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Unformatted text preview: Math 55  Homework 1 Solutions Friday June 24, 2011 1. Section 1.2, problems 46 through 51 For (46) and (48), here is a truth table including NAND (  ) and NOR ( ↓ ): p q ¬ ( p ∧ q ) p  q ¬ ( p ∨ q ) p ↓ q T T F F F F T F T T F F F T T T F F F F T T T T For (47) and (49), notice that the third and fourth columns and fifth and sixth columns are the same. (50) (a) p ↓ p ⇐⇒ ¬ ( p ∨ p ) ⇐⇒ ¬ p (b) By apply part (a), we get: ( p ↓ q ) ↓ ( p ↓ q ) ⇐⇒ ¬ ( p ↓ q ) ⇐⇒ ¬ ( ¬ ( p ∨ q )) ⇐⇒ p ∨ q (c) By parts (a) and (b), we can use ↓ to make ¬ and ∨ , which are functionally complete by (45). (51) We can use the rules from (50) to get: p → q ⇐⇒ ( ¬ p ) ∨ q ⇐⇒ ( p ↓ p ) ∨ q ⇐⇒ (( p ↓ p ) ↓ q ) ↓ (( p ↓ p ) ↓ q ) 2. Section 1.4, problem 52 We want to say that there is an x so that P ( x ) , and that if P ( y ) and P ( z ) , then y and z are the same: ( ∃ x.P ( x )) ∧ ( ∀ y. ∀ z. ( P ( y ) ∧ P ( z )) → y = z ) 3. The Weston Watermelon Company has managed to develop an especially durable wa termelon. They have hired you to determine the exact durability of these super water melons. Weston’s offices are in the 100story Pasteque building in San Francisco. They want to know the least number k , so that their watermelons will break when dropped from the k th floor. Unfortunately, they only have have two test watermelons for you. Come up with a sequence of experiments which is guaranteed to determine k using at most 14 drops. Prove that your answer always works....
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 Summer '08
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