hw1sol - CSE 20: Discrete Mathematics Fall 2010 Problem Set...

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CSE 20: Discrete Mathematics Fall 2010 Problem Set 1 Solutions Instructor: Daniele Micciancio Problem 1 1. ( q r ) → ¬ p 2. ¬ p ( q r ) 3. If you can’t vote, you are under 18 or you are from Mars. 4. p → ¬ ( q r ) (By DeMorgan’s law, this is the same as p ( ¬ q ∧ ¬ r ) ) 5. If you can vote, you are not under 18 and you are not from Mars. Problem 2 1. p ( q r ) ? ⇐⇒ ( p q ) r This is not a valid equivalence, since p ( q r ) is false and ( p q ) r is true when p is false and r is true. (see the first truth table) 2. p ( q r ) ? ⇐⇒ ( p q ) ( p r ) This equivalence is valid, since p ( q r ) and ( p q ) ( p r ) have the same value for every row in the table. (see the first truth table) 3. ¬ ( p q ) ? ⇐⇒ ( ¬ p ∨ ¬ q ) This is not a valid equivalence. For example, when p is true and q is false, ¬ ( p q ) is false, but ( ¬ p ∨ ¬ q ) is true. (see the second truth table)
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hw1sol - CSE 20: Discrete Mathematics Fall 2010 Problem Set...

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