DB lossless join and dependency preserving decomposition

Question

Consider the following relational schema R(ABCDEFG) with FD set {AB → C, BC → A, AC → B, B → D, D → E}. The minimum relations required to decompose R into BCNF which satisfy lossless join and dependency preserving decomposition are ________.

Comments

https://gateoverflow.in/190982/bcnf-lossless-functional-dependency-preserving#c191099

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ABFG / BCFG /ACFG

ABC

BD

DE

Answer 1

relation R(ABCDEFG)

because  closure of AB,BC,AC giving all elements of relation except F and G so Candidate keys are ABFG , BCFG, ACFG

 AB->C , BC->A  AC->B giving all elements except F and G so it be in one  relation R1(A,B,C) because of candidate keys another relation R2(A,B,F,G) or R2(B,C,F,G) or R2(A,C,F,G) and because in B->D  functional dependency B is not Candidate key so we have to seperate it out to R3(B,C) relation and R4(D,E) for same reason.

these 4 relations ensures Lossless Join ,Dependency preservation as well as BCNF