Question

Given the relation: \[ R = \{A, B, C, D, E, F, G\} \] With the following functional dependencies (FDs): \[ A \rightarrow BCEF, \quad E \rightarrow DG, \quad BC \rightarrow A \] Which of the following statements is correct?

• A. A is the only candidate key of \( R \)
• B. A, B, C, D, E are candidate keys of \( R \)
• C. \( R \) is not in BCNF
• D. A, B, C are the candidate keys of \( R \)

Hint:

A candidate key is the minimal set of attributes that uniquely determine all attributes of the relation.

Answer 1

The Correct Option is D

To determine the candidate keys, we analyze the closure of attributes using the given functional dependencies: 1. Find the closure of (A): \[ A^+ = \{A\} \cup \{B, C, E, F\} = \{A, B, C, E, F\} \] Since it does not include all attributes of \( R \), \( A \) alone is not a candidate key. 2. Find the closure of (BC): \[ BC^+ = \{B, C\} \cup \{A\} \cup \{B, C, E, F\} = \{A, B, C, E, F\} \] Again, it does not cover all attributes. 3. Find the closure of (A, B, C): \[ (A, B, C)^+ = \{A, B, C, E, F\} \cup \{D, G\} = \{A, B, C, D, E, F, G\} \] Since this set includes all attributes of \( R \), \( A, B, C \) together form a candidate key. Conclusion: The candidate keys of \( R \) are A, B, and C, making the correct answer (4) A, B, C are the candidate keys of \( R \).