GATE CS 2026 Set 1 Q20 — Normal Forms

We can evaluate the correctness of each option using Armstrong's axioms for functional dependencies:

• Option A: False. If $P \rightarrow R$ holds, then by augmentation, $PQ \rightarrow R$ also holds. However, this does not imply $Q \rightarrow R$. For example, in the relation instance with tuples $(P:1, Q:1, R:5)$ and $(P:2, Q:1, R:6)$, we have $P \rightarrow R$ and $PQ \rightarrow R$ holding, but $Q \rightarrow R$ is violated.
• Option B: False. A composite key $PQ \rightarrow R$ does not imply that either of the individual attributes determines $R$. For example, if $P$ is First Name, $Q$ is Last Name, and $R$ is Roll Number, $PQ \rightarrow R$ can hold without $P \rightarrow R$ or $Q \rightarrow R$ holding.
• Option C: True.
  • From $P \rightarrow R$, by augmenting with $Q$, we get $PQ \rightarrow RQ$.
  • From $Q \rightarrow S$, by augmenting with $R$, we get $RQ \rightarrow RS$.
  • By transitivity, $PQ \rightarrow RQ$ and $RQ \rightarrow RS$ imply $PQ \rightarrow RS$, which is equivalent to $\{P, Q\} \rightarrow \{R, S\}$.
• Option D: True. This is the standard augmentation rule. If $P \rightarrow R$, then adding any attribute $Q$ to the left-hand side still determines $R$, i.e., $PQ \rightarrow R$.