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Determining the truth value of this quantified statement

  1. May 13, 2007 #1
    1. The problem statement, all variables and given/known data

    I am trying to determine the truth value of this statement:

    [tex] \forall \mbox{ }x \mbox{ and } \forall \mbox{ }y, \exists \mbox{ }z \mbox{ such that } y-z=x. [/tex]


    2. Relevant equations

    N/A

    3. The attempt at a solution

    Here is how I arrived at my answer of false. The textbook lists the correct answer as true.

    Consider x=0 and y=1.

    We have,

    1-z=0.

    For this statement to be true, z must be equal to 1.

    Now, leaving z=1, switch to x=2 and y=0. This gives

    0-1=2. (1)

    Equation (1) is a false statement. Therefore, there exists no z such that for all x and for all y, y-z=x.

    Where is my logic wrong?

    Thanks
     
    Last edited: May 13, 2007
  2. jcsd
  3. May 13, 2007 #2
    Hi opticaltempest,

    I believe the problem is that you're fixing z, then fixing x and y. The statement is, given some x and y, there exists z.

    If you fix all three, then it's easy to find a contradiction to the statement for all x,y, z, we have y - z = x, but this isn't what your original statement said.

    Hope that helps!
     
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