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$R(A,B,C,D)$ is a relation. Which of the following does not have a lossless join, dependency preserving $BCNF$ decomposition?

  1. $A \rightarrow B, B \rightarrow CD$
  2. $A \rightarrow B, B \rightarrow C, C \rightarrow D$
  3. $ AB \rightarrow C, C \rightarrow AD$
  4. $A \rightarrow BCD$
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ans C)
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Option C: R(A, B, C, D) and the Functional Dependency set = {AB→C, C→AD}

Candidate keys are AB, CB & C→ AD violates BCNF, so decompose it using BCNF decomposition algorithm (Source: Navathe). C+ = CAD. Decomposition is R1(BC) and R2(CAD).

C→AD is preserved but AB → C can’t be preserved. So decomposition is in BCNF but not dependency preserving hence option C is the right answer.
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