The question you raise has to do with the definition of first normal form (1NF). Whether the answer directly involves functional dependencies depends in part on the definitions you accept. Wikipedia has a fairly simple article about 1NF.
title author year category
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An Introduction to Database Systems CJ Date 2003 databases, modeling, storage, retrieval
If you look at the column "category" one way, it contains a single value. Depending on your dbms and your design, that value might be the string "databases, modeling, storage, retrieval", or it might be the array "{databases, modeling, storage, retrieval}".
If you look at the column "category" another way, it contains four values. Those values are the four strings "databases", "modeling", "storage", and "retrieval".
In database design, the solution is to use two tables. But I don't think you can decompose the "bad" table by projection (which CJ Date identifies as the decomposition operator), because projection doesn't split the content of a column into multiple rows. (Projection doesn't give you four rows from the single value "databases, modeling, storage, retrieval", which is what you need to do. "Join", the recomposition operator, doesn't yield a single value like "databases, modeling, storage, retrieval", either.)
The inability to decompose by projection suggests that the solution to this problem doesn't have to do with functional dependencies. The resulting table would have three attributes, {title, author, category}, the only key would also be {title, author, category}, and that table would be in 5NF.
Yes, it's in 1NF.
You can't side-step the often hard work of determining all the candidate keys by hanging a number off the end of the table and saying, "There. I've got a primary key." One natural candidate key for this table is {Name, Bought from, Date bought}. Consider using "Time bought" instead of "Date bought".
Your definition of 2NF is wrong. Instead of
Second Normal Form: A relation that is in First Normal Form and every
non-primary-key attribute is fully functionally dependent on the
primary key.
you need something more like this.
Second Normal Form: a relation that is in First Normal Form, and every non-prime attribute is fully functionally dependent on every candidate key.
The term non-prime attribute doesn't mean quite what non-primary-key attribute means.
Your definition of 3NF is wrong, and it's wrong for the same reasons as your definition for 2NF was wrong.
Instead of this
Third Normal Form: A relation that is in First and Second Normal Form
and in which no non-primary-key attribute is transitively dependent on
the primary key.
you need something closer to this.
Third Normal Form: A relation that is in Second Normal Form and in which every non-prime attribute is nontransitively dependent on every candidate key. (There isn't a really good way to express all those negative in one sentence.)
Best Answer
For convenience, define the (implied) numbering of your question:
Then substituting 1 and 3 into 4 gives
(A)(AC) -> B
which reduces to just
5. AC -> B.