📘 Attribute Closure (X⁺)

🔹 Definition

For a set of attributes X, the closure of X (denoted X⁺) is:

The set of all attributes that can be functionally determined from X using the given functional dependencies (F).

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Why Closure is Important

Attribute closure helps us:

• ✅ Find all attributes determined by X
• ✅ Check if X is a superkey / candidate key
• ✅ Understand hidden relationships
• ✅ Perform normalization

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⚙️ Algorithm : Finding X⁺

📌 Input:

• Set of FDs → F
• Set of attributes → X

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🔄 Steps of the Algorithm

1. Start with:
   X⁺ = X

2. Repeat:
   - For each FD (Y → Z) in F:
       If Y ⊆ X⁺:
           Add Z to X⁺

3. Stop when:
   No new attributes are added

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💡 Key Idea

• Start with what you know (X)
• Keep adding attributes that can be derived
• Stop when nothing new can be added

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🧾 Given Example: CLASS Relation

Relation:

CLASS (Classid, Course#, Instr_name, Credit_hrs, Text, Publisher, Classroom, Capacity)

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📌 Functional Dependencies (F)

1. FD1:

   Classid → Course#, Instr_name, Credit_hrs, Text, Publisher, Classroom, Capacity
   

2. FD2:

   Course# → Credit_hrs
   

3. FD3:

   {Course#, Instr_name} → Text, Classroom
   

4. FD4:

   Text → Publisher
   

5. FD5:

   Classroom → Capacity
   

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🔍 Closure Calculations

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🔹 1. Closure of {Classid}⁺

Step-by-step:

• Start:

  X⁺ = {Classid}
  

• Apply FD1:

  Classid → all attributes
  

✅ Result:

{Classid}⁺ = {Classid, Course#, Instr_name, Credit_hrs, Text, Publisher, Classroom, Capacity}

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🎯 Interpretation:

• Classid determines all attributes
• 👉 So, Classid is a candidate key

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🔹 2. Closure of {Course#}⁺

Step-by-step:

• Start:

  X⁺ = {Course#}
  

• Apply FD2:

  Course# → Credit_hrs
  

✅ Result:

{Course#}⁺ = {Course#, Credit_hrs}

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🎯 Interpretation:

• Course# determines only Credit hours
• ❌ Not a key
• ❌ Cannot determine instructor, text, etc.

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🔹 3. Closure of {Course#, Instr_name}⁺

Step-by-step:

• Start:

  X⁺ = {Course#, Instr_name}
  

• Apply FD3:

  → Text, Classroom
  

• Apply FD4:

  Text → Publisher
  

• Apply FD5:

  Classroom → Capacity
  

• Apply FD2:

  Course# → Credit_hrs
  

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✅ Final Result:

{Course#, Instr_name}⁺ =
{Course#, Instr_name, Credit_hrs, Text, Publisher, Classroom, Capacity}

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🎯 Interpretation:

• Determines many attributes
• ❌ But does NOT determine Classid

👉 Therefore:

• ❌ Not a candidate key

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Key Takeaways

• Closure (X⁺) = all attributes determined by X
• Use algorithm iteratively
• Helps identify:
  • Candidate keys
  • Redundant attributes
• Interpretation gives real-world meaning