Functional Dependency (FD) and Multi-valued Dependency (MVD) are two important concepts in database normalization that help to reduce data redundancy and maintain data consistency.

Functional Dependency (FD):In a relational database, a functional dependency is a relationship between two attributes in a table such that the value of one attribute uniquely determines the value of the other attribute. For example, consider a table of students that includes attributes such as StudentID, Name, and Email. If we assume that each student has a unique email address, then we can say that the Email attribute is functionally dependent on the StudentID attribute, since the value of StudentID uniquely determines the value of Email.

In mathematical notation, we can represent this as StudentID -> Email, where the arrow represents the functional dependency. This means that for any given value of StudentID, there can only be one corresponding value of Email. Functional dependencies are important for database normalization because they help to ensure data consistency and reduce data redundancy.

Multi-valued Dependency (MVD):In a relational database, a multi-valued dependency is a relationship between three or more attributes in a table such that the values of one set of attributes determine the values of another set of attributes, independent of the values of any other attributes. For example, consider a table of employees that includes attributes such as EmployeeID, ProjectID, and Task. If we assume that each employee can work on multiple projects, and each project can have multiple tasks, then we can say that the ProjectID and Task attributes are multi-valued dependent on the EmployeeID attribute.

In mathematical notation, we can represent this as EmployeeID ->-> ProjectID, Task, where the double arrow represents the multi-valued dependency. This means that for any given value of EmployeeID, there can be multiple corresponding values of ProjectID and Task. Multi-valued dependencies are important for database normalization because they help to reduce data redundancy and improve data consistency.

To summarize, FDs and MVDs are two important concepts in database normalization that help to ensure data consistency and reduce data redundancy. Functional dependency is a relationship between two attributes in a table such that the value of one attribute uniquely determines the value of the other attribute, while multi-valued dependency is a relationship between three or more attributes in a table such that the values of one set of attributes determine the values of another set of attributes, independent of the values of any other attributes.