GATE CS 2012 Q2 — Normal Forms

To determine which statement is true, let us analyze the definitions and relationships between 3NF (Third Normal Form) and BCNF (Boyce-Codd Normal Form):

• BCNF Definition: A relation $R$ is in BCNF if, for every non-trivial functional dependency $X \rightarrow A$, $X$ is a superkey of $R$.
• 3NF Definition: A relation $R$ is in 3NF if, for every non-trivial functional dependency $X \rightarrow A$, either:
  1. $X$ is a superkey of $R$, or
  2. $A$ is a prime attribute (part of some candidate key).

Since the condition for BCNF ($X$ must be a superkey) is a stricter subset of the conditions for 3NF, any relation that satisfies BCNF automatically satisfies 3NF. Thus, BCNF is a stronger normal form than 3NF, meaning:

Therefore, every relation in BCNF is also in 3NF.

Analysis of other options:

• Option A is False: There are relations in 3NF that are not in BCNF. For example, a relation $R(A, B, C)$ with functional dependencies $AB \rightarrow C$ and $C \rightarrow B$ is in 3NF but not in BCNF.
• Option B is False: The condition that "every non-prime attribute is fully functionally dependent on every key" is the definition of 2NF (Second Normal Form), not 3NF.
• Option D is False: Since BCNF is a subset of 3NF, any relation in BCNF is simultaneously in both BCNF and 3NF.