# 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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