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Unformatted text preview: a. True. If x + y rational, then also y = ( x + y )-x is rational. b. False. Take e.g. x =-√ 2 and y = √ 2. c. True. If xy rational, so is x = xy y . Page 20 Problem 3. First solution: By Theorem 2.4.1 we can rind a rational number r such that x < r < y . Now by the Archimedean property we can ﬁnd n ∈ N such that √ 2 < n ( y-r ). Let t = r + √ 2 n . Then by the previous problem t is irrational and x < r < t < y . Second Solution: Let x < y . Then x-√ 2 < y-√ 2, so there exists a rational number r with x-√ 2 < r < y-√ 2. Take now t = r + √ 2....
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This note was uploaded on 02/05/2012 for the course MATH 554 taught by Professor Girardi during the Fall '10 term at South Carolina.
- Fall '10