##
**Intuitionistic fuzzy sets. Theory and applications.**
*(English)*
Zbl 0939.03057

Studies in Fuzziness and Soft Computing. 35. Heidelberg: Physica-Verlag. xviii, 323 p. (1999).

The book is devoted to the intuitionistic fuzzy set theory and its applications. The exposition is organized in five chapters: 1. Intuitionistic fuzzy sets; 2. Interval-valued intuitionistic fuzzy sets; 3. Other extensions of intuitionistic fuzzy sets; 4. Elements of intuitionistic fuzzy logic; 5. Applications of intuitionistic fuzzy sets.

An intuitionistic fuzzy set (IFS) \(A\) in \(E\) is defined as an object of the form \[ A= \{\langle x,\mu_A(x), \nu_A(x)\rangle/x\in E\}, \] where the functions \(\mu_A:E\to [0,1]\) and \(\nu_A: E\to[0, 1]\) define the degree of membership and the degree of non-membership for the elements \(x\in E\) satisfying the condition \(0\leq \mu_A(x)+ \nu_A(x)\leq 1\). The relationship between IFSs and ordinary fuzzy sets, as well as between IFSs and crisp sets is mentioned. The IFSs are natural extensions with higher describing abilities than the ordinary fuzzy sets. Operations (intersection, union, sum, etc.) and various relations (equality, inclusion, etc.) for IFSs are introduced and their properties are studied. The exposition is enriched with suitable geometric interpretations of operations between IFSs. The algebraic properties of the algebra \(\langle E^*,\cup, \cap\rangle\) are investigated, where \(E^*\) stays for all IFSs over the universe \(E\). Five kinds of Cartesian products are defined, their geometric interpretation is supplied, De Morgan’s laws are given. The basic notion of Cartesian product leads to a family of related notions: intuitionistic fuzzy relation, index matrix, intuitionistic fuzzy graph. Various operators are defined on the IFSs: modal-like operators, topological operators, various \(\alpha\)- and \(\alpha, \beta\)-operators (equipped with geometric interpretations), identifying and unary operators, level operators. Their properties are investigated. The most important property of IFS, which have no counterpart in the ordinary fuzzy sets, is that modal-like operators can be defined over IFSs. At the end of Chapter 1 a second type of IFS is introduced.

Interval-valued intuitionistic fuzzy sets are defined and their properties are investigated. It is proved that the interval-valued fuzzy sets and IFSs are equipotent generalizations of the notion of fuzzy set. Interval-valued intuitionistic fuzzy sets arise as an extension of both IFSs and interval-valued fuzzy sets. Operations over interval-valued intuitionistic fuzzy sets are introduced and their geometric interpretation is given. Operators of modal type are defined similarly as in Chapter 1.

Four extensions of the IFSs are presented in Chapter 3: intuitionistic \(L\)-fuzzy sets; IFSs over different universes; temporal IFSs; IFSs of second type. Future extensions of IFSs are discussed.

Elements of intuitionistic fuzzy logic are the subject of the next chapter: intuitionistic fuzzy propositional calculus (using the notations of the classical propositional calculus), intuitionistic fuzzy predicate logic, intuitionistic fuzzy modal logic, intuitionistic fuzzy modal types of operators and temporal intuitionistic fuzzy logic.

Chapter 5 covers various applications of intuitionistic fuzzy models. Intuitionistic fuzzy generalized nets are described with applications to modeling of a pneumatic transportation process; in medicine: to simulation, control, forecast of different processes, to education and organization. Intuitionistic fuzzy expert systems are discussed. Intuitionistic fuzzy models of neural networks are described. Various intuitionistic fuzzy systems are presented.

Open problems, listed at the end of the book, mark the next investigations. References include 513 items.

The book is well organized, lucid and interesting for specialists in fuzzy sets and their applications.

An intuitionistic fuzzy set (IFS) \(A\) in \(E\) is defined as an object of the form \[ A= \{\langle x,\mu_A(x), \nu_A(x)\rangle/x\in E\}, \] where the functions \(\mu_A:E\to [0,1]\) and \(\nu_A: E\to[0, 1]\) define the degree of membership and the degree of non-membership for the elements \(x\in E\) satisfying the condition \(0\leq \mu_A(x)+ \nu_A(x)\leq 1\). The relationship between IFSs and ordinary fuzzy sets, as well as between IFSs and crisp sets is mentioned. The IFSs are natural extensions with higher describing abilities than the ordinary fuzzy sets. Operations (intersection, union, sum, etc.) and various relations (equality, inclusion, etc.) for IFSs are introduced and their properties are studied. The exposition is enriched with suitable geometric interpretations of operations between IFSs. The algebraic properties of the algebra \(\langle E^*,\cup, \cap\rangle\) are investigated, where \(E^*\) stays for all IFSs over the universe \(E\). Five kinds of Cartesian products are defined, their geometric interpretation is supplied, De Morgan’s laws are given. The basic notion of Cartesian product leads to a family of related notions: intuitionistic fuzzy relation, index matrix, intuitionistic fuzzy graph. Various operators are defined on the IFSs: modal-like operators, topological operators, various \(\alpha\)- and \(\alpha, \beta\)-operators (equipped with geometric interpretations), identifying and unary operators, level operators. Their properties are investigated. The most important property of IFS, which have no counterpart in the ordinary fuzzy sets, is that modal-like operators can be defined over IFSs. At the end of Chapter 1 a second type of IFS is introduced.

Interval-valued intuitionistic fuzzy sets are defined and their properties are investigated. It is proved that the interval-valued fuzzy sets and IFSs are equipotent generalizations of the notion of fuzzy set. Interval-valued intuitionistic fuzzy sets arise as an extension of both IFSs and interval-valued fuzzy sets. Operations over interval-valued intuitionistic fuzzy sets are introduced and their geometric interpretation is given. Operators of modal type are defined similarly as in Chapter 1.

Four extensions of the IFSs are presented in Chapter 3: intuitionistic \(L\)-fuzzy sets; IFSs over different universes; temporal IFSs; IFSs of second type. Future extensions of IFSs are discussed.

Elements of intuitionistic fuzzy logic are the subject of the next chapter: intuitionistic fuzzy propositional calculus (using the notations of the classical propositional calculus), intuitionistic fuzzy predicate logic, intuitionistic fuzzy modal logic, intuitionistic fuzzy modal types of operators and temporal intuitionistic fuzzy logic.

Chapter 5 covers various applications of intuitionistic fuzzy models. Intuitionistic fuzzy generalized nets are described with applications to modeling of a pneumatic transportation process; in medicine: to simulation, control, forecast of different processes, to education and organization. Intuitionistic fuzzy expert systems are discussed. Intuitionistic fuzzy models of neural networks are described. Various intuitionistic fuzzy systems are presented.

Open problems, listed at the end of the book, mark the next investigations. References include 513 items.

The book is well organized, lucid and interesting for specialists in fuzzy sets and their applications.

Reviewer: Ketty Peeva (Sofia)

### MSC:

03E72 | Theory of fuzzy sets, etc. |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

68T35 | Theory of languages and software systems (knowledge-based systems, expert systems, etc.) for artificial intelligence |