.4 - Express each of these statements using predicates...Ch. 1.4 - Exercises 40-44 deal the translation between...Ch. 1.4 - Exercises 40-44 deal the translation between...Ch. 1.4 - Exercises 40-44 deal the translation between...Ch. 1.4 - Exercises 40-44 deal the translation between...Ch. 1.4 - Exercises 40-44 deal the translation between...Ch. 1.4 - Determine whether SSx(P(x)Q(x))andxP(x)xQ(x)are...Ch. 1.4 - Determine whetherx(P(x)Q(x))andxP(x)xQ(x)are...Ch. 1.4 - Show thatx(P(x)Q(x))andxP(x)xQ(x)are logically...Ch. 1.4 - Exercises 4851 establish rules fornull...Ch. 1.4 - Exercises 4851 establish rules fornull...Ch. 1.4 - Exercises 4851 establish rules fornull...Ch. 1.4 - Exercises 4851 establish rules fornull...Ch. 1.4 - Show thatxP(x)xQ(x)andx(P(x)Q(x))are not logically...Ch. 1.4 - Show thatxP(x)xQ(x)andx(P(x)Q(x))are not logically...Ch. 1.4 - As mentioned in the text, the...Ch. 1.4 - What are the truth values of these statements?...Ch. 1.4 - Write out!xP(x), where the domain consists of the...Ch. 1.4 - Given the Prolog facts inExample 28, what would...Ch. 1.4 - Given the Prolog facts inExample 28, what would...Ch. 1.4 - Prob. 59ECh. 1.4 - Suppose that Prolog facts are used to define the...Ch. 1.4 - Prob. 61ECh. 1.4 - Exercises 61-64 are based on questions found in...Ch. 1.4 - Exercises 61-64 are based on questions found in...Ch. 1.4 - Exercises 61-64 are based on questions found in...Ch. 1.5 - Translate these statements into English, where the...Ch. 1.5 - Translate these statements into English, where the...Ch. 1.5 - LetQ(x,y) be the statement "xhas sent an e-mail...Ch. 1.5 - LetP(x,y) be the statement "Studentxhas taken...Ch. 1.5 - Let W(x,y) mean that studentxhas visited websitey,...Ch. 1.5 - LetC(x,y) mean that studentxis enrolled in classy,...Ch. 1.5 - LetT(x,y) mean that studentxlikes cuisiney, where...Ch. 1.5 - LetQ(x,y) be the statement "Studentxhas been a...Ch. 1.5 - LetL(x,y) be the statement "xlovesy," where the...Ch. 1.5 - LetF(x,y) be statement “xcan fooly,” where the...Ch. 1.5 - LetS(x) be predicate “xis a student,”F(x) the...Ch. 1.5 - LetI(x) be the statement “xhas an Internet...Ch. 1.5 - LetM(x,y) be “xhas sentyan e-mail message”...Ch. 1.5 - Prob. 14ECh. 1.5 - Prob. 15ECh. 1.5 - A discrete mathematics class 1 mathematics major...Ch. 1.5 - Express each of these system specifications using...Ch. 1.5 - Express each of these system specifications using...Ch. 1.5 - Express each of these statements using...Ch. 1.5 - Express each of these statements using predicates,...Ch. 1.5 - Use predicates, quantifiers, logical connectives,...Ch. 1.5 - Use predicates, quantifiers, logical connectives,...Ch. 1.5 - Express each of these mathematical statements...Ch. 1.5 - Translate each of these nested quantifications...Ch. 1.5 - Translate each of these nested quantifications...Ch. 1.5 - LetQ(x,y) be the statement "x+y=xy .” If the...Ch. 1.5 - Determine the truth value of each of these...Ch. 1.5 - Determine the truth value of each of these...Ch. 1.5 - Suppose the domain of the propositional...Ch. 1.5 - Rewrite each of these statements so that negations...Ch. 1.5 - Express the negations of each of these statements...Ch. 1.5 - Express the negations of each of these statements...Ch. 1.5 - Rewrite each of these statements so that negations...Ch. 1.5 - Find a common domain for the variablex,y, andzfor...Ch. 1.5 - Find a common domain for the variablesx,y,z,...Ch. 1.5 - Express each of these statements using...Ch. 1.5 - Express each of these statements using...Ch. 1.5 - Express the negations of these propositions using...Ch. 1.5 - Prob. 39ECh. 1.5 - Find a counterexample, if possible, to these...Ch. 1.5 - Use quantifiers to express the associative law for...Ch. 1.5 - Prob. 42ECh. 1.5 - Use quantifiers and logical connectives to express...Ch. 1.5 - Use quantifiers and logical connectives to express...Ch. 1.5 - Determine the truth value of the statementxy(xy=1)...Ch. 1.5 - Determine the truth value of the statement xy(xy2)...Ch. 1.5 - Show that the two statementsxyP(x,y)andxyP(x,y),...Ch. 1.5 - Show thatxP(x)xQ(x)andxy(P(x)Q(y)), where all...Ch. 1.5 - a) Show thatxP(x)xQ(x)is logically equivalent...Ch. 1.5 - Put these statements in prenex normal form....Ch. 1.5 - Show how to transform an arbitrary statement to a...Ch. 1.5 - Express the quantification!P(x), introduced...Ch. 1.6 - Find the argument form for the following argument...Ch. 1.6 - Find the argument form for the following argument...Ch. 1.6 - What rule of inference is used in each of these...Ch. 1.6 - What rule of inference is used in each of these...Ch. 1.6 - Use rules of inference to show that the hypotheses...Ch. 1.6 - Use rules of inference to show that the hypotheses...Ch. 1.6 - What rules of inference are used in this famous...Ch. 1.6 - What rules of inference are used in this argument?...Ch. 1.6 - For each of these collections of premises, what...Ch. 1.6 - For each of these sets of premises, what relevant...Ch. 1.6 - Show that the argument form with premisesp1,p2,...Ch. 1.6 - Show that the argument...Ch. 1.6 - For each of these arguments, explain which rules...Ch. 1.6 - For each of these arguments, explain which rules...Ch. 1.6 - For each of these arguments determine whether the...Ch. 1.6 - For each of these arguments whether the argument...Ch. 1.6 - What is wrong this argument? LetH(x) be is “xis...Ch. 1.6 - What is wrong with this argument? LetS(x,y) be...Ch. 1.6 - Determine whether each of these arguments is...Ch. 1.6 - Determine whether these are valid arguments. a)...Ch. 1.6 - Which rules of inference are used to establish the...Ch. 1.6 - Which rules of inference are used to establish the...Ch. 1.6 - Identify the error or errors in argument that...Ch. 1.6 - Identify the error or errors in argument that...Ch. 1.6 - Justify the rule of universal modus tollens by...Ch. 1.6 - Justify the rule ofuniversal transitivity, which...Ch. 1.6 - Use rules of inference to show that...Ch. 1.6 - Use rules of inference to show that ifx(P(x)Q(x))...Ch. 1.6 - Use rules of inference to show...Ch. 1.6 - Use resolution to show the hypotheses “Allen is a...Ch. 1.6 - Use resolution to show that the hypotheses “It is...Ch. 1.6 - Prob. 32ECh. 1.6 - Use resolution to show that the compound...Ch. 1.6 - The Logic Problem, taken from WFPN PROOF, The Game...Ch. 1.6 - Determine whether this argument, taken from Kalish...Ch. 1.7 - Use a direct proof to show that the sum of two odd...Ch. 1.7 - Use a direct proof to show that the sum of two...Ch. 1.7 - Show that the square of an even number is an even...Ch. 1.7 - Show that the additive inverse, or negative, of an...Ch. 1.7 - Prove that ifm+n andn+p are even integers,...Ch. 1.7 - Use a direct proof to show that the product of two...Ch. 1.7 - Use a direct proof to show that every odd integer...Ch. 1.7 - Prove that ifnis a perfect square, thenn+2 is not...Ch. 1.7 - Use a proof by contradiction to prove that the sum...Ch. 1.7 - Use a direct proof to show that the product of two...Ch. 1.7 - Prove or disprove that the product of two...Ch. 1.7 - Prove or disprove that the product of a nonzero...Ch. 1.7 - Prove that ifxis irrational, then 1/xis...Ch. 1.7 - Prove that ifxis rational andx0 , then1/x is...Ch. 1.7 - Prove that ifxis an irrational number andx0 ,...Ch. 1.7 - Prove that ifx,y, andzare integers andx+y+z is...Ch. 1.7 - Use a proof by contraposition to show that ifx+y2...Ch. 1.7 - Prove that ifmandnare integers andmnis even,...Ch. 1.7 - Show that ifnis an integer and n3+5 is odd,...Ch. 1.7 - Prove that ifnis an integer and3n+2 is even,...Ch. 1.7 - Prove the propositionP(0), whereP(n) is the...Ch. 1.7 - Prove the propositionP(1), whereP(n) is the...Ch. 1.7 - LetP(n) be the proposition “Ifaandbare positive...Ch. 1.7 - Show that if you pick three socks from a drawer...Ch. 1.7 - Show that at least ten of any 64 days chosen must...Ch. 1.7 - Show that at least three of any 25 days chosen...Ch. 1.7 - Use a proof by contradiction to show that there is...Ch. 1.7 - Prove that ifnis a positive integer, thennis even...Ch. 1.7 - Prove that ifnis a positive integer, thennis odd...Ch. 1.7 - Prove that m2=n2 if and only ifm=n orm=n .Ch. 1.7 - Prove or disprove that ifmandnare integers such...Ch. 1.7 - Show that these three statements are equivalent,...Ch. 1.7 - Show that these statements about the integerxare...Ch. 1.7 - Show that these statements about the real...Ch. 1.7 - Show that these statements about the real...Ch. 1.7 - Is this reasoning for finding the solutions of the...Ch. 1.7 - Is this reasoning for finding the solutions...Ch. 1.7 - Show that the propositionsp1,p2,p3, andp4can be...Ch. 1.7 - Show that the propositionsp1,p2,p3,p4, andp5can be...Ch. 1.7 - Find a counterexample to the statement that every...Ch. 1.7 - Prove that at least real numbersa1,a2,…,anis...Ch. 1.7 - Use Exercise 41 to show that if the first 10...Ch. 1.7 - Prove that ifnis an integer, these four statements...Ch. 1.7 - Prove that these four statements about the...Ch. 1.8 - Prove thatn2+12n whennis a positive integer...Ch. 1.8 - Use a proof by cases to show that 10 is not the...Ch. 1.8 - Use a proof by cases to show that 100 is not the...Ch. 1.8 - Prove that there are no positive perfect cubes...Ch. 1.8 - Prove that ifxandyare real numbers,...Ch. 1.8 - Use a proof by cases to show...Ch. 1.8 - Prove using the notion of without loss of...Ch. 1.8 - Prove using the notion of without loss of...Ch. 1.8 - Prove the triangle inequality, which states that...Ch. 1.8 - Prove that there is a positive integer that equals...Ch. 1.8 - Prove that there are 100 consecutive positive...Ch. 1.8 - Prove that either210500+15 or210500+16 is not a...Ch. 1.8 - Prove that there exists a pair of consecutive...Ch. 1.8 - Show that the product of two of the...Ch. 1.8 - Prove or disprove that there is a rational...Ch. 1.8 - Prove or disprove that ifaandbare rational numbers...Ch. 1.8 - Show that each of these statements can be used to...Ch. 1.8 - Show that ifa,b, andcare real numbers anda0 , then...Ch. 1.8 - Suppose thataandbare odd integers with ab . Show...Ch. 1.8 - Show that ifris an irrational number, there is a...Ch. 1.8 - Show that ifnis an odd integer, then there is a...Ch. 1.8 - Prob. 22ECh. 1.8 - Prob. 23ECh. 1.8 - Use forward reasoning to show that ifxis a nonzero...Ch. 1.8 - Prob. 25ECh. 1.8 - Thequadratic meanof two real...Ch. 1.8 - Write the numbers 1, 2, …,2non the black board,...Ch. 1.8 - Suppose that five ones and four zeros are arranged...Ch. 1.8 - Prob. 29ECh. 1.8 - Formulate a conjecture about the final two decimal...Ch. 1.8 - Prove that there is no positive integernsuch...Ch. 1.8 - Prove that there are no solutions in...Ch. 1.8 - Prove that there are no solutions in positive...Ch. 1.8 - Prove that there are infinitely many solutions in...Ch. 1.8 - Prob. 35ECh. 1.8 - Prove that 23 is irrational.Ch. 1.8 - Prob. 37ECh. 1.8 - Prove that between every rational number and every...Ch. 1.8 - LetS=x1y1+x2y2++xnyn , wherex1,x2...,xn...Ch. 1.8 - Prove or disprove that if you have an 8-gallon jug...Ch. 1.8 - Verify the3x+1 conjecture for these integers. a) 6...Ch. 1.8 - Verify the3x+1 conjecture for these integers. a)...Ch. 1.8 - Prove or disprove that you can use to tile the...Ch. 1.8 - Prove or disprove that you can use dominoes to...Ch. 1.8 - Prove that you can use dominoes to tile a...Ch. 1.8 - Prove or disprove that you can use dominoes to...Ch. 1.8 - Use a proof by exhaustion to show that a tiling...Ch. 1.8 - Prove that when a white square and a black square...Ch. 1.8 - Show that by removing two white squares and two...Ch. 1.8 - Prob. 50ECh. 1.8 - Prob. 51ECh. 1.8 - Prove or disprove that you can tile a1010...Ch. 1 - a) Define the negation of a proposition. b) What...Ch. 1 - a) Define (using truth tables) the disjunction,...Ch. 1 - a) Describe at least five different ways to the...Ch. 1 - a) What does it mean for two propositions to be...Ch. 1 - (Depends on the Exercise Set inSection 1.3) a)...Ch. 1 - What are the universal and existential...Ch. 1 - a) What is the difference between the...Ch. 1 - Describe what is meant by a valid argument in...Ch. 1 - Prob. 9RQCh. 1 - a) Describe what is meant by a direct proof, a...Ch. 1 - a) Describe away to prove the bi-conditionalpq ....Ch. 1 - To prove that the statementp1,p2,p3, andp4are...Ch. 1 - a) Suppose that a statement of the formxP(x) is...Ch. 1 - What is the difference between a constructive and...Ch. 1 - What are the elements of a proof that there is a...Ch. 1 - Prob. 16RQCh. 1 - Letpbe the proposition "I do every exercise in...Ch. 1 - Find the truth table of the compound...Ch. 1 - Prob. 3SECh. 1 - Prob. 4SECh. 1 - Prob. 5SECh. 1 - Prob. 6SECh. 1 - Prob. 7SECh. 1 - Prob. 8SECh. 1 - Show that these statements are inconsistent: "If...Ch. 1 - Suppose that in a three-round obligato game, the...Ch. 1 - Suppose that in a four-round obligato game, the...Ch. 1 - Explain why every obligato game has a winning...Ch. 1 - Prob. 13SECh. 1 - Suppose that you meet three people, Anita, Boris,...Ch. 1 - (Adapted from [Sm78]) Suppose that on an island...Ch. 1 - Show that ifSis a proposition, whereSis the...Ch. 1 - Show that the argument premises "The tooth fry is...Ch. 1 - Suppose that the truth value of the...Ch. 1 - Model1616 Sudoku puzzles (with44 blocks) as...Ch. 1 - Let P(x) be the statement “Student x knows...Ch. 1 - LetP(m,n) be the statement “mdividesn," where the...Ch. 1 - Find a domain for the quantifiers in...Ch. 1 - Prob. 23SECh. 1 - Prob. 24SECh. 1 - Use existential and universal quantifiers to...Ch. 1 - The quantifiern denotes "there exists exactlyn,"...Ch. 1 - Express each of these statements using existential...Ch. 1 - Prob. 28SECh. 1 - Prob. 29SECh. 1 - IfyxP(x,y) is true, does it necessarily follow...Ch. 1 - Prob. 31SECh. 1 - Find the negations of these statements. a) If it...Ch. 1 - Express this statement using quantifiers: "Every...Ch. 1 - Express statement using quantifiers: "There is a...Ch. 1 - Prob. 35SECh. 1 - Prob. 36SECh. 1 - Prob. 37SECh. 1 - Prove that ifx3is irrational, thenxis irrational.Ch. 1 - Prob. 39SECh. 1 - Prob. 40SECh. 1 - Prove that there exists an integermsuch...Ch. 1 - Prob. 42SECh. 1 - Disprove the statement that every positive integer...Ch. 1 - Disprove the statement that every positive integer...Ch. 1 - Prob. 45SECh. 1 - Assuming the truth of the theorem that states...Ch. 1 - Given the truth values of the propositionspandq,...Ch. 1 - Prob. 2CPCh. 1 - Prob. 3CPCh. 1 - Prob. 4CPCh. 1 - Prob. 5CPCh. 1 - Given a portion of a checkerboard, look for...Ch. 1 - Look for positive integers that are not the sum of...Ch. 1 - Look for positive integers greater than 79 that...Ch. 1 - Prob. 3CAECh. 1 - Try to find winning strategies for the game of...Ch. 1 - Prob. 5CAECh. 1 - Find all the rectangles of 60 squares that can be...Ch. 1 - Discuss logical paradoxes, including the paradox...Ch. 1 - Describe how fuzzy logic is being applied to...Ch. 1 - Describe some of the practical problems that can...Ch. 1 - Prob. 4WPCh. 1 - Describe some of the techniques that have been...Ch. 1 - Describe the basic rules ofWFFN PROOF, The Game of...Ch. 1 - Read some of the writings of Lewis Carroll on...Ch. 1 - Extend the discussion of Prolog given inSection...Ch. 1 - Discuss some of the techniques used in...Ch. 1 - "Automated theorem proving" is the task of using...Ch. 1 - Describe how DNA computing has been used to solve...Ch. 1 - Look up some of the incorrect proofs of famous...Ch. 1 - Prob. 13WPCh. 1 - Describe various aspects of proof strategy...Ch. 1 - Describe a few problems and results about tilings...
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- Prove that 23 is not a rational number.arrow_forwardUse the fact that 3 is a prime to prove that there do not exist nonzero integers a and b such that a2=3b2. Explain how this proves that 3 is not a rational number.arrow_forwardProve that is irrational. (That is, prove there is no rational number such that .)arrow_forward
- Graph each subset of the real numbers on a number line: The integers between -5 to 0arrow_forwardGraph each subset of the real numbers on a number line: The composite numbers less than 10arrow_forwardIf x and y are positive rational numbers, prove that there exists a positive integer n such that nxy. This property is called the Archimedean Property of the rational numbers. (Hint: Write x=a/b and y=c/d with each of a,b,c,d+.)arrow_forward
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