Which one of the following statements about normal forms is FALSE?

Answer: The false statement is: "Lossless, dependency-preserving decomposition into BCNF is always possible."

Explanation: BCNF is stricter than 3NF, and while you can obtain lossless decompositions into BCNF, those decompositions do not always preserve all functional dependencies. In contrast, there is a standard synthesis algorithm that produces a decomposition into 3NF that is both lossless and dependency-preserving.

• BCNF vs 3NF: BCNF requires every nontrivial functional dependency X → Y to have X as a superkey. 3NF relaxes this by allowing Y to be a prime attribute, so every BCNF relation is in 3NF but not every 3NF relation is in BCNF.

• 3NF decomposition: The 3NF synthesis algorithm (using a minimal cover) guarantees a lossless-join decomposition that is dependency-preserving.

• BCNF decomposition may lose dependencies. Example: For R(A,B,C) with functional dependencies AB → C and C → B, decomposing into relations (B,C) and (A,C) yields a lossless BCNF decomposition, but the dependency AB → C is not preserved in the decomposed relations.

• Two-attribute relations: Any relation with exactly two attributes is in BCNF because any nontrivial dependency between the attributes makes the determinant a key for the relation.

Conclusion: The statement that a lossless, dependency-preserving decomposition into BCNF is always possible is false because BCNF decompositions can require sacrificing dependency preservation. The other statements are correct.