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Math Help - Small-o and Theta questions

  1. #1
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    Small-o and Theta questions

    I have these questions from a book:

    show that if f(n)=theta(log2 n) then f(n)=theta (log10 n)

    show that if f(n)=o(g(n)) and g(n)=o(f(n)) then f(n)=theta(g(n))

    how to prove? can anyone help me with it?
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: please help me with these quetions

    The first proposition, do you mean:
    f(n)=\theta(\log_{2}n) \Rightarrow f(n)=\theta(\log10n)
    ?
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  3. #3
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    Re: please help me with these quetions

    To show the first statement, write the definition of \Theta and note that \log_2n=\log_210\log_{10}n.

    The second statement is a little odd because I believe that if f(n) = o(g(n)) and g(n) = o(f(n)), then f and g must be 0 from some point. However, the fact that g(n) = o(f(n)) implies that g(n) = O(f(n)), and this is almost the same as f(n)=\Theta(g(n)) (up to absolute value).
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  4. #4
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    Re: Small-o and Theta questions

    The first proposition, do you mean:

    ?
    yes, that's what i meant

    To show the first statement, write the definition of and note that .

    The second statement is a little odd because I believe that if f(n) = o(g(n)) and g(n) = o(f(n)), then f and g must be 0 from some point. However, the fact that g(n) = o(f(n)) implies that g(n) = O(f(n)), and this is almost the same as (up to absolute value).
    thanks for help
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