if you are given 2 propositions with quantifiers, what is the method to find out that they are equivalent? For example the three propositions:
$\exists c : \exists n_0 : \forall n : n \geq n_0 \rightarrow f(n) \geq cg(n)$
$\forall n_0 : \exists c : \forall n : n \geq n_0 \rightarrow f(n) \geq cg(n)$
$\forall c : \exists n_0 : \forall n : n \geq n_0 \rightarrow f(n) \geq cg(n)$
How would we determine any of these propositions are equivalent?
Given the different quantifiers, most likely these three statements are all different. To demonstrate that, come up with a domain and interpretation for the functions so that the statements get different truth-values.
OK, for our domain, let's use the natural numbers
For the functions, let's first pick $f(n)=g(n)=n$
Then the three statements become:
$\exists c : \exists n_0 : \forall n : n \geq n_0 \rightarrow n \geq cn$
$\forall n_0 : \exists c : \forall n : n \geq n_0 \rightarrow n \geq cn$
$\forall c : \exists n_0 : \forall n : n \geq n_0 \rightarrow n \geq cn$
Now, the first two statements are true, since for $c$ we can just pick $c=0$, and it will always be true that $n \geq 0$
The third statement, however, is not true, because for $c=2$ then, no matter what you pick for $n_0$, the conditional will evaluate to false for $n=n_0+1$, since then $n\geq n_0$, but not $n \geq cn$
OK, so now we know the third statement is not equivalent to either the first or the second statement.
Now, can you think of a choice for $f(n)$ and $g(n)$ that differentiate between the first two statements?