Let $P, Q, R$ and $S$ be the attributes of a relation in a relational schema. Let $X \rightarrow Y$ indicate functional dependency in the context of a relational database, where $X, Y \subseteq \{P, Q, R, S\}$. Which of the following options is/are always true?

We can verify each option using Armstrong's Axioms and standard inference rules for functional dependencies:

• Option (A): If $\{P, Q\} \rightarrow \{R\}$ and $\{P\} \rightarrow \{R\}$, then $\{Q\} \rightarrow \{R\}$. This is not always true. For example, in relation $R(P, Q, R)$ where $P$ is a primary key, $P \rightarrow R$ holds, but attribute $Q$ may be completely independent of $R$.

• Option (B): If $\{P, Q\} \rightarrow \{R\}$, then $(\{P\} \rightarrow \{R\} \text{ or } \{Q\} \rightarrow \{R\})$. This is not always true. A composite determinant does not imply that any proper subset of it determines the attribute (e.g., in a minimal superkey consisting of $\{P, Q\}$).

• Option (C): If $(\{P\} \rightarrow \{R\} \text{ and } \{Q\} \rightarrow \{S\})$, then $\{P, Q\} \rightarrow \{R, S\}$. This is always true (Composition Rule):

  • By augmentation of $P \rightarrow R$ with $Q$: $\{P, Q\} \rightarrow \{R, Q\}$

  • By augmentation of $Q \rightarrow S$ with $R$: $\{R, Q\} \rightarrow \{R, S\}$

  • By transitivity: $\{P, Q\} \rightarrow \{R, S\}$.

• Option (D): If $\{P\} \rightarrow \{R\}$, then $\{P, Q\} \rightarrow \{R\}$. This is always true (Augmentation / Generalization Rule):

  • Augmenting $P \rightarrow R$ with $Q$ yields $\{P, Q\} \rightarrow \{P, R\}$ or $\{P, Q\} \rightarrow \{R, Q\}$.

  • By decomposition, $\{P, Q\} \rightarrow \{R\}$.

Thus, statements C and D are always true.