GATE CS 2020 Q36 — Normal Forms

To find the correct statement, let us analyze the definitions of 3NF and BCNF for any non-trivial functional dependency $X \rightarrow A$ in a relation $R$:

• BCNF: For every non-trivial functional dependency $X \rightarrow A$, $X$ must be a superkey.
• 3NF: For every non-trivial functional dependency $X \rightarrow A$, either $X$ is a superkey, or $A$ is a prime attribute (part of some candidate key).

Given that the relation $R$ is in 3NF but not in BCNF:

1. Since $R$ is not in BCNF, there must exist at least one non-trivial functional dependency $X \rightarrow A$ where $X$ is not a superkey.
2. Since $R$ is in 3NF, this dependency $X \rightarrow A$ must satisfy the 3NF condition. Since $X$ is not a superkey, the other condition must hold: $A$ must be a prime attribute.

Thus, there must exist a non-trivial functional dependency $X \rightarrow A$ where $X$ is not a superkey and $A$ is a prime attribute. This matches Option A.