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?

Here is a more intuitive explanation. Suppose that $A$ and $B$ are unrelated. For example, $A$ could be "France is a country in Europe" and $B$ could be "I will win the lottery". It is certainly the case that we know $A$ does not imply $B$ for these sentences: knowing that France is in Europe tells me nothing about the future! But we also do not know that $A \Rightarrow \lnot B$, for the same reason. 
So in this case, we only know that $A \not  \Rightarrow B$.  This shows there is a difference between $A \not  \Rightarrow B$ (which just says that $A$ does not imply $B$) and $A  \Rightarrow \lnot B$ (which says $A$ does imply the negation of $B$). 
This kind of intuition is important when you move from propositional logic to more general mathematics. For example, suppose we are looking at natural numbers. Let $A$ say "$x$ is an even natural number" and let $B$ say "$x$ is a natural number that is a multiple of $6$". Neither $A$ nor $B$ is true or false on its own, because the $x$ has no fixed value. Sometimes $A$ is true and sometimes it is false. But we still have that $A \not \Rightarrow B$ (e.g., $2$ is even but not a multiple of 6) and $B \Rightarrow A$ (because every multiple of 6 is even). We also do not have $A \Rightarrow \lnot B$ (because some even numbers are multiples of 6). 
Rather than trying to analyze this more complicated kind of implication in terms of truth values, it is helpful to develop the right intuition: $A \Rightarrow B$ says that knowing $A$ alone is enough to know $B$. From that intuition, you can work out the formalities involving truth values, but more importantly you can do mathematics from that intuition. 
