week 6 -- solving negation problem analysis

This week we get out first midterm back, overall, I did OK on this test since I got 85%. But I found myself keep having the same problem on how to negate an implication without hesitation.

For an implication like:
                       ∀a∈ N, P(a) ⇒ Q(a)

P(a) ⇒ Q(a) equals:
                      ∀a∈ N, ¬P(a) ∨ Q(a)

hence the negation:

                       ∃ a∈ N, P(a) ∧ ¬Q(a)


Let's try another way(this is a tricky False statement):

the only false of it :
                       ∃ a∈ N, P(a) ⇒ ¬Q(a)
P(a) ⇒ ¬Q(a) equals:
                       ∃ a∈ N,  ¬P(a) ∨ ¬Q(a)

Let's use "negation" of implication:(False)
so the negation of ∀a∈ N, P(a) ⇒ Q(a)
                       is ∃a∈ N, ¬P(a) ⇒ ¬Q(a)
equals:
                   ∃a∈ N, P(a) ∨ ¬Q(a)