C__Documents_and_Settings_Raghu_Desktop_CS_2800_hw1-soln -...

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CS 2800 – Fall 2008 Homework #1 Solution Section 1.1 Problem 8 (c) If you miss the final exam, then you do not pass the course. (f) Either you have the flu and miss the final exam, or you do not miss the final exam and pass the course. Problem 10 (a) r ∧ ¬ q (b) p q r (c) r p (d) p ∧ ¬ q r (e) ( p q ) r (f) r ( q p ) Problem 28 (c) Truth table: p q p q p ( p q ) T T T F T F T F F T T T F F F F (f) Truth table: p q ¬ q p q p ↔ ¬ q ( p q ) ( p ↔ ¬ q ) T T F T F T T F T F T T F T F F T T F F T T F T 1
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Problem 44 No – this is the classic Barber’s Paradox. We will use the male pronoun in what follows, assuming that we are talking about males shaving their beards here, and assuming that all men have facial hair. If we restrict ourselves to beards and allow female barbers, then the barber could be female with no contradiction. If such a barber existed, then who shaves the barber? If the barber shaved himself, then he would be violating the rule that he shaves only those people who do not shave themselves. On the other hand, if he does not shave himself, then the rule says that he must shave himself. Neither is possible, so there can be no such barber. Section 1.2 Problem 12 (a) Assume the hypothesis is true. Then, p must be false. But ( p q ) needs to be true and hence q must be true. Alternately, one can make an algebraic argument as follows: [ ¬ p ( p q )] q [( ¬ p p ) ( ¬ p q )] q (distributing over ) [ F ( ¬ p q )] q (negation law) ( ¬ p q ) q (identity law) ≡ ¬ ( ¬ p q
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C__Documents_and_Settings_Raghu_Desktop_CS_2800_hw1-soln -...

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