# Fuzzy sets and systems

## Fuzzy sets and systems

A fuzzy set is a generalized set to which objects can belong with various degrees (grades) of memberships over the interval [0,1]. Fuzzy systems are processes that are too complex to be modeled by using conventional mathematical methods. In general, fuzziness describes objects or processes that are not amenable to precise definition or precise measurement. Thus, fuzzy processes can be defined as processes that are vaguely defined and have some uncertainty in their description. The data arising from fuzzy systems are, in general, soft, with no precise boundaries. Examples of such systems are large-scale engineering complex systems, social systems, economic systems, management systems, medical diagnostic processes, and human perception.

The mathematics of fuzzy set theory was originated by L. A. Zadeh in 1965. It deals with the uncertainty and fuzziness arising from interrelated humanistic types of phenomena such as subjectivity, thinking, reasoning, cognition, and perception. This type of uncertainty is characterized by structures that lack sharp (well-defined) boundaries. This approach provides a way to translate a linguistic model of the human thinking process into a mathematical framework for developing the computer algorithms for computerized decision-making processes. The theory has grown very rapidly. Many fuzzy algorithms have been developed for application to process control, medical diagnosis, management sciences, engineering design, and many other decision-making processes where soft data are generated. Thus, fuzzy mathematics provides a modeling link between the human reasoning process, which is vague, and computers, which accept only precise data.

For example, in the design of many engineering systems, process information is not available both because it is difficult to understand precisely the complexity of the phenomena and because human reasoning is inexact and is based upon subjective perception. However, by virtue of knowledge and experience, which is inexact, it is possible to build increasingly good systems. In fact, fuzziness in thinking and reasoning processes is an asset since it makes it possible to convey a large amount of information with a very few words. However, in order to emulate this experience and these reasoning processes on a computer, for example, for intelligent robotics applications and medical diagnosis, a mathematical precision must be given to the vagueness of the information so that a computer can accept it. This is done by using the theory of fuzzy sets. Probability theory deals with the uncertainty or randomness that arises in mechanistic systems, whereas fuzzy set theory has been created to deal with the uncertainty that arises in human cognitive processes.

The premise of fuzzy set theory is that the key elements in human reasoning processes are not numbers but labels of fuzzy sets. The degree of membership is specified by a number between 1 (full membership) and 0 (full nonmembership). An ordinary set is a special case of a fuzzy set, where the degree of membership is either 0 or 1. By virtue of fuzzy sets, human concepts like small, big, rich, old, very old, and beautiful can be translated into a form usable by computers.