Proving Soundness of Armstrong’s Axioms

Prerequisite – Armstrong’s Axioms in Functional Dependency in DBMS
Armstrong mentioned that rules 1 through 3 have completeness along with soundness. Armstrong axioms are sound as they do not generate any incorrect Functional Dependencies and it allows us to generate the F⁺ closure. 

The proof of Soundness –
Set of functional dependencies that can be derived with logic from a given set of functional dependencies (F_closure) and the set of functional dependencies that can be inferred from Armstrong’s Axioms (F_armstrong) are same. In terms of sets, if we want to show equality, then we mention that one is a proper subset of the other. Hence, we need to prove that, 

 Farmstrong  is a proper subset of Fclosure 
    for soundness.

This means that all things derived from F after applying Armstrong’s Axioms are correct functional dependencies and this property is called Soundness. 

Consideration for Soundness –
has been derived from F using Armstrong’s Axioms after m number of steps. 

In each step we would apply the rules of reflexivity, augmentation and transitivity. 
(If we derive from Axioms which are only 3 in number, the repeated application of them would also give similar results.) 

Rule of Reflexivity –
This rule states that if E is a proper subset of G, then G implies E (). This rule always results in correct Functional Dependencies because it is a trivial functional dependency. 

Rule of Augmentation –
This rule states that for , if we add Z on both sides, then we would get , which would be a correct functional dependency. 
If t_a and t_b have the same corresponding values in G and Z (GZ), 
It would mean that they have same corresponding value for G, 
which would mean that they have same corresponding value for E, since G implies E, 
it would also mean that they correspond to E and Z (EZ). 

Rule of Transitivity –
This rule states that if , then G determines Z () 
If t_a and t_b have the same corresponding values in G 
which would mean that they have same corresponding value for E, since G implies E holds, 
it would also mean that they correspond to E, since E implies Z holds. 

So, applying these 3 rules will always give rise to correct functional dependencies (even if we apply them m number of times). 
Hence proved.