Consider relation R(A,B,C) with the MVD A↠B. Which of the
Consider a relation R(A,B,C) with the MVD A↠B. Which of the following decompositions is guaranteed to be a lossless join?
- A.
R into R1(A,B) and R2(A,C)
- B.
R into R1(A,B,C) and R2(B,C)
- C.
R into R1(A,B) and R2(B,C)
- D.
None
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Correct answer: A
Key insight: For a relation R(X,Y,Z), decomposing into R1(X,Y) and R2(X,Z) is lossless if X → Y or X ↠ Y holds in R.
Apply the condition with X = A, Y = B, Z = C: A ↠ B is given, so the decomposition into R1(A,B) and R2(A,C) is lossless.
Why the other decompositions are not guaranteed lossless:
Decomposing into R1(A,B,C) and R2(B,C): the common attributes are B and C. There is no given functional or multivalued dependency making BC determine A, so the join is not guaranteed to be lossless.
Decomposing into R1(A,B) and R2(B,C): the common attribute is B. A ↠ B does not imply any dependency from B to A or C, so B does not guarantee reconstruction of R on its own and the join may be lossy.
Conclusion: The decomposition into R1(A,B) and R2(A,C) is guaranteed lossless because A ↠ B satisfies the lossless condition; the other listed decompositions are not guaranteed lossless.