Functional Dependency (FDs)

It is an association between two attributes of the same table (relation).

X→Y states that if two tuples agree on the values in attribute 'X' they must also agree on value in attribute 'Y'.

If it can take only one value for a given value of attribute upon which it is functionally dependent.

Classification of Functional Dependency :-

(i) Trivial Functional Dependency :-
         Functional Dependency X→Y is trivial if and only if Y⊆X. (Here, Y is a subset of X).
              Example :- AB→B   (It means AB is X and B is Y)

(ii) Non-Trivial Functional Dependency :-
        If there is at-least one attribute in R.H.S that is not past of the L.H.S.
            (AB is L.H.S and B is R.H.S)

(iii) Fully Functional Dependency :-
         Given R and Functional Dependency X→Y, then Y is fully functional dependent on X if there is no Z, where Z is proper subset of X such that Z→Y.

(iv) Partial Dependency :- 
           Given a relation R with Functional Dependency F defined on the attributes of R and K as a candidate key. if X is a proper subset of K and if Functional Dependency X→A, then A is said to be partially dependent on K.

(v) Transitive Dependency :-
        X→Y , Y→A , then A is transitively dependent on X.


Inference Rules / Closure Properties / Armstrong's Axioms

(i) Reflexivity:-
          if X ⊇ Y, then X→Y
                     AB→B
(ii) Augmentation:-
          if  X→Y, then
             XZ→YZ
(iii) Transitivity:-
          if X→Y, and Y→Z, then 
                X→Z
(iv) Union:-
         if X→Y and X→Z, then 
               X→YZ
(v) Decomposition:-
        if X→YZ, then
           X→Y
           X→Z

Closure Set of Functional Dependency's:-
  
     F D is the set of all FDs that can be determined using the given set of functional dependency (F).
   Total No. of Possible FDs in R (n attributes)
                = 2 to the power 2n