Preface

The book is designed for researchers and students working in the field of fuzzy set, rough set, soft set, and their hybrid models. In our real-life problems, there are situations with the uncertain data that may not be successfully modelled by the classical mathematics. There are some mathematical tools for dealing with uncertainties: fuzzy set theory, rough set theory, and soft set theory. The book is written in order to accumulate all the contents of generalised fuzzy theory and all hybrid structures of fuzzy set, rough set, and soft set, so that the researchers get all the information at one place. The primary goal of this book is to help bridge the gap to provide a textbook on the hybrid structures in fuzzy mathematics and their applications in social science. The concept of ‘fuzzy set theory’ was first introduced by Lotfi A. Zadeh in 1965 (Information and Control, vol. 8, pp. 338–353) and thereafter by C.L. Chang (in Fuzzy topological spaces, J. Math. Anal. Appl., vol. 24, pp. 182–190), paved the way of subsequent development of numerous fuzzy topological concepts. In 1983, Atanassov introduced the concept of ‘intuitionistic fuzzy set’ as a generalisation of the notion of a fuzzy set. Intuitionistic fuzzy sets give both a degree of membership and a degree of non-membership, which are independent of each other. The only requirement is that the sum of these two degrees is not greater than 1. Using intuitionistic fuzzy sets, not only vagueness but also uncertainty is modelled. The concept of ‘rough set theory’, which was first introduced by Z. Pawlak in 1981/1982, deals with the approximation of sets that are difficult to describe with the available information. Rough set introduced by Z. Pawlak is expressed by a boundary region of a set. It is also an approach to vagueness. Thus, fuzzy sets and rough sets are two different approaches to vagueness or impreciseness of the reallife problems. The ‘soft set theory’, which was introduced by Molodtsov in 1999, takes care of the problem that involves such vagueness. In 2001, Maji et al. introduced the idea of intuitionistic fuzzy soft set theory and established some results on them. Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov [2] initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy set, rough set, and soft set are closely related