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Suppose we have a relation R(A, B, C, D, E) with FDs {A → B, AB → C, D → AC, D → E}. R is decomposed as R1(ACD), R2(DE), R3(ABC). Which of the following statement is true regarding the decomposition?

1. lossless and depending preserving

2. lossless but not depending preserving

3. lossy and dependency preserving

4. lossy and not dependency preserving

Made easy says option 1 but i got option 3. Please guide.

1 Answer

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R(A, B, C, D, E) with FDs {A → B, AB → C, D → AC, D → E}. R is decomposed as
R1(ACD)                     , R2(DE),                                  R3(ABC).
D → AC                        D → E                                     AB → C,A → B
so dependency is preserved.
two rln x nd y r lossless decomposition if  x∩y is super key of x or y
merge R1(ACD), R2(DE) common is D ie. s.k. for R2(DE): ---->R12(ACDE) 
Now merge R12(ACDE)  nd R3(ABC). common is AC  ie. s.k. for R3(ABC).
 hence it is lossless decomposition 

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