Prove or disprove that:
a) $\forall c\in \mathbb{N} $ $2^{cn} = 2^n $ ( disproved )
b) $\forall f,g: \mathbb{N} \rightarrow \mathbb{N}$ $ g= \Omega(f) \cup O(f) $ (disproved)
c)$\forall f: \mathbb{N} \rightarrow \mathbb{N}$ $\omega(\Theta(f))= \Theta(\omega(f))$
As you can see the last claim is my problem. I'm pretty sure that the last one is true. So
$$\omega(\Theta(f)) := \cup_{g \in \Theta(f)} \omega(g) $$ and
$$\Theta(\omega(f)) := \cup_{g \in \omega(f)} \Theta(g) $$
My idea is to show that $\omega(\Theta(f)) = \omega(f)$ and $\Theta(\omega(f)) = \omega(f)$, which would prove the claim, but failed.
* Definition in general for Landau notation: *
$ O(f) $ = { $g : \mathbb{N} \rightarrow \mathbb{N} | \exists c > 0 , n_0 \in\mathbb{N}, \forall n \geq n_0 : g(n) \leq c \cdot f(n)$ }
$ \Omega(f) $ = { $g : \mathbb{N} \rightarrow \mathbb{N} | \exists c > 0 , n_0 \in\mathbb{N}, \forall n \geq n_0 : g(n) \geq c \cdot f(n)$ }
$ \Theta(f) = O(f) \cup \Omega(f) $
$ \omega(f) $ = { $ g | \lim_{ n \rightarrow \infty} \frac{f(n)}{g(n)} = 0 $ }