Discussion
Consider a schema R(A, B, C, D) and functional dependencies A → B and C → D. Then the decomposition R1(A, B) and R2(C, D) is
Answer: A
The decomposition R1(A, B) and R2(C, D) keeps both dependencies, A to B in R1 and C to D in R2, so it is dependency preserving. It is not lossless, because R1 and R2 have no attribute in common, so their natural join is a cross product that gives spurious tuples, and no dependency tells how to join them. So it is dependency preserving but not lossless join, option (A). Options (B), (C) and (D) all claim it is lossless.
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