# Quiz5 - CSE 260 QUIZ-5 Proofs(20 minutes(30 points NAME 1(10 points Use rules of inference(table given at the end to show that if x(P(x Q(x and

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CSE 260 QUIZ-5– Proofs (20 minutes) (30 points) NAME: 1. (10 points) Use rules of inference (table given at the end) to show that if x ( P ( x ) Q ( x )) and x (( ¬ P ( x ) Q ( x )) R ( x )) are true, then x ( ¬ R ( x )) P ( x )) is also true, where the domains of all quantiﬁers are the same. Justify every step. Step Reason 1) ( ¬ P ( a ) Q ( a )) R ( a ) by universal instantiation of the second premise. 2) ¬ ( ¬ P ( a ) Q ( a )) R ( a ) by implication law from (1). 3) ( P ( a ) ∨ ¬ Q ( a )) R ( a ) by De Morgan’s law from (2). 4) ( R ( a ) P ( a )) ∨ ¬ Q ( a ) by associative and commutative laws from (3). 5) P ( a ) Q ( a ) by universal instantiation from the ﬁrst premise. 6) ( R ( a ) P ( a )) P ( a )) by resolution from (4) and (5). 7) R ( a ) ( P ( a ) P ( a )) by associative law from (6). 8) R ( a ) P ( a ) by idempotent law from (7). 9)

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## This note was uploaded on 04/28/2008 for the course CSE 260 taught by Professor Saktipramanik during the Fall '08 term at Michigan State University.

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Quiz5 - CSE 260 QUIZ-5 Proofs(20 minutes(30 points NAME 1(10 points Use rules of inference(table given at the end to show that if x(P(x Q(x and

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