2. Introduction to Functional Dependency
• A functional dependency (FD) in a database occurs when one
attribute uniquely determines. The functional dependency is a
relationship that exists between two attributes. It typically exists
between the primary key and non-key attribute within a tables
another.
• Importance in DBMS:
• Essential for normalization
• Ensures data integrity and minimizes redundancy
• Example - Emp_ID → Emp_Name (An employee ID determines the
employee name)
10. Types of Functional Dependencies
• 1. Trivial Functional Dependency - A dependency where an attribute
is functionally dependent on itself or a subset of attributes.
• Example:Emp_ID, Proj_ID → Emp_ID
Here, Emp_ID is trivially dependent on (Emp_ID, Proj_ID).
SELECT Emp_ID, Emp_ID
FROM Employee
GROUP BY Emp_ID;
11. Non-Trivial Functional Dependency
• When an attribute is functionally dependent on a combination of
other attributes, and it’s not a trivial dependency.
• Example: Emp_ID → Dept
• Employee ID determines the department, which is non-trivial because
Dept is not part of the primary key.
• SELECT Emp_ID, Emp_Name
• FROM Employee
• GROUP BY Emp_ID, Emp_Name;
12. Partial Functional Dependency
• A non-prime attribute is functionally dependent on part of a
candidate key.
• Example:(Emp_ID, Proj_ID) → Emp_NameIn the Works On table, if
Emp_ID is part of the composite key, but Emp_Name depends only on
Emp_ID.
• SELECT Proj_ID, SUM(Hours_Worked)
• FROM Works_On
• GROUP BY Proj_ID;
13. Full Functional Dependency
• A non-prime attribute depends on the full composite key and not just
a part of it.
• Example: (Emp_ID, Proj_ID) → Hours_Worked
• Hours worked by an employee depends on both the employee and
the project.
• SELECT Emp_ID, Proj_ID, Hours_Worked
• FROM Works_On
• GROUP BY Emp_ID, Proj_ID, Hours_Worked;
14. Transitive Functional Dependency
• When one attribute depends on another through a third attribute.
• Example: Emp_ID → Dept and Dept → Manager
• Therefore, Emp_ID → Manager (transitive dependency).
• CREATE TABLE Department
• ( Dept_ID VARCHAR2(10), Dept_Name VARCHAR2(50), Manager_ID
VARCHAR2(10) );
• INSERT INTO Department VALUES ('D01', 'Development', 'M001’);
• INSERT INTO Department VALUES ('D02', 'Testing', 'M002');
15. Transitive Functional Dependency
• -- Transitive Dependency Query
• SELECT E.Emp_ID, D.Manager_ID
• FROM Employee E
• JOIN Department D ON E.Dept_ID = D.Dept_ID;
16. Inference Rules (Armstrong’s Axioms)
• Reflexivity Rule If Y is a subset of X, then X → Y.
• Example: Emp_ID → Emp_ID
SELECT Emp_ID, Emp_Name
FROM Employee
GROUP BY Emp_ID, Emp_Name;
17. Inference Rules (Armstrong’s Axioms)
• Augmentation Rule If X → Y, then XZ → YZ
(adding additional attributes does not affect the dependency).
Example: Emp_ID → Emp_Name implies (Emp_ID, Proj_ID) →
(Emp_Name, Proj_ID)
• SELECT Emp_ID, Emp_Name, Proj_ID
• FROM Employee E
• JOIN Works_On W ON E.Emp_ID = W.Emp_ID;
18. Inference Rules
• Transitivity Rule If X → Y and Y → Z, then X → Z.
• Example: Emp_ID → Dept and Dept → Manager implies
• Emp_ID → Manager
• SELECT E.Emp_ID, D.Manager_ID
• FROM Employee E
• JOIN Department D ON E.Dept_ID = D.Dept_ID;
19. Inference Rules
• Decomposition Rule If X → YZ, then X → Y and X → Z.
• Example: (Emp_ID, Proj_ID) → (Emp_Name, Hours_Worked) implies
Emp_ID → Emp_Name and (Emp_ID, Proj_ID) → Hours_Worked
• SELECT Emp_ID, Emp_Name, Salary
• FROM Employee
• GROUP BY Emp_ID, Emp_Name, Salary;
20. Inference Rules
• Union Rule If X → Y and X → Z, then X → YZ.
• Example: Emp_ID → Emp_Name and Emp_ID → Dept implies Emp_ID
→ (Emp_Name, Dept)
• SELECT Emp_ID, Emp_Name, Salary
• FROM Employee
• GROUP BY Emp_ID, Emp_Name, Salary;
21. Inference Rules
• Pseudo-Transitivity Rule If X → Y and WY → Z, then WX → Z.
• Example: Emp_ID → Dept and (Proj_ID, Dept) → Manager
• implies (Emp_ID, Proj_ID) → Manager
• SELECT W.Emp_ID, W.Proj_ID, P.Budget
• FROM Works_On W
• JOIN Project P ON W.Proj_ID = P.Proj_ID;
22. Case Study- Normalizing Works_On Table
• Issue: Emp_Name repeats for the same Emp_ID.
• Solution: First Normal Form (1NF): Already satisfied (atomic values).
• Second Normal Form (2NF): Remove partial dependencies.
• Separate Emp_Name into Employee table.
