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Explain the inference rules for functional dependencies in DBMS
Functional dependencies are the constraints that are derived from the meaning and interrelationship of the data. Let F is a set of all functional dependencies. The set of all dependencies that include F as well as all dependencies that can be inferred from F is called CLOSURE of F denoted by F+.
Example 1
Given below is an example of functional dependency in database management system (DBMS) −
F= { SSN-> {ENMAE,BDATE,ADDRESS,DNUMBER}, DNUMBER-> {DNAME,DMGRSSN} }Output
You will get the following result −
Example 2
Given below is another example of functional dependency in the DBMS −
F+ = { SSN-> {ENAME,BDATE,ADDRESS,DNUMBER},
DNUMBER-> {DNAME, DMGRSSN},
SSN -> {DNAME, DMGRSSN},
ENO->ENO,
DNUMBER-> DNAME }To determine a systematic way to infer dependencies, we must discover a set of interface rules that can be used to infer new dependencies from a given set of dependencies.
Rules of functional dependencies
There are six inference rules which are as follows −
- Reflexivity: X->X // An attribute(s) determines itself
- Augmentation: if X->Y then XZ->YZ
- Transitivity: if X->Y & Y->Z then X->Z
- Additivity or Union : if X->Y & X->Z then X->YZ
- Projectivity or Decomposition: If X->YZ then X-> Y & X->Z
- Pseudo-Transitivity: If X->Y, YZ->W then XZ->W
Example
Let’s take a relation R with attributes R(A,B,C,D,E,F)
F: AB->C, BC->AD, D->E, E->F, CF->B, then prove that F logically implies CD->B
Solutions
D->E, E->F THEN D=>F { Transitivity property}.
D->E, CF-> THEN D=>B {Pseudo-Transitivity}
D->E, E->F THEN D=>F {Transitivity}
D->F THEN CD->CF {Augmentation}
CD->CF, CF-> B THEN => CD-> B {Transitivity}
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