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Math Help - Mathematical proofs

  1. #1
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    Mathematical proofs

    For all real numbers x and y,  sqrt(xy) is less than or equal to

    X + Y
    -----
    2

    provide a counter example for statements that are false and provide a complete proof for those that are true
    Last edited by treetheta; January 30th 2009 at 07:42 AM.
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  2. #2
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    Quote Originally Posted by treetheta View Post
    For all real numbers x and y,  sqrt(xy) is less than or equal to

    X + Y
    -----
    2

    provide a counter example for statements that are false and provide a complete proof for those that are true
    you mean, \sqrt{xy} \leq \dfrac{x+y}{2}?

    FALSE: take x=y=-1
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  3. #3
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    Start with this: (x-y)^2 \geq 0 \qquad \forall x, y \in \mathbb{R}^{\color{red}+}

    Playing around with it gives us:
    \begin{aligned}x^2 - 2xy + y^2 & \geq 0 \\ x^2 + 2xy + y^2 & \geq 4xy \\ (x+y)^2 & \geq 4xy \\ & \ \ \vdots\end{aligned}
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    oh wow thanks
    I was thinking so hard on that one too,

    can you help me out with something else too i got a test in 2 hours xD

    If we are asked to write the contrapostive of a conditional statement ( For each integer n, if n^2 is an odd integer, then n is an odd integer)

    would the contra positive be replacing the odd's with even?
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  5. #5
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    Quote Originally Posted by treetheta View Post
    oh wow thanks
    I was thinking so hard on that one too,

    can you help me out with something else too i got a test in 2 hours xD

    If we are asked to write the contrapostive of a conditional statement ( For each integer n, if n^2 is an odd integer, then n is an odd integer)

    would the contra positive be replacing the odd's with even?
    No!

    Let p = " n^2 is an odd integer" and q = "n is an odd integer"

    The contrapositive of "If p, then q" is: "If ~q, then ~p."

    The negation of p is "n^2 is an even integer", and the negation of q is "n is an even integer."
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  6. #6
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    so the contrapositive would be if n is even then n^2 is even?
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