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Thread: The Negation

  1. #1
    Newbie
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    The Negation

    The Question:
    Rewrite the definition in terms for logical symbols, then write the negation using the same symbols.

    The real number L is the limit of the function $\displaystyle f \rightarrow \mathbb{R} $ at the point c iff for each $\displaystyle \varepsilon > 0$ there exists $\displaystyle \delta > 0 $ such that $\displaystyle |f(x) - L|< \varepsilon $ whenever $\displaystyle x \in D $ and $\displaystyle 0 < |x - c| < \delta $.

    My Answer
    The definition:
    $\displaystyle \forall \varepsilon > 0, \exists \delta > 0, ( x \in D \cap 0 < |x - c| < \delta )\Rightarrow |f(x) - L| < \varepsilon$

    The negation:
    $\displaystyle \exists \varepsilon > 0, \forall \delta > 0, ( x \in D \cap 0 < |x - c| < \delta ) \cap |f(x) - L| \geq \varepsilon$

    Can any1 tell me if im right or wrong?
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  2. #2
    Junior Member
    Joined
    Oct 2009
    Posts
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    I think you need another quantifier for the x:

    $\displaystyle \forall\,\epsilon>0,\,\exists\,\delta>0,\,\forall\ ,x\in D,\,0<|x-c|<\delta\ \Rightarrow\ |f(x)-L|<\epsilon$
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