🔗 FUNCTIONAL DEPENDENCY

🔗 FUNCTIONAL DEPENDENCY

DBMS • Definition • Armstrong's Axioms • Examples • Step-by-Step

📘 Interactive DBMS Lesson

Click any button below to learn about Functional Dependency and Armstrong's Axioms.

Topics: Functional Dependency, Determinant, Dependent Attribute, Reflexivity, Augmentation, Transitivity, Union, Decomposition and step-by-step examples.

🔗 What is Functional Dependency?

A Functional Dependency (FD) describes a relationship between attributes of a database table.

If the value of attribute X uniquely determines the value of attribute Y, then we write:

X → Y

It is read as: "X functionally determines Y."

Simple idea:

If two records have the same value of X, they must have the same value of Y.

Example

Student_ID → Student_Name

If Student_ID is unique, it can determine the name of the student.

📊 Basic Example

Student_ID	Name	Course
101	Rahul	BCA
102	Anita	BCA
103	Rohan	BBA

1

Student_ID is unique for each student.

2

Student_ID 101 identifies Rahul.

3

Student_ID 102 identifies Anita.

4

Therefore Student_ID determines Student_Name.

Student_ID → Student_Name

Student_ID = Determinant
Student_Name = Dependent Attribute

🧠 Armstrong's Axioms

Armstrong's Axioms are inference rules used to derive new Functional Dependencies from existing dependencies.

Axiom	Formula	Simple Meaning
Reflexivity	X → Y	Y is already inside X
Augmentation	XZ → YZ	Add the same attribute
Transitivity	X → Z	Follow a dependency chain

Easy way to remember:

🔵 Reflexivity = Already contained
🟣 Augmentation = Add
🟢 Transitivity = Chain

1️⃣ Reflexivity

If Y is a subset of X, then:

X → Y

Example

{Student_ID, Name} → Student_ID

1

Start with {Student_ID, Name}.

2

Student_ID is already contained in the left side.

3

Therefore the complete set determines Student_ID.

{Student_ID, Name} → Student_ID

This follows the Reflexivity Axiom.

2️⃣ Augmentation

If:

X → Y

then adding the same attribute Z to both sides gives:

XZ → YZ

Example

Student_ID → Name

Add Course to both sides.

Student_ID, Course → Name, Course

1

Start with Student_ID → Name.

2

Choose Course as the additional attribute.

3

Add Course to the left side.

4

Add Course to the right side.

Student_ID, Course → Name, Course

3️⃣ Transitivity

If:

X → Y

Y → Z

then:

X → Z

Example

Student_ID → Department_ID

Department_ID → Department_Name

1

Student_ID determines Department_ID.

2

Department_ID determines Department_Name.

3

The dependency forms a chain.

4

Therefore Student_ID determines Department_Name.

Student_ID → Department_Name

Derived using Transitivity.

⚙️ Derived Rules

1. Union Rule

X → Y
X → Z

Therefore
X → YZ

Example

Student_ID → Name
Student_ID → Course

Therefore:
Student_ID → Name, Course

2. Decomposition Rule

X → YZ

Therefore:
X → Y
X → Z

3. Pseudotransitivity

X → Y
WY → Z

Therefore:
WX → Z

🧮 Step-by-Step FD Derivation

Given:

A → B
B → C
C → D

Find: A → D

1

Given A → B.

2

Given B → C.

3

Using Transitivity:

A → B and B → C

Therefore:
A → C

4

Given C → D.

5

Again use Transitivity:

A → C and C → D

Therefore:
A → D

🎯 FINAL ANSWER

A → D

Therefore A functionally determines D.

🎓 Student Database Example

Roll_No	Name	Course_ID	Course_Name
101	Rahul	C01	BCA
102	Anita	C01	BCA
103	Rohan	C02	BBA

Functional Dependencies

Roll_No → Name

Course_ID → Course_Name

1

Roll_No uniquely identifies a student.

2

Therefore Roll_No → Name.

3

Course_ID identifies a particular course.

4

Therefore Course_ID → Course_Name.

📋 Quick Summary

Concept	Rule	Meaning
Functional Dependency	X → Y	X determines Y
Reflexivity	X → Y	Y is contained in X
Augmentation	XZ → YZ	Add same attribute
Transitivity	X → Z	Use dependency chain
Union	X → YZ	Combine dependencies
Decomposition	X → Y, X → Z	Split dependency

❓ Short Questions & Answers

Q1. What is Functional Dependency?

Functional Dependency is a relationship where one attribute or group of attributes determines another attribute.

Example: Student_ID → Student_Name

Q2. What are Armstrong's Axioms?

The three basic Armstrong's Axioms are:

1. Reflexivity
2. Augmentation
3. Transitivity

Q3. What is Reflexivity?

If Y is a subset of X, then X → Y.

Q4. What is Augmentation?

If X → Y, then XZ → YZ. The same attribute is added to both sides.

Q5. What is Transitivity?

If X → Y and Y → Z, then X → Z.

Q6. In A → B, which is the determinant?

A is the determinant because it appears on the left side.

Q7. In A → B, which is the dependent attribute?

B is the dependent attribute because it appears on the right side.

B.K. PAUL

Bhairab Ganguly College

DBMS • Functional Dependency • Armstrong's Axioms

Interactive Learning Module