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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 100-story 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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