Finding Closure of Attributes and Attribute Sets

1. Introduction

In DBMS, attribute closure is a fundamental concept used in functional dependency analysis. It helps in:

• Identifying all attributes functionally determined by a given attribute set
• Checking candidate keys
• Verifying normal forms (like 2NF, 3NF, BCNF) 

2. What is Attribute Closure?

The closure of an attribute set X, denoted as X⁺ (X plus), is the set of all attributes that can be functionally determined from X using the given functional dependencies.

Definition

If X is a set of attributes, then:

X⁺ = { all attributes that can be derived from X using functional dependencies }

3. Why is Attribute Closure Important?

Attribute closure is used to:

• Determine whether an attribute set is a super key
• Find candidate keys
• Check whether a functional dependency is valid
• Assist in normalization 

4. Steps to Find Attribute Closure

To find X⁺ (closure of X):

1. Start with X⁺ = X
2. Apply all functional dependencies repeatedly
3. If a dependency A → B exists and A ⊆ X⁺, then add B to X⁺
4. Repeat until no new attributes can be added 

5. Example Problem

Given:

Relation:
R(A, B, C, D, E, F, G)

Functional Dependencies:

• A → BC
• BC → DE
• D → F
• CF → G 

Find:

• Closure of (A)
• Closure of (D)
• Closure of (B, C) 

6. Finding Closure of A (A⁺)

Step 1:

Start with:
A⁺ = {A}

Step 2:

Apply A → BC
A⁺ = {A, B, C}

Step 3:

Apply BC → DE
A⁺ = {A, B, C, D, E}

Step 4:

Apply D → F
A⁺ = {A, B, C, D, E, F}

Step 5:

Apply CF → G
Since C and F are in A⁺
A⁺ = {A, B, C, D, E, F, G}

Final Result:

A⁺ = {A, B, C, D, E, F, G}

Conclusion:

A⁺ contains all attributes → A is a candidate key

7. Finding Closure of D (D⁺)

Step 1:

D⁺ = {D}

Step 2:

Apply D → F
D⁺ = {D, F}

Step 3:

No further dependencies apply

Final Result:

D⁺ = {D, F}

Conclusion:

D is not a candidate key

8. Finding Closure of (B, C) ((BC)⁺)

Step 1:

(BC)⁺ = {B, C}

Step 2:

Apply BC → DE
(BC)⁺ = {B, C, D, E}

Step 3:

Apply D → F
(BC)⁺ = {B, C, D, E, F}

Step 4:

Apply CF → G
(C and F are present)
(BC)⁺ = {B, C, D, E, F, G}

Final Result:

(BC)⁺ = {B, C, D, E, F, G}

Conclusion:

(BC) is not a candidate key (A is missing)

9. Summary of Results

10. Key Observations

• Closure helps determine reachability of attributes
• If X⁺ contains all attributes → X is a super key
• Minimal super keys are candidate keys
• Closure calculation is iterative and rule-based 

11. Important Points for Exams

• Always start with X⁺ = X
• Apply functional dependencies step-by-step
• Clearly show each expansion
• Final closure must include all derived attributes
• Identify whether it forms a candidate key