GATE CS 2015 Set 3 Q20 — Normal Forms

A functional dependency $A \rightarrow B$ is defined as trivial if the right-hand side attribute set is a subset of the left-hand side attribute set, i.e., $B \subseteq A$.

Let's evaluate each option based on this definition:

• Option A: $\{P, R\} \rightarrow \{S, T\}$. Here, $\{S, T\} \not\subseteq \{P, R\}$. This is a non-trivial dependency.
• Option B: $\{P, R\} \rightarrow \{R, T\}$. Here, $\{R, T\} \not\subseteq \{P, R\}$. This is a semi-trivial (non-trivial) dependency.
• Option C: $\{P, S\} \rightarrow \{S\}$. Here, $\{S\} \subseteq \{P, S\}$. This is a trivial dependency. By the reflexivity rule of Armstrong's axioms, trivial dependencies always hold and are guaranteed to be in $F^+$.
• Option D: $\{P, S, U\} \rightarrow \{Q\}$. Here, $\{Q\} \not\subseteq \{P, S, U\}$. This is a non-trivial dependency.

Therefore, the only trivial functional dependency is $\{P, S\} \rightarrow \{S\}$.