Book Structure on Fuzzy Sets Extension and Application

A book on Fuzzy Sets Extension and Application typically follows a structure that first establishes the fundamental concepts, then delves into extensions and applications, according to O'Reilly's "Fuzzy Set and Its Extension" and a book on engineering applications of fuzzy logic. 

This includes introductory sections on fuzzy sets, membership functions, and fuzzy relations, followed by more advanced topics like fuzzy numbers, fuzzy integrals, and fuzzy logic. The book then transitions to practical applications, demonstrating how these concepts are used in decision-making, image processing, control systems, and other fields. 

Here's a more detailed breakdown of a typical book structure:

Part 1: Fundamentals of Fuzzy Sets

• Chapter 1: Introduction to Fuzzy Sets:

  This chapter would define fuzzy sets, contrast them with classical (crisp) sets, and introduce the concept of membership functions.
• Chapter 2: Mathematical Representation and Operations:

  This would cover the mathematical notation and operations on fuzzy sets, including union, intersection, complement, and other fuzzy set operators.
• Chapter 3: Fuzzy Relations:

  This chapter would discuss fuzzy relations, their properties, and how they relate to fuzzy sets.
• Chapter 4: Fuzzy Numbers and Arithmetic:

  This section would introduce fuzzy numbers, operations on fuzzy numbers (like addition, subtraction, etc.), and the extension principle. 

Part 2: Extensions of Fuzzy Set Theory

• Chapter 5: Fuzzy Measures and Integrals:

  This chapter would delve into fuzzy measures, fuzzy integrals, and their application in various fields. 
• Chapter 6: Fuzzy Logic:

  This section would cover fuzzy logic principles, fuzzy inference systems, and their use in decision-making and control. 
• Chapter 7: Extensions to Fuzzy Sets:

  This chapter could explore extensions like intuitionistic fuzzy sets, Pythagorean fuzzy sets, or other advanced types. 

Part 3: Applications of Fuzzy Sets

• Chapter 8: Fuzzy Decision Making:

  This would demonstrate how fuzzy sets can be used in decision-making processes, particularly when dealing with incomplete or imprecise information. 
• Chapter 9: Fuzzy Control Systems:

  This section would cover the application of fuzzy sets in control systems, including examples of fuzzy controllers and their design. 
• Chapter 10: Fuzzy Image Processing:

  This would explore how fuzzy sets can be used in image processing tasks, such as image segmentation and feature extraction. 
• Chapter 11: Other Applications:

  This chapter could cover applications in various other domains, such as mathematics, engineering, and other fields. 

Overall Structure:

The book would likely be structured to gradually increase in complexity, starting with basic concepts and building up to more advanced topics. It would also aim to provide a balance between theoretical understanding and practical applications, making it suitable for both students and practitioners in the field.