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