Question

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

• A. \( (X, Y) \rightarrow (Z, W) \) implies \( X \rightarrow (Z, W) \)
• B. \( (X, Y) \rightarrow (Z, W) \) implies \( (X, Y) \rightarrow Z \)
• C. \( ((X, Y) \rightarrow Z \text{ and } W \rightarrow Y) \) implies \( (X, W) \rightarrow Z \)
• D. \( (X \rightarrow Y \text{ and } Y \rightarrow Z) \) implies \( X \rightarrow Z \)

Hint:

Functional dependency follows reflexivity, augmentation, and transitivity. Carefully apply these properties to analyze dependencies.

Answer 1

The Correct Option is B

Step 1: Analyze each functional dependency statement. Option (1): \( (X, Y) \rightarrow (Z, W) \) does not necessarily imply \( X \rightarrow (Z, W) \). For example, if \( X \) does not uniquely determine \( Z \) or \( W \) without \( Y \), this fails. Hence, this is FALSE. Option (2): \( (X, Y) \rightarrow (Z, W) \) implies \( (X, Y) \rightarrow Z \), as \( Z \) is part of the attributes determined by \( (X, Y) \). This is TRUE. Option (3): From the given dependencies \( (X, Y) \rightarrow Z \) and \( W \rightarrow Y \), we can infer \( (X, W) \rightarrow Z \) by substitution. This is TRUE. Option (4): The transitivity property of functional dependency ensures \( X \rightarrow Y \text{ and } Y \rightarrow Z \) implies \( X \rightarrow Z \). This is TRUE. Final Answer: \[ \boxed{(2), (3), (4)} \]