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In functional dependency Armstrong inference rules refers to Reflexive, Augmentation and ... Decomposition Reflexive, Transitive and Decomposition

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ISRO | ISRO CS 2011 | Question 53

Last Updated : 18 Oct, 2019

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In functional dependency Armstrong inference rules refers to

(A) Reflexivity, Augmentation and Decomposition(B) Transitivity, Augmentation and Reflexivity(C) Augmentation, Transitivity, Reflexivity and Decomposition(D) Reflexivity, Transitivity and DecompositionAnswer: (B)Explanation: Armstrong inference rules refer to a set of inference rules used to infer all the functional dependencies on a relational database. It consists of the following axioms:Axiom of Reflexivity:

This axiom states: if Y is a subset of X, then X determines Y

Axiom of Augmentation:

The axiom of augmentation, also known as a partial dependency,

states if X determines Y, then XZ determines YZ, for any Z

Axiom of Transitivity:

The axiom of transitivity says if X determines Y, and Y

determines Z, then X must also determine Z.

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DBMS Inference Rule

DBMS Inference Rule with DBMS Overview, DBMS vs Files System, DBMS Architecture, Three schema Architecture, DBMS Language, DBMS Keys, DBMS Generalization, DBMS Specialization, Relational Model concept, SQL Introduction, Advantage of SQL, DBMS Normalization, Functional Dependency, DBMS Schedule, Concurrency Control etc.

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:

X = {a, b, c, d, e} Y = {a, b, c}

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

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.

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)

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