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Database normalization is a systematic approach to organizing data in a database to reduce redundancy and improve data integrity. The process involves dividing large tables into smaller, more manageable pieces, which can help prevent data anomalies and ensure that the database reflects real-world structures and relations.

Normalization typically progresses through several stages, called 'normal forms,' each with specific criteria that must be met. Starting from the First Normal Form (1NF), which eliminates duplicate columns and creates unique identifiers for each row, to higher levels such as the Second (2NF) and Third Normal Forms (3NF), which further decompose the data structure, normalization seeks to streamline database design.

The aim of normalization is not only to avoid unnecessary duplication but also to make the database more intuitive and easier to maintain. It helps in optimizing storage space and ensures that queries can be performed quickly and accurately.

Multi-valued dependencies occur in a database table when one attribute determines multiple other attributes independently of any other attributes. To put it more clearly, a multi-valued dependency exists if, for a single value of one attribute, called a determinant, there are multiple possible values of another attribute that are independent of other attributes.

Consider an example of a university course catalog where a course can have multiple textbooks and also multiple instructors. The textbooks for a course are not dependent on the instructors, and vice versa. This is a multi-valued dependency between the course attribute and both the textbooks and instructors attributes.

Multi-valued dependencies can complicate database design and lead to redundancy and maintenance challenges. Eliminating them, or structuring them properly, is critical and forms the basis of achieving Fourth Normal Form, which further refines database design to better handle complex real-world scenarios.

Data integrity refers to the accuracy, consistency, and reliability of data throughout its lifecycle. It is a critical aspect of database management and ensures that the data remains an accurate reflection of the real-world entities it represents. Data integrity encompasses various types of integrity constraints, such as entity integrity, referential integrity, and domain integrity.

Maintaining data integrity involves enforcing data validity through various constraint mechanisms. These can include primary and foreign key constraints that enforce uniqueness and proper relations among tables, check constraints that maintain valid data ranges or formats, and unique constraints that prevent duplicate values in a column.

Ensuring data integrity helps to prevent errors and inconsistencies that could corrupt the data and degrade the quality and reliability of information gleaned from the database. It plays a vital role in operational decision-making and strategic planning, ultimately affecting the trustworthiness of the system as a whole.