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Fuzzy sets and fuzzy logic. The foundations of application - from a mathematical point of view. (German) Zbl 0782.94025
Artificial Intelligence. Braunschweig: Vieweg. viii, 216 p. (1993).
The book focuses primarily on the theoretical aspects of fuzzy sets and addresses essential methodological issues of their applications in modelling and fuzzy controllers. This clearly identified objective becomes visible throughout the section of the material and its organization. The material is structured into five chapters. The first two of them are of an introductory nature as they contain some logical preliminaries and expose the fundamentals of fuzzy set theory. In particular, they are concentrated on set algebra and fuzzy relations including detailed studies on their types and operations – these are studied in the setting of triangular norms. In addition to the standard relation operations being commonly utilized, some graded properties of the relational calculus are also analyzed. Chapter 3 discusses various types of fuzzy relational equations viewed as set-to-set or relation-to- relation transformtions being realized with the aid of various triangular norms. The material includes solutions to these equations as well as introduces and studies the concept of solvability of the equations. The two remaining chapters are strongly oriented toward applications of fuzzy sets in fuzzy control and fuzzy modelling. Fuzzy controllers (Chapter 4) that are based upon “if-then” conditional control statements with linguistic components are conveniently represented and analyzed as a collection of relational constraints induced by the rules. The analysis contained in this chapter looks at several tasks of the static verification of the fuzzy controller including such crucial aspects ad interactivity of control rules. Finally, Chapter 5 concentrates on the methodological aspects of construction of fuzzy models with the aid of fuzzy relational equations. Here the discussion involves the topics central to the area such as an evaluation of the fuzzy models, approximate solutions to the equations of the model, and a quantification of the direct impact the quality of the model poses on the formulation and solutions of the prediction and control tasks emerging in this framework.
The organization of the material is conspicuous and well-thought. Some main results are summarized in a handy tabular form that makes them easily accessible. The book is definitely useful for all the readers being interested in fuzzy control and fuzzy modelling.

##### MSC:
 94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory) 93C42 Fuzzy control/observation systems 03E72 Theory of fuzzy sets, etc. 03B52 Fuzzy logic; logic of vagueness