21+PS2+Solutions

# 21+PS2+Solutions - Handout#21 CS103 Robert Plummer Problem...

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Handout #21 CS103 April 15, 2011 Robert Plummer Problem Set #2—Solutions 1. (Exercise 6, p. 72) Use rules of inference to show that the hypotheses “If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on,” “If the sailing race is held, then the trophy will be awarded,” and “The trophy was not awarded” imply the conclusion “It rained.” Use the following symbols: R = "It rains" S = "The sailing race will be held" G = "It is foggy" L = "The lifesaving demonstration will go on" 1. ( ¬ R ¬F) (S L) 2. S T 3. ¬T 4. ¬S Modus tollens, 2, 3 5. ¬(S L) ¬(¬R ¬F) Contrapositive 1 (Table 7, line 2) 6. ¬S ¬L Addition, 4 7. ¬(S L) DeMorgan's, 6 8. ¬(¬R ¬F) Modus ponens, 5, 7 9. R F DeMorgan's, 8 10. R Simplification, 9 2. Give a statement/reason proof for the following. Use a horizontal line to separate hypotheses from the rest of the proof. 1. E K 2. ¬G ¬K 3. H I 4. E H 5. ¬E K Table 7 (Line 1), 1 6. H K Resolution, 4, 5 7. G ¬K Table 7 (Line 1), 2 8. G H Resolution, 6, 7 9. ¬H I Table 7 (Line 1), 3 10. G I Resolution 8, 9

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2 Here is an alternate proof for #2, using proof by cases: 1. E K 2. ¬G ¬K 3. H I 4. E H Proof by cases on line 4 5. Case 1: E 6. K Modus ponens, 5, 1 7. K G Contrapositive (Table 7 Line 2), 2 8. G Modus ponens, 6, 7 9. G I Addition, 8 10. Case 2: H 11. I Modus ponens, 10, 3 12. G I Addition, 11 13. G I Proof by cases, 5 - 9, 10 - 12 3. (Exercise 34, p. 74) The Logic Problem, taken from WFF’N PROOF, The Game of Logic, has these two assumptions: 1. “Logic is difficult or not many students like logic.” 2. “If mathematics is easy, then logic is not difficult.” By translating these assumptions into statements involving propositional variables and logical connectives, determine whether each of the following are valid conclusions of these assumptions: a) That mathematics is not easy, if many students like logic. b) That not many students like logic, if mathematics is not easy.
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## This note was uploaded on 06/01/2011 for the course EE 103 taught by Professor Plummer during the Spring '11 term at Stanford.

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21+PS2+Solutions - Handout#21 CS103 Robert Plummer Problem...

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