GATE CS 2026 Set 1 — Question 31

Statement (A) is true: There exists a standard algorithm (Bernstein's synthesis algorithm) to decompose any relation schema into 3NF which is both lossless-join and dependency-preserving.Statement (B) is true: Since 3NF is a stricter form of normalization than 1NF (i.e., every relation in 3NF is also in 1NF), and we can always obtain a dependency-preserving 3NF decomposition, it follows that we can always obtain a dependency-preserving 1NF decomposition.Statement (C) is true: It is a known property of BCNF that a dependency-preserving decomposition is not always possible. For example, consider a relation R(A, B, C) with functional dependencies $F = \{AB \to C, C \to B\}$. The decomposition into BCNF requires splitting based on C → B, which results in relations where the dependency AB → C cannot be preserved.Statement (D) is false: Since 3NF implies 2NF, and a dependency-preserving 3NF decomposition is always possible, a dependency-preserving 2NF decomposition is also always possible.