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 22 n3 = 0 ⇒ ( n + 1)( n3) = 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.
 Spring '10
 Math

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