Functional Dependeny in DBMS

Functional dependency in DBMS is a constraint between two set of attributes. In a relational database schema having n attributes, there exists a universal relation R = {A1, A2, A3, ..., An}. A functional dependency is denoted by X→Y, where X and Y are subset of R. In relation R, attribute (or set of attributes) Y is functionally dependent to attribute X, means value of X can uniquely determine the value of Y.

The abbreviation of functional dependency is FD or f.d.

The left hand side is called determinant and right hand side is dependent.

Consider the below relation, in this each value of X uniquely determines value of Y

Now consider the below table, here there are two 1s in X attribute but both determines two different value of Y, this violates functional dependency X->Y.

Inference Rules for Functional Dependencies

In a database schema, F is a set of functional dependency. But are some other dependency that can be inferred or deduced from functional dependencies FD in F. 

The set of all dependencies including F as well as other dependencies inferred from F are known as closure of F and it is denoted by F*

For example: {Emp_id → {Emp_name, Emp_salary, Emp_mobile}}

Emp_mobile → {Emp_name, Emp_salary}

Following dependencies can be inferred from above two FD

Emp_mobile → Emp_name

Emp_id → Emp_id  

The following rules are inference rules for functional dependency.  

- Reflexive rule 

If Y ⊆ X, then X →Y

- Augmentation rule

If {X →Y},  then  XZ →YZ

- Transitive rule

If  {X →Y, Y →Z}, then X →Z

- Decomposition, or projective, rule

If {X →YZ}, then X →Y

- Union, or additive, rule

If {X →Y, X →Z}, then X →YZ 

- Pseudotransitive rule

If {X →Y, WY →Z}, then WX →Z