Which one of the following statements is FALSE?

Concept

For a candidate key K, the dependency of Z on K is transitive through Y when K → Y and the non-trivial dependency Y → Z hold, Y is not a superkey, and Z is not contained in Y. Thus Z depends on K indirectly through Y. A prime attribute is one that belongs to at least one candidate key.

For 3NF, every non-trivial dependency X → A is permitted when X is a superkey or A is prime. BCNF is stricter: X must be a superkey, regardless of whether A is prime. 2NF concerns only partial dependency on a proper subset of a composite candidate key.

Application

1. With only two attributes, a non-trivial determinant fixes the other attribute and therefore the entire schema. It is a superkey, so the relation meets BCNF.

2. When every candidate key has one attribute, no proper non-empty subset of a candidate key exists. A partial dependency cannot arise, so the relation meets 2NF.

3. In 3NF, a prime dependent attribute satisfies the second branch of the 3NF condition even when the determinant is not a superkey. Such a transitive dependency can therefore occur.

4. In BCNF, prime status gives no exception: the determinant of every non-trivial dependency must be a superkey. A transitive dependency through a non-superkey determinant therefore cannot occur.

Cross-check

The statement about a prime attribute in 3NF and the statement about a prime attribute in BCNF describe the same dependency pattern under different normal forms. The 3NF rule has an “attribute is prime” branch, whereas the BCNF rule does not; this single difference makes the 3NF statement possible and the BCNF statement impossible.

Result

The false statement is: “A prime attribute can be transitively dependent on a key in a BCNF relation.”