GATE CS 2019 Q32 — Normal Forms

To determine the correctness of the statements, let us analyze the decomposition of $X(PQRS)$ into $Y(PR)$ and $Z(QRS)$ under the functional dependencies $F = \{QR \rightarrow S, \: R \rightarrow P, \: S \rightarrow Q \}$.

Statement I: Both $Y$ and $Z$ are in BCNF

• For $Y(PR)$: The projected dependency is $R \rightarrow P$. Since $R$ is the candidate key of $Y$, $Y$ is in BCNF.
• For $Z(QRS)$: The projected dependencies are $QR \rightarrow S$ and $S \rightarrow Q$. The candidate keys of $Z$ are $QR$ and $SR$. For the dependency $S \rightarrow Q$, the left-hand side $S$ is not a superkey of $Z$. Thus, $Z$ is not in BCNF.

Therefore, Statement I is False.

Statement II: Decomposition is dependency preserving and lossless

• Lossless Join: The intersection of the two schemas is $Y \cap Z = \{PR\} \cap \{QRS\} = \{R\}$. Since $R \rightarrow P$ holds on $Y$, $R$ is a superkey of $Y$. Thus, the decomposition is lossless.
• Dependency Preserving:
  • $R \rightarrow P$ is preserved in $Y(PR)$.
  • $QR \rightarrow S$ and $S \rightarrow Q$ are preserved in $Z(QRS)$.
  Since all original FDs are preserved, the decomposition is dependency preserving.

Therefore, Statement II is True.