Closure of Functional Dependency in DBMS

Closure of a functional dependency in database design refers to the set of functional dependencies that can be inferred from a given set of functional dependencies. In other words, it is the process of finding all the functional dependencies that can be derived from the original set of functional dependencies, using the properties of functional dependencies. The closure of functional dependencies is important in normalization, as it helps identify redundant data and improve the efficiency of database operations.

Steps to Calculate the Closure of Functional Dependency

The steps to calculate the closure of functional dependencies are as follows:

Step 1: Start with the original set of functional dependencies.

Step 2: Repeat the following steps until there are no new functional dependencies to be added:

• For each functional dependency in the set, check if its right-hand side can be determined using the left-hand side and the functional dependencies in the set.
• If so, add the functional dependency to the set.

Step 3: The final set of functional dependencies is the closure of the original set.

Example:

Let A, B, C and D be attributes and the functional dependencies are as follows:

A -> B

B -> CD

The closure of this set of functional dependencies is {A -> B, B -> CD, A -> CD}.

How closure of functional dependencies is important in normalization, and improving the efficiency of database operations

Normalization is the process of organizing a database to minimize redundancy and dependency. The closure of functional dependencies is important in normalization because it allows for a more comprehensive analysis of the relationships between attributes in a database.

For example, if the closure of a functional dependency shows that two attributes are dependent on each other, it would not make sense to include both attributes in the same table, as they would be redundant. Instead, they would be split into separate tables, improving the efficiency of database operations by reducing the amount of redundant data.

In summary, the closure of functional dependencies is important in normalization because it helps identify redundant data and improves the efficiency of database operations by allowing for a more comprehensive analysis of the relationships between attributes in a database.

Calculating Candidate Key in Closure of Functional Dependency

A candidate key in the closure of functional dependencies refers to a set of attributes in a relational database that uniquely identifies a tuple (row) in a relation (table). A functional dependency is a relationship between two attributes in a relation, where the value of one attribute determines the value of another attribute. The process of finding candidate keys involves using functional dependencies to identify attributes that are necessary and sufficient to determine the values of all other attributes in the relation. Once these candidate keys are identified, the closure of functional dependencies is the complete set of functional dependencies that can be derived from these candidate keys.

Example of Calculating Candidate Key

Consider a relation R(A, B, C, D) with functional dependencies {A → B, B → C, C → D}. To calculate the candidate key in the closure of functional dependencies:

Start by finding the candidate keys. In this case, the candidate keys are {A}, {A, B}, and {A, C}.

Calculate the closure of functional dependencies for each candidate key.

1. For the candidate key {A}:
   • Start with {A}.
   • Apply the functional dependencies to find the attributes that are functionally dependent on {A}. In this case, {B} is functionally dependent on {A} (A → B), so add {B} to the closure.
   • Repeat the process until there are no more attributes to be added. In this case, {C} and {D} are functionally dependent on {B} (B → C, C → D), so add {C} and {D} to the closure.
   • The closure of {A} is {A, B, C, D}.
2. Repeat the process for the candidate keys {A, B} and {A, C} to find their closures.

The candidate key with the smallest closure is the primary key of the relation. In this case, {A} is the primary key since its closure is the smallest of the three closures calculated.

Conclusion

In summary, the candidate key in the closure of functional dependencies refers to the set of attributes that uniquely identify a tuple in a relation and its closure is the complete set of functional dependencies that can be derived from these attributes. The candidate key with the smallest closure is chosen as the primary key of the relation.