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section_3.02_answers

section_3.02_answers - PHIL12A Section questions 2 March...

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PHIL12A Section questions, 2 March 2011 Julian Jonker 1 How much do you know? 1. Show that the sentences in each pair are tautologically equivalent. (a) (Ex 7.1) A B and ¬ A B A B A B ¬ A B T T T T T F T T T T F T F F F T F F F T F T T T F T T F F F T F T F T F (b) (Ex 7.4) A B and ( A B ) ( ¬ A ∧ ¬ B ) A B A B ( A B ) ( ¬ A ¬ B ) T T T T T T T T T F T F F T T F T F F T F F F F T F T F F T F F T F F T F T F F F T F F F T F F F F T T F T T F (c) (Ex 7.7) A ( B ( C D )) and (( A B ) C ) D (Feel free to do this the short way.) I assumed this could be done the short way, but in fact it only helps with one direction. (For the shortcut, you would have to show that you reach a contradiction when assigning different truth values to each sentence: and there are two ways to assign different truth values to the sentences. The problem is that with one case there are many ways to assign those truth values to the complex sentences.) Let me know if you want to post the truth table! It was a bit too dull to type up. 1
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Give formal proofs for the following arguments. (Most of these are left over from the last problem set!) (a) (Ex 6.19) 1 A B 2 ¬ B C 3 A C Proof: 1 A B 2 ¬ B C 3 A 4 A C Intro: 3 5 B 6 ¬ B 7 Intro: 5,6 8 A C Elim: 7 9 C 10 B C Intro: 9 11 A C Elim: 2, 6-8, 9-10 12 A C Elim: 1, 3-4, 5-11 2
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(b) (Ex 6.24) 1 ¬ ( A B ) 2 ¬ A ∧ ¬ B Proof: 1 ¬ ( A B ) 2 A 3 A B Intro: 2 4 Intro: 1, 3 5 ¬ A ¬ Intro: 2-4 6 B 7 A B Intro: 6 8 Intro: 1, 7 9 ¬ B ¬ Intro: 6-8 10 ¬ A ∧ ¬ B Intro: 2, 6 3
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