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Write the negation

  1. Jan 20, 2004 #1
    How do I write the negation of:

    For all x > 0, x^2 + y^2 > 0 for all y.

    I thought it might be this:

    There exists x < or = to 0 such that x^2 + y^2 < or = to 0 for one y value.

  2. jcsd
  3. Jan 20, 2004 #2
    Re: Negations

    I think that was close but not exact, it is:

    There exist one x > 0 such that x^2+y^2 < or = 0 for one y.

    The thing is that there is no statement about x < 0. So that there must be no statement for x <0 in the negation.


    Maybe an better formulation (and equivalent) of the problem is:

    How do I write the negation of:

    For all x>0 and for all y, x^2 + y^2 > 0.


    There exist one x>0 and there one y such that x^2 + y^2 <= 0.


    I hope it did help...
  4. Jan 20, 2004 #3
    Whoops, I did that wrong. The actual statements are (for the problem, not the solution):

    For every x >0, x^2 + y^2 > 0 for all y.

    Close enough I guess?
  5. Jan 20, 2004 #4


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    [tex]\forall x>0 \exists y>0 \] s.t. \[ x^2+y^2 \leq 0[/tex]
  6. Jan 21, 2004 #5
    What is that in English? Thanks.
  7. Jan 21, 2004 #6


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    "For all x greater than 0, there exist a y> 0

    such that [tex]x^2+ y^2\leq0[/tex]"

    (It is, by the way, false.)
    Last edited by a moderator: Jan 21, 2004
  8. Jan 21, 2004 #7


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    Right, but it is the negation of the statement he made.
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