Functional Dependency and Normalization

Contents

**Recall the basics of Relational Schema** 1

**Describe design issues of Relational Schema** 2

**Recall the term Functional Dependency** 3

**Recall the rules of Inference and Armstrong’s Axioms** 4

**Describe Attribute Closure and its use to find: a) Membership set, b) Closure of FD set** 5

**Explain the equivalence of 2 FDs** 6

**Recall the way to find Minimal Cover** 7

**Recall the steps to find Candidate Key** 8

**Recall the term Normalisation** 9

**Describe First Normal Form (1NF) and Second Normal Form (2NF)** 10

**Recall Third Normal Form (3NF)** 14

**Explain Boyce-Codd Normal Form (BCNF)** 17

**Solve PYQs based on 3NF and BCNF** 18

**Describe additional properties of Normalisation: Lossless Decomposition and Dependency Preserving** 23

**Describe Higher Normal Forms** 24

**Recall Multivalued Dependency and Fourth Normal Form (4NF)** 25

**Recall Fifth Normal Form (5NF)** 26

**Recall the basics of Relational Schema**

A relational schema is a blueprint or design of the structure of a relational database. It defines the tables, columns, and relationships between tables in a database. A table in a relational schema is represented as a relation in mathematical terms, and it contains rows and columns that store data. Each column in a table is called an attribute, and each row is called a tuple. The relationships between tables are defined through foreign keys, which are attributes that reference the primary key of another table.

Key points about relational schema:

- Tables: A relational schema consists of multiple tables, each representing a specific entity or concept. Tables are composed of rows (tuples) and columns (attributes) that store data. Each table has a unique name.
- Attributes: Attributes are the columns of a table, and they define the specific data elements that can be stored for each record. Attributes have names and data types, such as integer, string, date, etc.
- Tuples: Tuples are the rows of a table, and each tuple represents a single record or instance of the entity being modeled. Each tuple consists of values for each attribute defined in the table.
- Primary Key: The primary key is a unique identifier for each tuple in a table. It ensures that each record in the table can be uniquely identified. The primary key can be composed of one or more attributes, and its values must be unique and non-null.
- Foreign Key: A foreign key is an attribute in one table that refers to the primary key of another table. It establishes relationships between tables and helps maintain data integrity. The foreign key values in one table match the primary key values in the related table, creating a link between the two tables.
- Relationships: Relationships between tables are defined through foreign keys. Common relationship types include one-to-one, one-to-many, and many-to-many. These relationships help represent the associations and dependencies between different entities.
- ER Diagram: An entity-relationship (ER) diagram is a graphical representation of a relational schema. It visually depicts the tables, their attributes, and the relationships between them. ER diagrams are useful for understanding the structure of the database and for designing the schema.

The relational schema serves as a foundation for creating and managing databases in a structured and organized manner. It provides a clear representation of the data model, allowing for efficient data storage, retrieval, and manipulation.

**Describe design issues of Relational Schema**

Designing a relational schema involves making several important decisions to ensure the efficiency, integrity, and scalability of the database. Here are some key design issues to consider when designing a relational schema:

- Entity Identification: Determine the entities that need to be represented in the database and identify the key attributes that uniquely identify each entity. This involves identifying primary keys for each table and ensuring they are unique and non-null.
- Attribute Specification: Define the attributes for each table, determining the data type, size, and constraints for each attribute. Consider the appropriate data types that accurately represent the data and ensure data integrity.
- Relationship Establishment: Establish relationships between tables to represent associations and dependencies between entities. Determine the relationship types (one-to-one, one-to-many, many-to-many) and use foreign keys to link related tables.
- Normalization: Apply normalization techniques to eliminate data redundancy and improve data integrity. Normalization involves breaking down tables into smaller, well-structured tables to minimize data duplication and maintain data consistency.
- Indexing: Determine the appropriate columns to be indexed for efficient data retrieval. Indexing improves query performance by creating data structures that allow for faster searching and sorting.
- Denormalization: Consider denormalization techniques to optimize query performance for complex queries involving multiple tables. Denormalization involves reintroducing redundancy to tables to improve performance at the cost of some data redundancy.
- Integrity Constraints: Define integrity constraints such as primary key constraints, foreign key constraints, and other constraints to enforce data consistency and prevent data inconsistencies or anomalies.
- Performance Optimization: Consider performance optimization techniques such as query optimization, proper indexing, partitioning, and caching to enhance the performance of the database system.
- Scalability: Design the schema with scalability in mind to accommodate future growth and increasing data volumes. Consider factors such as table partitioning, distributed databases, and replication strategies.
- Security: Implement appropriate security measures to protect the database, including user authentication, authorization, and data encryption. Define access controls and permissions to ensure data confidentiality and integrity.
- Documentation: Document the schema design, including the rationale behind design decisions, entity-relationship diagrams, and any specific business rules or constraints.

Designing a relational schema involves finding the right balance between data organization, data integrity, query performance, and scalability. It requires a thorough understanding of the business requirements, data relationships, and optimization techniques to create an efficient and robust database system.

**Recall the term Functional Dependency**

Functional dependency is a fundamental concept in database management systems (DBMS) that describes the relationship between attributes in a relation (table). It defines the dependency between two sets of attributes, where one set of attributes determines the values of another set of attributes.

In a functional dependency, if the values of one set of attributes (known as the determinant) uniquely determine the values of another set of attributes (known as the dependent), then it is said that there is a functional dependency between them. This means that for any given value of the determinant, there is only one possible value for the dependent.

Functional dependencies are represented using arrow notation: A -> B, where A represents the determinant and B represents the dependent. This notation indicates that the values of attributes in A uniquely determine the values of attributes in B.

Functional dependencies are important in database design and normalization. They help identify and eliminate data redundancy and anomalies by organizing data into properly structured tables. By understanding the functional dependencies, one can determine the appropriate table structures, keys, and relationships to ensure data integrity and efficiency in the database.

Functional dependencies can be categorized into different types, including:

- Full functional dependency: A functional dependency is considered full if removing any attribute from the determinant will break the dependency. In other words, all attributes in the determinant are necessary for determining the dependent attributes.
- Partial functional dependency: A functional dependency is considered partial if removing some attributes from the determinant will still maintain the dependency. In this case, only a subset of attributes in the determinant is necessary for determining the dependent attributes.
- Transitive functional dependency: A transitive functional dependency occurs when there is a chain of dependencies between attributes. If A -> B and B -> C, then it implies that A -> C. This transitive dependency allows for inferring the dependency between attributes indirectly.

Identifying and understanding functional dependencies is crucial for database design, normalization, and query optimization. It helps ensure data integrity, minimize data redundancy, and improve the efficiency of data retrieval and modification operations.

**Recall the rules of Inference and Armstrong’s Axioms**

Inference Rules:

- Reflexivity: If X is a set of attributes, then X -> X.
- Augmentation: If X -> Y, then XZ -> YZ for any set of attributes Z.
- Transitivity: If X -> Y and Y -> Z, then X -> Z.

Armstrong’s Axioms (Closure Rules):

- Union: If X -> Y and X -> Z, then X -> YZ.
- Decomposition: If X -> YZ, then X -> Y and X -> Z.
- Pseudotransitivity: If X -> Y and WY -> Z, then WX -> Z.

These rules and axioms are used in functional dependency inference, which is the process of determining additional functional dependencies from a given set of functional dependencies. By applying these rules, we can derive new functional dependencies based on the existing ones.

The rules of inference allow us to make logical deductions about functional dependencies. For example, the transitivity rule states that if A -> B and B -> C, then we can infer that A -> C. This is based on the logical transitivity property.

Armstrong’s axioms, also known as closure rules, are used to derive additional functional dependencies from a given set of functional dependencies. They allow us to determine all the implied functional dependencies based on the given ones.

By applying these rules and axioms, we can systematically analyze the functional dependencies in a database schema and identify all the relevant dependencies. This information is crucial for database normalization and ensuring data integrity and consistency.

**Describe Attribute Closure and its use to find: a) Membership set, b) Closure of FD set**

Attribute closure is a concept used in the field of database management to determine the complete set of attributes that can be inferred from a given set of functional dependencies. It helps in understanding the dependencies among the attributes and ensuring data integrity and consistency.

a) Membership Set:

The membership set refers to the set of attributes that can be determined directly or indirectly from a given set of attributes using the given functional dependencies. To find the membership set, we start with the given set of attributes and iteratively apply the closure rules (Armstrong’s axioms) until no new attributes can be added to the set.

Here’s a step-by-step process to find the membership set:

- Start with the given set of attributes.
- Apply the closure rules to generate new attributes based on the given functional dependencies.
- Repeat the previous step until no new attributes can be added to the set.
- The resulting set of attributes is the membership set.

b) Closure of FD Set:

The closure of a set of functional dependencies (FDs) refers to the complete set of functional dependencies that can be derived from the given set of FDs using the closure rules. It helps in understanding all the implicit dependencies in the database schema.

To find the closure of an FD set, follow these steps:

- Start with the given set of FDs.
- Apply the closure rules (Armstrong’s axioms) to generate new FDs based on the given FDs.
- Repeat the previous step until no new FDs can be added to the set.
- The resulting set of FDs is the closure of the given FD set.

By finding the closure of an FD set, we can identify all the dependencies that exist in the database schema. This information is crucial for tasks such as database normalization, where we aim to eliminate redundant data and ensure data integrity.

**Explain the equivalence of 2 FDs**

The equivalence of two functional dependencies (FDs) means that they have the same closure, i.e., they imply the same set of functional dependencies. In other words, if two FDs are equivalent, they provide the same information and can be used interchangeably.

To demonstrate the equivalence of two FDs, let’s consider the following example:

Given FDs:

- A -> B
- AB -> C

We want to determine if these two FDs are equivalent.

Step 1: Closure of FD 1 (A -> B)

Starting with the given FD, we can apply the closure rules to find the closure of FD 1:

A+ = {A} (initially)

Applying FD 1 (A -> B), we add B to the closure:

A+ = {A, B}

Step 2: Closure of FD 2 (AB -> C)

Similarly, we apply the closure rules to find the closure of FD 2:

AB+ = {A, B} (initially)

Applying FD 2 (AB -> C), we add C to the closure:

AB+ = {A, B, C}

Step 3: Comparing the closures

Now, we compare the closures of FD 1 and FD 2:

Closure of FD 1: {A, B}

Closure of FD 2: {A, B, C}

Since the closures are not the same, we can conclude that FD 1 (A -> B) and FD 2 (AB -> C) are not equivalent.

**Recall the way to find Minimal Cover**

The minimal cover of a set of functional dependencies is the smallest equivalent set of functional dependencies that can be derived from the original set. It is important to find the minimal cover because it reduces redundancy and simplifies the design of the database schema.

To find the minimal cover, we follow these steps:

- Remove any redundant dependencies: If we have a dependency X → Y and we can derive X → Z from the same set of dependencies, then we can remove X → Y.
- Remove any extraneous attributes: If we have a dependency X → Y and Y contains more than one attribute, we can remove any attributes in Y that are not strictly necessary to determine X.
- Eliminate transitive dependencies: If we have a dependency X → Y and Y → Z, we can eliminate the transitive dependency by replacing X → Y with X → Z.
- Repeat steps 1

Given FDs:

- A -> B
- B -> C

To find the minimal cover, we eliminate redundant dependencies and transitive dependencies.

Step 1: Eliminate Redundant Dependencies

In this case, no redundant dependencies exist.

Step 2: Eliminate Transitive Dependencies

By applying the transitivity property, we can infer a new dependency:

A -> C

Since we have A -> B and B -> C, we can eliminate the intermediate dependency B. Thus, the minimal cover is:

A -> C

**Recall the steps to find Candidate Key**

A candidate key is a set of attributes that can uniquely identify each tuple in a relation. To find the candidate key of a relation, we follow these steps:

- Identify all the functional dependencies in the relation.
- Find the closure of each attribute set. If the closure of an attribute set includes all the attributes in the relation, then that set is a candidate key.
- If we have multiple candidate keys, we can choose any one of them to be the primary key.

For example, suppose we have a relation R(A,B,C,D) with the following functional dependencies: AB → CD, C → B, D → A. To find the candidate key, we start by finding the closure of each attribute set. The closures are:

- Closure of A: AD
- Closure of B: BC
- Closure of C: BC
- Closure of D: AD

Since the closures of AB, AC, and AD include all the attributes in the relation, these are the candidate keys of the relation. We can choose any one of them to be the primary key.

**Recall the term Normalisation**

Normalization is the process of organizing the data in a database into tables that are free of redundancy and dependency. The main goal of normalization is to eliminate data redundancy, which occurs when the same information is stored in multiple places in the database. Redundancy can lead to data inconsistencies and anomalies, which can cause problems in data retrieval and maintenance.

Normalization is based on a set of rules known as normal forms, which specify how the data should be organized in the tables. The most commonly used normal forms are first normal form (1NF), second normal form (2NF), and third normal form (3NF).

First normal form (1NF) requires that each table has a primary key, and that each column in the table contains atomic values (i.e., values that cannot be further divided). This eliminates repeating groups and ensures that each record in the table can be uniquely identified.

Second normal form (2NF) requires that each non-key attribute in the table is dependent on the entire primary key, and not just on part of it. This eliminates partial dependencies and ensures that each piece of data in the table is only stored once.

Third normal form (3NF) requires that each non-key attribute in the table is dependent only on the primary key, and not on other non-key attributes. This eliminates transitive dependencies and ensures that each piece of data in the table is only stored in one place.

Normalization is an iterative process, and a database may need to be normalized to different levels depending on the specific requirements of the application. Normalization can help to improve the performance, reliability, and maintainability of a database.

**Describe First Normal Form (1NF) and Second Normal Form (2NF)**

- First Normal Form (1NF):

First Normal Form (1NF) is the fundamental level of database normalization. It requires that each column in a table contains only atomic values, meaning that each value should be indivisible. Additionally, each column must have a unique name, and the order of the rows is generally unimportant.

Example:

Consider a table called “Students” with the following structure:

Student ID | Student Name | Subjects |

1 | John | Math, Science |

2 | Emily | English, History |

3 | Mark | Math, English |

This table does not satisfy the 1NF because the “Subjects” column contains multiple values separated by commas. To bring it into 1NF, we need to create a separate row for each subject for each student:

Student ID | Student Name | Subject |

1 | John | Math |

1 | John | Science |

2 | Emily | English |

2 | Emily | History |

3 | Mark | Math |

3 | Mark | English |

Now each column contains atomic values, and the table satisfies the 1NF.

2. Second Normal Form (2NF):

Second Normal Form (2NF) builds upon the requirements of 1NF. In addition to satisfying 1NF, a table must also ensure that each non-key attribute is fully functionally dependent on the entire primary key, not just part of it. This means that any attribute that depends on only a portion of the primary key should be moved to a separate table.

Example

Original Table: Employees

Employee ID |
Employee Name |
Department |
Department Head |
Department Location |

1 | John Doe | Sales | Jane Smith | New York |

2 | Jane Smith | Marketing | Mark Johnson | Los Angeles |

3 | Mike Johnson | Sales | Jane Smith | New York |

4 | Sarah Brown | Human Resources | Tom Davis | Chicago |

Normalized Tables:

Table 1: Employees

Employee ID | Employee Name | Department |

1 | John Doe | Sales |

2 | Jane Smith | Marketing |

3 | Mike Johnson | Sales |

4 | Sarah Brown | Human Resources |

Table 2: Departments

Department | Department Head | Department Location |

Sales | Jane Smith | New York |

Marketing | Mark Johnson | Los Angeles |

Human Resources | Tom Davis | Chicago |

In this normalized form, the original “Employees” table has been split into two separate tables: “Employees” and “Departments.” The “Employees” table contains the employee-specific information, and the “Departments” table contains department-related information.

By doing so, we eliminate the partial dependency where the Department Head and Department Location attributes are dependent only on the Department column. Now, each table represents a single entity, and we can retrieve the necessary information by performing joins between the tables using the appropriate foreign keys.

**Recall Third Normal Form (3NF)**

Third Normal Form (3NF) is a level of database normalization that builds upon the concepts of the First Normal Form (1NF) and Second Normal Form (2NF). It eliminates transitive dependencies and ensures that every non-key attribute depends only on the primary key of the table.

To satisfy the requirements of 3NF, a table must meet the following criteria:

- It should be in 2NF.
- There should be no transitive dependencies, which means that non-key attributes should not depend on other non-key attributes.

In other words, 3NF requires that all non-key attributes in a table are dependent only on the primary key and not on other non-key attributes.

If a table violates 3NF, it can be further decomposed into multiple tables to remove the transitive dependencies and achieve 3NF.

It’s important to note that achieving higher levels of normalization, such as 3NF, can sometimes result in more complex database structures and additional join operations when retrieving data. Therefore, the level of normalization should be determined based on the specific requirements and trade-offs of the application or system.

Here’s the example in tabular form:

Original Table: Orders

Order ID | Customer Name | Customer Address | Product Name | Product Price | Product Category |

1 | John Doe | 123 Main St | Laptop | $1000 | Electronics |

2 | Jane Smith | 456 Elm St | Smartphone | $800 | Electronics |

3 | John Doe | 123 Main St | Printer | $200 | Office Supplies |

4 | Mike Johnson | 789 Oak St | Headphones | $100 | Electronics |

Normalized Tables:

Table 1: Orders

Order ID | Customer Name | Product Name |

1 | John Doe | Laptop |

2 | Jane Smith | Smartphone |

3 | John Doe | Printer |

4 | Mike Johnson | Headphones |

Table 2: Customers

Customer Name | Customer Address |

John Doe | 123 Main St |

Jane Smith | 456 Elm St |

Mike Johnson | 789 Oak St |

Table 3: Products

Product Name | Product Price | Product Category |

Laptop | $1000 | Electronics |

Smartphone | $800 | Electronics |

Printer | $200 | Office Supplies |

Headphones | $100 | Electronics |

In this normalized form, the original “Orders” table has been split into three separate tables: “Orders,” “Customers,” and “Products.” The “Orders” table contains order-specific information, with foreign keys referencing the “Customers” and “Products” tables. The “Customers” table contains customer-related information, and the “Products” table contains product-related information.

By doing so, we eliminate the transitive dependencies where the Customer Address depends on the Customer Name and the Product Category depends on the Product Name. Each table represents a single entity, and non-key attributes depend only on the primary key.

To retrieve information about an order, including customer details and product information, you can perform join operations between the “Orders,” “Customers,” and “Products” tables using the appropriate foreign keys.

**Explain Boyce-Codd Normal Form (BCNF)**

Boyce-Codd Normal Form (BCNF) is a higher level of database normalization that builds upon the concepts of the First Normal Form (1NF), Second Normal Form (2NF), and Third Normal Form (3NF). It further eliminates anomalies and ensures that every determinant is a candidate key.

To satisfy the requirements of BCNF, a table must meet the following criteria:

- It should be in 3NF.
- For every non-trivial functional dependency (X -> Y) in the table, X must be a superkey.

In other words, BCNF requires that every determinant (the attribute or set of attributes on the left side of the functional dependency arrow) must be a candidate key, meaning it uniquely determines all other attributes in the table.

If a table violates BCNF, it can be further decomposed into multiple tables to remove the dependencies that do not satisfy the BCNF conditions.

Achieving BCNF helps eliminate all types of anomalies, such as insertion, deletion, and update anomalies, and ensures a well-structured and dependency-free database design. However, it’s important to note that achieving BCNF may result in additional tables and more complex relationships, which can impact data retrieval and maintenance operations. Therefore, the level of normalization should be determined based on the specific requirements and trade-offs of the application or system.

**Solve PYQs based on 3NF and BCNF**

Q: Consider a relation R (A, B, C, D) and the following functional dependencies:

A → B, B → C, and C → D. Is the relation R in 3NF?

A: The relation R is not in 3NF because it contains transitive dependencies. Specifically, C → D is a transitive dependency, with C being the intermediate attribute. To bring R into 3NF, we can split the table into two tables: one containing the attributes A, B, and C and the other containing the attributes C and D.

Q: Consider a relation R (A, B, C, D, E) and the following functional dependencies:

A → B, B → C, C → D, and D → E. Is the relation R in BCNF?

A: The relation R is not in BCNF because the determinant D → E is not a candidate key. To bring R into BCNF, we can split the table into two tables: one containing the attributes A, B, and C and the other containing the attributes C, D, and E.

Q: Consider a relation R (A, B, C, D) and the following functional dependencies:

A → B, B → C, and A → D. Is the relation R in BCNF?

A: The relation R is not in BCNF because the determinant A → D is not a candidate key. To bring R into BCNF, we can split the table into two tables: one containing the attributes A and D and the other containing the attributes A, B, and C.

Example 1:

Consider the following relation R(A, B, C, D) with the functional dependencies:

A → B

B → C

C → D

a) Determine the candidate keys for relation R.

b) Is relation R in 3NF? If not, decompose it into 3NF.

c) Is relation R in BCNF? If not, decompose it into BCNF.

Solution:

a) To determine the candidate keys for relation R, we need to find the attribute or combination of attributes that uniquely identify each tuple in the relation. In this case, we have the functional dependencies A → B, B → C, and C → D.

The candidate keys for relation R can be found by determining the closure of each attribute:

A+ = {A, B, C, D}

B+ = {B, C, D}

C+ = {C, D}

D+ = {D}

Since A is the only attribute that determines all other attributes, the candidate key for relation R is {A}.

b) To check if relation R is in 3NF, we need to ensure that it is in 2NF and that there are no transitive dependencies. In this case, R is already in 2NF because it does not have any partial dependencies.

However, there is a transitive dependency present: A → B → C → D. To decompose R into 3NF, we need to break it down into separate tables to remove the transitive dependency.

Decomposed Tables:

Table 1: R1(A, B)

Table 2: R2(B, C)

Table 3: R3(C, D)

c) To check if relation R is in BCNF, we need to ensure that for every non-trivial functional dependency X → Y, X is a superkey. In this case, the functional dependencies A → B, B → C, and C → D are all non-trivial.

However, the attribute B is not a superkey since A is a proper subset of B+. Therefore, relation R is not in BCNF.

To decompose R into BCNF, we need to break it down into separate tables based on the functional dependencies.

Decomposed Tables:

Table 1: R1(A, B)

Table 2: R2(B, C)

Table 3: R3(C, D)

In this example, we found the candidate key for relation R, identified that it was not in 3NF due to a transitive dependency, and decomposed it into 3NF. We also identified that it was not in BCNF due to a non-trivial dependency and decomposed it into BCNF using the same decomposition as for 3NF.

Example 2:

Consider the following relation R(A, B, C, D, E) with the functional dependencies:

A → B

B → C

CD → E

a) Determine the candidate keys for relation R.

b) Is relation R in 3NF? If not, decompose it into 3NF.

c) Is relation R in BCNF? If not, decompose it into BCNF.

Solution:

a) To determine the candidate keys for relation R, we need to find the attribute or combination of attributes that uniquely identify each tuple in the relation. In this case, we have the functional dependencies A → B, B → C, and CD → E.

The candidate keys for relation R can be found by determining the closure of each attribute:

A+ = {A, B, C}

B+ = {B, C}

C+ = {C}

D+ = {D}

E+ = {E}

Since A is the only attribute that determines all other attributes, the candidate key for relation R is {A}.

b) To check if relation R is in 3NF, we need to ensure that it is in 2NF and that there are no transitive dependencies. In this case, R is already in 2NF because it does not have any partial dependencies.

However, there is a transitive dependency present: CD → E. To decompose R into 3NF, we need to break it down into separate tables to remove the transitive dependency.

Decomposed Tables:

Table 1: R1(A, B, C)

Table 2: R2(C, D, E)

c) To check if relation R is in BCNF, we need to ensure that for every non-trivial functional dependency X → Y, X is a superkey. In this case, the functional dependencies A → B, B → C, and CD → E are all non-trivial.

The attribute CD is not a superkey since D is not functionally dependent on CD. Therefore, relation R is not in BCNF.

To decompose R into BCNF, we need to break it down into separate tables based on the functional dependencies.

Decomposed Tables:

Table 1: R1(A, B)

Table 2: R2(B, C)

Table 3: R3(C, D, E)

In this example, we determined the candidate key for relation R, identified that it was not in 3NF due to a transitive dependency, and decomposed it into 3NF. We also identified that it was not in BCNF due to a non-trivial dependency and decomposed it into BCNF using the same decomposition as for 3NF.

These examples demonstrate how to analyze a given relation, identify its normal form violations, and decompose it accordingly to achieve higher levels of normalization.

**Describe additional properties of Normalisation: Lossless Decomposition and Dependency Preserving**

In addition to reducing data redundancy and ensuring data consistency, normalisation also has two important properties: lossless decomposition and dependency preservation.

- Lossless Decomposition:

A decomposition of a relation is said to be lossless if the original relation can be reconstructed from the decomposed relations without any loss of data. This means that no information is lost when the original relation is decomposed into smaller relations. In other words, if a relation is decomposed into smaller relations, and the smaller relations are joined back together, the result should be exactly the same as the original relation.

- Dependency Preservation:

A decomposition of a relation is said to be dependency preserving if all the functional dependencies that held in the original relation still hold in the decomposed relations. In other words, if a relation is decomposed into smaller relations, the functional dependencies that existed in the original relation should still exist in the smaller relations. This is important because functional dependencies determine the relationships between attributes in a table and play a critical role in maintaining data consistency.

It is possible for a decomposition to be lossless but not dependency preserving, or dependency preserving but not lossless. However, the ideal situation is when a decomposition is both lossless and dependency preserving. By achieving both properties, we can ensure that the decomposed relations are a valid representation of the original relation and that we have not introduced any new anomalies or inconsistencies in the data.

**Describe Higher Normal Forms**

In addition to the commonly known normal forms (1NF, 2NF, 3NF, and BCNF), there are higher normal forms that aim to further refine the design and eliminate more complex anomalies. These higher normal forms include Fourth Normal Form (4NF), Fifth Normal Form (5NF), and Domain/Key Normal Form (DK/NF). Let’s briefly describe each of these higher normal forms:

- Fourth Normal Form (4NF):

4NF deals with multivalued dependencies, which occur when an attribute set determines multiple independent sets of values. In 4NF, a relation is in 4NF if it is in BCNF and has no non-trivial multivalued dependencies. It eliminates redundancy by separating out sets of multivalued attributes into their own relations, thereby reducing data duplication.

- Fifth Normal Form (5NF):

5NF, also known as Project-Join Normal Form (PJ/NF), addresses join dependencies. A relation is in 5NF if it is in 4NF and has no non-trivial join dependencies. Join dependencies arise when a relation can be reconstructed by joining multiple other relations. In 5NF, relations are decomposed into smaller relations to avoid redundant joins and to preserve data integrity.

- Domain/Key Normal Form (DK/NF):

Domain/Key Normal Form focuses on the integrity of the domain and key constraints within a relation. A relation is in DK/NF if it is in 5NF and every constraint on the relation is a logical consequence of the domains of the attributes and the set of candidate keys. DK/NF ensures that the constraints are fully based on the attribute domains and the key structures, providing a high level of data integrity.

These higher normal forms are progressively more stringent in terms of the anomalies they address and the dependencies they eliminate. However, achieving these higher normal forms may result in a more complex database design and may require careful consideration of trade-offs between normalization and query performance. It is important to analyze the specific requirements and characteristics of the database before deciding on the appropriate level of normalization.

**Recall Multivalued Dependency and Fourth Normal Form (4NF)**

A multivalued dependency (MVD) is a type of dependency that occurs when there is a relationship between sets of attributes in a relation. It represents a situation where one set of attributes can have multiple independent sets of values associated with it.

More formally, a multivalued dependency is defined as follows: Given a relation R with attributes A, B, and C, a multivalued dependency A →→ B holds if for every value of A, there is a set of values for B that is independent of C. This means that for each value of A, there can be multiple sets of values for B that are not influenced or determined by the values of C.

Fourth Normal Form (4NF) is a higher level of database normalization that addresses multivalued dependencies. A relation is in 4NF if it is in Boyce-Codd Normal Form (BCNF) and has no non-trivial multivalued dependencies.

The purpose of achieving 4NF is to eliminate redundancy and anomalies that arise from multivalued dependencies. By decomposing the relation into separate relations, each representing an independent multivalued dependency, we can ensure that the data is organized efficiently and avoids duplication.

To summarize, a multivalued dependency occurs when one set of attributes can have multiple independent sets of values associated with it. 4NF is a normal form that ensures a relation is free from non-trivial multivalued dependencies, leading to a more optimized and normalized database design.

**Recall Fifth Normal Form (5NF)**

Fifth Normal Form (5NF), also known as Project-Join Normal Form (PJ/NF), is a higher level of database normalization that addresses join dependencies. It builds upon the concepts of previous normal forms, such as Boyce-Codd Normal Form (BCNF) and Fourth Normal Form (4NF), to further eliminate redundancy and ensure data integrity.

In 5NF, a relation is considered to be in 5NF if it is in 4NF and has no non-trivial join dependencies. A join dependency arises when a relation can be reconstructed by joining multiple other relations. It represents a relationship between two or more independent relations that need to be joined together to form the original relation.

By decomposing a relation into smaller relations that do not exhibit join dependencies, 5NF helps eliminate data duplication and redundancy. It ensures that data integrity is preserved, as the decomposed relations can be joined back together to reconstruct the original relation without any loss of information.

Achieving 5NF can be a complex task, as it involves carefully analyzing the dependencies and relationships within the database schema. It often requires the identification and decomposition of join dependencies into separate relations, each representing an independent subset of data.

5NF provides a high level of normalization and ensures a well-structured and optimized database design. However, it’s important to note that achieving 5NF may result in a more complex database schema and potentially impact query performance. Therefore, the decision to normalize a relation into 5NF should be based on the specific requirements and trade-offs of the application or system.