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Let 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?
  • a)
    II only
  • b)
    Both I and II
  • c)
    Neither I nor II
  • d)
    I only
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Let the set of functional dependencies F = {QR → S, R → P, S...

' QR ' is key
'S → Q ' violates
BCNF since
S is not superkey
So Z is not in BCNF
Since no depending is lost when we combine Y and Z.
So it is dependency preserving
Also Y ∩ Z= R
R is key in Y(P, R) So Decomposition is lossless.
So only II statement is correct.
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Most Upvoted Answer
Let the set of functional dependencies F = {QR → S, R → P, S...
Decomposition of X into Y and Z:
- The original relation schema X=(PQRS) is not in BCNF due to the functional dependency QR → S.
- After decomposition into Y=(PR) and Z=(QRS), the problematic dependency QR → S is no longer present in either Y or Z.

Statement I: Both Y and Z are in BCNF
- Y=(PR) is in BCNF because P → R holds and there are no non-trivial dependencies involving P.
- Z=(QRS) is in BCNF because both S → Q and R → S are trivial dependencies.

Statement II: Decomposition is dependency preserving and lossless
- The decomposition of X into Y and Z is dependency preserving because the original functional dependencies hold in Y and Z.
- The decomposition is lossless because a natural join of Y and Z will result in the original relation X, without losing any information.
Therefore, the correct answer is option 'A) II only' as Statement II is correct while Statement I is not entirely accurate.
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Let 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 BCNFII. Decomposition of X into Y and Z is dependency preserving and losslessWhich of the above statements is/are correct?a)II onlyb)Both I and IIc)Neither I nor IId)I onlyCorrect answer is option 'A'. Can you explain this answer?
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