Question

Consider a database relation \( R \) with attributes \( A, B, C, D, E, F, G \), and having the following functional dependencies: \[ A \rightarrow BCEF \quad E \rightarrow DG \quad BC \rightarrow A \] Which of the following statements is/are correct?

• A. \( A \) is the only candidate key of \( R \)
• B. \( A, BC \) are the candidate keys of \( R \)
• C. \( A, BC, E \) are the candidate keys of \( R \)
• D. Relation \( R \) is not in Boyce-Codd Normal Form (BCNF)

Hint:

When solving database normalization problems, always check if the given functional dependencies can uniquely determine all attributes and analyze candidate keys carefully. For BCNF, ensure the determinant is always a superkey.

Answer 1

The Correct Option is B, D

Step 1: Analyzing the given functional dependencies.
From \( A \rightarrow BCEF \), we know that if we know \( A \), we can determine \( B, C, E, F \).
From \( E \rightarrow DG \), knowing \( E \) gives us \( D \) and \( G \).
From \( BC \rightarrow A \), knowing \( B \) and \( C \) gives us \( A \).

Thus, the combination of \( A \) and \( BC \) will allow us to determine all the attributes in the relation, making \( A \) and \( BC \) valid candidate keys.

Step 2: Checking other options.
Option (A): \( A \) alone is not the only candidate key because \( BC \) also forms a candidate key.
Option (C): \( E \) can be considered a candidate key along with \( A \) and \( BC \), because knowing \( E \) gives us \( D \) and \( G \), and with \( BC \), we can determine \( A \). This leads to all the attributes in the relation.
Option (D): The relation is in BCNF because the determinant in all functional dependencies is either a superkey or part of a superkey.