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l4soln

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

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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

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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 It’s 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 text’s 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 ⇚⇛ ( p q ) ( ¬ p r ) Impl ⇚⇛ ( p q ) ( p r ) Impl 3

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2. We have changed the annotation of the third line of the proof in the text to read contradiction law
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l4soln - CS 245 Winter 2009 Solution Set for Lecture 4...

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