GATE DA 2025 | Question 57

MSQ · 2 marks

Consider a database relation R with attributes ABCDEFG, and having the following functional dependencies:

\[ \mathrm{A} \rightarrow \mathrm{BCEF} \qquad \mathrm{E} \rightarrow \mathrm{DG} \qquad \mathrm{BC} \rightarrow \mathrm{A} \]

Which of the following statements is/are correct?

  1. Option A: A is the only candidate key of R
  2. Option B: A, BC are the candidate keys of R (correct)
  3. Option C: A, BC, E are the candidate keys of R
  4. Option D: Relation R is not in Boyce-Codd Normal Form (BCNF) (correct)

Answer

(B), (D)

Explanation

Find the keys with attribute closures.

  • \(\mathrm{A}^{+}\): A \(\rightarrow\) BCEF, then E \(\rightarrow\) DG, so \(\mathrm{A}^{+} = \mathrm{ABCDEFG}\) and A is a key.
  • \(\mathrm{BC}^{+}\): BC \(\rightarrow\) A, and everything follows as above. \(\mathrm{B}^{+} = \mathrm{B}\) and \(\mathrm{C}^{+} = \mathrm{C}\), so BC is a candidate key.
  • \(\mathrm{E}^{+} = \mathrm{DEG}\), so E is not a key.

A appears on a right side only in BC \(\rightarrow\) A, and B and C are determined only by A, so every key contains A or both B and C. The candidate keys are exactly A and BC: B holds, while A and C fail.

In E \(\rightarrow\) DG the left side E is not a superkey, so R is not in BCNF and D holds.

Answer: B, D