ucdavishw2soln - Homework 2 - For 10/18/07 Exercise 1) p T...

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Homework 2 - For 10/18/07 Exercise 1) p q r ( p q ) ( p q ) ( q r ) ( p q ) ( p q ) ( q r ) [( p q ) ( p q ) ( q r )] r T T T T T T T T T T F T F F F T T F T T T T T T T F F T F T F T F T T T T T T T F T F T T F F T F F T F T T F T F F F F T T F T Exercise 2) By truth table: p q r ¬ p ( p q ) ( ¬ p r ) ( q r ) (( p q ) ( ¬ p r )) (( p q ) ( ¬ p r )) ( q r ) T T T F T T T T T T T F F T F T F T T F T F T T T T T T F F F T F F F T F T T T T T T T T F T F T T T T T T F F T T F T T F T F F F T F T F F T Using logical equivalences ( p q ) ( ¬ p r ) ( q r ) (( p q ) ( ¬ p r )) ( q r ) Commutativity 2 (( ¬ p r ) ( p q )) ( q r ) Distributivity 2 ((( ¬ p r ) p ) (( ¬ p r ) q )) ( q r ) Distributivity 2 ((( ¬ p p ) ( r p )) (( ¬ p q ) ( r q ))) ( q r ) Distributivity 2 (( F ( r p )) (( ¬ p q ) ( r q ))) ( q r ) Complement law 4 (( r p ) ( ¬ p q ) ( r q )) ( q r ) Implication law ¬ (( r p ) ( ¬ p q ) ( r q )) ( q r ) De Moran’s law (( ¬ r ∨ ¬ p ) ( p ∨ ¬ q ) ( ¬ r ∨ ¬ q )) ( q r ) Distributivity 1 (( ¬ r ∨ ¬ p ) ( q r )) (( p ∨ ¬ q ) ( q r )) (( ¬ r ∨ ¬ q ) ( q r )) Associativity 1 ( ¬ r ∨ ¬ p q r ) [ p ∨ ¬ q q r ) ( ¬ r ∨ ¬ q q r ) Complement law 4 ( T ∨ ¬ p q ) ( p T r ) ( T T ) Absorption law 1 T T T T Hence it is a tautology 1
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Exercise 3 Determine whether these are valid arguments: a) if x 2 is irrational, then x is irrational. Therefore, if
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This note was uploaded on 10/14/2009 for the course MATH 55 taught by Professor Strain during the Spring '08 term at University of California, Berkeley.

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ucdavishw2soln - Homework 2 - For 10/18/07 Exercise 1) p T...

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