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

Answer: The statement "Achieving a lossless and dependency-preserving decomposition into BCNF is always guaranteed." is false.

• BCNF imposes stricter conditions compared to 3NF. This is true: BCNF requires every non-trivial functional dependency to have a superkey on the left-hand side, which is stronger than 3NF.

• It is always possible to achieve a lossless and dependency-preserving decomposition into 3NF. This is true: the 3NF synthesis algorithm produces a decomposition that is lossless and preserves the original functional dependencies.

• Achieving a lossless and dependency-preserving decomposition into BCNF is always guaranteed. This is false: although you can always decompose to BCNF while keeping the decomposition lossless, dependency preservation is not guaranteed. For example, with attributes A, B, C and functional dependencies A -> B and B -> C, a BCNF decomposition can separate attributes so that the transitive dependency A -> C is not preserved directly in any single decomposed relation.

• Every relation with exactly two attributes satisfies BCNF. This is true: in a two-attribute relation, any non-trivial dependency makes the determinant a key (or there are no non-trivial dependencies), so the BCNF condition holds.

Summary: The false statement is the claim that BCNF decomposition always preserves dependencies. Lossless BCNF decompositions are possible, but they may not preserve all functional dependencies, whereas 3NF has algorithms that ensure both losslessness and dependency preservation.