INTRODUCTION

Basics

The concept of fuzziness was first discovered and introduced in the seminal article written by Lotfi A. Zadeh in 1965, see [103].

So in our description next we follow [103].

Frequently classes of objects encountered in the real natural world do not have exactly defined criteria of membership. For example, the class of animals clearly includes lions, tigers, horses, birds, fish, etc. as its members and obviously excludes objects such as trees, gases, cars, stones, houses, metals, etc. However there are objects such as starfish, bacteria, etc. that have an ambiguous status in comparison to the class of animals.

Similar ambiguity arises when we compare the number 20 to the class of real numbers much greater than zero. Clearly, “the class of real numbers much greater than zero”, or “the class of beautiful women”, or “the class of tall men” or “the class of smart students” are not defined precisely, thus they do not constitute sets of objects in the usual mathematical sense where each element of a set is 100% there. However such imprecisely considered “classes” of objects exist frequently and play an important role in every aspect of our lives,they show up a lot especially in engineering, computer science, pattern recognition, industry, etc. So the concept under consideration is the fuzzy set, which is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership function which assigns to each object a grade of membership varying from zero to one. The notions of inclusion, union, intersection, complementation.