Closure of Attribute Sets in DBMS – Definition, Steps, and Example

• 📌 What is Closure of an Attribute Set?
• ✍️ Steps to Find Closure (X⁺)
• 📘 Example: Closure of {Student\_ID}
• 💡 Why Attribute Closure is Important
• ✅ Conclusion

In relational database design, the closure of an attribute set is a fundamental concept used to determine all attributes that can be functionally derived from a given set using known functional dependencies. This helps verify candidate keys, enforce data integrity, and apply normalization rules effectively.

📌 What is Closure of an Attribute Set?

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

✍️ Steps to Find Closure (X⁺)

To compute X⁺:

1. Start with X Add all attributes from X into the closure set.
2. Apply Functional Dependencies For each FD A → B, if A ⊆ X⁺, add B to X⁺.
3. Repeat Until Stable Continue applying FDs until no new attributes can be added.
4. Result The final X⁺ includes all attributes functionally dependent on X.

📘 Example: Closure of {Student\_ID}

Let's use a table called Student_Course with the following attributes and FDs:

Functional Dependencies:

• Student_ID → Course_ID
• Course_ID → Instructor
• Instructor → Department

Calculate: {Student\_ID}⁺

Start with: Closure = {Student_ID}

Apply:

• Student_ID → Course_ID → {Student_ID, Course_ID}
• Course_ID → Instructor → {Student_ID, Course_ID, Instructor}
• Instructor → Department → {Student_ID, Course_ID, Instructor, Department}

✅ Result: {Student\_ID}⁺ = {Student\_ID, Course\_ID, Instructor, Department}

This means Student_ID can functionally determine all attributes in the table — thus, it's a candidate key.

💡 Why Attribute Closure is Important

✅ Conclusion

Understanding how to find the closure of attribute sets is a crucial skill for database designers. It not only helps validate functional dependencies and candidate keys, but also plays a central role in normalization to avoid redundancy and ensure data consistency.

Would you like to explore how to find all candidate keys from a given FD set? Let me know, and I'll walk you through it.