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Consider the following collection of relations and dependencies. Assume that each relation is obtained through decomposition from a relation with attributes ABCDEFGHI and that all the known dependencies over relation ABCDEFGHI are listed for each question. (The questions are independent of each other, obviously, since the given dependencies over ABCDEFGHI are different.) For each (sub)relation:

(a) State the strongest normal form that the relation is in. (b) If it is not in BCNF,decompose it into a collection of BCNF relations.

1. R1(A,C,B,D,E), A → B, C → D

2. R2(A,B,F), AC → E, B → F

3. R3(A,D), D → G, G → H

4. R4(D,C,H,G), A → I, I → A

5. R5(A,I,C,E)

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2. R2(A,B,F), AC → E, B → F

actually, the FD AC → E is not belongs to the relation R$_2$(A,B,F) due to C is not belongs to this relation.

the B → F is the only FD which belongs to the relation R$_2$(A,B,F)

So, R$_2$(A,B,F) have partial functional dependency ==> highest = 1 NF only.

decomposed into R$_{i}$(B,F) R$_{ii}$(AB)

remaining attributes = R$_{iii}$(ABCDEGHI) here we have AC → E.

decomposed into R$_{iv}$(AC,E), R$_{v}$(ACBDGHI)

total relations to satisfy BCNF : R$_{i}$(B,F) , R$_{ii}$(AB), R$_{iv}$(AC,E), R$_{v}$(ACBDGHI)

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So for 2) the answer would be BF,AB,ACE,ABCDGHI instead of BF & AB ?

the B → F is the only FD which belongs to the relation R2(A,B,F)

But sir why we are considering AC-->E when we already removed that FD’s from consideration.

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Given fd's aee for original relation, so you should preserve them after decomposition
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https://gateoverflow.in/968/gate2003-85

In this question also if they have asked for BCNF decomposition then we must have to considered the removed FD’s also ?

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No, note that question is asking about the given relation only but not the whole.

If it is asking about the whole relation then Yes. Those fd's applied on some other relations of original relation decomposition.