This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ) 2c 2 . Hint : First simplify ( a + b ) 2c 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...
View
Full
Document
This note was uploaded on 11/09/2011 for the course CS 173 taught by Professor Erickson during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Erickson

Click to edit the document details