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**Introduction to fuzzy arithmetic. Theory and applications.**
*(English)*
Zbl 0588.94023

Van Nostrand Reinhold Electrical/Computer Science and Engineering Series. New York: Van Nostrand Reinhold Company. XVII, 351 p. £49.45 (1985).

This book follows the well known style of other books of the first author [for instance, see ”Introduction à la théorie des sous-ensembles flous”, Vol. I (1973; Zbl 0302.02023), Vol. II (1975; Zbl 0456.68081), Vol. III (1975; Zbl 0314.94001), and Vol. IV (1977; Zbl 0346.94002)]. The second author is a well-known editor of several books on Fuzzy Set Theory (see cited bibliography).

This book is devoted mainly to give a graduate approach to Fuzzy Arithmetic, based on the notion of fuzzy number by means of the extension of the concept of interval of confidence, that has its background in statistics. Namely, the authors take for an interval of confidence a closed interval \([a_ 1,a_ 2]\) of the set of real numbers in which a determined information available locates an uncertain value. \(a_ 1\) and \(a_ 2\) can be eventually infinite.

The book has four chapters. In Chapter 1, the authors firstly introduce between two intervals of confidence the operations of addition, subtraction, multiplication and division. Successively, these operations are extended to the fuzzy numbers, level by level. The operators of minimum and maximum, convolutions and deconvolutions are also defined and analyzed in detail. The L-R fuzzy numbers of D. Dubois and H. Prade [”Fuzzy sets and systems. Theory and applications” (1980; Zbl 0444.94049)] are also studied, with particular attention on triangular fuzzy numbers.

Chapter 2 deals with novel concepts such as hybrid numbers, that are viewed in the twofold, possibilistic and probabilistic aspect. Related operations between hybrid numbers and several results are established. Some applications are performed, such as the simulation of an hybrid number using a Monte Carlo method. Further, this chapter develops the theory of random fuzzy numbers, fuzzy numbers of type 2 and of higher dimension.

In Chapter 3, certain properties of arithmetic and combinatorics of uncertain numbers, such as fuzzy relative integers modulo \(n\), fuzzy real numbers modulo 1, are presented. Further, factorials, sequences, series and functions of fuzzy numbers are defined and studied. An approach to fuzzy trigonometric and hyperbolic functions and to fuzzy complex numbers is presented in the last section.

Chapter 4 deals mainly with catastrophe theory using fuzzy numbers. Three appendices complete the book. All the arguments are illustrated with the help of suitable numerical examples, diagrams and figures. The bibliography is essential. (A rich list of references can be found in the good survey of D. Dubois and (INVALID INPUT) H. Prade [The analysis of fuzzy information, J. C. Bezdek (Ed.), CRC Press (1985)].)

This book is devoted mainly to give a graduate approach to Fuzzy Arithmetic, based on the notion of fuzzy number by means of the extension of the concept of interval of confidence, that has its background in statistics. Namely, the authors take for an interval of confidence a closed interval \([a_ 1,a_ 2]\) of the set of real numbers in which a determined information available locates an uncertain value. \(a_ 1\) and \(a_ 2\) can be eventually infinite.

The book has four chapters. In Chapter 1, the authors firstly introduce between two intervals of confidence the operations of addition, subtraction, multiplication and division. Successively, these operations are extended to the fuzzy numbers, level by level. The operators of minimum and maximum, convolutions and deconvolutions are also defined and analyzed in detail. The L-R fuzzy numbers of D. Dubois and H. Prade [”Fuzzy sets and systems. Theory and applications” (1980; Zbl 0444.94049)] are also studied, with particular attention on triangular fuzzy numbers.

Chapter 2 deals with novel concepts such as hybrid numbers, that are viewed in the twofold, possibilistic and probabilistic aspect. Related operations between hybrid numbers and several results are established. Some applications are performed, such as the simulation of an hybrid number using a Monte Carlo method. Further, this chapter develops the theory of random fuzzy numbers, fuzzy numbers of type 2 and of higher dimension.

In Chapter 3, certain properties of arithmetic and combinatorics of uncertain numbers, such as fuzzy relative integers modulo \(n\), fuzzy real numbers modulo 1, are presented. Further, factorials, sequences, series and functions of fuzzy numbers are defined and studied. An approach to fuzzy trigonometric and hyperbolic functions and to fuzzy complex numbers is presented in the last section.

Chapter 4 deals mainly with catastrophe theory using fuzzy numbers. Three appendices complete the book. All the arguments are illustrated with the help of suitable numerical examples, diagrams and figures. The bibliography is essential. (A rich list of references can be found in the good survey of D. Dubois and (INVALID INPUT) H. Prade [The analysis of fuzzy information, J. C. Bezdek (Ed.), CRC Press (1985)].)

Reviewer: Salvatore Sessa (Napoli)

### MSC:

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

03E72 | Theory of fuzzy sets, etc. |

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

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |