Question

5c. Consider the relation R(ABCDE) with the set of functional dependencies F={BE→D, DE→A, AD→C, B→E}.

Using decomposition, find a lossless, dependency preserving, BCNF set of relations for R, if such exists. Be sure to identify the projections of the functional dependencies onto the resulting relations at each stage of the decomposition.

Answer 1

Answer:

The R.H.S of the functional dependencies contains the attributes A, C, D, and E. Hence the candidate key must contain the attribute B.

If we take the closure of B or (B)+ we get:-

(B)+ = {BE} [ as B->E]

or,

(B)+ = {BDE} [ as BE->D]

or,

(B)+ = {ABDE} [ as DE->A]

or,

(B)+ = {ABCDE} [ as AD->C]

As (B)+ contains all the attributes in its closure so, it is a candidate key.

As B is a single attribute candidate key and every candidate key must include it so, it is the only candidate key.

This relation is in 2NF and not in 3NF because there are non-prime attributes that are transitively dependent on the candidate ke