GATE CS 2024 Set 1 Q34 — Normal Forms

We analyze each option using Armstrong's axioms for functional dependencies:

• Option A: $(X, Y) \rightarrow (Z, W)$ implies $X \rightarrow (Z, W)$ is False. Removing attributes from the left-hand side (left-reduction) is invalid. For example, $\{\text{Roll\_No}, \text{Course}\} \rightarrow \text{Grade}$ does not imply $\text{Roll\_No} \rightarrow \text{Grade}$.
• Option B: $(X, Y) \rightarrow (Z, W)$ implies $(X, Y) \rightarrow Z$ is True. By the Decomposition Rule, if a set of attributes determines a set, it also determines any of its subsets.
• Option C: $((X, Y) \rightarrow Z$ and $W \rightarrow Y)$ implies $(X, W) \rightarrow Z$ is True. By augmenting $W \rightarrow Y$ with $X$, we get $(X, W) \rightarrow (X, Y)$. Since $(X, Y) \rightarrow Z$, by transitivity, we obtain $(X, W) \rightarrow Z$.
• Option D: $(X \rightarrow Y$ and $Y \rightarrow Z)$ implies $X \rightarrow Z$ is True. This is the standard Transitivity Rule.