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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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- Spring '08
- Logic, Euclidean algorithm, particular proof technique, various diﬀerent proof