p -> q is represented as "If you have a library card, then you can check out books"
~q -> ~p is represented as "If you cannot check out books, then you do not have a library card" This is contrapositive by definition.
q -> p is represented as "If you can check out books, then you have a library card" This is the converse
~p -> ~q is represented as "If you do not have library card, then you cannot check out books". This is the inverse
p <-> q is represented as "You have library card if and only if you can check out books". This is the biconditional.