Fuzzy Sets and Fuzzy Reasoning: An Introduction

Conclusions

Fuzzy set theory was initiated by Zadeh in 1965 xc1 and permits the treatment of vague, uncertain, imprecise and ill defined knowledge and concepts in an exact mathematical way. Throughout the years this theory was fully studied and used for the analysis, modelling and control of technological and nontechnological systems xc2–21. Actually, our life and world obey the priciple of compatibility of Zadeh, according to which "the closer one looks at a ‘real’ problem, the fuzzier becomes its solution". Stated informally, the essence of this principle is that, as the complexity of a system increases, out ability to make precise and yet significant statements about its behavior diminishes until a threshold beyond which precision and significance (relevance) become almost exclusive characteristics.

Fuzzy controllers and fuzzy reasoning have found particular applications in industrial systems which are very complex and cannot be modelled precisely even under various assumptions and approximations. The control of such systems by experienced human operators was proved to be in many cases more successful and efficient than by classical automatic controller. The human controllers employ experiential rules which can cast into the fuzzy logic framework. These observations inspired many investigations to work in this area with result being the development of the so called fuzzy logic and fuzzy rule- based control xc3,8,9,14,18,20.

The purpose of this chapter is to provide a short account of fuzzy set and fuzzy reasoning theory in order to help the unfamiliar reader to study and understand easier the rest of the book. The reader who is familiar with the fuzzy sets can probably find here a ready-to-use material for his(her) applications. Section 2 presents the basic concepts and definitions of fuzzy sets. Section 3 reviews the three fundamental fuzzy logic operations of Zadeh and section 4 provides a set of other fuzzy operations and relations. Section 5 presents a generalization of the three fundamental operations, and section 6 introduces the concepts of hypercube as used in fuzzy theory. Section 7 presents the representation theorem, discusses the fuzzy functions (domain of definition and domain of values) and states the fuzzy extension principle that helps in the fuzzification of mathematical concepts and laws. Section 8 provides a brief discussion of categories and lattices in the framework of fuzzy sets, and section 9 examines the theory of fuzzy reasoning (linguistic variables, linguistic modifiers, generalized modus ponens rule, max-min composition rule, and estimation of the membership function). Finally, section 10 gives a quick look at fuzzy or linguistic control.