GATE CS 1990 Q3-ii — Normal Forms

A relation is in Third Normal Form (3NF) if, for every non-trivial functional dependency $X \rightarrow Y$, at least one of the following conditions holds:

1. $X$ is a superkey.
2. $Y$ is a prime attribute (i.e., each attribute in $Y$ is part of some candidate key).

A relation is in Boyce-Codd Normal Form (BCNF) if, for every non-trivial functional dependency $X \rightarrow Y$, $X$ is a superkey. BCNF eliminates all redundancy arising from functional dependencies. Therefore, any redundancy present in a 3NF relation is due to dependencies allowed in 3NF but forbidden in BCNF (i.e., where $X$ is not a superkey, but $Y$ consists of prime attributes).

Let us analyze each option based on this:

• Option A (True): 3NF prevents transitive dependencies of non-prime attributes on candidate keys. However, it allows transitive dependencies of prime attributes. For example, if $K \rightarrow X$ and $X \rightarrow A$ where $K$ is a candidate key, $X$ is not a superkey, and $A$ is a prime attribute, this transitive dependency is allowed in 3NF but causes redundancy.
• Option B (True): Dependencies of the form $X \rightarrow A$ where $A$ is a prime attribute and $X$ is not a superkey are allowed in 3NF. Since $X$ is not a superkey, this causes data redundancy.
• Option C (False): If prime attributes are only on the left-side (e.g., $X \rightarrow Y$ where $Y$ contains non-prime attributes), then for the relation to be in 3NF, $X$ must be a superkey. If $X$ is a superkey, this dependency is also valid in BCNF and does not introduce BCNF-level redundancy.
• Option D (True): If a dependency $X \rightarrow Y$ involves only prime attributes and $X$ is not a superkey, it is allowed in 3NF (since $Y$ consists of prime attributes) but violates BCNF, leading to redundancy.

Example: Consider a relation $R(A, B, C)$ with candidate keys $AB$ and $BC$. All attributes $A, B, C$ are prime. The dependency $C \rightarrow A$ is non-trivial, involves only prime attributes (Option D), has a prime attribute on the right-side (Option B), and represents a transitive dependency of $A$ on $AB$ (Option A). This relation is in 3NF but not in BCNF, and contains redundant data.