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Thread: BIG O proof

  1. #1
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    Question BIG O proof

    How do I prove the following:
    O(n^2 logn) is also O(n^3)?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by taurus View Post
    How do I prove the following:
    O(n^2 logn) is also O(n^3)?
    $\displaystyle f(n)\in O(n^2 \log(n))$

    means there exists a constant $\displaystyle k>0$ and an $\displaystyle N>0$ such that for all $\displaystyle n>N$:

    $\displaystyle |f(n)|<k n^2 \log(n)$

    Now for $\displaystyle n>0$ we have $\displaystyle n>\log(n)$ so for $\displaystyle n>N>0$:

    $\displaystyle |f(n)|<k n^3$

    etc.

    CB
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  3. #3
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    You will have to expand on that, I have no clue how to go on...
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    $\displaystyle f(n)\in O(n^2 \log(n))$

    means there exists a constant $\displaystyle k>0$ and an $\displaystyle N>0$ such that for all $\displaystyle n>N$:

    $\displaystyle |f(n)|<k n^2 \log(n)$

    Now for $\displaystyle n>0$ we have $\displaystyle n>\log(n)$ so for $\displaystyle n>N>0$:

    $\displaystyle |f(n)|<k n^3$

    etc.

    CB
    Quote Originally Posted by taurus View Post
    You will have to expand on that, I have no clue how to go on...
    So there exists a constant $\displaystyle k>0$ and an $\displaystyle N>0$ such that for all $\displaystyle n>N$:

    $\displaystyle |f(n)|<k n^3$

    hence $\displaystyle f(n)\in O(n^3)$

    CB
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