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Unformatted text preview: 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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This note was uploaded on 09/24/2011 for the course CSE 20 taught by Professor Foster during the Fall '08 term at UCSD.
- Fall '08