Project ForgeLet the set of functional dependencies F = {QR → S, R → P, S → Q} hold on a relation schema X = (PQRS). X is not in BCNF. Suppose X is decomposed into two schemas Y and Z, where Y = (PR) and Z = (QRS). Consider the two statements given below:
I. Both Y and Z are in BCNF. II. Decomposition of X into Y and Z is dependency preserving and lossless.
Which of the above statements is/are correct?
- ABoth I and II
- BI only
- II only
- DNeither I nor II
Explanation
Y=(PR) has only the FD R→P (from R→P restricted to PR), so R is trivially a key and Y IS in BCNF. Z=(QRS) has FDs QR→S and S→Q (projected from F); candidate keys of Z are QR and SR, but the FD S→Q has S alone as determinant, which is NOT a superkey of Z — this violates BCNF, so Z is NOT in BCNF. Statement I is therefore FALSE. For statement II: the decomposition is LOSSLESS because Y∩Z={R} and R→P holds (R is a determinant covering Y's other attribute) — satisfying the lossless-join test. It is also DEPENDENCY-PRESERVING because the union of the projected FDs, {R→P} ∪ {QR→S, S→Q}, exactly reproduces the original F = {QR→S, R→P, S→Q} with nothing lost. So statement II is TRUE — the answer is 'II only'.
Source: GATE CS 2019 (Set-2) (2019)
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