In relational database design, Attribute Closure is a technique used to find all attributes that can be determined from a given set of attributes using a set of functional dependencies. It is denoted as $X^+$ and is read as “the closure of $X$”.

Attribute closure is very useful for:

• Finding candidate keys and super keys from a given set of functional dependencies.

• Verifying whether a functional dependency $X \to Y$ logically follows from a given set of FDs.

What is Attribute Closure?

For a relation schema and a set of functional dependencies $F$:

• The attribute closure of $X$, written $X^+$, is the set of all attributes that can be functionally determined by $X$ using the FDs in $F$.

• In practice, you start with $X$ and repeatedly apply FDs to add more attributes as long as new attributes can be determined.

Algorithm to Compute Attribute Closure

Given a set of functional dependencies $F$ and an attribute set $X$:

1. Initialize $X^+ = X$.

2. For each functional dependency $A \to B$ in $F$:

   • If $A \subseteq X^+$, then add $B$ to $X^+$.

3. Repeat step 2 until no more attributes can be added to $X^+$.

The final result is $X^+$, the attribute closure of $X$.

Example of Attribute Closure

Consider a relation $R(A, B, C, D)$ with functional dependencies:

• $A \to B$

• $B \to C$

• $C \to D$

Find $A^+$.

1. Start with: $A^+ = \{A\}$.

2. $A \to B$ and $A \subseteq A^+$ ⇒ add $B$:

   • $A^+ = \{A, B\}$.

3. $B \to C$ and $B \subseteq A^+$ ⇒ add $C$:

   • $A^+ = \{A, B, C\}$.

4. $C \to D$ and $C \subseteq A^+$ ⇒ add $D$:

   • $A^+ = \{A, B, C, D\}$.

So $A^+ = \{A, B, C, D\}$. This means $A$ is a super key for this relation.

Using Attribute Closure to Check Functional Dependencies

To check whether a functional dependency $X \to Y$ follows from a set of FDs $F$:

1. Compute $X^+$.

2. If $Y \subseteq X^+$, then $X \to Y$ holds.

If $X^+$ contains all the attributes of the relation, then $X$ is a super key.

Why Attribute Closure Matters in Design?

• It helps identify candidate keys from a set of functional dependencies.

• It supports normalization by clarifying which attributes determine others.

• It is used in dependency preservation and lossless decomposition algorithms.

Summary

Attribute Closure in DBMS is the process of finding all attributes that can be functionally determined by a given set of attributes using the given functional dependencies. It is denoted as $X^+$ and is a fundamental tool for finding super keys, verifying FDs, and supporting normalization in relational database design.