Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

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MAT102S - Introduction to Mathematical Proofs - UTM - Spring 2010 Solutions to Selected Problems from Problem Set D For the first two questions (about fields) see solutions to Quiz #2. 2.2. Take a = 0 and b = 1. The statement claims that there are integers m,n for which 0 = m + n and 1 = m - n . This implies that m = - n and thus 1 = m + m = 2 m . But there is no integer m for which 1 = 2 m . Therefore the statement is false. To correct the statement, note that the equations a = m + n and b = m - n imply that m = ( a + b ) / 2 and n = ( a - b ) / 2. To guarantee that both m and n are integers, we can require that a and b have the same parity (i.e., they are both even, or both odd). This will make the statement true. 2.4. (a) “There is an x A such that for all b B , b x ”. (b) “For all x A there is a b B for which b x ”. (c) “There exist x,y R for which f ( x ) = f ( y ) and x 6 = y ”. (e) “There exist x,y R and an ± P such that for all δ P , | x - y | < δ and | f ( x ) - f ( y ) | ≥ ± ”. 2.28. (a) Take a = - 1 and x = 1. Then the equation x 4 y + ay + x = 0 becomes y - y + 1 = 0, or 1 = 0 which is false for any y . Therefore the original statement is false.
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Unformatted text preview: (b) Note that the given equation can be written as ( x 4 + a ) · y =-x . In order to have a unique solution for the given equation (for all x ∈ R ), we must make sure that x 4 + a is never zero. This happens if and only if a > 0. Therefore, if a > 0, the given statement is true. 2.34(a) The equation n 2 + ( n + 1) 2 = ( n + 2) 2 can be simplified: n 2 + n 2 + 2 n + 1 = n 2 + 4 n + 4 ⇒ n 2-2 n-3 = 0 ⇒ ( n + 1)( n-3) = 0 . Since n is a natural number, we must have n = 3 . Therefore, the statement is true . 2.48. (a) For any integer x , if x is odd, then it cannot be even (i.e., twice an integer). Hence P ( x ) ⇒ Q ( x ) will be false for any odd x , and thus the given statement is false . (b) The statement ( ∀ x ∈ Z )( P ( x )) is false, since not every integer is odd. Therefore, the given statement, which is a conditional statement, is true ....
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This document was uploaded on 03/25/2011.

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