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HW2 Solutions

HW2 Solutions - CSE 20 Discrete Mathematics Spring 2010...

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CSE 20: Discrete Mathematics Spring 2010 Problem Set 2 Instructor: Daniele Micciancio Due on: Thu. April 15, 2010 Problem 1 (6 points) Formulate each of the following statements in English (without using mathematical symbols) and assert which one is true or false: 1. x Z . y Z .x < y For all integers there exist another integer which is larger than them. True. 2. y Z . x Z .x < y There exist an integer which is larger than all integers. False. Problem 2 (10 points) Complete the proof of the following theorem by filling the gaps. Theorem 1. If p → ¬ q and q r , then p r . Proof. Given p → ¬ q and q r , we need to prove p r . In order to prove p r , we assume p is true, and show that r follows. We give a proof by cases. Since q r is given, either q or r is true. We consider the following two cases: Case 1 ( r is true): In this case r is true and there is nothing to be proved. Case 2 ( q is true): Since p → ¬ q is given and p is true by assumption, we can deduce ¬ q . But we also have q . This proves q ∧ ¬ q which is a contradiction. So, also in this case r follows. In both cases we proved that r is true, and this concludes the proof of the theorem.
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