Armstrong’s Axioms in Functional Dependency in DBMS

• Difficulty Level : Basic
• Last Updated : 23 Jun, 2021

Prerequisite – Functional Dependencies 

The term Armstrong axioms refer to the sound and complete set of inference rules or axioms, introduced by William W. Armstrong, that is used to test the logical implication of functional dependencies. If F is a set of functional dependencies then the closure of F, denoted as , is the set of all functional dependencies logically implied by F. Armstrong’s Axioms are a set of rules, that when applied repeatedly, generates a closure of functional dependencies. 

Axioms –

1. Axiom of reflexivity – 
   If is a set of attributes and is subset of , then holds . If then This property is trivial property.
2. Axiom of augmentation – 
   If holds and is attribute set, then also holds. That is adding attributes in dependencies, does not change the basic dependencies. If , then for any .
3. Axiom of transitivity – 
   Same as the transitive rule in algebra, if holds and holds, then also holds. is called as functionally that determines . If and , then 

Secondary Rules –

These rules can be derived from the above axioms. 

1. Union – 
   If holds and holds, then holds. If and then 
2. Composition – 
   If and holds, then holds.
3. Decomposition – 
   If holds then and hold. If then and 
4. Pseudo Transitivity – 
   If holds and holds, then holds. If and then .

Why armstrong axioms refer to the Sound and Complete ? 
By sound, we mean that given a set of functional dependencies F specified on a relation schema R, any dependency that we can infer from F by using the primary rules of Armstrong axioms holds in every relation state r of R that satisfies the dependencies in F. 
By complete, we mean that using primary rules of Armstrong axioms repeatedly to infer dependencies until no more dependencies can be inferred results in the complete set of all possible dependencies that can be inferred from F.