BCNF project FD on the result relations

Question

So i have a relation R(A, B, C, D, E), with FD

{A->C, CE->B, BC->AD, D-E}.

And i want to normalize it to BCNF.

Since A->C and D->E violate BCNF. i substract C and E from ABCDE which will be ABD. But how do i project the FD of the original Relation to the new ABD relation? I was told that the resulted FD on ABD will be

{AB->D, AD->B}

I don't quite understand it.

Any help would be appreciated!

Answer 1

The algorithm to project a set of dependencies over a decomposed schema is exponential, since one should calculate the closure of each subset of the attributes of the decomposed schema, and see which attributes of the closure belong to the decomposed relation. In simple cases, however, like that of your example, it is easy to check all the possible subsets (six):

A+ = AC
B+ = B
D+ = DE
AB+ = ABCDE (D is present in ABD, so AB → D is in the projection)
AD+ = ABCDE (B is present in ABD, so AD → B is in the projection)
BD+ = BDE