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Unformatted text preview: 1 CS/MA320 HW1 Solutions--Parts 1 & 2 1.1 Reading. All of this is relatively important. A lot of it is how to translate between Logic and English sentences -- I will not spend much time on this in class, but will answer questions. When in doubt as to importance of coverage, consider the Exercises with Solutions for which you're responsible (odd-numbered ones are solved starting on page S-1), and of cours Exercises For You To Solve (even-numbered ones). Exercises With Solutions in 1.1: p 16: 5,7,11,15,23,27,29,33,51,63. NOTE: There is nothing to turn in, but these Exercises are fair game for Quizzes and Exams. You don't have to look at solutions for all parts of an Exercise if you know the answers without looking. Up to you. 1.2 Reading. Example 3 is important, and you should mark with a sticky Table 6 for Open-Book Quizzes and Exams. Exercises With Solutions in 1.2: p 28: 1,7,9,13,15,19,23,27,31 Exercises For You To Solve in 1.1 and 1.2: As stated, not in Rosen. 1 . State the converse, contrapositive, and inverse (labeling clearly) of the following implication: If it snows tonight then I will stay at home. Which of pairs of those forms mean the same thing (two pairs of implications are identical in meaning). Converse: If I stay at home then it snows tonight. Inverse: If it doesn't snow tonight then I don't stay at home. Contrapositive: If I don't stay at home then it doesn't snow tonight. The last of these is equivalent to the original. 2 . Construct a truth table for each of these compound propositions (you can use the same frame for all truth tables). Are any of them Contradictions or Tautologies? (a) p ÀÂ p p Â p p ÀÂ p T F F F T F (b) (p À q) ª (p Á q) (c) p ⊕ (p Á q) p q p À q p Á q (p À q) ª (p Á q) p ⊕ (p Á q) T T T T T F T F F T T F F T F T T T F F F F T F Note that (a) is a Contradiction and (b) is a Tautology. 2 (d) p Á (q À r) ® (p Á q) À (p Á r) (e) (p À q) À r ® p À ( q À r) (1) (2) (3) (4) (5) (d) (6) (7) (8) (e) p q r q À r p Á (1) p Á q p Á r (3) À (4) (2) ® (5) p À q (6) À r p À (1) (7) ® (8) T T T T T T T T T T T T T T T F F T T T T T T F F T T F T F T T T T T F F F T T F F F T T T T T F F F T F T T T T T T T T F F F T F T F F F T F F T F F F T F F T F F F T F T F F F T F F F F F F F F T F F F T Note that (d) and (e) are both Tautologies. Can you find either or both of them in Table 6, pg. 24? 3 . Note precedence diagram below. (a) Could the parentheses be left out of 2.(b) above without changing the meaning? (b) Same question for 2.(c). Note that the symbol Ä we've been using for tautological equivalence will be ¡ from now on. 2.(b) is: (p À q) ª (p Á q); if we leave the parentheses out we get: p À q ª p Á q. The higher precedence operations are À , then Á , then ª . So À binds first and the result is as if we put parentheses around the À term to force it to evaluate first: ( p À q) ª p Á q; then Á binds, so the result looks like: ( p À q) ª (p...
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This note was uploaded on 04/30/2011 for the course MECHANICAL MSC taught by Professor Dr.rahula during the Spring '10 term at University of Moratuwa.
- Spring '10