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Recent questions and answers in Mathematical Logic
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Kenneth Rosen Edition 7th Exercise 1.7 Question 1 (Page No. 91)
Use a direct proof to show that the sum of two odd integers is even.
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Kenneth Rosen Edition 7th Exercise 1.7 Question 5 (Page No. 91)
Prove that if $m+n$ and $n+p$ are even integers, where $m, n$,and $p$ are integers, then $m+p$ is even. What kind of proof did you use?
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Kenneth Rosen Edition 7th Exercise 1.6 Question 33 (Page No. 80)
Use resolution to show that the compound proposition $(p \vee q) \wedge (\sim p \vee q) \wedge (p \vee \sim q) \wedge (\sim p \vee \sim q)$ is not satisfiable.
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Kenneth Rosen Edition 7th Exercise 1.7 Question (Page No. 91)
Use a direct proof to show that every odd integer is the difference of two squares.
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Kenneth Rosen Edition 7th Exercise 1.7 Question 6 (Page No. 91)
Use a direct proof to show that the product of two odd numbers is odd.
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Kenneth Rosen Edition 7th Exercise 1.7 Question 4 (Page No. 91)
Show that the additive inverse, or negative, of an even number is an even number using a direct proof.
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Kenneth Rosen Edition 7th Exercise 1.7 Question 3 (Page No. 91)
Show that the square of an even number is an even number using a direct proof
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Kenneth Rosen Edition 7th Exercise 1.7 Question 2 (Page No. 91)
Use a direct proof to show that the sum of two even integers is even.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 35 (Page No. 80)
Determine whether this argument, taken from Kalish and Montague [KaMo64], is valid. If Superman were able and willing to prevent evil,he would do so. If Superman were unable to prevent evil, he would be impotent; ... does not prevent evil. If Superman exists, he is neither impotent nor malevolent. Therefore, Superman does not exist.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 34 (Page No. 80)
The Logic Problem, taken from WFF'N PROOF, The Game of Logic, has these two assumptions:1. Logic is difficult or not many students like logic. 2. If mathematics is easy, then logic is not difficult. By translating ... not easy. That if not many students like logic, then either mathematics is not easy or logic is not difficult.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 32 (Page No. 80)
Show that the equivalence $p \wedge \sim p \equiv F$ can be derived using resolution together with the fact that a conditional statement with a false hypothesis is true. [Hint:Let $q=r=F$ in resolution.]
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Kenneth Rosen Edition 7th Exercise 1.6 Question 31 (Page No. 80)
Use resolution to show that the hypotheses “It is not raining or Yvette has her umbrella,” “Yvette does not have her umbrella or she does not get wet,” and “It is raining or Yvette does not get wet” imply that “Yvette does not get wet.”
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Kenneth Rosen Edition 7th Exercise 1.6 Question 30 (Page No. 80)
Use resolution to show the hypotheses “Allen is a bad boy or Hillary is a good girl” and “Allen is a good boy or David is happy” imply the conclusion “Hillary is a good girl or David is happy.”
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Kenneth Rosen Edition 7th Exercise 1.6 Question 29 (Page No. 80)
Use rules of inference to show that if $\forall x (P(x) \vee Q(x))$, $\forall x (\sim Q(x) \vee S(x)), \forall x (R(x) \rightarrow \sim S(x)),$ and $\exists x \sim P(x)$ are true, then $\exists x \sim R(x)$ is true.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 28 (Page No. 80)
Use rules of inference to show that if $\forall x (P(x) \vee Q(x))$ and $\forall x ((\sim P(x) \wedge Q(x)) \rightarrow R(x))$ are true, then $\forall x (\sim R(x) \rightarrow P(x))$ is also true, where the domains of all quantifiers are the same.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 27 (Page No. 80)
Use rules of inference to show that if $\forall x (P(x) \rightarrow (Q(x) \wedge S(x)))$ and $\forall x ( P(x) \wedge R(x))$ are true, then $\forall x (R(x) \wedge S(x))$ is true.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 26 (Page No. 80)
Justify the rule of universal transitivity, which states that if $\forall x (P(x) \rightarrow Q(x))$ and $\forall x(Q(x) \rightarrow R(x))$ are true, then $\forall x (P(x) \rightarrow R(x))$ is true, where the domains of all quantifiers are the same.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 25 (Page No. 80)
Justify the rule of universal modus tollens by showing that the premises $ \forall x (P(x) \rightarrow Q(x)) $ and $\sim Q(a)$for $a$ particular element $a$ in the domain, imply $\sim P(a)$
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Kenneth Rosen Edition 7th Exercise 1.6 Question 18 (Page No. 79)
What is wrong with this argument? Let $S(x, y)$ be “$x$ is shorter than $y$.” Given the premise $\exists s S(s, Max)$, it follows that $S(Max, Max)$. Then by existential generalization it follows that $\exists x S(x,x)$, so that someone is shorter than himself.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 17 (Page No. 79)
What is wrong with this argument? Let $H(x)$ be “$x$ is happy.” Given the premise $\exists x H(x)$, we conclude that $H(Lola)$. Therefore, Lola is happy.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 16 (Page No. 79)
For each of these arguments determine whether the argument is correct or incorrect and explain why. Everyone enrolled in the university has lived in a dormitory. Mia has never lived in a dormitory. Therefore,Mia is not enrolled ... set at least a dozen traps. Hamilton is a lobsterman. Therefore, Hamilton sets at least a dozen traps
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Kenneth Rosen Edition 7th Exercise 1.6 Question 15 (Page No. 79)
For each of these arguments determine whether the argument is correct or incorrect and explain why. All students in this class understand logic. Xavier is a student in this class. Therefore, Xavier understands logic. Every ... granola every day is healthy. Linda is not healthy. Therefore, Linda does not eat granola every day.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 14 (Page No. 79)
For each of these arguments, explain which rules of inference are used for each step. Linda, a student in this class, owns a red convertible.Everyone who owns a red convertible has gotten at least one speeding ticket. ... France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 13 (Page No. 79)
For each of these arguments, explain which rules of inference are used for each step. Doug, a student in this class, knows how to write programs in JAVA. Everyone who knows how to write programs in JAVA can get a high ... has never seen the ocean. Therefore, someone who lives within 50 miles of the ocean has never seen the ocean.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 9 (Page No. 78)
For each of these collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. If I take the day off, it either rains or snows. I took ... gnaw their food. Mice are rodents. Rabbits do not gnaw their food. Bats are not rodents.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 8 (Page No. 78)
What rules of inference are used in this argument? “No man is an island. Manhattan is an island. Therefore, Manhattan is not a man.”
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Kenneth Rosen Edition 7th Exercise 1.6 Question 7 (Page No. 78)
What rules of inference are used in this famous argument? “All men are mortal. Socrates is a man. Therefore,Socrates is mortal.”
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Kenneth Rosen Edition 7th Exercise 1.6 Question 5 (Page No. 78)
Use rules of inference to show that the hypotheses “Randy works hard,” “If Randy works hard, then he is a dull boy,”and “If Randy is a dull boy, then he will not get the job”imply the conclusion “Randy will not get the job.”
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Kenneth Rosen Edition 7th Exercise 1.6 Question 4 (Page No. 78)
What rule of inference is used in each of these arguments? Kangaroos live in Australia and are marsupials. Therefore, kangaroos are marsupials. It is either hotter than 100 degrees today or the pollution is dangerous. It ... the material . Therefore ,If I work all night on this homework, Then I will understand the material.
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Kenneth Rosen Edition 7th Exercise 1.6 Question 3 (Page No. 78)
What rule of inference is used in each of these arguments? Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major. Jerry is a mathematics major and a computer science major. Therefore, ... stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will sunburn
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Kenneth Rosen Edition 7th Exercise 1.5 Question 49 (Page No. 68)
Show that $\forall x P(x) \wedge \exists x Q(x)$ is logically equivalent to $\forall x \exists y (P(x) \wedge Q(y))$ ... from propositional variables,predicates,T, and F using logical connectives and quantifiers is equivalent to a statement in prenex normal form.
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Kenneth Rosen Edition 7th Exercise 1.5 Question 48 (Page No. 68)
Show that $\forall x P(x) \vee \forall x Q(x)$ and $\forall x \forall y (P(x) \vee Q(y))$, where all quantifiers have the same nonempty domain, are logically equivalent . (The new variable $y$ is used to combine the quantifications correctly.)
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Kenneth Rosen Edition 7th Exercise 1.5 Question 47 (Page No. 68)
Show that the two statements $\sim \exists x \forall y P(x,y)$ and $\forall x \exists y \sim P(x,y)$ , where both quantifiers over the first variable in $P(x,y)$ have the same domain , and both quantifiers over the second variable in $P(x,y)$ have the same domain, are logically equivalent.
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Kenneth Rosen Edition 7th Exercise 1.5 Question 46 (Page No. 68)
Determine the truth value of the statement $\exists x \forall y (x<=y^2)$ if the domain for the variables consists of he positive real numbers. the integers. the nonzero real numbers.
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Kenneth Rosen Edition 7th Exercise 1.5 Question 45 (Page No. 68)
Determine the truth value of the statement $\forall x \exists y (xy=1)$ if the domain for the variables consists of the nonzero real numbers. the nonzero integers. the positive real numbers.
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Kenneth Rosen Edition 7th Exercise 1.5 Question 40 (Page No. 68)
Find a counterexample, if possible ,to these universally quantified statements, where the domain for all variables consists of all integers. $\forall x \exists y (x=1/y)$ $\forall x \exists y (y^2 x <100)$ $\forall x \forall y (x^2 \neq y^3)$
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Kenneth Rosen Edition 7th Exercise 1.5 Question 39 (Page No. 68)
Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. $\forall x \forall y (x^2 = y^2 \rightarrow x =y)$ $\forall x \exists y (y^2=x)$ $\forall x \forall y (xy>=x)$
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Kenneth Rosen Edition 7th Exercise 1.5 Question 37 (Page No. 68)
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase It is not ... a movie with Kevin Bacon or has been in a movie with someone who has been in a movie with Kevin Bacon
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Kenneth Rosen Edition 7th Exercise 1.5 Question 36 (Page No. 68)
Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase ... solved every exercise in this book. No student has solved at least one exercise in every section of this book.
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propositional logic
which of the following is tautology? (¬P^(P>q))>¬q ¬(p>q)>¬q [(¬p^q)^[q>(p>q)]]>¬r Both (B) and(C) please explain in detail how to check for especially for condition (C) Because “r” is only in RHS but not in LHS of this implication.
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