l4soln - CS 245 Winter 2009 Solution Set for Lecture 4...

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Unformatted text preview: CS 245 Winter 2009 Solution Set for Lecture 4 Nissanke Ch. 4, Extra Problems on CNF and DNF Exercise 4.1 (p. 47) 1. ( p ) p 1 2 3 p p ( p ) T F T F T F Columns 1 and 3 are identical. 2. p q p q 1 2 3 4 p q p q p q T T T T T F F F F T T T F F T T Columns 3 and 4 are identical. 3. ( p q ) p q In text. 1 4. p q ( p q ) ( q p ) 1 2 3 4 5 6 p q p q p q q p ( p q ) ( p q ) T T T T T T T F F F T F F T F T F F F F T T T T Columns 3 and 6 are identical. Exercise 4.2 (p. 48-49) 1. p q q p (commutativity of ) Proof: p q ( p q ) ( q p ) Equiv ( q p ) ( p q ) Comm q p Equiv Its okay to skip the middle line of the proof based on Commutativity. 2. p q p p (contrapositive law) Proof: p q p q Impl q p Comm ( q ) p Neg q p Impl 3. In text. 2 Exercise 4.3 (p. 53-54) Note: The proofs presented here differ slightly from the intended proofs of the text. The differences between our proofs and the texts proof will be indicated. 1. We have removed the fourth to last line of the proof in the text. p ( q r ) ( p q ) ( p r ) Proof: p ( q r ) p ( q r ) Impl p ( q r ) Impl ( p q ) r Assoc ( true ( p q )) r Simp I (( p p ) ( p q )) r EM ( p ( p q )) r Distr (( p q ) p ) r Comm ( p q ) ( p r ) Assoc ( p q ) ( p r ) Neg ( p q ) ( p r ) DM ( p q ) ( p r ) Neg ///////////////////////////////// Neg ///// - remove this line ( p q ) ( p r ) Impl...
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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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l4soln - CS 245 Winter 2009 Solution Set for Lecture 4...

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