91影视

For each of these compound propositions, use the conditional-disjunction equivalence (Example 3\()\) to find an equivalent compound proposition that does not involve conditionals. a) \(p \rightarrow \neg q\) b) \((p \rightarrow q) \rightarrow r\) c) \((\neg q \rightarrow p) \rightarrow(p \rightarrow \neg q)\)

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. a) 鈥淚f I take the day off, it either rains or snows.鈥� 鈥淚 took Tuesday off or I took Thursday off.鈥� 鈥淚t was sunny on Tuesday.鈥� 鈥淚t did not snow on Thursday.鈥� b) 鈥淚f I eat spicy foods, then I have strange dreams.鈥� 鈥淚 have strange dreams if there is thunder while I sleep.鈥� 鈥淚 did not have strange dreams.鈥� c) 鈥淚 am either clever or lucky.鈥� 鈥淚 am not lucky.鈥� 鈥淚f I am lucky, then I will win the lottery.鈥� d) 鈥淓very computer science major has a personal computer.鈥� 鈥淩alph does not have a personal computer.鈥� 鈥淎nn has a personal computer.鈥� e) 鈥淲hat is good for corporations is good for the United States.鈥� 鈥淲hat is good for the United States is good for you.鈥� 鈥淲hat is good for corporations is for you to buy lots of stuff.鈥� f ) 鈥淎ll rodents gnaw their food.鈥� 鈥淢ice are rodents.鈥� 鈥淩abbits do not gnaw their food.鈥� 鈥淏ats are not ro- dents.鈥�

Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational.

Let F(x, y) be the statement 鈥渪 can fool y,鈥� where the do- main consists of all people in the world. Use quantifiers to express each of these statements. a) Everybody can fool Fred. b) Evelyn can fool everybody. c) Everybody can fool somebody. d) There is no one who can fool everybody. e) Everyone can be fooled by somebody. f ) No one can fool both Fred and Jerry. g) Nancy can fool exactly two people. h) There is exactly one person whom everybody can fool. i) No one can fool himself or herself. j) There is someone who can fool exactly one person besides himself or herself.

For each of these sets of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. a) 鈥淚f I play hockey, then I am sore the next day.鈥� 鈥淚 use the whirlpool if I am sore.鈥� 鈥淚 did not use the whirlpool.鈥� b) 鈥淚f I work, it is either sunny or partly sunny.鈥� 鈥淚 worked last Monday or I worked last Friday.鈥� 鈥淚t was not sunny on Tuesday.鈥� 鈥淚t was not partly sunny on Friday.鈥� c) 鈥淎ll insects have six legs.鈥� 鈥淒ragonflies are insects.鈥� 鈥淪piders do not have six legs.鈥� 鈥淪piders eat dragon-flies.鈥� d) 鈥淓very student has an Internet account.鈥� 鈥淗omer does not have an Internet account.鈥� 鈥淢aggie has an Internet account.鈥� e) 鈥淎ll foods that are healthy to eat do not taste good.鈥� 鈥淭ofu is healthy to eat.鈥� 鈥淵ou only eat what tastes good.鈥� 鈥淵ou do not eat tofu.鈥� 鈥淐heeseburgers are not healthy to eat.鈥� f ) 鈥淚 am either dreaming or hallucinating.鈥� 鈥淚 am not dreaming.鈥� 鈥淚f I am hallucinating, I see elephants running down the road.鈥�

Let \(p\) and \(q\) be the propositions \(p :\) I bought a lottery ticket this week. \(q :\) I won the million dollar jackpot. Express each of these propositions as an English sentence. a) \(\neg p\) b) \(p \vee q\) c) \(p \rightarrow q\) d) \(p \wedge q\) e) \(p \leftrightarrow q\) f) \(\neg p \rightarrow \neg q\) g) \(\neg p \wedge \neg q\) h) \(\neg p \vee(p \wedge q)\)

Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?

Let \(C(x)\) be the statement " \(x\) has a cat," let \(D(x)\) be the statement " \(x\) has a dog," and let \(F(x)\) be the statement "x has a ferret." Express each of these statements in terms of \(C(x), D(x), F(x),\) quantifiers, and logical connectives. Let the domain consist of all students in your class. a) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret, but not a dog. d) No student in your class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.

Use a direct proof to show that the product of two rational numbers is rational.

Are these system specifications consistent? 鈥淲henever the system software is being upgraded, users cannot access the file system. If users can access the file system, then they can save new files. If users cannot save new files, then the system software is not being upgraded.鈥�