Is this relation in Boyce-Codd's Normal Form? I thought I understood BCFN until I bumped into this example from my course slides.

A relation in a library database contains what books are currently borrowed by what users.
The relation contains the unique ID of the user, their name and the title of the book.
Different users might share the same name.
A book can only be borrowed by one user.
A user can have borrowed more than one book.

An example of this relation could be:




USER_ID
USER_NAME
BOOK_TITLE




U012
Charles
Hamlet


U034
Alex
Blue Moon


U491
Diane
Capital


U034
Alex
The Crow


U491
Diane
Economics 2


U012
Charles
Lost Girl


U034
Alex
Chess basics


U012
Charles
Greek History


U491
Diane
Red Mars




According to the slide, this relation is in Boyce-Codd's Normal Form,as:

all functional dependencies of the form $\alpha \rightarrow \beta $
are either trivial or satisfy that $\alpha$ is a super-key of the
relation

While it is my understanding that those are the requirements for BCNF, I don't think this relation satisfies them. In particular, the functional dependency USER_ID $\rightarrow$ USER_NAME is not trivial (as $\beta$ is not a subset of $\alpha$) and $\alpha$ is not a super-key (since by itself we cannot distinguish between entries representing different books borrowed by the same user).
Am I missing something?
Assuming that each book has a different title, you are correct in thinking that the relation is not in BCNF.
In fact, the (only) candidate key of the relation is  Book_title, so the functional dependency User_id -&gt; User_name violates the definition of BCNF. Assuming that a cover of the dependencies are:
User_id -&gt; User_name    (each user_id identifies a user with a certain name)
Book_title -&gt; User_id   (each book can be borrowed only by a certain user)

a decomposition of the original relation in BCNF is:
R1(User_id, User_name)  with candidate key User_id
R2(Book_title, User_id) with candidate key Book_title