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hw2_solutions

hw2_solutions - Problem Set 2 Solutions Problem 1 1 p...

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Problem Set 2 Solutions Problem 1 1. p q [given] 2. p r [given] 3. q r [given] 4. ¬ p r [2, implication] 5. ¬ q r [3, implication] 6. ( ¬ p r ) ( ¬ q r ) [4, 5, conjunction] 7. ( r ¬ p ) ( r ¬ q ) [6, commutativity] 8. r ( ¬ p ¬ q ) [7, distributive property] 9. ( ¬ p ¬ q ) r ) [8, commutativity] 10. ¬ ( p q ) r [9, De Morgan’s law] 11. ( p q ) r [10, implication] 12. r [1, 11, modus ponens] Problem 2 1. p r [given] 2. q s [given] 3. ¬ p r [1, implication] 4. ( ¬ p r ) s [3, addition] 5. ¬ p ( r s ) [4, associativity] 6. ¬ q s [2, implication] 7. ( ¬ q s ) r [6, addition] 8. ¬ q ( s r ) [7, associativity] 9. ¬ q ( r s ) [8, commutativity] 10. ( ¬ p ( r s )) ( ¬ q ( r s )) [5, 9, conjunction] 11. ( ¬ p ¬ q ) ( r s ) [10, distributive property] 12. ¬ ( p q ) ( r s ) [11, De Morgan’s law] 13. ( p q ) ( r s ) [12, implication] Problem 3 The following assignments all work. p = T, q = T, r = T, s = F p = F, q = T, r = T, s = F p = T, q = T, r = F, s = T p = T, q = F, r = F, s = T 1
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Problem 4 (a) A = ”this house is next to a lake” B = ”the treasure is in the kitchen” C = ”the tree in the front yard is an elm”
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