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Understanding functional dependencies is fundamental to grasping relational database design. In essence, a functional dependency exists when the value of one set of attributes precisely determines the value of another set of attributes within a relation. For example, in a classroom database, if each student has a unique ID, then the student ID functionally determines the student's name, as each ID is associated with exactly one name.

To put it mathematically, let's say we have attributes X and Y within a relation R, a functional dependency X → Y means for any two tuples (rows) in R, if they have the same value for X, they must also have the same value for Y.

Why is this important? Well, functional dependencies are the building blocks for database normalization, which aims to minimize redundancy and avoid undesirable characteristics like update anomalies. Identifying these dependencies is essential when you're organizing data in tables to make sure that each piece of information is stored logically and efficiently.

Database normalization is a process applied to relational databases to reduce redundancy and dependency by organizing fields and table of data in order to minimize the potential for data anomalies. The main goal is to divide large tables into smaller, and less redundant, tables and link them using relationships.

The normalization process involves several normal forms, beginning with the first normal form (1NF) and going up to the fifth normal form (5NF). Each 'normal form' is like a level of cleanliness in organizing the database's structure. The basic idea is that by the time you reach the third normal form (3NF), you've eliminated most of the redundancy and update anomalies.

In our example, we determined the closure of functional dependencies to normalize our database. This can help prevent update anomalies such as insertion, update, and deletion anomalies, which can otherwise lead to inconsistent data if not handled properly. Without normalization, you might end up with duplicate data across different tables or have a tough time updating records without affecting others incorrectly.

Relational database design is an intricate process that entails defining a database so that it faithfully represents all aspects of the organization, including its interactions with external entities, its operational requirements, and its possible areas for growth. The aim is to create a detailed blueprint of how data should be structured and interrelated, which is done through a set of rigorous design phases.

The design process is typically iterative and can be broken down into several key stages: requirement analysis, conceptual design, logical design, schema refinement, and physical design. Each stage has its own tools and methodologies to guide the design, such as entity-relationship diagrams (ERDs) for conceptual design or normalization during logical design.

Clear understanding and application of functional dependencies and normalization, as discussed earlier, are paramount during the logical design stage. These concepts ensure that when we eventually create our tables and define relationships among them, each table serves a clear purpose, avoiding unnecessary duplication of data and ensuring the integrity of our database.