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Unformatted text preview: SOLUTIONS FOR HOMEWORK 4 2.23. We say that g S if ( ( a,c ) (0 , ) 2 )( x ( a, ))(  g ( x )  c  f ( x )  ) . Negating this statement, we see that g / S if and only if ( ( a,c ) (0 , ) 2 )( x ( a, ))(  g ( x )  > c  f ( x )  ) , or, in plain English, for any positive real numbers a and c , there exists x > a with the property that  g ( x )  > c  f ( x )  . 2.37. (a) The statement A C is false , for instance, for x = 2. In this case, A is true, but C is false. (c) The statement ( A B ) C is false , for instance, for x = 2. In this case, A and B are true, hence A B is true, but C is false. (d) The statement ( A B ) C is true for any x . Indeed, A B is equivalent to x { 1 , 2 } . C is true when x = 1, while D holds when x = 2. (f) The statement D [ A B ( C )] is true for any x . Indeed, if x negationslash = 2, then the hypothesis of this conditional statement (that is, D ) is false, which makes the whole conditional statement true. If, on the other hand, x = 2, then both the hypothesis and the conclusion (both D and A B ( C )) are true, and once again, the conditional statement is true. (g) The statement ( A C ) B is false , for instance, for x = 3 / 2....
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This note was uploaded on 01/27/2010 for the course MATH 347 taught by Professor ? during the Fall '09 term at University of Illinois at Urbana–Champaign.
 Fall '09
 ?
 Math, Real Numbers

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