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**Fuzzy sets, uncertainty, and information.**
*(English)*
Zbl 0675.94025

Englewood Cliffs, NJ: Prentice Hall. xi, 355 p. $ 57.95 (1988).

This comprehensive book deals with the application of fuzzy set theory and fuzzy measures as a suitable mathematical environment for characterizing the concept of uncertainty and its relationship to the increasingly important concepts of information and complexity. The presentation of the different concepts is partioned into six chapters each of them supplemented with lots of step-by-step examples and additional notes for an in-depth look at the issues of the developed theory.

The first three chapters give a pleasant introduction to fuzzy set theory and its connection with fuzzy logic. Chapter 4 contains a discussion of fuzzy measures, especially the dual classes of belief and plausibility measures including some of their special subclasses which are measures of possibility, necessity, and probability. A detailed investigation of classical and new measures of uncertainty and their relationship to information and complexity is given in chapter 5. So Hartley information, Shannon entropy, and Boltzmann entropy are overviewed to generalize them for handling various aspects of uncertainty like dissonance, confusion, and nonspecifity of evidence within the framework of belief and plausibility measures. The last chapter shows numerous interesting examples for successful applications of the considered theory. The applications range from civil engineering, medicine, and computer science to social sciences, management, and decision making.

In presenting the theoretical concepts the authors have attained a carefully chosen synthesis between a pure mathematical and an application-related point of view. Furthermore it should be emphasized that the book contains some original results, particularly in the areas of uncertainty and information measures.

The first three chapters give a pleasant introduction to fuzzy set theory and its connection with fuzzy logic. Chapter 4 contains a discussion of fuzzy measures, especially the dual classes of belief and plausibility measures including some of their special subclasses which are measures of possibility, necessity, and probability. A detailed investigation of classical and new measures of uncertainty and their relationship to information and complexity is given in chapter 5. So Hartley information, Shannon entropy, and Boltzmann entropy are overviewed to generalize them for handling various aspects of uncertainty like dissonance, confusion, and nonspecifity of evidence within the framework of belief and plausibility measures. The last chapter shows numerous interesting examples for successful applications of the considered theory. The applications range from civil engineering, medicine, and computer science to social sciences, management, and decision making.

In presenting the theoretical concepts the authors have attained a carefully chosen synthesis between a pure mathematical and an application-related point of view. Furthermore it should be emphasized that the book contains some original results, particularly in the areas of uncertainty and information measures.

Reviewer: R.Kruse

### MSC:

94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |

94-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory |

94A17 | Measures of information, entropy |