91影视

Translate these statements into English, where \(R(x)\) is "x is a rabbit" and \(H(x)\) is " \(x\) hops" and the domain consists of all animals. $$ \begin{array}{ll}{\text { a) } \forall x(R(x) \rightarrow H(x))} & {\text { b) } \forall x(R(x) \wedge H(x))} \\ {\text { c) } \quad \exists x(R(x) \rightarrow H(x))} & {\text { d) } \exists x(R(x) \wedge H(x))}\end{array} $$

Use De Morgan's laws to find the negation of each of the following statements. a) Kwame will take a job in industry or go to graduate school. b) Yoshiko knows Java and calculus. c) James is young and strong. d) Rita will move to Oregon or Washington.

Express these system specifications using the propositions p: 鈥淭he user enters a valid password,鈥� q: 鈥淎ccess is granted,鈥� and r: 鈥淭he user has paid the subscription fee鈥� and logical connectives (including negations). a) 鈥淭he user has paid the subscription fee, but does not enter a valid password.鈥� b) 鈥淎ccess is granted whenever the user has paid the subscription fee and enters a valid password.鈥� c) 鈥淎ccess is denied if the user has not paid the subscription fee.鈥� d) 鈥淚f the user has not entered a valid password but has paid the subscription fee, then access is granted.鈥�

Suppose that Smartphone A has 256 MB RAM and 32 GB ROM, and the resolution of its camera is 8 MP; Smartphone B has 288 MB RAM and 64 GB ROM, and the resolution of its camera is 4 MP; and Smartphone C has 128 MB RAM and 32 GB ROM, and the resolution of its camera is 5 MP. Determine the truth value of each of these propositions. a) Smartphone B has the most RAM of these three smartphones. b) Smartphone C has more ROM or a higher resolution camera than Smartphone B. c) Smartphone B has more RAM, more ROM, and a higher resolution camera than Smartphone A. d) If Smartphone B has more RAM and more ROM than Smartphone C, then it also has a higher resolution camera. e) Smartphone A has more RAM than Smartphone B if and only if Smartphone B has more RAM than Smartphone A.

What rules of inference are used in this argument? 鈥淣o man is an island. Manhattan is an island. Therefore, Manhattan is not a man.鈥�

Prove using the notion of without loss of generality that \(5 x+5 y\) is an odd integer when \(x\) and \(y\) are integers of opposite parity.

Are these system specifications consistent? 鈥淭he 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 is in interrupt mode. The system is not in interrupt mode.鈥�

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 111 billion dollars and its net profit was 13 billion dollars. Determine the truth value of each of these propositions for the most recent fiscal year. a) Quixote Media had the largest annual revenue. b) Nadir Software had the lowest net profit and Acme Computer had the largest annual revenue. c) Acme Computer had the largest net profit or Quixote Media had the largest net profit. d) If Quixote Media had the smallest net profit, then Acme Computer had the largest annual revenue. e) Nadir Software had the smallest net profit if and only if Acme Computer had the largest annual revenue.

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�

Let L(x, y) be the statement 鈥渪 loves y,鈥� where the domain for both x and y consists of all people in the world. Use quantifiers to express each of these statements. a) Everybody loves Jerry. b) Everybody loves somebody. c) There is somebody whom everybody loves. d) Nobody loves everybody. e) There is somebody whom Lydia does not love. f ) There is somebody whom no one loves. g) There is exactly one person whom everybody loves. h) There are exactly two people whom Lynn loves. i) Everyone loves himself or herself. j) There is someone who loves no one besides himself or herself.