Negate the following: $\mathbb{P}$ is the set of prime numbers.
1) $ \forall a \in \mathbb{P} : \exists b \in \mathbb{P}: a|b$
Neg: $ \exists a \in \mathbb{P}: \forall b \in \mathbb{P}: a \not| b$
(I am having problems with the first sentence because both the sentence and the negation would be true?)
2) $ \forall a \in \mathbb{P} : \forall b \in \mathbb{P}: (a|b \rightarrow a=b)$
Neg: $\exists a \in \mathbb{P}: \exists b \in \mathbb{P}: (a|b \land a \neq b)$
3) $\exists a,b \in \mathbb{R}: (a \neq 0 \land b \neq 0 \land a \cdot b =0)$
Neg: $\forall a,b \in \mathbb{R}: (a = 0 \lor b=0 \lor a \cdot b \neq 0)$
I have tried to come up with the solutions myself, it would be great if some could check whether those are fine.