Is the decomposition preserving dependencies?

Question

a relational schema R (A, B, C, D) is decomposed into (A, B), (B, C), (B, D). If functional dependencies A → B, B → C, C → D and D → B were present in R, is the decomposition preserving dependencies?

Answer 1

It is dependency preserving:

Ref: https://gateoverflow.in/3371/gate2008-it-61

Given FDs are transitive. B → C, C → D = B → D. Hence keeping B → D preserves B → C and C → D similarly C → B preserves C → D and D → B

• R1 (A, B), gives A → B
• R2 (B, C) gives C → B                              // C → D and D → B
• R3 (B, D) gives B → D                               //Hence B → C and C → D

Hence it is Dependency preserving.

Answer 2

functional dependency C->D is not possible in decomposed realtions R1(A,B) , R2(B,C) and R3(B,D). So decomposition is not preserving dependencies

Answer 3

$A\rightarrow B$ dependency preserves

$B\rightarrow C$ dependency also preserves

$C \rightarrow D and D \rightarrow B$

i.e.$C \rightarrow B$

But there is no cyclic dependency.

So, this one not preserving dependency

Comments

https://gateoverflow.in/3371/gate2008-it-61

Sir, I wonder this link says, it is dependency preserving decomposition..

Please explain which is correct!!!