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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. Westons 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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