Inference Rules / Armstrong's Axioms

1. Armstrong's Axioms are the complete set of basic rules used to infer all the functional dependencies on the relational database.
2. Developed by the William W.Armstrong' in 1974

Primary Inference Rules 

a) Reflexive rule

If 'X' is a subset of 'Y' then Y-->X is always true

Example:-

Y= {1,2,3,4,}, X={1,3}

Then Y-->X is true

b) Transitivity Rule

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

c) Augmentation Rule

If X-->Y then XZ--->YZ always true

Secondary Rules

d) Decomposition Rule

If X-->YZ the X-->Y and X-->Z always holds

e) Composition Rules

If X-->Y and Z-->P then XZ-->YP always true.

f) Additive Rule ( Union Rule)

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

g) Pseudo Transitivity Rule

If A-->B and BC-->D then AC-->D is always true

Why Armstrong's Axioms refer to the sound and complete

i) By sound we mean that given a set of FD's F specified on a relation schema R, any dependency that we can infer from F by using The primary rules of Armstrong's Axioms in every relation state 'r' of R that satisfies the dependencies F.

ii) By complete we mean that using primary rules of Armstrong's Axioms repeatedly to infer Armstrong's Axioms  until no more dependencies can be interfered results in the complete set of all possible dependencies that can be inferred from 'F'