Assignment 2 Fuzzy Logic

Assignment 2 Fuzzy Logic

[Name of the Institute]

Assignment 2 Fuzzy Logic

Introduction

In general, fuzzy set theory and fuzzy logic are intended to transform precise, crisp, and discrete values into measurements of degrees of membership or degrees of truth. The premise of this approach is that it is a good model of human perception and reasoning. It suggests that fuzzy logic based machines might behave more naturally and react more “humanly” due to their ability to emulate the human way of thinking. In recent years, fuzzy logic has been applied in many areas including fuzzy neural networks, fuzzy clustering, data mining, and software testing.

Fuzzy set theory can be framed as a mapping of the values {?, ?} into the real numbers interval. In a similar way, fuzzy logic is a mapping of the Boolean logic values {0, 1} into the real numbers interva.Ramot et al. proposed an extension of fuzzy set theory and fuzzy logic where the range of degrees of membership and the range of truth values is the complex unit circle. In the formalism of complex fuzzy sets proposed by Ramot et al., fuzzy membership is represented in polar coordinates and only the absolute value of the complex membership function conveys fuzzy information (Bone, 2006, 125).

The current paper provides further generalization of these concepts and uses a Cartesian complex fuzzy membership function where both the real part and the imaginary part can be fuzzy functions. Alternatively, polar representation where both the absolute value and the phase value of the complex membership function convey fuzzy information can be utilized. Furthermore, a new interpretation of complex fuzzy grades of membership as a representation of a complex fuzzy class along with complex fuzzy set/class operations is provided.

The main advantage of the new interpretation over the interpretation provided by Ramot et al. relates to the expressive power of complex fuzzy classes. The new interpretation is more comprehensive, and in general the fuzzy content of each complex fuzzy set can be represented by a complex fuzzy class. Yet, in most of the cases, the expressive power of complex fuzzy sets is not sufficient for representing the information contained in a complex fuzzy class. A related and highly practical advantage is the fact that the phase component of a complex fuzzy class represented in polar coordinates is a fuzzy function. This enables inference about fuzzy events with fuzzy cycles, e.g., stock values.

Answer 1)

According to Ramot et al.,a complex fuzzy set S on a universe of discourse U is a set defined by a complex-valued grade of membership function µS(x):

where equation image. The function µS(x) maps U into the unit disk of the complex plane. This definition utilizes polar representation of complex numbers along with a conventional fuzzy set definition, where rS(x) is the amplitude part of the grade of membership and is a fuzzy function defined in the interval [0,1]. And ?S(x) is a real number standing for the phase part of the grade of membership.

The absolute value of complex fuzzy grade of membership ...