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# hw3 - Pythagorean if a 2 b 2 = c 2 Consider the claim For...

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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. When a problem specifies a particular proof technique, you must use that technique in your solution, even if it’s not the only reasonable approach to the mathematical problem. This is because the main point of these problems is to learn how to use the various different proof techniques. 1. [8 points] Thinking about negations Using precise mathematical notation, give the negations of the following statements. (a) ! x R , P ( x ) (b) p Z and m Z and n Z and p is not the greatest common divisor of m and n You can use any combination of mathematical shorthand or mathematical English (e.g., for the logical operations) but be clear about the order in which operations are applied using parentheses, indentation or line breaks. 2. [22 points] A triple ( a, b, c ) of positive integers is

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Unformatted text preview: 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) State the contrapositive of (*). (b) Use proof by contrapositive to prove (*). (c) Use (*) to show that 4 | ( a + b ) 2-c 2 . Hint : First simplify ( a + b ) 2-c 2 using the deﬁnition of Pythagorean triple. (d) Prove that c > a in any Pythagorean triple ( a,b,c ). You can use the following results without proof: • the square of a positive integer is positive • k is odd if and only if k 2 is odd ( k is odd ↔ k 2 is odd) • k is even if and only if k 2 is even ( k is even ↔ k 2 is even) 1 3. [10 points] Proof by contradiction Consider the following claim: Claim: √ 6-√ 2 > 1 (a) State the negation of the claim. (b) Use proof by contradiction to prove the claim. 2...
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