Answer: C
It is given that $\{\text{Author},\text{Title}\}$ is the key for both schemas.
The given dependencies are :
- $\{\text{Title}, \text{Author}\}\to \text{Catalog_no}$
- $\text{Catalog_no} \to \{\text{Title},\text{Author}, \text{Publisher}, \text{Year}\}$
- $\{\text{Publisher}, \text{Title}, \text{Year}\} \to \{\text{Price}\}$
First, let's take schema Collection (Title, Author, Catalog_no) :
- $\{\text{Title}, \text{Author}\} \to \text{Catalog_no}$
$\{\text{Title}, \text{Author}\}$ is a candidate key and hence super key also and by definition of $\text{BCNF}$ this is in $\text{BCNF}$.
Now, let's see Book (Title, Author, Catalog_no, Publisher, Year , Price):
- $\{\text{Title}, \text{Author}\}^+ \to \{\text{Title}, \text{Author}, \text{Catalog_no}, \text{Publisher}, \text{Year}, \text{Price}\}$
- $\{\text{Catalog_no}\}^+ \to \{\text{Title}, \text{Author}, \text{Publisher}, \text{Year}, \text{Price}, \text{Catalog_no}\}$
So candidate keys are : $\text{Catalog_no}, \{\text{Title}, \text{Author}\}$
But in the given set of dependencies we have $\{\text{Publisher}, \text{Title}, \text{Year}\} \to \text{Price},$ which has a Transitive Dependency. So, Book is not in 3NF but is in 2NF.