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Math Help - Analysis question - statements

  1. #1
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    Analysis question - statements

    I'm trying to wrap my mind around the concept of mathematical statements with qualifiers and such, and I am just not seeming to get it.

    Check this statement:

     \forall x \exists y \ni y > x

    The book says it's true because there is always a real number y that is greater than x.

     \exists y \ni \forall x, y > x

    This is apparently false "since there is no fixed real number y that is greater than every real number."

    At first glance, they appear identical, but the notation is different. What is it that I'm not understanding about the notation that's misleading me?
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  2. #2
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    \left( {\forall x} \right)\left( {\exists y} \right)\left[ {y > x} \right] translates as “For each real number, x, there is a real number y, such that x is greater than y.”
    OR: Every real number is greater that some real number.

    On the other hand, \left( {\exists y} \right)\left( {\forall x} \right)\left[ {y > x} \right] translates as
    “There is a real number such that it is greater than any real number x.”
    “Some real is greater than any other real number”.

    Do you see a difference there?
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  3. #3
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    I think I understand now! And I see the difference.

    The first one is obviously true because every real number has another real number that is greater.

    The second one is false because it is not possible for some specific real number to be greater than every real number.

    That makes sense. Thanks a lot!
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  4. #4
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    well, I thought I understood...

    Switching the order in which you write the quantifiers can affect the truth value?

    You're either saying

    "for all real numbers x there exists a real number y such that y is greater than x"

    or

    "There exists a real number y such that for all real numbers x, y is greater than x"

    They still appear to be saying the same thing. Why is the last one false?

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  5. #5
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    Let me give you the same example but use different variables.

    \left( {\forall a} \right)\left( {\exists b} \right)\left[ {a > b} \right] says Every real number is greater than some real number..

    \left( {\exists a} \right)\left( {\forall b} \right)\left[ {a > b} \right] says Some real number is greater than every real number.
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  6. #6
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    Okay, I got it now. I really do got it now.

     \left( {\exists a} \right)\left( {\forall b} \right)\left[ {a > b} \right]

    Some real number is greater than every real number.


    This is false because There is an infinite number of real numbers, and to say that there is some real number that is larger than all of them is impossible.

    <br /> <br />
\left( {\forall a} \right)\left( {\exists b} \right)\left[ {a > b} \right]<br />

    Every real number is greater than some real number.

    This is true because it is possible for every real number to some real number smaller than itself.

    I'm happy I got that cleared up, thanks for your patience.
    Last edited by Skinner; January 27th 2009 at 01:19 PM.
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