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Logic, negation of a statement containing quantifiers

  1. Dec 20, 2012 #1
    Hi, I've got another answer I'd like checked. I'm pretty sure it works out, but I want to be certain.

    1. The problem statement, all variables and given/known data
    Write a sentence in everyday English that properly communicates the negation of each statement.

    "Some differentiable functions are bounded."

    2. Relevant equations

    3. The attempt at a solution
    First, I wrote the statement symbolically:
    [itex](\exists f(x) \in X) (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})[/itex],
    where I let [itex]X[/itex] be the set of differentiable functions.

    [itex]\neg (\exists f(x) \in X) (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})[/itex]

    [itex](\forall f(x) \in X) \neg (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})[/itex]

    [itex](\forall f(x) \in X) (f(x) \; \mbox{is not bounded} \vee f(x) \; \mbox{is not differentiable})[/itex]

    My question is, can I simplify this sentence to
    [itex](\forall f(x) \in X) (f(x) \; \mbox{is not bounded})[/itex],
    since all [itex]f(x) \in X[/itex] are differentiable and therefore cannot be differentiable?

    I just want to make sure my reasoning works here. Thanks!

    As for the English translation:
    "Every differentiable function is not bounded."
     
    Last edited: Dec 20, 2012
  2. jcsd
  3. Dec 20, 2012 #2

    pasmith

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    Homework Helper

    So the condition "f(x) is differentiable" is equivalent to [itex]f(x) \in X[/itex] and so redundant:
    [tex](\exists f(x) \in X) (f(x)\mbox{ is bounded})[/tex]

    Yes: "P or false" is equivalent to P.
     
  4. Dec 20, 2012 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    If you are defining X to be the set of all differentiable functions, then there is no need for "[itex]\and \text{f is differentiable}[/itex]" in your original statement. With that definition of X, your statement is simply "[itex](\exist f\in X)(f \text{is bounded})[/itex]" and it negation is "[itex](\all f\in X) (f \text{is not bounded})[/itex]".

    In any case, the negation of "some differentiable functions are bounded", in "every day English" is "no differentiable functions are bounded".
     
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