Rules for Derivation

In Database Management System, Armstrong’s Axioms are a set of rules used to:

Derive new functional dependencies from existing ones

These rules are the foundation for:

• Checking equivalence
• Finding closure
• Database normalization

The Three Main Axioms

1. Reflexivity Rule

Concept

If Y is a subset of X, then:

Example

Because A is part of (A, B)

Real Life Example

If you know:
“Full student profile”

You automatically know:
“Student name”

(No extra effort needed)

2. Augmentation Rule

If:

Then:

You can add same attribute to both sides

Example

Real Life Example

If:
Roll_No → Name

Then:
(Roll_No, Course) → (Name, Course)

Adding extra info doesn’t break dependency

3. Transitivity Rule

If:

Then:

Example

Real Life Example

If:

• Roll_No → Student_ID
• Student_ID → Name

Then:

• Roll_No → Name

Chain relationship

Derived Rules

Using above axioms, we also get:

1. Union Rule

2. Decomposition Rule

3. Pseudo Transitivity

Technical Example

Given:

Using transitivity:

Derived dependencies help in equivalence checking

Common Mistakes

• Forgetting subset rule in reflexivity
• Applying augmentation incorrectly
• Missing chained dependencies

Real Life Use Cases

1. Database Design

Helps identify hidden relationships

2. Normalization

Used to remove redundancy

3. Query Optimization

Reduces unnecessary joins

4. Data Integrity

Ensures consistent relationships