l17-midterm-review-exercises-soln

l17-midterm-review-exercises-soln - CS 245 Winter 2009...

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CS 245 Winter 2009 Review Problems on Propositional and Predicate Logic Natural Deduction and Semantic Tableaux Prove the following arguments are valid or show that they are invalid by providing a coun- terexample and demonstrating that your counterexample shows the argument is invalid. Use both natural deduction and semantic tableaux. Do not use any logical laws from transfor- mational proof in your proofs. 1 Propositional Logic 1. ( P ( Q R )) (( P Q ) R ) 1. P ( Q R ) assumption 2. P Q assumption 3. P E 2 4. Q E 2 5. Q R E 1 , 3 6. R E 4 , 5 7. P Q R I 2 6 8. ( P ( Q R )) ( P Q R ) I 1 7 9. ( P Q ) R assumption 10. P assumption 11. Q assumption 12. P Q I 10 , 11 13. R E 9 , 12 14. Q R I 11 13 15. P ( Q R ) I 10 14 16. (( P Q ) R ) ( P ( Q R )) I 9 15 17. ( P ( Q R )) ( P Q R ) I 8 , 16 1
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1. ¬ ( P ( Q R )) (( P Q ) R ) 2. ( P ( Q R )) ∧ ¬ (( P Q ) R ) NOT-IFF 1 3. P ( Q R ) 4. ¬ (( P Q ) R ) AND 2 5. P Q 6. ¬ R NOT-IMPLIES 4 7. P 8. Q AND 5 9. ¬ P CLOSED 7,9
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This note was uploaded on 10/23/2009 for the course CS 245 taught by Professor A during the Spring '08 term at Waterloo.

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l17-midterm-review-exercises-soln - CS 245 Winter 2009...

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