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Unformatted text preview: Neal Whittington (whittin3) Tuesdays, at 4:00PM AD9 CS 173: Discrete Structures, Spring 2010 Homework 3 This homework contains 3 problems worth a total of 40 points. It is due on Friday, 12 February at 4pm. 1. [8 points] Thinking about negations (a) x R , P ( x ) (b) p Z m Z n Z p is the GCD of ( m n ) 2. [22 points] A triple ( a,b,c ) of positive integers is Pythagorean if a 2 + b 2 = c 2 . Consider the claim: (*) For any Pythagorean triple ( a,b,c ), if c is odd than either a or b is even. (a) Contrapositive: There exists a Pythagorean triple ( a,b,c ), if a and b are odd then c is even. (b) Proof by Contrapositive: Suppose a and b are odd which by definition means a = 2 p + 1 and b = 2 q + 1, where p,q are both integers. This means that we can write a 2 + b 2 = c 2 as (2 p + 1) 2 + (2 q + 1) 2 = c 2 . By expanding this, we get (4 p 2 + 4 p + 1) + (4 q 2 + 4 q + 1) = c 2 . By the associative property we get 4 p 2 +4 q 2 +4 p +4 q +2 = c 2 2(2p...
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