The Fifth Normal Form (5NF) is a concept in database theory aimed at structuring database tables to minimize redundancy and anomalies. The 5NF builds upon the previous normal forms, particularly the Fourth Normal Form (4NF).

In 5NF, join dependencies are taken into account. A join dependency occurs when two or more attributes in a table depend on each other, but not directly; rather, they are connected through another table via a join operation.

A table is in 5NF if it is in 4NF and does not have any non-trivial join dependencies. Trivial join dependencies are those that are already implied by the primary key or superkeys. Non-trivial join dependencies indicate an additional relationship between the attributes that is not determined by the keys.

Applying 5NF helps further normalize databases and optimize their structure, leading to better data integrity and consistency.

The Fourth Normal Form (4NF) is a concept in database theory aimed at structuring database tables to reduce redundancy and anomalies. It builds upon the principles of the first three normal forms (1NF, 2NF, and 3NF).

The 4NF aims to address Multivalued Dependency (MVD), which occurs when a table contains attributes that do not depend on a primary key but are related to each other beyond the primary key. When a table is in 4NF, it means it is in 3NF and does not contain MVDs.

In practice, this means that in a 4NF table, each non-key attribute combination is functionally dependent on every one of its superkeys, where a superkey is a set of attributes that uniquely identifies a tuple in the table. Achieving 4NF can make databases more efficiently designed by minimizing redundancies and maximizing data integrity.

The Boyce-Codd Normal Form (BCNF) is a normalization form in relational database theory that aims to eliminate redundancy and anomalies in a database. It is a stricter form of the Third Normal Form (3NF) and is often considered an extension of it.

A relation (table) is in Boyce-Codd Normal Form if it meets the following conditions:

1. The relation is in Third Normal Form (3NF): This means it is already in First and Second Normal Form, and there are no transitive dependencies between the attributes.

2. Every non-trivial functional dependency $X\to Y$ has a superkey as the determinant: This means that for every functional dependency where $X$ is the set of attributes determining $Y$, $X$ must be a superkey. A superkey is a set of attributes that can uniquely identify the entire relation.

Differences from Third Normal Form (3NF)

While Third Normal Form requires that any attribute not part of the primary key must be directly dependent on it (not transitively through another attribute), BCNF goes a step further. It requires that all determinants (the left-hand side of functional dependencies) must be superkeys.

Example

Consider a relation R with attributes A, B, and C, and the following functional dependencies:

• $A\to B$
• $B\to C$

To check if this relation is in BCNF, we proceed as follows:

• We observe that $A\to B$ is not problematic if $A$ is a superkey.
• However, $B\to C$ is problematic if $B$ is not a superkey, as $B$ in this case cannot uniquely identify the entire relation.

If $B$ is not a superkey, the relation is not in BCNF and must be decomposed into two relations to meet BCNF requirements:

• One relation containing $B$ and $C$
• Another relation containing $A$ and $B$

Summary

The Boyce-Codd Normal Form is stricter than the Third Normal Form and ensures that there are no functional dependencies where the left-hand side is not a superkey. This helps to avoid redundancy and anomalies in the database structure and ensures data integrity.

The Third Normal Form (3NF) is a stage in database normalization aimed at minimizing redundancies and ensuring data integrity. A relation (table) is in Third Normal Form if it satisfies the following conditions:

1. The relation is in Second Normal Form (2NF):

   • This means the relation is in First Normal Form (1NF) (all attribute values are atomic, no repeating groups).
   • All non-key attributes are fully functionally dependent on the entire primary key, not just part of it.

2. No transitive dependencies:

   • No non-key attribute depends transitively on a candidate key. This means a non-key attribute should not depend on another non-key attribute.

In detail, for a relation $R$ to be in 3NF, for every non-key attribute $A$ and every candidate key $K$ in $R$, the following condition must be met:

$\text{If }A\text{ depends on }K,\text{ then }A\text{ should depend directly on }K$$\text{and not through another attribute}B\text{ that itself depends on }K.$

Example:

Suppose we have a Students table with the following attributes:

• Student_ID (Primary Key)
• Name
• Course_ID
• Course_Name
• Instructor

In this table, the attributes Course_Name and Instructor might depend on Course_ID, not directly on Student_ID. This is an example of a transitive dependency because:

• Student_ID → Course_ID
• Course_ID → Course_Name, Instructor

To convert this table to 3NF, we eliminate transitive dependencies by splitting the table. We could create two tables:

1. Students:

   • Student_ID (Primary Key)
   • Name
   • Course_ID

2. Courses:

   • Course_ID (Primary Key)
   • Course_Name
   • Instructor

Now, both tables are in 3NF because each non-key attribute depends directly on the primary key and there are no transitive dependencies.

By achieving Third Normal Form, data consistency is increased, and redundancies are reduced, which improves the efficiency of database operations.