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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.

Example:

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.

Example:

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.

Proof:

1. X → Y (given)
2. X → Z (given)
3. X → XY (using IR2 on 1 by augmentation with X. Where XX = X)
4. XY → YZ (using IR2 on 2 by augmentation with Y)
5. X → YZ (using IR3 on 3 and 4)

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.

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.

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)

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