Discussion
Schema I: Registration (rollno, courses)
Field 'courses' is a set-valued attribute containing the set of courses a student has registered for.
Non-trivial functional dependency:
rollno → courses
Schema II: Registration (rollno, courseid, email)
Non-trivial functional dependencies:
rollno, courseid → email
email → rollno
Schema III: Registration (rollno, courseid, marks, grade)
Non-trivial functional dependencies:
rollno, courseid → marks, grade
marks → grade
Schema IV: Registration (rollno, courseid, credit)
Non-trivial functional dependencies:
rollno, courseid → credit
courseid → credit
Which one of the relational schemas above is in 3NF but not in BCNF?
Answer: B
The schema in 3NF but not in BCNF is Schema II.
In Schema II, email → rollno means {email, courseid} is also a key, so rollno is a prime attribute. The dependency email → rollno has a determinant that is not a superkey, which breaks BCNF, but its right side is prime, which 3NF allows.
The other schemas fail 3NF or even earlier normal forms. Schema I has a set-valued attribute, so it is not even in 1NF. In Schema III, marks → grade makes the non-prime grade depend on the non-key marks, a transitive dependency that breaks 3NF. In Schema IV, courseid → credit makes the non-prime credit depend on part of the key, a partial dependency that breaks 2NF.
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