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Functional Dependencies, Keys and Normalization PYQs for GATE CS

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Question 1

2026 Slot Set2 PYQ

In the context of schema normalization in relational DBMS, consider a set F of functional dependencies. The set of all functional dependencies implied by F is called the closure of F. To compute the closure of F, Armstrong’s Axioms can be applied. Consider X, Y, and Z as sets of attributes over a relational schema. The three rules of Armstrong’s Axioms are described as follows. Reflexivity: If Y \subseteq X, then X \to Y Augmentation: If X \to Y, then X Z \to Y Z for any Z Transitivity: If X \to Y and Y \to Z, then X \to Z The additional rule of Union is defined as follows. Union: If X \to Y and X \to Z, then X \to Y Z It can be proved that the additional rule of Union is also implied by the three rules of Armstrong’s Axioms. Listed below are four combinations of these three rules. Which one of these combinations is both necessary and sufficient for the proof ?

Question 2

2026 Slot Set1 PYQ

Consider a relational database schema with a relation R (A, B, C, D) . If {A, B} and {A, C} are the only two candidate keys of the relation R, then the number of superkeys of relation R is ______. (answer in integer)

Question 3

2026 Slot Set1 PYQ

Let P, Q, R and S be the attributes of a relation in a relational schema. Let X ⟶ Y indicate functional dependency in the context of a relational database, where X, Y \subseteq {P, Q, R, S} . Which of the following options is/are always true?

Question 4

2026 Slot Set1 PYQ

In the context of relational database normalization, which of the following statements is/are true?

Question 5

2025 Slot Set2 PYQ

Consider the following relational schema along with all the functional dependencies that hold on them. R 1 (A, B, C, D, E) : {D \to E, E A \to B, E B \to C} R 2 (A, B, C, D) : {A \to D, A \to B, C \to A} Which of the following statement(s) is/are TRUE?

Question 6

2025 Slot Set1 PYQ

Consider a relational schema t e am (nam e, c i t y, o w n er), with functional dependencies {nam e \to c i t y, nam e \to o w n er} . The relation t e am is decomposed into two relations, t 1 (nam e, c i t y) and t 2 (nam e, o w n er) . Which of the following statement(s) is/are TRUE ?

Question 7

2024 Slot Set2 PYQ

A functional dependency F : X \to Y is termed as a useful functional dependency if and only if it satisfies all the following three conditions: • X is not the empty set. • Y is not the empty set. • Intersection of X and Y is the empty set. For a relation R with 4 attributes, the total number of possible useful functional dependencies is __________

Question 8

2024 Slot Set1 PYQ

The symbol \to indicates functional dependency in the context of a relational database. Which of the following options is/are TRUE?

Question 9

2024 Slot Set1 PYQ

Which of the following statements about a relation R in first normal form (1NF) is/are TRUE ?

Question 10

2022 PYQ

Consider a relation R (A, B, C, D, E) with the following three functional dependencies. A B \to C; B C \to D; C \to E; The number of superkeys in the relation R is ____________.

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Functional Dependencies, Keys and Normalization PYQs for GATE CS

Solve 13+ Functional Dependencies, Keys and Normalization previous year questions for GATE CS with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Functional Dependencies, Keys and Normalization

Databases › Functional Dependencies, Keys and Normalization

Chapter Journey

Three topics that form the backbone of relational database theory.

Functional Dependency Inference and Armstrong's Axioms

The logical rules for deriving new dependencies from a given set. You are here.

5 PYQs

Candidate Keys, Closures and Superkeys

Using attribute closures to find all candidate keys and count superkeys.

2 PYQs

Normal Forms, Lossless Decomposition and Dependency Preservation

1NF through BCNF, checking whether a decomposition loses data or dependencies.

6 PYQs

What Is a Functional Dependency?

A **Functional Dependency (FD)** is a constraint between two sets of attributes in a relation.

Definition

X \to Y

holds on a relation

R

if, for every valid instance of

R

, any two tuples that agree on all attributes of

X

also agree on all attributes of

Y

In plain English: If you know the value of

X

, you can uniquely determine the value of

Y

Determinant (

X

)

The attribute(s) you already know

Dependent (

Y

)

The attribute(s) you can deduce

Key point: An FD is a statement about all possible valid instances of the relation, not just the rows you currently see in a table.

Functional Dependencies, Keys and Normalization: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Databases · 2026_Set2 MCQ

In the context of schema normalization in relational DBMS, consider a set F of functional dependencies. The set of all functional dependencies implied by F is called the closure of F. To compute the closure of F, Armstrong’s Axioms can be applied. Consider

X

Y

, and

Z

as sets of attributes over a relational schema. The three rules of Armstrong’s Axioms are described as follows.

Reflexivity: If

Y \subseteq X

, then

X \to Y

Augmentation: If

X \to Y

, then

X Z \to Y Z

for any Z
Transitivity: If

X \to Y

and

Y \to Z

, then

X \to Z

The additional rule of Union is defined as follows.

Union: If

X \to Y

and

X \to Z

, then

X \to Y Z

It can be proved that the additional rule of Union is also implied by the three rules of Armstrong’s Axioms. Listed below are four combinations of these three rules. Which one of these combinations is both necessary and sufficient for the proof ?

A.
Reflexivity, Augmentation, and Transitivity
B.
Reflexivity and Augmentation
C.
Transitivity
D.
Augmentation and Transitivity

Question 2 · Databases · 2026_Set1 NAT

Consider a relational database schema with a relation $R (A, B, C, D)$ . If ${A, B}$ and ${A, C}$ are the only two candidate keys of the relation $R$ , then the number of superkeys of relation $R$ is ______. (answer in integer)

Question 3 · Databases · 2026_Set1 MSQ

Let

P

Q

R

and

S

be the attributes of a relation in a relational schema. Let

X ⟶ Y

indicate functional dependency in the context of a relational database, where

X, Y \subseteq {P, Q, R, S}

.

Which of the following options is/are always true?

A.
If ( ${P, Q} ⟶ {R}$ and ${P} ⟶ {R}$ ), then ${Q} ⟶ {R}$
B.
If ${P, Q} ⟶ {R}$ , then ( ${P} ⟶ {R}$ or ${Q} ⟶ {R}$ )
C.
If ( ${P} ⟶ {R}$ and ${Q} ⟶ {S}$ ), then ${P, Q} ⟶ {R, S}$
D.
If ${P} ⟶ {R}$ , then ${P, Q} ⟶ {R}$

Question 4 · Databases · 2026_Set1 MSQ

In the context of relational database normalization, which of the following statements is/are true?

A.
It is always possible to obtain a dependency-preserving 3NF decomposition of a relation
B.
It is always possible to obtain a dependency-preserving 1NF decomposition of a relation
C.
It is not always possible to obtain a dependency-preserving BCNF decomposition of a relation
D.
It is not always possible to obtain a dependency-preserving 2NF decomposition of a relation

Question 5 · Databases · 2025_Set2 MSQ

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

R 1 (A, B, C, D, E) : {D \to E, E A \to B, E B \to C}

R 2 (A, B, C, D) : {A \to D, A \to B, C \to A}

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

A.
$R 1$ is in 3NF
B.
$R 2$ is in 3NF
C.
$R 1$ is NOT in 3NF
D.
$R 2$ is NOT in 3NF

Question 6 · Databases · 2025_Set1 MSQ

Consider a relational schema

t e am (nam e, c i t y, o w n er)

, with functional dependencies

{nam e \to c i t y, nam e \to o w n er}

.

The relation

t e am

is decomposed into two relations,

t 1 (nam e, c i t y)

and

t 2 (nam e, o w n er)

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

A.
The relation $t e am$ is NOT in BCNF.
B.
The relations $t 1$ and $t 2$ are in BCNF.
C.
The decomposition constitutes a lossless join.
D.
The relation $t e am$ is NOT in 3NF.

Question 7 · Databases · 2024_Set2 NAT

A functional dependency

F : X \to Y

is termed as a useful functional dependency if and only if it satisfies all the following three conditions:

•

X

is not the empty set.
•

Y

is not the empty set.
• Intersection of

X

and

Y

is the empty set.

For a relation

R

with 4 attributes, the total number of possible useful functional dependencies is __________

Question 8 · Databases · 2024_Set1 MSQ

The symbol $\to$ indicates functional dependency in the context of a relational database. Which of the following options is/are TRUE?

A.
$(X, Y) \to (Z, W)$ implies $X \to (Z, W)$
B.
$(X, Y) \to (Z, W)$ implies $(X, Y) \to Z$
C.
$((X, Y) \to Z and W \to Y)$ implies $(X, W) \to Z$
D.
$(X \to Y and Y \to Z)$ implies $X \to Z$

Question 9 · Databases · 2024_Set1 MSQ

Which of the following statements about a relation $R$ in first normal form (1NF) is/are TRUE ?

A.
$R$ can have a multi-attribute key
B.
$R$ cannot have a foreign key
C.
$R$ cannot have a composite attribute
D.
$R$ cannot have more than one candidate key

Question 10 · Databases · 2022 NAT

Consider a relation $R (A, B, C, D, E)$ with the following three functional dependencies.

$A B \to C; B C \to D; C \to E;$

The number of superkeys in the relation $R$ is ____________.

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Unlock the whole course

Functional Dependencies, Keys and Normalization PYQs for GATE CS

Chapter Roadmap: Functional Dependencies, Keys and Normalization

Chapter Journey

What Is a Functional Dependency?

What Is a Functional Dependency?

Functional Dependencies, Keys and Normalization: Solved Questions with Step-by-Step Explanations (10 Problems)

More previous year questions (pyqs) in this unit