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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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 Fall '09
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 Math, Logic, Real Numbers, J. Bertrand, P. Chebyshev

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