# Can I disprove the following statement with this example?

I do not know much about O notation so could you please help me, I have to prove or disprove the following statment:

$\forall f,g: \mathbb{N} \rightarrow \mathbb{R}_{\geq 0} \text{ }f=O(g) \text{ or } f=\Omega(g)$

Can I disprove this statement with $f = \sin(x) + 1$ and $g = \cos(x) + 1$?

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Are you taking $R_+$ to mean the positive reals or the nonnegative reals? –  only Oct 13 '12 at 21:32
Yes, this is essential, it is about nonnegative reals. Already edited –  haemhweg Oct 13 '12 at 21:34

If you are working in radians, which is the usual convention in mathematics unless one says otherwise, it can be done. But we need to find out quite a lot about how sine and cosine behave at the integers. Not easy! Replacing the $n$ in $\sin n$ and $\cos n$ by $\frac{\pi n}{2}$ will work.
There are simpler examples that disprove the assertion. We can let $f(n)=0$ when $n$ is even, $f(n)=1$ when $n$ is odd, and let $g(n)=1-f(n)$.
Or for an example that is slightly more realistic in a computing context, let $f(n)=n$ when $n$ is even, $f(n)=n^2$ when $n$ is odd, and let $g(n)=n^2$ when $n$ is even, $g(n)=n$ when $n$ is odd.