Big Oh notation [closed]

I've recently learned about the Big Oh notation and heard that the following aren't true:

1. $f(n)\in O(f(n)^2)$.
2. Either $f(n)\in O(g(n))$ or $f(n)\in\Omega(g(n))$ or both.
3. $f(n)\in\omega(g(n))$ implies $\log(f(n)) = \omega(\log(g(n))$.

I'm trying to find a good counter-example for each but it's proving to be a bit challenging. What would be a good one for these?

• can you give more context?
– Joe
Mar 19 '14 at 2:31
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– Raphael
Mar 19 '14 at 8:13

For (1), take $f(n) = 1/n$.
For (2), take $f(n) = \begin{cases} 1 & n \text{ even} \\ n & n \text{ odd} \end{cases}$ and $g(n) = \sqrt{n}$.
For (3), take $f(n) = n^2$ and $g(n) = n$.