Functional Dependency

Functional Dependency (FD)

A Functional Dependency is a relationship between attributes in a table.

👉 If one attribute determines another → it is called Functional Dependency

Notation: A → B

👉 If we know A, we can uniquely determine B

📊 Example

STUD_ID	NAME	DEPT
101	Amit	CSE
102	Ravi	ECE

🔍 Analysis

STUD_ID → NAME
STUD_ID → DEPT

👉 STUD_ID uniquely determines NAME and DEPT

📊 Functional Dependency Example (GATE Level)

Given Table:

A	B	C
1	4	2
3	5	6
3	4	6
7	3	8
9	1	0

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🎯 Goal

Check whether the following functional dependencies hold:

AB → C ?
BC → A ?
AC → B ?

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⚙️ Rule to Check Functional Dependency

👉 If two rows have same LHS values, then RHS must also be same

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🔍 Check AB → C

Check rows where (A, B) are same:

No two rows have same (A, B)

👉 No violation found

✅ AB → C holds

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🔍 Check BC → A

Check rows where (B, C) are same:

No repeated (B, C) pairs

👉 No violation found

✅ BC → A holds

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🔍 Check AC → B

Check rows where (A, C) are same:

(3, 6) appears twice

Check corresponding B values:

(3, 6) → B = 5
(3, 6) → B = 4

Since same (A, C) gives different B values → violation occurs

❌ AC → B does NOT hold

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💣 Final Answer

AB → C ✔ Holds
BC → A ✔ Holds
AC → B ❌ Does NOT Hold

👉 Final Conclusion: Only AC → B is violated

Types of Functional Dependency

1. Trivial Functional Dependency

A functional dependency A → B is called trivial if B is a subset of A.

👉 In simple words, if the right-hand side attribute is already present in the left-hand side, then it is trivial.

Condition: B ⊆ A

Example:

(A, B) → A
(Roll No, Name) → Name

👉 No new information is added 👉 Always true

2. Non-Trivial Functional Dependency

A functional dependency A → B is called non-trivial if B is NOT a subset of A.

👉 This means the dependency gives new and useful information.

Condition: B ⊄ A

Example:

Roll No → Name
Roll No → Department

👉 Provides useful information 👉 Helps in finding keys

3. Completely Non-Trivial Dependency

If A and B have no common attributes, then it is called completely non-trivial dependency.

Example: Roll No → Name

Difference Between Trivial and Non-Trivial

Trivial	Non-Trivial
B ⊆ A	B ⊄ A
No new information	Provides new information
Always true	Depends on data

Armstrong’s Inference Rules

1. Reflexivity (Basic Rule)

If Y ⊆ X, then X → Y

Example: (Roll No, Name) → Name

2. Augmentation (Basic Rule)

If X → Y, then XW → YW

Example: If Roll No → Name, then (Roll No, Dept) → (Name, Dept)

3. Transitivity (Basic Rule)

If X → Y and Y → Z, then X → Z

Example: Roll No → Dept and Dept → HOD, so Roll No → HOD

Derived Rules

4. Pseudotransitivity

If X → Y and WY → Z, then WX → Z

Example: If A → B and CB → D, then CA → D

5. Union (Additivity)

If X → Y and X → Z, then X → YZ

Example: If Roll No → Name and Roll No → Dept, then Roll No → (Name, Dept)

6. Decomposition (Projectivity)

If X → YZ, then X → Y and X → Z

Example: If Roll No → (Name, Dept), then Roll No → Name and Roll No → Dept

👉 These rules are used to find attribute closure, candidate keys, and normalization.

💣 Important Concept

Functional Dependency shows relationship between attributes, not rows.

👉 It is used in Normalization and finding Candidate Keys

🤯 Common Confusion

Does A → B mean every value must be different?

❌ NO

👉 It means: same A value → same B value

Why Functional Dependency is Important?

Helps in database normalization
Reduces data redundancy
Used to find candidate keys
Improves database design

🎯 GATE Important Points

A → B means A determines B
Used in normalization
Helps find candidate keys

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