Ok, the problem is this:

Provide a specific counterexample that shows the statement is false.

Statement: for all function f and g on N, if $\displaystyle f(n)$ $\displaystyle \in$ $\displaystyle O(g(n))$, then $\displaystyle g(n)$ $\displaystyle \in$ $\displaystyle O(f(n))$.

I've chosen $\displaystyle f(n) = n^2$ and $\displaystyle g(n) = n^3$ as this seems fairly obvious.

It is easy to show that $\displaystyle n^2$ is in $\displaystyle O(n^3)$...though this is not accurate, it still holds. I much pick values for N and K such that:

$\displaystyle n^2 < K*n^3$

$\displaystyle 1 < K*n,$ so I choose K = 1 and N = 2

The second one, I am trying a proof by contradiction (though I was always told to assume the negation of the hypothesis). If we simply do the same thing we get:

$\displaystyle n^3 < K*n^2$

$\displaystyle n < K$ , choose n = 2K and we get

$\displaystyle 2K < K$ which is false, so the original statement must be false. Is this a correct way to do this?