Why is the negation of $A \Rightarrow B$ not $A \Rightarrow \lnot B$? The book I am reading says that the negation of "$A$ implies $B$" is "$A$ does not necessarily imply $B$" and not "$A$ implies not $B$". I understand the distinction between the two cases but why is the first one considered true?

Consider this simple example:
$$x>0\Rightarrow x>1.$$
This is clearly a false statement.
Among the next two ones, which is true ?
$$x>0\nRightarrow x>1$$
$$x>0\Rightarrow x\le1?$$
