This article is a summary of Normalization from the book 'Database Systems' by Thomas Connolly (Publisher: Addison Wesley)

Why Needed?
Normalization is needed in designing Relational database system to minimize data redundancy and update anomalies.

- Minimizes cost (Reduce file storage space)
- Updates can be performed with minimal number of operations

Functional Dependency

• A relationship between attributes

Describes the relationship between attributes in a relation. For example, if A and B are attributes of relation R, B is functionally dependent on A (denoted A -> B), if each value of A is associated with exactly one value of B. (A and B may each consist of  one or more attributes.)

•  A -> B (A is determinant of B)

Type of functional dependency

• Full functional dependency : Indicates that if A and B are attributes of a relation, B is fully functionally dependent on A if B is functionally dependent on A,but not on any proper subset of A.
• Transitive dependency : A condition where A, B, and C are attributes of a relation such that if A->B and B->C, then C is transitively dependent on A via B (provided that A is not functionally dependent on B or C).
• Multi-Valued Dependency (MVD): Represents a dependency between attributes (for example, A, B, and C) in a relation, such that for each value of A there is a set of values for B and a set of values for C. However, the set of values for B and C are independent of each other.
• Join Dependency: Describes a type of dependency. For example, for a relation R with subsets of the attributes of R denoted as A,B, ... , Z, a relation R satisfies a join dependency if and only if every legal value of R is equal to the join of its projections on A, B, ..., Z.

The process of Normalization

Unnormalized Form (UNF): A table that contains one or more repeating groups.

First Normal Form (1NF): A relation in which the intersection of each row and column contains one and only one value.

Second Normal Form (2NF): A relation that is in First Normal Form and every non-primary-key attribute is fully functionally dependent on the primary key.

Third Normal Form (3NF): A relation that is in First and Second Normal Form and in which no non-primary-key attribute is transitively dependent on the primary key.

Fourth Normal Form (4NF): A relation that is in Boyce-Codd normal form and does not contain nontrivial multi-valued dependencies.

Fifth Normal Form (5NF): A relation that has no join dependency.

Armstrong's Axioms

1. Reflexivity : If B is a subset of A, then A -> B
2. Augmentation : If A -> B, then A,C -> B,C
3. Transitivity : If A -> B and B -> C, then A -> C
4. Self-determination : A -> A
5. Decomposition : If A -> B,C, then A -> B and A -> C
6. Union : If A -> B and A -> C, then A -> B,C
7. Composition : If A -> B and C -> D then A,C -> B,D