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Help negate this statement

  1. Aug 31, 2011 #1
    1. The problem statement, all variables and given/known data

    Give a useful negation for:

    [itex]\forall x > 0, \exists y > 0 s.t. y^2 = x[/itex]

    2. Relevant equations



    3. The attempt at a solution

    I'm not sure how to do this, I have

    [itex]\exists y > 0 s.t. \forall x > 0, y^2 = x[/itex]

    Which says "there exists a y that for all x, y^2 = x"....which is obviously incorrect, as there is no y that when squared equals EVERYTHING.
     
  2. jcsd
  3. Aug 31, 2011 #2
    How about, "there exists an x greater than zero such that for all y greater than zero, y squared is not equal to x"
     
  4. Aug 31, 2011 #3

    SammyS

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    If you were looking for an example to counter this (a counter example), what would need to be true for that example ? ... and, wouldn't it take just one example (at least as regards x)?
     
  5. Aug 31, 2011 #4
    What do you mean counter it? I can't counter the first statement, it's true.
     
  6. Aug 31, 2011 #5
    This supports my suggestion.
     
  7. Aug 31, 2011 #6
    A negation of "for all x, this is true" is "there exists an x such that this is not true." You say "there exists an x such that for all (for any arbitrary) y this is not true" because it has to be not true for all y. if it's only not true for some y, then there exists a y such that it is true, and you've lost it.
     
  8. Aug 31, 2011 #7

    SammyS

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    Well yes, it is true for x & y being real numbers. That doesn't mean that you can't coming up with criteria that would need to hold for a counter-example, if such existed.
     
  9. Aug 31, 2011 #8
    Okay, I get it! So what you're saying, is that if there WERE a counter example, it would satisfy Arcana's negation statement, correct?
     
  10. Aug 31, 2011 #9

    SammyS

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    That's what I'm saying if indeed, Arcana's negation statement is correct.
     
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