The GATE Grind

GATE 2026 DA – Question 17

Database Management and Warehousing · Integrity constraints and normal forms · 1 mark · Multiple choice

Let R(A,B,C,D,E) be a relational schema with functional dependency set $F = \{A \to BC,\ CD \to E,\ E \to A\}$.

Which of the following statements is correct?

  1. AD, ED and CD are the only candidate keys of R.
  2. AD and ED are the only candidate keys of R.
  3. A, E and CD are the only candidate keys of R.
  4. A and CD are the only candidate keys of R.

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Correct answer: (A) AD, ED and CD are the only candidate keys of R.

Explanation

D does not occur on the right of any dependency, so every key contains D. The closure of $AD$ is $A, D, B, C$ and then $CD \to E$ gives $E$, so $AD$ determines everything. $ED$: $E \to A$ gives $A$, and then $A \to BC$, so $ED$ is also a key. $CD$: $CD \to E$, $E \to A$, $A \to BC$, so $CD$ is a key too. None of them can be shortened, since $D$ alone has closure $\{D\}$ and none of $A$, $E$, $C$ alone contains $D$. So the candidate keys are exactly $AD$, $ED$ and $CD$.