55PS1 - SOLUTION SET 1—DUE 1/30/2008 Please report any...

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Unformatted text preview: SOLUTION SET 1—DUE 1/30/2008 Please report any errors in this document to Ian Sammis ( isammis@math.berkeley.edu ). Problem 1 (#1.1.8) . Let p,q, and r be the propositions p : You have the flu. q : You miss the final examination. r : You pass the course. Express each of these propositions as an English sentence. a) p → q b) ¬ q ↔ r c) q → ¬ r d) p ∨ q ∨ r e) ( p → ¬ r ) ∨ ( q → ¬ r ) f) ( p ∧ q ) ∨ ( ¬ q ∧ r ) Solution. Note that the tense of the sentences below is entirely optional and is only intended to make the sentences render as naturally as possible into natural English language. Natural language being what it is, the below are merely suggested solutions—there are many possible correct solutions to this problem. a) If you have the flu, you miss the final examination. b) You did not miss the final examination if and only if you passed the course. c) If you miss the final examination, you will not pass the course. d) You had the flu or you missed the final examination or you missed the course. e) Either your having the flu implies you didn’t pass the course or your missing the final examination implies you didn’t pass the course. f) Either you had the flu and missed the final examination, or you didn’t miss the final examination and passed the course. Problem 2 (#1.1.10) . Let p,q, and r be the propositions p : You get an A on the final exam.. q : You do every exercise in this book. r : You get an A in this class. Write these propositions using p,q and r and logical connectives. a) You get an A in this class, but you do not do every exercise in this book. b) You get an A on the final, you do every exercise in this book, and you get an A in this class. c) To get an A in this class, it is necessary for you to get an A on the final. d) You get an A on the final, but you don’t do every exercise in this book; nevertheless, you get an A in this class. e) Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class. f) You will get an A in this class if and only if you either do every exercise in this book or you get an A on the final. 1 2 SOLUTION SET 1—DUE 1/30/2008 Solution. a) r ∧¬ q . b) p ∧ q ∧ r . c) r → p . d) p ∧ ( ¬ q ) ∧ r . The parentheses aren’t necessary by the precedence rules, but add clarity to the expression. e) ( p ∧ q ) → r . f) r ↔ ( p ∨ q ). Problem 3 (#1.1.28) . Construct a truth table for each of these compound propo- sitions a) p → ¬ p b) p ↔ ¬ p c) p ⊕ ( p ∨ q ) d) ( p ∧ q ) → ( p ∨ q ) e) ( q → ¬ p ) ↔ ( p ↔ q ) f) ( p ↔ q ) ⊕ ( p ↔ ¬ q ) Solution. In each table, we also tabulate important partial results....
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55PS1 - SOLUTION SET 1—DUE 1/30/2008 Please report any...

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