GATE CS 2026 Set 1 Q21 — Normal Forms

Let us analyze each statement individually:

• Statement A: It is always possible to obtain a dependency-preserving $3\text{NF}$ decomposition of a relation.
  True. By using the $3\text{NF}$ synthesis algorithm, we can always decompose any relation schema into a set of relations that are in $3\text{NF}$ while guaranteeing both the lossless-join property and dependency preservation.


• Statement B: It is always possible to obtain a dependency-preserving $1\text{NF}$ decomposition of a relation.
  True. $1\text{NF}$ only requires that all attribute values are atomic. Since any relation can be represented in $1\text{NF}$ without losing any functional dependencies (the original relation itself with atomic attributes is already in $1\text{NF}$ and preserves all its dependencies), a dependency-preserving $1\text{NF}$ decomposition is always trivially possible.


• Statement C: It is not always possible to obtain a dependency-preserving BCNF decomposition of a relation.
  True. While a lossless-join BCNF decomposition is always possible, dependency preservation is not guaranteed.
  Counterexample: Consider the relation $R(A, B, C)$ with functional dependencies:$$AB \rightarrow C, \quad C \rightarrow A$$Here, the candidate keys are $AB$ and $BC$. The dependency $C \rightarrow A$ violates BCNF because $C$ is not a superkey. Any BCNF decomposition (such as $R_1(A, C)$ and $R_2(B, C)$) will fail to preserve the dependency $AB \rightarrow C$.


• Statement D: It is not always possible to obtain a dependency-preserving $2\text{NF}$ decomposition of a relation.
  False. Every relation in $3\text{NF}$ is also in $2\text{NF}$ by definition (since $3\text{NF}$ is a stricter normal form). Since we can always obtain a dependency-preserving $3\text{NF}$ decomposition (as established in Statement A), that same decomposition is guaranteed to be in $2\text{NF}$ and preserve all dependencies. Thus, it is always possible to obtain a dependency-preserving $2\text{NF}$ decomposition.

Therefore, statements A, B, and C are true.