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Math Help - Proof

  1. #1
    No one in Particular VonNemo19's Avatar
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    Proof

    Using any property of numbers that may be needed, show that

    \{x\in\mathbb{R}|\text{for a real number y}, x=y^2\}=\{x\in\mathbb{R}|x\geq0\}

    OK, I'm very new to this, so bear with me. I think that I have one direction of the proof done...

    For the sake of LaTeX, Let A equal the left hand side.

    Let x\in{A}

    Since x=y^2\Longrightarrow{x}\geq0 \forall{y}\in\mathbb{R}\Longrightarrow{x}\in{B}\Lo  ngrightarrow{A}\subseteq{B}.

    So, I know that we've gotta do the other direction...

    Did I even do the first part right?
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  2. #2
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    that first part looks right, depending on how rigorous you need it to be, but thats the exact idea, the squares of every real number is greater than or equal to zero, so if x\in A then x\in B

    For the other direction, Let y be a real number and let x\in B

    Since x \geq 0, there is some y\in\mathbb{R} s.t. y^2=x

    Since this works for all x, we get x\in A
    Last edited by artvandalay11; December 21st 2009 at 05:42 AM.
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  3. #3
    Senior Member Shanks's Avatar
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    The above reply has logic error. (what are we supposed to prove?)
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  4. #4
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    where's the logic error? the exercise is to prove to two sets are equal, meaning show 2 inclusions
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  5. #5
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by Shanks View Post
    The above reply has logic error. (what are we supposed to prove?)
    That A=B. What else?
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  6. #6
    Senior Member Shanks's Avatar
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    Quote Originally Posted by artvandalay11 View Post
    that first part looks right, depending on how rigorous you need it to be, but thats the exact idea, the squares of every real number is greater than or equal to zero, so if x\in A then x\in B

    For the other direction, Let y be a real number and let x\in B

    Since y^2\geq 0, there is some x\in B s.t. x=y^2

    Since this works for all y, we get x\in B
    For the other direction, we need to show If x is in B, then x is in A.
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  7. #7
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    good call, i fixed it... always remember which part you are assuming
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