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  1. Fuzzy sets

The concept of fuzziness as described by L.A. Zadeh in 1965 includes imprecision, uncertainty and degree of truthfulness of values. A fuzzy set 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 ranging between zero and one.

A fuzzy set  in a universe of discourse X is a function of the form  : X → [0, 1].

  1.  Fuzzy logic

Fuzzy logic has two different meaning. Firstly, fuzzy logic is a logical system, which is an extension of multivalued logic. Secondly, it is almost synonymous with the theory of fuzzy sets, i.e. a theory which relates to classes of objects with unsharp boundaries in which membership is a matter of degree. Fuzzy logic is a sort of computer logic that is different from Boolean algebra. It is different in the way that it allows values to be more accurate than on or off. While Boolean algebra only allows true and false, fuzzy logic allows all things in between.

New operations for the calculus of logic have been proposed in principles which are at least a generalization of classic logic. Fuzzy logic provides an inference morphology that enables approximate human reasoning capabilities to be applied to knowledge-based systems. The theory of fuzzy logic provides a mathematical strength to capture the uncertainties associated with human cognitive processes such as thinking and reasoning. The conventional approaches to knowledge representation lack means for representing the meaning of fuzzy concepts. As a consequence, the approaches based on first order logic and classical probability theory does not provide an appropriate conceptual framework for dealing with the representation of common-sense knowledge. Such knowledge by nature is lexically imprecise and non- categorical. The development of fuzzy logic motivated large measure by generating need for a conceptual framework which can address issue of uncertainty and lexical imprecision.

The essential characteristics of fuzzy logic relate to the following:

  1. Exact reasoning is viewed as a limiting case of approximate reasoning.
  2. Everything is a matter of degree.
  3. Knowledge is interpreted as a collection of elastic or equivalently, fuzzy constraint on a collection of variables.
  4. Inference is viewed as a process of propagation of elastic constraint.

Some general observations about fuzzy logic are:

  • Fuzzy logic is conceptually easy to understand.
  • Fuzzy logic is flexible.
  • Fuzzy logic is tolerant of imprecise data.
  • Fuzzy logic is based on natural language.
  1. Membership functions

We already know that fuzzy logic is not logic that is fuzzy but logic that is used to describe fuzziness. This fuzziness is best characterized by its membership function. In other words, we can say that membership function represents the degree of truth in fuzzy logic. A fuzzy set is completely characterized by its membership function (MF). Since most fuzzy sets in use have a universe of discourse X consisting of the real line R, it would be impractical to list all the pair defining a membership function. Membership functions were first introduced in 1965 by Lofti A. Zadeh in his first research paper “fuzzy sets”.

Membership functions characterize fuzziness (i.e., all the information in fuzzy set), whether the elements in fuzzy sets are discrete or continuous. Membership functions can be defined as a technique to solve practical problems by experience rather than knowledge. Membership functions are represented by graphical forms. Rules for defining fuzziness are fuzzy too.

Fig 1.1 A fuzzy set

The function is called a membership function, and for any x in X,  (x) in [0, 1] represents the grade of membership of x in .

Two important types of membership functions are

1. Triangular MFs

2. Trapezoidal MFs


Teacher’s Name: Dr. Seema Mishra

Designation: Assistant Professor

Department: Mathematics

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