Consider the relation $X(P, Q, R, S, T, U)$ with the following set of functional dependencies $$F = \{ \{P, R\} \rightarrow \{S, T\}, \{P, S, U\} \rightarrow \{Q, R\} \}$$ Which of the following is the trivial functional dependency in $F^+$, where $F^+$ is closure of $F$ ?

A functional dependency $Y \rightarrow Z$ is defined as trivial if and only if the right-hand side is a subset of the left-hand side ($Z \subseteq Y$).

Let us check each option:

• (A) $\{P, R\} \rightarrow \{S, T\}$: Here, $\{S, T\}

ot\subseteq \{P, R\}$, so it is non-trivial.

• (B) $\{P, R\} \rightarrow \{R, T\}$: Here, $\{R, T\}

ot\subseteq \{P, R\}$(since$T \notin \{P, R\}$), so it is semi-trivial / non-trivial.

• (C) $\{P, S\} \rightarrow \{S\}$: Here, $\{S\} \subseteq \{P, S\}$, which satisfies the exact definition of a trivial functional dependency.

• (D) $\{P, S, U\} \rightarrow \{Q\}$: Here, $\{Q\}

ot\subseteq \{P, S, U\}$, so it is non-trivial.

Therefore, option (C) is the trivial functional dependency in $F^+$.

Hence, the correct option is (C).