DBMS 3.pdf
- 1. Functional Dependency • The functional dependency is a relationship that exists between two attributes. It typically exists between the primary key and non-key attribute within a table. X → Y • The left side of FD is known as a determinant, the right side of the production is known as a dependent. For example: • Assume we have an employee table with attributes: Emp_Id, Emp_Name, Emp_Address. • Here Emp_Id attribute can uniquely identify the Emp_Name attribute of employee table because if we know the Emp_Id, we can tell that employee name associated with it. • Functional dependency can be written as: • Emp_Id → Emp_Name • We can say that Emp_Name is functionally dependent on Emp_Id.
- 3. Types of Functional dependency 1. Trivial functional dependency A → B has trivial functional dependency if B is a subset of A OR EQUAL TO A i.e A ⊇ B The following dependencies are also trivial like: A → A, B → B Example: Consider a table with two columns Employee_Id and Employee_Name. {Employee_id, Employee_Name} → Employee_Id is a trivial functional dep endency as Employee_Id is a subset of {Employee_Id, Employee_Name}. Also, Employee_Id → Employee_Id and Employee_Name → Employee_Na me are trivial dependencies too.
- 4. 2. Non-trivial functional dependency • A → B has a non-trivial functional dependency if B is not a subset of A. • When A intersection B is NULL i.e. A∩B=Ф then A → B is called as complete non-trivial. Example: • ID → Name, • Name → DOB 3.Semi Non trivial functional dependency Employee_Id → Employee_Id Employee_Name Employee_Id → Employee_Id Employee_Id → Employee_Name
- 6. Inference Rule (IR): • The Armstrong's axioms are the basic inference rule. • Armstrong's axioms are used to conclude functional dependencies on a relational database. • The inference rule is a type of assertion. It can apply to a set of FD(functional dependency) to derive other FD. • Using the inference rule, we can derive additional functional dependency from the initial set. • The Functional dependency has 6 types of inference rule: • 1. Reflexive Rule (IR1 ) • In the reflexive rule, if Y is a subset of X, then X determines Y. • If X ⊇ Y then X → Y • Example:
- 7. 2. Augmentation Rule (IR2 ) • The augmentation is also called as a partial dependency. In augmentation, if X determines Y, then XZ determines YZ for any Z. • If X → Y then XZ → YZ Example: • For R(ABCD), if A → B then AC → BC 3. Transitive Rule (IR3 ) • In the transitive rule, if X determines Y and Y determine Z, then X must also determine Z. • If X → Y and Y → Z then X → Z 4. Union Rule (IR4 ) • Union rule says, if X determines Y and X determines Z, then X must also determine Y and Z. • If X → Y and X → Z then X → YZ
- 8. 5. Decomposition Rule (IR5 ) • Decomposition rule is also known as project rule. It is the reverse of union rule. • This Rule says, if X determines Y and Z, then X determines Y and X determines Z separately. • If X → YZ then X → Y and X → Z Proof: 1. X → YZ (given) 2. YZ → Y (using IR1 Rule) 3. X → Y (using IR3 on 1 and 2) 6. Pseudo transitive Rule (IR6 ) • In Pseudo transitive Rule, if X determines Y and YZ determines W, then XZ determines W. • If X → Y and YZ → W then XZ → W Proof: 1. X → Y (given) 2. WY → Z (given) 3. WX → WY (using IR2 on 1 by augmenting with W) 4. WX → Z (using IR3 on 3 and 2)
- 22. Relational Decomposition • When a relation in the relational model is not in appropriate normal form then the decomposition of a relation is required. • In a database, it breaks the table into multiple tables. • If the relation has no proper decomposition, then it may lead to problems like loss of information. • Decomposition is used to eliminate some of the problems of bad design like anomalies, inconsistencies, and redundancy. Types of Decomposition 1.Lossless Decomposition 2.Dependency Preserving
- 24. Dependency Preserving • It is an important constraint of the database. • In the dependency preservation, at least one decomposed table must satisfy every dependency. • If a relation R is decomposed into relation R1 and R2, then the dependencies of R either must be a part of R1 or R2 or must be derivable from the combination of functional dependencies of R1 and R2. • For example, suppose there is a relation R (A, B, C, D) with functional dependency set (A->BC). The relational R is decomposed into R1(ABC) and R2(AD) which is dependency preserving because FD A->BC is a part of relation R1(ABC).
- 29. Lossless Decomposition • If the information is not lost from the relation that is decomposed, then the decomposition will be lossless. • The lossless decomposition guarantees that the join of relations will result in the same relation as it was decomposed. • The relation is said to be lossless decomposition if natural joins of all the decomposition give the original relation. Example: EMPLOYEE_DEPARTMENT table: EMP_ID EMP_NAME EMP_AGE EMP_CITY DEPT_ID DEPT_NAME 22 Denim 28 Mumbai 827 Sales 33 Alina 25 Delhi 438 Marketing 46 Stephan 30 Bangalore 869 Finance 52 Katherine 36 Mumbai 575 Production 60 Jack 40 Noida 678 Testing
- 30. The above relation is decomposed into two relations EMPLOYEE and DEPARTMENT EMPLOYEE table: EMP_ID EMP_NAME EMP_AGE EMP_CITY 22 Denim 28 Mumbai 33 Alina 25 Delhi 46 Stephan 30 Bangalore 52 Katherine 36 Mumbai 60 Jack 40 Noida DEPARTMENT table: DEPT_ID EMP_ID DEPT_NAME 827 22 Sales 438 33 Marketing 869 46 Finance 575 52 Production
- 31. • Now, when these two relations are joined on the common column "EMP_ID", then the resultant relation will look like: • Employee ⋈ Department EMP_ID EMP_NAME EMP_AGE EMP_CITY DEPT_ID DEPT_NAME 22 Denim 28 Mumbai 827 Sales 33 Alina 25 Delhi 438 Marketing 46 Stephan 30 Bangalore 869 Finance 52 Katherine 36 Mumbai 575 Production 60 Jack 40 Noida 678 Testing Hence, the decomposition is Lossless join decomposition.