How do I interpret this with logic notation? "not A doesn't mean not B" $$
\neg A \rightarrow \neg B \equiv B \rightarrow A
$$
Is this correct?
As negation have more high priority, then implication, we have
$$\neg A \Rightarrow \neg B \equiv \\
(\neg A) \Rightarrow (\neg B) \equiv \\ 
\neg(\neg A) \lor (\neg B) \equiv\\
A \lor (\neg B) \equiv\\ 
B \Rightarrow A$$
So, yes, you are correct.
This is an implication so if A is true and b is true it is true
if not A implies not B then A is False implies B is False so the implication is still true.
A implies B is the converse of the statement and the truth of a statement does not imply the truth of the converse