Inference Rule (IR):
The Functional dependency has 6 types of inference rule: 1. Reflexive Rule (IR_{1})In the reflexive rule, if Y is a subset of X, then X determines Y. Example: 2. Augmentation Rule (IR_{2})The augmentation is also called as a partial dependency. In augmentation, if X determines Y, then XZ determines YZ for any Z. Example: 3. Transitive Rule (IR_{3})In the transitive rule, if X determines Y and Y determine Z, then X must also determine Z. 4. Union Rule (IR_{4})Union rule says, if X determines Y and X determines Z, then X must also determine Y and Z. Proof:
1. X → Y (given)
2. X → Z (given) 3. X → XY (using IR_{2} on 1 by augmentation with X. Where XX = X) 4. XY → YZ (using IR_{2} on 2 by augmentation with Y) 5. X → YZ (using IR_{3} on 3 and 4) 5. Decomposition Rule (IR_{5})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. Proof:
1. X → YZ (given)
2. YZ → Y (using IR_{1} Rule) 3. X → Y (using IR_{3} on 1 and 2) 6. Pseudo transitive Rule (IR_{6})In Pseudo transitive Rule, if X determines Y and YZ determines W, then XZ determines W. Proof:
1. X → Y (given)
2. WY → Z (given) 3. WX → WY (using IR_{2} on 1 by augmenting with W) 4. WX → Z (using IR_{3} on 3 and 2)
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