I am trying to prove some logical stuff using the definitions of BIG O, BIG Theta and BIG Omega.
Unfortunately I am a bit confused.
And how can we represent Θ in terms of of those other notations?
Answer: f(n) ∈ Θ(g(n)) <=> f(n) ∈ O(g(n)) and f(n) ∈ Ω(g(n))
We know that $O$ is the family of functions for which the basis $g(n)$ serves as an upper bound. In other words, every element of $O(g(n))$ has that property that $g(n)$ always grows faster than that element.
We also know $\Omega$ is the family of functions for which the basis $h(n)$ serves as a lower bound. In other words,e very element of $\Omega(h(n))$ has the property that $h(n)$ always grows slower than that element.
So, if the same function is a basis for both $O$ and $\Omega$ (shown as $O(g(n))$ and $\Omega(g(n))$, then the elements that are common between $O$ and $\Omega$ have the property that $g(n)$ grows faster than or equal to and slower than or equal to that element. You may think of the function $g(n)$ "sandwiching" (if you will) the elements of both $O$ and $\Omega$.
So $f(n) \in O(g(n) \land f(n) \in \Omega(g(n))$ means that $g$ serves as both and upper bound and lower bound to $f$ and thus the growth of $g$ is directly comparable to that of $f$ which is to say that they are scalar multiples of each other (which is, by no coincidence, precisely what membership of $\Theta$ means!).