DBMS Armstrong’s Axioms for functional dependencies

Armstrong’s Axioms for functional dependencies

Once a set S of functional dependencies for any relation R is given, it may be possible that it contains  redundant functional dependencies. sometimes it is necessary to find out closure of a set of functional dependencies.
Armstrong provided a set of inference rules,generally known as Armstrong's Axioms , to infer new FDs from other FDs  . these are given below:
Let us assume that a relation R contains attribute-sets W, X ,Y ,and z.

Rule 1: Reflexivity (or inclusion)

 If , Y ⊆ X, then X -> Y.

Rule 2: Augmentation

 If , X -> Y, then XZ -> Y, and XZ -> YZ.

Rule 3: Transitivity

 If , X -> Y and Y -> Z, then X -> Z.

Rule 4: Self-determination

 X -> X.

Rule 5: Psuedo-transitivity

 If , X -> Y and YW -> Z, then XW -> Z.

Rule 6: Union (or Additive)

 If , X -> Z and X -> Y, then X -> YZ.

Rule 7: Decomposition (or Projective)

 If , X -> YZ, then X -> Y and X -> Z.

Rule 8: Composition

 If , X -> Y and Z -> W, then XZ -> YW.

Rule 9: Self-accumulation

 If, X -> YZ and Z -> W, then X -> YZW .


These rules are used to find out redundant functional dependencies as well as closure of a set of functional dependencies.



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