# GATE | GATE-CS-2001 | Question 48

• Difficulty Level : Medium
• Last Updated : 28 Jun, 2021

R(A,B,C,D) is a relation. Which of the following does not have a lossless join, dependency preserving BCNF decomposition?
(A) A->B, B->CD
(B) A->B, B->C, C->D
(D) A ->BCD

Explanation: Background :

• Lossless-Join Decomposition:
Decomposition of R into R1 and R2 is a lossless-join 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 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 :
We know that for lossless decomposition common attribute should be candidate key in one of the relation.

A) A->B, B->CD

R1(AB) and R2(BCD)

B is the key of second and hence decomposition is lossless.

B) A->B, B->C, C->D

R1(AB) , R2(BC), R3(CD)

B is the key of second and C is the key of third, hence lossless.

R1(ABC), R2(CD)

C is key of second, but C->A violates BCNF condition in ABC as C is not a key. We cannot decompose ABC further as AB->C dependency would be lost.

D) A ->BCD