Assessing statement about third normal form
The exercises says:


  A relation R is in 3NF, if every non-prime attribute of R, is fully functionally dependent on some key of R.


As you have noted, that definition is imprecise. Let’s try to correct it:


  A relation R is in 3NF, if every non-prime attribute of R is fully functionally dependent on some (candidate) key and on nothing else.


Of course such definition should also exclude trivial dependencies or derived dependencies in which the left part contains superflous attributes.