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l12soln - CS 245 Winter 2009 Solution Set for Lecture 12...

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CS 245 Winter 2009 Solution Set for Lecture 12 Nissanke Ch. 9 1 Exercise 8.3 (p. 112) 1. (The text asks for informal proofs. Here formal proofs are provided using natural deduction.) Here we prove the material equivalence ( ) of the two formulas. The soundness of natural deduction allows us to conclude they are logically equivalent ( ). (a) | = ND x p ( x ) q ( x ) ( x p ( x )) ( x q ( x )) 1. x p ( x ) q ( x ) assumption 2. x u p ( x u ) q ( x u ) assumption 3. p ( x u ) assumption 4. x p ( x ) I 3 5. ( x p ( x )) ( x q ( x )) I 4 6. q ( x u ) assumption 7. x q ( x ) I 6 8. ( x p ( x )) ( x q ( x )) I 7 9. ( x p ( x )) ( x q ( x )) cases 2 , 3 5 , 6 8 10. ( x p ( x )) ( x q ( x )) E 1 , 2 9 11. ( x p ( x ) q ( x )) (( x p ( x )) ( x q ( x ))) I 1 10 12. ( x p ( x )) ( x q ( x )) assumption 13. x p ( x ) assumption 14. x u p ( x u ) assumption 15. p ( x u ) q ( x u ) I 14 16. x p ( x ) q ( x ) I 15 17. x p ( x ) q ( x ) E 13 , 14 16 18. x q ( x ) assumption 19. x u q ( x u ) assumption 20. p ( x u ) q ( x u ) I 19 21. x p ( x ) q ( x ) I 20 22. x p ( x ) q ( x ) E 18 , 19 21 23. x p ( x ) q ( x ) cases 12 , 13 17 , 18 22 24. (( x p ( x )) ( x q ( x ))) ( x p ( x ) q ( x )) I 12 23 25. (( x p ( x )) ( x q ( x ))) ( x p ( x ) q ( x )) I 11 , 24 1
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(b) | = ND x • ¬ p ( x ) ⇔ ¬∃ x p ( x ) 1. x • ¬ p ( x ) assumption 2. x p ( x ) assumption 3. x u p ( x u ) assumption 4. ¬ p ( x u ) I 1 5. false ¬ E 3 , 4 6. false E 2 , 3 5 7. ¬∃ x p ( x ) RAA 2 6 8. x • ¬ p ( x ) ⇒ ¬∃ x p ( x ) I 1 7 9. ¬∃ x p ( x ) assumption 10. x g 11. p ( x g ) assumption 12. x p ( x ) I 11 13. false ¬ E 9 , 12 14. ¬ p ( x g ) RAA 11 13 15. x • ¬ p ( x ) I 10 14 16. ( ¬∃ x p ( x )) ( x • ¬ p ( x )) I 9 15 17. ( ¬∃ x p ( x )) ( x • ¬ p ( x )) I 8 , 16 (c) Are the following formulas logically equivalent? y •∃ x p ( x,y ) and x •∀ y p ( x,y ) No, they are not logically equivalent. To be logically equivalent, they must have
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