A relation R(P,Q,R,S) has functional dependencies {P→QR,QS→R}. If R is…
A relation R(P,Q,R,S) has functional dependencies {P→QR,QS→R}. If R is decomposed into R1(P,Q,R) and R2(Q,R,S), How many statements are correct?
S1: Lossy Decomposition
S2: Dependency not preserving decomposition
Answer: 1 — Answer: Exactly one statement is correct. Check for lossless decomposition: Common attributes: R1 ∩ R2 = {Q, R}. Compute closure: {Q,R}+ under the given FDs…
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Correct answer: 1
Answer: Exactly one statement is correct.
Check for lossless decomposition:
Common attributes: R1 ∩ R2 = {Q, R}.
Compute closure: {Q,R}+ under the given FDs (P → Q R, Q S → R) is {Q, R}.
Since {Q,R}+ does not include P or S, {Q,R} does not determine the remaining attributes of R1 (which is P) or R2 (which is S). For a decomposition to be lossless the intersection must functionally determine one side; that is not the case here.
Therefore the decomposition is lossy. The statement asserting a lossy decomposition is correct.
Check for dependency preservation:
The FD P → Q R is entirely contained in R1(P, Q, R), so it is preserved in R1.
The FD Q S → R is entirely contained in R2(Q, R, S), so it is preserved in R2.
Hence both FDs are preserved by the decomposition; the statement claiming the decomposition is not dependency preserving is incorrect.
Conclusion: One statement is correct — the decomposition is lossy; dependency preservation holds.