section_2.16_answers

section_2.16_answers - PHIL12A Section answers, 16 February...

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PHIL12A Section answers, 16 February 2011 Julian Jonker 1 How much do you know? 1. Show that the following sentences are equivalent. (a) (Ex 4.16) A B A and A B A B ( A B ) A A B T T T T T T T T T T T F T T F T T T T F F T F T T T F F T T F F F F F F F F F F (b) (Ex 4.18) A ( B C ) and ( A B ) ( A C ) A B C A ( B C ) ( A B ) ( A C ) T T T T T T T T T T T T T T T T T F T T T F F T T T T T T F T F T T T F F T T T F T T T T T F F T T F F F T T F T T T F T F F T T F F F T T F T T T F F T T F T T T T F T T T F T T F T F F F T F F F T T F F F F F F T F F F F T F F F F F T T F F F F F F F F F F F F F F F 1
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2. For each of the following arguments, show that the conclusion is a tautological consequence of the premises using truth tables. (I’ll use the shortcut method in each case, starting by placing a T under the main connective of each premise and an F under the main connective of the conclusion, and the working backwards until I hit a contradiction, at which point I will leave an x under the sentence to which I cannot assign a consistent truth value. (a) (Ex 4.20) 1 ( A B ) C 2 ( B C ) ( A B ) C B C T F T F F F T Fx T F F F F (b) (Ex 4.22) 1 A 2 B C 3 (( B A ) ( C A ) A B C (( B A ) ( C A ) T T F T T F T F F T T Fx T Fx F F T F F F T 2
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1 A B 2 B C 3 ¬ B 4 A C A B B C ¬ B A C T T T F T F F T T F F T T F F T T T F Tx F Tx (d) The following conclusion is in fact not a tautological consequence of the premises, as the truth assignment below illustrates. 1 A B 2 C D 3 ¬ A ∨ ¬ C 4 B D A B C D ¬ A ∨ ¬ C B D F T T T T F T F T F T T F F 3. Recall the following equivalences: Associativity of : P ( Q R ) ( P Q ) R P Q R Associativity of : P ( Q R ) ( P Q ) R P Q R Commutativity of : P Q Q P Commutativity of : P Q Q P Idempotence of : P P P Idempotence of : P P P Double negation:
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This note was uploaded on 09/19/2011 for the course PHILOS 12A taught by Professor Fitelson during the Spring '08 term at Berkeley.

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section_2.16_answers - PHIL12A Section answers, 16 February...

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