Prerequisite – Armstrong’s Axioms in Functional Dependency in DBMS

Armstrong mentioned that rules 1 through 3 have completeness along with soundness. Armstrong’s axioms are complete because for a given set of functional dependencies, F, all functional dependencies implied by F^{+} can be derived from F using these rules.

**What we need to prove –**

The 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 write it as, one is a proper subset of the other. Hence, we need to show that,

Fis a proper subset of_{closure }F_{armstrong }

Correctness means that whatever can be derived from F can also be derived using Armstrong’s Axioms.

F_{closure}^{+} represents the closure of F. Mathematically we can represent, F_{closure}^{+} = {A| F_{closure}^{+} implies A can be derived from F using Armstrong’s Axioms} where, A belongs to Real Numbers.

**Claim-1 :**

is derivable from F using Armstrong’s Axioms if and only if Y is a proper subset of F_{closure}^{+}.

To prove ‘if’ part :

Let Y={A_{0}, A_{1}, ……, A_{m}, } where Y is a proper subset of F_{closure}^{+}. F_{closure}^{+} implies A_{j} can be derived from F using Armstrong’s Axioms} where, A belongs to 0<j<=m

If we follow union rule then, is derivable from F

To prove the ‘only if’ part :

is derivable from F using Armstrong's Axioms

by projection rule we know that F_{closure}^{+} implies A_{j}where, A belongs to 0<j<=m

Thus by the property of F_{closure}^{+}, A_{j} belongs to F_{closure}^{+}.

This implies that Y is a proper subset of F_{closure}^{+}.

X determines Y follows from F using Armstrong’s Axioms.

We shall try to prove the contrapositive of the above statement.

We would prove that X determines Y cannot be determined from F using Armstrong’s Axioms. In terms of tuple Relations we can say that, There exists an instance r belonging to Real numbers such that, all the functional dependencies of F hold on r. However, X implies Y does not hold.

This mentions that our assumption was wrong and the claim stands correct.

Hence proved that Claim 01 is correct.

Let us consider that r has only 2 rows and is represented by r{F_{closure}^{+} attributes, other attributes}

**Claim-2 :**

All FDs of F are satisfied by r. Let us again take the negative of it. Let W implies R belonging to F does not satisfy r, then W is a proper subset of F_{closure}^{+} and Z is not a proper subset of F_{closure}^{+}.

Let G belongs to Z- F_{closure}^{+}

We get :

because W is a proper subset of F_{closure}^{+},

by rule of transitivity,

by rule of reflexivity as G belongs to Z and

by rule of transitivity.

By definition of closures G must belong to F_{closure}^{+}

We get a contradiction.

We had taken a contradiction which was proven wrong, which means that the statement is true.

Hence proved that Claim 02 is correct.

**Claim-3 :**

X implies Y is not satisfied by r.

We assume the contradiction again (X does not imply Y). Due to the Structure of r, Y is a proper subset of F_{closure}^{+} which means that X implies Y can be proved using Armstrong’s Axioms.

Our assumption is proved false hence the claim is correct.

Hence proved that Claim 03 is correct.

These mean that when X does not imply Y, using Armsrong’s Axioms, then F doesn’t logically imply that X implies Y.

We can also say that when X implies Y, using Armsrong’s Axioms (mechanically), then F logically implies that X implies Y as well.

Hence proved that Armstrong’s Axioms are complete.

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