cs2022a16quiz02.pdf - Discrete Mathematics Ma2201/CS2022...

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Ma2201/CS2022 Discrete Mathematics A Term, 2016 Quiz 0010 Print Name: Sign: 1. ( 6 pts ) Let p , q and r be statements. Show by any method we have learned that ( p r ) ( q r ) ( p q r ) Method 1: You can make a truth table: (every entry must be correct.) p q r p r q r p r q r p q p q r T T T T T T T T T T F F F F T F T F T T T T F T T F F F T T F T F T T T T T F T F T F T F T F T F F T T T T F T F F F T T T F T Method 2: We can use logical identities: ( p r ) ( q r ) ( r ∨ ¬ p ) ( r ∨ ¬ q ) definition of implies r ( ¬ p ∨ ¬ q ) associativity, and r duplicated r ∨ ¬ ( p q ) Demorgans law p q r definition of implies Method 3: Double inplication: Proof: First show ( p r ) ( q r ) ( p q r ). Let ( p r ) ( q r ). We want to show ( p q r ), so let p q . Since ( p r ) ( q r ) there are two cases: If ( p r ), then we already have p since p q , so r is true as required.
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