GATE CS 2015 Set 3 Q.30 (CS Q.20): Functional Dependencies and Normalization

Consider the relation $X(P, Q, R, S, T, U)$ with the following set of functional dependencies

F = \{\{P, R\} \to \{S, T\}, \; \{P, S, U\} \to \{Q, R\}\}

Which of the following is the trivial functional dependency in $F^+$ , where $F^+$ is closure of $F$ ?

{P, R} -> {S, T}

{P, R} -> {R, T}

{P, S} -> {S}

{P, S, U} -> {Q}

Official answer and solution

Solution

Recall the formal definition of a trivial functional dependency.

By definition, a functional dependency $A \to B$ is trivial if and only if the right-hand side is a subset of the left-hand side:

B \subseteq A

Evaluate the choices:

Option A: $\{P, R\} \to \{S, T\}$ . Here $\{S, T\} \not\subseteq \{P, R\}$ , so it is non-trivial.
Option B: $\{P, R\} \to \{R, T\}$ . Here $\{R, T\} \not\subseteq \{P, R\}$ (since $T \notin \{P, R\}$ ), so it is semi-trivial / non-trivial.
Option C: $\{P, S\} \to \{S\}$ . Here $\{S\} \subseteq \{P, S\}$ . Thus, $\{P, S\} \to \{S\}$ is a trivial functional dependency.
Option D: $\{P, S, U\} \to \{Q\}$ . Here $\{Q\} \not\subseteq \{P, S, U\}$ , so it is non-trivial.

Therefore, Option C is the trivial functional dependency.

Final answer

Common trap

Confusing functional dependencies given in F with trivial dependencies, which hold universally by reflexivity regardless of F.

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