Skip to content
Related Articles
Open in App
Not now

Related Articles

Armstrong’s Axioms in Functional Dependency in DBMS

Improve Article
Save Article
  • Difficulty Level : Basic
  • Last Updated : 23 Jun, 2021
Improve Article
Save Article

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 F^+    , 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 {\displaystyle A}    is a set of attributes and {\displaystyle B}    is subset of {\displaystyle A}    , then {\displaystyle A}    holds {\displaystyle B}    . If {\displaystyle B\subseteq A}    then {\displaystyle A\to B}    This property is trivial property.
  2. Axiom of augmentation – 
    If {\displaystyle A\to B}    holds and {\displaystyle Y}    is attribute set, then {\displaystyle AY\to BY}    also holds. That is adding attributes in dependencies, does not change the basic dependencies. If {\displaystyle A\to B}    , then {\displaystyle AC\to BC}    for any {\displaystyle C}    .
  3. Axiom of transitivity – 
    Same as the transitive rule in algebra, if {\displaystyle A\to B}    holds and {\displaystyle B\to C}    holds, then {\displaystyle A\to C}    also holds. {\displaystyle A\to B}    is called as {\displaystyle A}    functionally that determines {\displaystyle B}    . If {\displaystyle X\to Y}    and {\displaystyle Y\to Z}    , then {\displaystyle X\to Z}

Secondary Rules –

These rules can be derived from the above axioms. 

  1. Union – 
    If {\displaystyle A\to B}    holds and {\displaystyle A\to C}    holds, then {\displaystyle A\to BC}    holds. If {\displaystyle X\to Y}    and {\displaystyle X\to Z}    then {\displaystyle X\to YZ}
  2. Composition – 
    If {\displaystyle A\to B}    and {\displaystyle X\to Y}    holds, then {\displaystyle AX\to BY}    holds.
  3. Decomposition – 
    If {\displaystyle A\to BC}    holds then {\displaystyle A\to B}    and {\displaystyle A\to C}    hold. If {\displaystyle X\to YZ}    then {\displaystyle X\to Y}    and {\displaystyle X\to Z}
  4. Pseudo Transitivity – 
    If {\displaystyle A\to B}    holds and {\displaystyle BC\to D}    holds, then {\displaystyle AC\to D}    holds. If {\displaystyle X\to Y}    and {\displaystyle YZ\to W}    then {\displaystyle XZ\to W}    .

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. 

References – 

This article is contributed by Samit Mandal. If you like GeeksforGeeks and would like to contribute, you can also write an article using or mail your article to See your article appearing on the GeeksforGeeks main page and help other Geeks. 

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

My Personal Notes arrow_drop_up
Related Articles

Start Your Coding Journey Now!