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You can upgrade your operating system only if you have a 32-bit processor running at 1 GHz or faster, at least 1 GB RAM, and 16 GB free hard disk space, or a 64-bit processor running at 2 GHz or faster, at least 2 GB RAM, and at least 32 GB free hard disk space. Express your answer in terms of u: 鈥淵ou can upgrade your operating system,鈥� b32: 鈥淵ou have a 32-bit processor,鈥� b64: 鈥淵ou have a 64-bit processor,鈥� g1: 鈥淵our processor runs at 1 GHz or faster,鈥� g2: 鈥淵our processor runs at 2 GHz or faster,鈥� r1: 鈥淵our processor has at least 1 GB RAM,鈥� r2: 鈥淵our processor has at least 2 GB RAM,鈥� h16: 鈥淵ou have at least 16 GB free hard disk space,鈥� and h32: 鈥淵ou have at least 32 GB free hard disk space.鈥�

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

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�

Let \(T(x, y)\) mean that student \(x\) likes cuisine \(y,\) where the domain for \(x\) consists of all students at your school and the domain for \(y\) consists of all cuisines. Express each of these statements by a simple English sentence. a) \(\neg T(\text { Abdallah Hussein, Japanese })\) b) \(\exists x T(x, \text { Korean }) \wedge \forall x T(x, \text { Mexican })\) c) \(\exists y(T(\text { Monique Arsenault, } y) \vee\) \(T(\text { Jay Johnson, } y) )\) d) \(\forall x \forall z \exists y((x \neq z) \rightarrow \neg(T(x, y) \wedge T(z, y)))\) e) \(\exists x \exists z \forall y(T(x, y) \leftrightarrow T(z, y))\) f) \(\forall x \forall z \exists y(T(x, y) \leftrightarrow T(z, y))\)

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.

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. \(]\)

What rules of inference are used in this famous argument? 鈥淎ll men are mortal. Socrates is a man. Therefore, Socrates is mortal.鈥�