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?
To show the first statement, write the definition of $\displaystyle \Theta$ and note that $\displaystyle \log_2n=\log_210\log_{10}n$.
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 $\displaystyle f(n)=\Theta(g(n))$ (up to absolute value).
yes, that's what i meantThe first proposition, do you mean:
?
thanks for helpTo 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).