GATE CS 2024 Set 1 — Question 44

We evaluate each option based on Armstrong's Axioms and derived rules for functional dependencies:(A) $(X, Y) \rightarrow (Z, W)$ implies $X \rightarrow (Z, W)$
This is FALSE. We cannot simply remove attributes from the determinant (left side) of a functional dependency. For example, if $X$ is StudentName and $Y$ is CourseID, together they might determine Grade ($Z$), but StudentName alone does not determine Grade.(B) $(X, Y) \rightarrow (Z, W)$ implies $(X, Y) \rightarrow Z$
This is TRUE by the Decomposition Rule. If a set of attributes determines a set of attributes, it determines any subset of those attributes.(C) $((X, Y) \rightarrow Z \text{ and } W \rightarrow Y)$ implies $(X, W) \rightarrow Z$
This is TRUE by the Pseudotransitivity Rule.
Proof:(D) $(X \rightarrow Y \text{ and } Y \rightarrow Z)$ implies X → Z
This is TRUE by the Transitivity Rule (one of Armstrong's Axioms).Therefore, options (B), (C), and (D) are correct.