Question :
Explain the inference rules for functional dependencies in DBMS
Subject
Database Management System
Standard
Computer Science Engineering
Views
1766
Asked By
Kirti
Knowledge
Answer / Solution
Answer / Solution
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.
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.
3. Transitive Rule (IR3)
In the transitive rule, if X determines Y and Y determine Z, then X must also determine Z.
4. Union Rule (IR4)
Union rule says, if X determines Y and X determines Z, then X must also determine Y and Z.
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.
6. Pseudo transitive Rule (IR6)
In Pseudo transitive Rule, if X determines Y and YZ determines W, then XZ determines W.
Knowledge
Answer / Solution
Answer / Solution
The Functional dependency has 6 types of inference rule:
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
X = {a, b, c, d, e}
Y = {a, b, c}
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
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
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
Proof:
X → Y (given) X → Z (given) X → XY (using IR2 on 1 by augmentation with X. Where XX = X) XY → YZ (using IR2 on 2 by augmentation with Y) X → YZ (using IR3 on 3 and 4)
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:
X → YZ (given) YZ → Y (using IR1 Rule) 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:
X → Y (given) WY → Z (given) WX → WY (using IR2 on 1 by augmenting with W) WX → Z (using IR3 on 3 and 2)
Knowledge
Answer / Solution
Answer / Solution
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
Knowledge
Answer / Solution
Answer / Solution
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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