Let R (A, B, C, D) be a relational schema with the following functional dependencies:
A → B, B → C, C → D and D → B. The decomposition of R into (A, B), (B, C), (B, D)
(A) gives a lossless join, and is dependency preserving
(B) gives a lossless join, but is not dependency preserving
(C) does not give a lossless join, but is dependency preserving
(D) does not give a lossless join and is not dependency preserving
Answer: (A)
Explanation: Background :

LosslessJoin Decomposition:
Decomposition of R into R1 and R2 is a losslessjoin decomposition if at least one of the following functional dependencies are in F+ (Closure of functional dependencies)R1 ∩ R2 → R1 OR R1 ∩ R2 → R2
 Dependency Preserving Decomposition:
Decomposition of R into R1 and R2 is a dependency preserving decomposition if closure of functional dependencies after decomposition is same as closure of of FDs before decomposition.
A simple way is to just check whether we can derive all the original FDs from the FDs present after decomposition.
Question :
Let R (A, B, C, D) be a relational schema with the following functional dependencies:
A > B, B > C, C > D and D > B. The decomposition of R into (A, B), (B, C), (B, D)
Note that A, B, C and D are all key attributes. We can derive all attributes from every attribute.
Since Intersection of all relations is B and B derives all other attributes, relation is lossless.
The relation is dependency preserving as well as all functional dependencies are preserved directly or indirectly. Note that C > D is also preserved with following two C > B and B > D.
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