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HW3 Solutions

# HW3 Solutions - CSE 20 Discrete Mathematics Spring 2010...

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CSE 20: Discrete Mathematics Spring 2010 Problem Set 3 Solutions Instructor: Daniele Micciancio Due on: Thu. April 29, 2010 Problem 1 (25 points) For each of the following statements, Formulate the statement in the language of prepositional logic. You may use all the standard logic connectives and quantifiers. You may also use the predicate Odd(n) and the relation “a|b”. (As a reminder, the definition of Odd(n) is m.n = 2 m + 1 , and the definition of a|b is c.b = ca .) State if the statement is true or false. Prove your answer correct, i.e., prove either statement or the negation of the statement. In all statements, the variables range of the set of nonnegative integers. 1. The product of any two odd integers is odd x, y N .Odd ( x ) Odd ( y ) Odd ( xy ) [TRUE] Proof. let x = 2 n + 1 , y = 2 m + 1 , xy = (2 n + 1)(2 m + 1) = 4 nm + 2 n + 2 m + 1 = 2(2 nm + n + m ) + 1 . let k = 2 nm + n + m xy = 2 k + 1 . 2. For any two numbers x and y , the sum x + y is even if and only if x and y are either both even or both odd.

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