Attribute Closure (X⁺)
Attribute Closure of a set X is the set of all attributes that can be determined from X using functional dependencies.
👉 X⁺ = all attributes that X can determine (direct + indirect)
Formula: X⁺ = { A | X → A }
🎯 Why Attribute Closure is Important?
- Find Super Keys & Candidate Keys
- Check Functional Dependencies
- Used in Normalization
- Helps in Minimal Cover
👉 If X⁺ contains all attributes → X is a Super Key
Learn more about
Super Key
⚙️ Steps to Find Attribute Closure
- Start with X⁺ = X
- Check all Functional Dependencies
- If LHS ⊆ X⁺ → add RHS
- Repeat until no change occurs
👉 Continue until closure stops growing
📊 Example (GATE Level)
R(A, B, C, D, E)
F = { A → BC, B → D, CD → E }
A⁺ = {A}
→ {A, B, C}
→ {A, B, C, D}
→ {A, B, C, D, E}
Final: A⁺ = {A, B, C, D, E}
👉 A is a Super Key
📊 Attribute Closure Example (GATE Level)
Given Functional Dependencies:
F = { A → B, C → D, AB → E, BE → C, EF → G }
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🎯 Goal
Find the following closures:
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⚙️ Step-by-Step Rule
👉 Start with given attributes → keep adding new attributes using FDs until no change
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🔍 Find A⁺
Start: A⁺ = {A}
A⁺ = {A, B}
No more dependencies can be applied
👉 A⁺ = {A, B}
---
🔍 Find AF⁺
Start: AF⁺ = {A, F}
- A → B → add B
- AB → E → add E
- BE → C → add C
- C → D → add D
- EF → G → add G
AF⁺ = {A, F, B, E, C, D, G}
👉 AF⁺ = {A, B, C, D, E, F, G}
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🔍 Find AE⁺
Start: AE⁺ = {A, E}
- A → B → add B
- BE → C → add C
- C → D → add D
AE⁺ = {A, E, B, C, D}
👉 AE⁺ = {A, B, C, D, E}
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🔍 Find CD⁺
Start: CD⁺ = {C, D}
CD⁺ = {C, D}
No new attribute can be added
👉 CD⁺ = {C, D}
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💣 Final Answer
- A⁺ = {A, B}
- AF⁺ = {A, B, C, D, E, F, G}
- AE⁺ = {A, B, C, D, E}
- CD⁺ = {C, D}
👉 AF⁺ gives all attributes → AF is a candidate key
Learn more about
Candidate Key
🧠 Key Observations
- Closure only increases (monotonic)
- Order of FD application does not matter
- Captures indirect (transitive) dependencies
❌ Common Mistakes
- Ignoring transitive dependencies
- Stopping early
- Not rechecking after adding attributes
Prime and Non-Prime Attributes (Deep Concept)
In DBMS, attributes are classified based on their participation in candidate keys.
This concept is extremely important for normalization (2NF, 3NF, BCNF).
1. Prime Attribute
A prime attribute is any attribute that is part of at least one candidate key.
👉 If an attribute appears in ANY candidate key → it is PRIME
Example 1
- Relation: R(A, B, C)
- Candidate Keys: (A, B) and (A, C)
👉 Prime Attributes = A, B, C
⚠️ Important: Even if an attribute appears in only one candidate key, it is still PRIME.
2. Non-Prime Attribute
A non-prime attribute is an attribute that is NOT part of any candidate key.
👉 If attribute is NOT in any candidate key → NON-PRIME
Example 2
- Relation: R(A, B, C, D)
- Candidate Key: (A, B)
👉 Prime = A, B
👉 Non-Prime = C, D
🔥 Tricky GATE Concept
If a relation has multiple candidate keys, then all attributes appearing in any candidate key are PRIME.
- Relation: R(A, B, C)
- Candidate Keys: (A, B) and (B, C)
👉 Prime = A, B, C
👉 Non-Prime = None
💡 GATE Trick: If all attributes are part of some candidate key → No non-prime attributes
Relation with Functional Dependency
Functional Dependency helps identify candidate keys, and candidate keys determine prime attributes.
- Find closure → find candidate key
- Candidate key → identify prime attributes
🔥 Role in Normalization
➡️ Second Normal Form (2NF)
A relation is in 2NF if:
- It is in 1NF
- No partial dependency of NON-PRIME attribute on candidate key
👉 Partial dependency = Non-prime depends on part of key
➡️ Third Normal Form (3NF)
A relation is in 3NF if:
- No transitive dependency
- Non-prime attribute should not depend on another non-prime
👉 X → Y allowed if Y is PRIME
📊 GATE-Level Question Example
Relation: R(A, B, C, D)
FD: A → B, B → C
Step 1: Find candidate key
👉 A+ = A, B, C → Not complete
👉 Add D → (A, D) is candidate key
👉 Prime = A, D
👉 Non-Prime = B, C
Difference Table
| Prime Attribute |
Non-Prime Attribute |
| Part of candidate key |
Not part of candidate key |
| Used in normalization rules |
Causes dependency issues |
| Helps identify records |
Stores extra data |
🚀 Final Shortcut:
Candidate Key ke andar → PRIME
Candidate Key ke bahar → NON-PRIME