Let $\Omega$ be a set and $a(x)$ any statement about $x \in \Omega$. Proof that
$$\forall x \in \Omega: \neg a(x) \Leftrightarrow \neg(\exists x \in \Omega: a(x))$$
I think there is no way in proving this without giving an example or describing, right?
I tried:
What the left side is saying is that for all x in the set omega, we have that "not a(x)".
This is equivalent to: There does not exist an x in the set omega so that we have a(x).
That's what the task is saying in maths language.
So left side: No matter (=all) what x we choose, we will always get "not a(x)".
Right side: There doesn't exist an x to get a(x).
This is really the same but how can you prove this? This is very frustrating task because it makes sense but you cannot prove it / prove it easily : /