GATE CS 2026 Set 1 Q.30: Functional Dependencies and Normalization

Let $P, Q, R$ and $S$ be the attributes of a relation in a relational schema. Let $X \longrightarrow Y$ 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?

If ( $\{P, Q\} \longrightarrow \{R\}$ and $\{P\} \longrightarrow \{R\}$ ), then $\{Q\} \longrightarrow \{R\}$

If $\{P, Q\} \longrightarrow \{R\}$ , then ( $\{P\} \longrightarrow \{R\}$ or $\{Q\} \longrightarrow \{R\}$ )

If ( $\{P\} \longrightarrow \{R\}$ and $\{Q\} \longrightarrow \{S\}$ ), then $\{P, Q\} \longrightarrow \{R, S\}$

If $\{P\} \longrightarrow \{R\}$ , then $\{P, Q\} \longrightarrow \{R\}$

Official answer and solution

Official answer: C and D
From: Official GATE 2026 answer key

Solution

Apply Armstrong's axioms (augmentation and union) to test each implication.

Option (A) is false: If $P \rightarrow R$ , then $\{P, Q\} \rightarrow R$ holds automatically by augmentation regardless of $Q$ , so $Q$ need not determine $R$ .
Option (B) is false: A composite determinant $\{P, Q\} \rightarrow R$ cannot be split into $\{P\} \rightarrow R$ or $\{Q\} \rightarrow R$ .
Option (C) is true: By augmentation, $P \rightarrow R \implies PQ \rightarrow R$ and $Q \rightarrow S \implies PQ \rightarrow S$ ; by union, $PQ \rightarrow RS$ .
Option (D) is true: By augmentation of $P \rightarrow R$ with $Q$ , we get $\{P, Q\} \rightarrow \{R\}$ .

Final answer

Options (C), (D)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2026 Computer Science and Information Technology - Session 1 (CS1)
Question: 30 of 65
Section: Computer Science and Information Technology

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