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Math Help - Write formula using quantifiers

  1. #1
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    Write formula using quantifiers

    Hello there.
    I need to write a following formula using quantifiers, but it has to be in prenex normal form.

    The formula is : function f is not weakly decreasing and not weakly increasing.

    I got to here:

    \exists x \exists y(x \le y \wedge f(x)>f(y))  \wedge \exists x\exists y(x \le y \wedge f(x)<f(y))

    by negating :


    (\forall x \forall y(x \le y \wedge f(x)\le f(y))  \vee (\forall x \forall y(x \le y \wedge f(x)\le f(y))

    But I have no idea how to proceed, since my solution is not in prenex normal form.

    Please help.
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  2. #2
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    Re: Write formula using quantifiers

    How are those terms, weakly decreasing and weakly increasing defined?
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  3. #3
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    Re: Write formula using quantifiers

    Weakly increasing- (\forall x \forall y(x \le y \wedge f(x)\le f(y))

    Weakly decreasing-  (\forall x \forall y(x \le y \wedge f(x) \geq f(y))


    I came up with this:

    \exists x \exists y\exists a \exists b((x \le y \wedge (f(x)>f(y)))  \wedge (a \le b \wedge (f(a)<f(b)))

    But is it correct? If so then why and how can I derive it from those two formulas above?
    Last edited by MachinePL1993; November 12th 2012 at 11:53 AM.
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  4. #4
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    Re: Write formula using quantifiers

    Quote Originally Posted by MachinePL1993 View Post
    Weakly increasing- (\forall x \forall y(x \le y \wedge f(x)\le f(y))

    Weakly decreasing-  (\forall x \forall y(x \le y \wedge f(x) \geq f(y))


    I came up with this:
    \exists x \exists y\exists a \exists b((x \le y \wedge (f(x)>f(y)))  \wedge (a \le b \wedge (f(a)<f(b)))
    But is it correct? If so then why and how can I derive it from those two formulas above?
    Are you sure that it is not an implication?
    Weakly increasing- (\forall x \forall y(x \le y  \Rightarrow f(x)\le f(y))
    Weakly decreasing-  (\forall x \forall y(x \le y \Rightarrow f(x) \geq f(y))
    Then the negation you have would be correct.
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  5. #5
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    Re: Write formula using quantifiers

    Yes, I'm sorry, there should be an implication.

    Weakly increasing- (\forall x \forall y(x \le y \rightarrow f(x)\le f(y))

    Weakly decreasing-  (\forall x \forall y(x \le y \rightarrow f(x) \geq f(y))


    But now I have to bring it to prenex normal form. Is there some other way to arrive to a proper formula without converting my negation from above using an algorithm to prenex normal form like I did two messages prior? Is my answer from above even correct?
    Last edited by MachinePL1993; November 12th 2012 at 11:55 AM.
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  6. #6
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    Re: Write formula using quantifiers

    Quote Originally Posted by MachinePL1993 View Post
    But now I have to bring it to prenex normal form. Is there some other way to arrive to a proper formula without converting my negation from above using an algorithm to prenex normal form like I did two messages prior? Is my answer from above even correct?
    Yes, I would accept that. But I don't know the level of rigor expected of you.
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