Consider the relation \(R(P,Q,S,T,X,Y,Z,W)\) with the following functional dependencies:
\[ PQ \rightarrow X; P \rightarrow YX; Q \rightarrow Y; Y \rightarrow ZW \] Consider the decomposition of the relation \(R\) into the constituent relations according to the following two decomposition schemes. \[ \begin{aligned} D_1 &: R = [(P,Q,S,T);\; (P,T,X);\; (Q,Y);\; (Y,Z,W)] \\ D_2 &: R = [(P,Q,S);\; (T,X);\; (Q,Y);\; (Y,Z,W)] \end{aligned} \] Which one of the following options is correct?
Step 1: Determine the candidate key of relation R.
Using the given functional dependencies:
P → YX and Y → ZW
From these, we obtain:
P → X, Y, Z, W
Also, Q → Y, and since Y → ZW, this further determines Z and W.
The attributes S and T are not functionally determined by any dependency, so they must be included explicitly.
Hence, the attribute set (P, Q, S, T) functionally determines all attributes of R and forms a candidate key.
Step 2: Analyze decomposition D1.
One of the relations in D1 contains the attribute set (P, Q, S, T),
which is a candidate key of R.
By the lossless-join criterion, if at least one decomposed relation contains a key of the original relation, the decomposition is lossless.
Therefore, D1 is a lossless decomposition.
Step 3: Analyze decomposition D2.
In D2, none of the resulting relations includes a candidate key of R.
As a result, joining these relations can introduce spurious tuples, implying that the decomposition is lossy.
Final Conclusion:
Decomposition D1 is lossless, whereas D2 is lossy.
Final Answer: (A)