91影视

In Exercises 1鈥�6, translate the given statement into propositional logic using the propositions provided. You are eligible to be President of the U.S.A. only if you are at least 35 years old, were born in the U.S.A., or at the time of your birth both of your parents were citizens, and you have lived at least 14 years in the country. Express your answer in terms of e: 鈥淵ou are eligible to be President of the U.S.A.,鈥� a: 鈥淵ou are at least 35 years old,鈥� b: 鈥淵ou were born in the U.S.A.,鈥� p: 鈥淎t the time of your birth, both of your parents were citizens,鈥� and r: 鈥淵ou have lived at least 14 years in the U.S.A.鈥�

Use a proof by cases to show that \(\min (a, \min (b, c))=\) \(\min (\min (a, b), c)\) whenever \(a, b,\) and \(c\) are real numbers.

Use rules of inference to show that the hypotheses 鈥淚f it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on,鈥� 鈥淚f the sailing race is held, then the trophy will be awarded,鈥� and 鈥淭he trophy was not awarded鈥� imply the conclusion 鈥淚t rained.鈥�

Use a direct proof to show that every odd integer is the difference of two squares. [Hint: Find the difference of the squares of \(k+1\) and \(k\) where \(k\) is a positive integer. \(]\)

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.

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. $$ \begin{array}{ll}{\text { a) } \forall x(C(x) \rightarrow F(x))} & {\text { b) } \forall x(C(x) \wedge F(x))} \\ {\text { c) } \quad \exists x(C(x) \rightarrow F(x))} & {\text { d) } \exists x(C(x) \wedge F(x))}\end{array} $$

Prove using the notion of without loss of generality that \(\min (x, y)=(x+y-|x-y|) / 2\) and \(\max (x, y)=(x+y+\) \(|x-y| ) / 2\) whenever \(x\) and \(y\) are real numbers.

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 system." b) 鈥淭he message was sent from an unknown system but it was not scanned for viruses.鈥� c) 鈥淚t is necessary to scan the message for viruses when- ever it was sent from an unknown system.鈥� d) 鈥淲hen a message is not sent from an unknown system it is not scanned for viruses.鈥�

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.