GATE CS 2025 Set 2 Q.46: Functional Dependencies and Normalization

Consider the following relational schema along with all the functional dependencies that hold on them.

$R_1(A, B, C, D, E): \{ D \rightarrow E, \; EA \rightarrow B, \; EB \rightarrow C \}$
$R_2(A, B, C, D): \{ A \rightarrow D, \; A \rightarrow B, \; C \rightarrow A \}$

Which of the following statement(s) is/are TRUE?

$R_1$ is in 3NF

$R_2$ is in 3NF

$R_1$ is NOT in 3NF

$R_2$ is NOT in 3NF

Official answer and solution

Official answer: C and D
From: Official GATE 2025 answer key

Solution

Find all candidate keys and prime attributes of $R_1$ and $R_2$ , then check whether every non-trivial FD $X \rightarrow Y$ has either $X$ a superkey or $Y$ prime.

Analysis of $R_1(A, B, C, D, E)$ :
- Attributes $A$ and $D$ never appear on the RHS of any FD, so every candidate key must contain $AD$ .
- Since $(AD)^+ = \{A, D, E, B, C\}$ , $AD$ is the unique candidate key of $R_1$ .
- Prime attributes: $\{A, D\}$ ; Non-prime attributes: $\{B, C, E\}$ .
- In the FD $D \rightarrow E$ , $D$ is not a superkey and $E$ is non-prime (similarly for $EA \rightarrow B$ and $EB \rightarrow C$ ). Thus $R_1$ is NOT in 3NF (C).
Analysis of $R_2(A, B, C, D)$ :
- Attribute $C$ never appears on the RHS of any FD, and $C^+ = \{C, A, B, D\}$ , so $C$ is the unique candidate key of $R_2$ .
- Prime attribute: $\{C\}$ ; Non-prime attributes: $\{A, B, D\}$ .
- In the FDs $A \rightarrow D$ and $A \rightarrow B$ , $A$ is not a superkey and $B, D$ are non-prime. Thus $R_2$ is NOT in 3NF (D).

Final answer

Options (C), (D)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 2 (CS2)
Question: 46 of 65
Section: Computer Science and Information Technology

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