Boyce Codd normal form

Boyce Codd normal form (BCNF)

• BCNF is the advance version of 3NF. It is stricter than 3NF.
• A table is in BCNF if every functional dependency X → Y, X is the super key of the table.
• For BCNF, the table should be in 3NF, and for every FD, LHS is super key.

Example: Let's assume there is a company where employees work in more than one department.

EMPLOYEE table:

EMP_ID	EMP_COUNTRY	EMP_DEPT	DEPT_TYPE	EMP_DEPT_NO
264	India	Designing	D394	283
264	India	Testing	D394	300
364	UK	Stores	D283	232
364	UK	Developing	D283	549

In the above table Functional dependencies are as follows:

1. EMP_ID  →  EMP_COUNTRY  
2. EMP_DEPT  →   {DEPT_TYPE, EMP_DEPT_NO}  

Candidate key: {EMP-ID, EMP-DEPT}

The table is not in BCNF because neither EMP_DEPT nor EMP_ID alone are keys.

To convert the given table into BCNF, we decompose it into three tables:

EMP_COUNTRY table:

EMP_ID	EMP_COUNTRY
264	India
264	India

EMP_DEPT table:

EMP_DEPT	DEPT_TYPE	EMP_DEPT_NO
Designing	D394	283
Testing	D394	300
Stores	D283	232
Developing	D283	549

EMP_DEPT_MAPPING table:

EMP_ID	EMP_DEPT
D394	283
D394	300
D283	232
D283	549

Functional dependencies:

1. EMP_ID   →    EMP_COUNTRY  
2. EMP_DEPT   →   {DEPT_TYPE, EMP_DEPT_NO}  

Candidate keys:

For the first table: EMP_ID
For the second table: EMP_DEPT
For the third table: {EMP_ID, EMP_DEPT}

Now, this is in BCNF because left side part of both the functional dependencies is a key.

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Third Normal Form (3NF)

• A relation will be in 3NF if it is in 2NF and not contain any transitive partial dependency.
• 3NF is used to reduce the data duplication. It is also used to achieve the data integrity.
• If there is no transitive dependency for non-prime attributes, then the relation must be in third normal form.

A relation is in third normal form if it holds atleast one of the following conditions for every non-trivial function dependency X → Y.

1. X is a super key.
2. Y is a prime attribute, i.e., each element of Y is part of some candidate key.

Example:

EMPLOYEE_DETAIL table:

EMP_ID	EMP_NAME	EMP_ZIP	EMP_STATE	EMP_CITY
222	Harry	201010	UP	Noida
333	Stephan	02228	US	Boston
444	Lan	60007	US	Chicago
555	Katharine	06389	UK	Norwich
666	John	462007	MP	Bhopal

Super key in the table above:

1. {EMP_ID}, {EMP_ID, EMP_NAME}, {EMP_ID, EMP_NAME, EMP_ZIP}....so on  

Candidate key: {EMP_ID}

Non-prime attributes: In the given table, all attributes except EMP_ID are non-prime.

Here, EMP_STATE & EMP_CITY dependent on EMP_ZIP and EMP_ZIP dependent on EMP_ID. The non-prime attributes (EMP_STATE, EMP_CITY) transitively dependent on super key(EMP_ID). It violates the rule of third normal form.

That's why we need to move the EMP_CITY and EMP_STATE to the new <EMPLOYEE_ZIP> table, with EMP_ZIP as a Primary key.

EMPLOYEE table:

EMP_ID	EMP_NAME	EMP_ZIP
222	Harry	201010
333	Stephan	02228
444	Lan	60007
555	Katharine	06389
666	John	462007

EMPLOYEE_ZIP table:

EMP_ZIP	EMP_STATE	EMP_CITY
201010	UP	Noida
02228	US	Boston
60007	US	Chicago
06389	UK	Norwich
462007	MP	Bhopal

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Second Normal Form (2NF)

• In the 2NF, relational must be in 1NF.
• In the second normal form, all non-key attributes are fully functional dependent on the primary key

Example: Let's assume, a school can store the data of teachers and the subjects they teach. In a school, a teacher can teach more than one subject.

TEACHER table

TEACHER_ID	SUBJECT	TEACHER_AGE
25	Chemistry	30
25	Biology	30
47	English	35
83	Math	38
83	Computer	38

In the given table, the non-prime attribute TEACHER_AGE is dependent on TEACHER_ID which is a proper subset of a candidate key. That's why it violates the rule for 2NF.

To convert the given table into 2NF, we decompose it into two tables:

TEACHER_DETAIL table:

TEACHER_ID	TEACHER_AGE
25	30
47	35
83	38

TEACHER_SUBJECT table:

TEACHER_ID	SUBJECT
25	Chemistry
25	Biology
47	English
83	Math
83	Computer

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First Normal Form

First Normal Form (1NF)

• A relation will be 1NF if it contains an atomic value.
• It states that an attribute of a table cannot hold multiple values. It must hold only The first single-valued attribute.
• First normal form disallows the multi-valued attribute, composite attribute, and their combinations.

Example: Relation EMPLOYEE is not in 1NF because of the multi-valued attribute EMP_PHONE.

EMPLOYEE table:

EMP_ID	EMP_NAME	EMP_PHONE	EMP_STATE
14	John	7272826385, 9064738238	UP
20	Harry	8574783832	Bihar
12	Sam	7390372389, 8589830302	Punjab

The decomposition of the EMPLOYEE table into 1NF has been shown below:

EMP_ID	EMP_NAME	EMP_PHONE	EMP_STATE
14	John	7272826385	UP
14	John	9064738238	UP
20	Harry	8574783832	Bihar
12	Sam	7390372389	Punjab
12	Sam	8589830302	Punjab

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Normalization

• Normalization is the process of organizing the data in the database.
• Normalization is used to minimize the redundancy from a relation or set of relations. It is also used to eliminate, undesirable characteristics like Insertion, Update and Deletion Anomalies.
• Normalization divides the larger table into the smaller table and links them using relationships.
• The normal form is used to reduce redundancy from the database table.

Types of Normal Forms

There are the four types of normal forms:

Normal Form	Description
1NF	A relation is in 1NF if it contains an atomic value.
2NF	A relation will be in 2NF if it is in 1NF and all non-key attributes are fully functional dependent on the primary key.
3NF	A relation will be in 3NF if it is in 2NF and no transition dependency exists.
4NF	A relation will be in 4NF if it is in Boyce Codd's normal form and has no multi-valued dependency.
5NF	A relation is in 5NF if it is in 4NF and does not contain any join dependency and joining should be lossless.

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Inference Rule

 

Inference Rule (IR):

• The Armstrong's axioms are the basic inference rule.
• Armstrong's axioms are used to conclude functional dependencies on a relational database.
• The inference rule is a type of assertion. It can apply to a set of FD(functional dependency) to derive other FD.
• Using the inference rule, we can derive additional functional dependency from the initial set.

The Functional dependency has 6 types of inference rule:

1. Reflexive Rule (IR₁)

In the reflexive rule, if Y is a subset of X, then X determines Y.

1. If X ⊇ Y then X  →    Y  

Example:

1. X = {a, b, c, d, e}  
2. Y = {a, b, c}  

2. Augmentation Rule (IR₂)

The augmentation is also called as a partial dependency. In augmentation, if X determines Y, then XZ determines YZ for any Z.

1. If X    →  Y then XZ   →   YZ   

Example:

1. For R(ABCD),  if A   →   B then AC  →   BC  

3. Transitive Rule (IR₃)

In the transitive rule, if X determines Y and Y determine Z, then X must also determine Z.

1. If X   →   Y and Y  →  Z then X  →   Z    

4. Union Rule (IR₄)

Union rule says, if X determines Y and X determines Z, then X must also determine Y and Z.

1. If X    →  Y and X   →  Z then X  →    YZ     

Proof:

1. X → Y (given)
2. X → Z (given)
3. X → XY (using IR₂ on 1 by augmentation with X. Where XX = X)
4. XY → YZ (using IR₂ on 2 by augmentation with Y)
5. X → YZ (using IR₃ on 3 and 4)

5. Decomposition Rule (IR₅)

Decomposition rule is also known as project rule. It is the reverse of union rule.

This Rule says, if X determines Y and Z, then X determines Y and X determines Z separately.

1. If X   →   YZ then X   →   Y and X  →    Z   

Proof:

1. X → YZ (given)
2. YZ → Y (using IR₁ Rule)
3. X → Y (using IR₃ on 1 and 2)

6. Pseudo transitive Rule (IR₆)

In Pseudo transitive Rule, if X determines Y and YZ determines W, then XZ determines W.

1. If X   →   Y and YZ   →   W then XZ   →   W   

Proof:

1. X → Y (given)
2. WY → Z (given)
3. WX → WY (using IR₂ on 1 by augmenting with W)
4. WX → Z (using IR₃ on 3 and 2)

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Functional Dependency

Functional dependency is a relationship that exists between two attributes. It typically exists between the primary key and non-key attribute within a table.

1. X   →   Y  

The left side of FD is known as a determinant, and the right side of the production is known as a dependent.

For example:

Assume we have an employee table with Emp_Id, Emp_Name, Emp_Addressand  attributes.

Here Emp_Id attribute can uniquely identify the Emp_Name attribute of the employee table because if we know the Emp_Id, we can tell that the employee name is associated with it.

Functional dependency can be written as:

1. Emp_Id → Emp_Name   

We can say that Emp_Name is functionally dependent on Emp_Id.

Types of Functional dependency

1. Trivial functional dependency

• A → B has trivial functional dependency if B is a subset of A.
• The following dependencies are also trivial: A → A, B → B

Example:

1. Consider a table with two columns Employee_Id and Employee_Name.  
2. {Employee_id, Employee_Name}   →    Employee_Id is a trivial functional dependency as   
3. Employee_Id is a subset of {Employee_Id, Employee_Name}.  
4. Also, Employee_Id → Employee_Id and Employee_Name   →    Employee_Name are trivial dependencies too.  

2. Non-trivial functional dependency

• A → B has a non-trivial functional dependency if B is not a subset of A.
• When A intersection B is NULL, then A → B is called a complete non-trivial.

Example:

1. ID   →    Name,  
2. Name   →    DOB  

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