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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

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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

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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.

    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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