this again is the reply i got from authors of the book

I am glad that you are taking databases seriously. With regard to your
question. Assume you have the relation R with the attributes that you
have defined. The relation R is in 2NF if, first, it is in 1NF. We can
assume that none of the attributes V, W, X, and Y have attributes of
their own. Therefore, the relation is in 1NF. Now, the 2NF requires
that you have a key but it also assume that there are non prime
attributes. These non prime attributes cannot be partially dependent
on the key. Yes, you have a key. It is obvious from the reflexivity
axiom that the key determines all other attributes. In fact, {V, W, X,
Y} -> V; {V, W, X, Y}-> W; {V, W, X, Y} -> X; {V, W, X, Y} ->Y. You
also have that {V, W, X, Y} -> {W,X,Y} by the axiom of augmentation.
Finally, {V, W, X, Y} -> {X,Y} by the same axiom. The apparent
contradiction that you have found to the definition of 3NFcomes from
not having nonprime attributes which are not partially dependent upon
ANY key.  Take into account also that the def of 3NF requires that the
relation be previously in 2NF (no nonprime attribute can be partially
dependent upon any key).Remember the definitions (2NF, 3NF) assume
that the key A determines every other attribute but it also assumes
the existence of nonprime attributes which are not partially dependent
upon the key. All the attributes that you have here, B and C are
partially dependent upon the key. In the def of transitive dependence
(refer to the graph in the book) A is assumed to be a single key. B
is a nonprime attribute and it cannot be partially dependent upon the
key. Finally, C must also be a nonprime attribute. In case, you
forgot, nonprime attributes are those attributes that do no
participate in any key.

If there are no partial dependencies you are in 2NF. If you do not
have transitive dependencies you are in 3NF. Obviously, I am assuming
that we have a key for the relation as determined by the set of  given
FDs. Finally, if you have a relation with attributes A, B, and C and
they all can be used as key and they do not overlap then you are
already in 3NF.

By the way, in the book all attributes A,B, and C are assumed to be
simple attributes unless defined otherwise. This is to simplify the
explanations.

If you want to take a more formal book, take a look at The Theory of
Relational Databases by David Maier. It is heavy in mathematics.

Before, I forgot, Trivial FDs are those that are determined by the
Axiom of Reflexivity. A comment about the use of notations. In the DB
world, there are definitions and definitions. Zaniolo is top expert in
DB but if you read some of his papers you will find that the same
concept may have more than one interpretation. This is typical of DB.
That is, why it is very important to know what definitions you are
using. Take for example, the word set, I believe that the last time I
checked there were more than 200 definitions of this word in different
DBMSs alone. When working with FD and non redundant FDs you need to
exclude trivial FDs.

The Definition that I used in the book is the same as Maier which in
turn is the definition that Date uses. I do a lot of consulting in DB
across the world in ORACLE. It is fun to work with set of FDs
developed by the designers who are using the IBM concepts and
notations and sometimes notations and definitions of their own. Take
into account that the same thing happens sometimes in mathematics. One
author considers the set of natural numbers beginning with one. Others
may assume that it includes zero. That is, sometimes you need to check
what the definitions are to be consistent. Hope this helps ;if not I
hope to hear from you again soon.

I have enjoyed the discussion across the ocean.
Be well,