+1 vote
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R(A,B,C) is a relation.Which of the following don't have dependency preserving BCNF decomposition?

1. A->B, B->C
2. A->B,B->C,C->A
3. AB->C,C->A
4. None of these
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option c

See AB is the candidate key and C which is a non prime attribute is able to generate part of prime attribute A.

So ans should be C

In option, A relation can be clearly divided into AB and BC which would be both dependency preserving as well as in BCNF.

option B is already in BCNF .

option C has no BCNF decomposition that can preserve dependencies because to satisfy AB->C  you will need all three attributes in relation which in turn will create C->A dependency which would violate BCNF normal form

example :- one possible decomposition that is in BCNF is  AC and BC  but this does not satisfy AB->C dependency.

So answer should be option C.

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by (23 points) 1 flag:
✌ Edit necessary (Chirag Shilwant)
In BCNF, NP->P and P->P relations are not allowed. In option C, C->A is NP->P. Hence it is not in BCNF.

Hence Option C.
by (277 points)

+1 vote