GATE CS 2024 Set 1 Q.44: Functional Dependencies and Normalization

The symbol $\rightarrow$ indicates functional dependency in the context of a relational database. Which of the following options is/are TRUE?

$(X, Y) \rightarrow (Z, W)$ implies $X \rightarrow (Z, W)$

$(X, Y) \rightarrow (Z, W)$ implies $(X, Y) \rightarrow Z$

$((X, Y) \rightarrow Z \text{ and } W \rightarrow Y)$ implies $(X, W) \rightarrow Z$

$(X \rightarrow Y \text{ and } Y \rightarrow Z)$ implies $X \rightarrow Z$

Official answer and solution

Solution

Check each inference rule using Armstrong's axioms (decomposition, pseudotransitivity, and transitivity).

(A) is FALSE: Knowing $(X, Y) \rightarrow (Z, W)$ does not imply that $X$ alone functionally determines $(Z, W)$ .
(B) is TRUE: By the decomposition rule, $(X, Y) \rightarrow (Z, W) \implies (X, Y) \rightarrow Z$ and $(X, Y) \rightarrow W$ .
(C) is TRUE: By augmentation, $W \rightarrow Y \implies (X, W) \rightarrow (X, Y)$ ; then by transitivity with $(X, Y) \rightarrow Z$ , we get $(X, W) \rightarrow Z$ (pseudotransitivity).
(D) is TRUE: This is Armstrong's transitivity axiom.

Final answer

Options (B), (C), (D)

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

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