section_3.07_answers - ( D C D ) Idempotency of ( A B ) ( C...

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PHIL12A Section answers, 7 March 2011 Julian Jonker 1 Revision problems 1. Translate the following sentences into FOL, devising your own names and predicates where necessary: (a) Block c is a tetrahedron just in case block a is in between block c and block d. Tet(c) Between(a,c,d) (b) Neither block a nor block b is to the left of e or to the right of c. ¬ (LeftOf(a,e) RightOf(a,c)) ∧¬ (LeftOf(b,e) RightOf(b,c)) or ¬ LeftOf(a,e) ∧¬ LeftOf(b,e) ∧¬ (RightOf(a,c) ∧¬ RightOf(b,c) (c) John is his father’s son, provided he is a human being. Human(john) john=son(father(john)) 2. Is ¬ ( A B ) C a tautological consequence of ¬ ( ¬ A ∧ ¬ B ∧ ¬ C ) ? Using the short cut: ¬ ( ¬ A ¬ B ) C ¬ ( A B ) C T F T T F T F F T F T F T F F F F F T Tx F F Tx F T F T F F F F F 1
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3. Put the following sentence into Negation Normal Form, showing your working through a chain of equiv- alences: ( ¬ ( ¬ A ∨ ¬ B ) A ) ( ¬¬ D ∧ ¬ ( C ∨ ¬ D )) . ( ¬ ( ¬ A ∨ ¬ B ) A ) ( ¬¬ D ∧ ¬ ( C ∨ ¬ D )) ( ¬¬ A ∧ ¬¬ B A ) ( ¬¬ D ∧ ¬ ( C ∨ ¬ D )) De Morgan’s law ( ¬¬ A ∧ ¬¬ B A ) ( ¬¬ D ∧ ¬ C ∧ ¬¬ D ) De Morgan’s law ( A B A ) ( D ∧ ¬ C D ) Double negation ( A B )
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Unformatted text preview: ( D C D ) Idempotency of ( A B ) ( C D ) Idempotency of A B C D Associativity of 4. Give a formal proof for the following argument: 1 A B 2 B 3 A ( D E ) Proof: 1 A B 2 B 3 A 4 A ( D E ) Intro: 3 5 B 6 Intro: 2, 5 7 A ( D E ) Elim: 6 8 A ( D E ) Elim: 1, 3-4, 5-7 2 5. Give a formal proof for the following argument: 1 A B C 2 A B C Proof: 1 A B C 2 A 3 A 4 Intro: 2, 3 5 A Intro: 2-4 6 A B C Intro: 5 7 B 8 A B C Intro: 7 9 C 10 C 11 Intro: 9, 10 12 C Intro: 10-11 13 C 14 Intro: 12, 13 15 C Intro: 13-14 16 A B C Intro: 15 17 A B C Elim: 1, 2-6, 7-8, 9-16 3...
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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 University of California, Berkeley.

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section_3.07_answers - ( D C D ) Idempotency of ( A B ) ( C...

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