4Fuzzy/Intuitionistic Fuzzy Measures and Fuzzy Integrals

4.1 Introduction

The problem of defining and measuring fuzziness in a fuzzy set is an important part in fuzzy mathematics. In mathematics, fuzzy measure theory is considered as a generalized measure and it was introduced by Choquet in 1953 and defined by Sugeno in 1974 in the context of fuzzy integrals. Fuzzy set deals with membership grades whereas fuzzy measure deals with measures of fuzzy set. It considers degree of possibility that a given element belongs to a fuzzy set or a non‐fuzzy set. They have many applications in engineering and their main characteristic is additivity. Classical measure holds additive property. Additive property can be very useful in many applications but may not be adequate in real‐time world problems such as in approximate reasoning, fuzzy logic, artificial intelligence, data mining, etc. For example, if we measure the efficiency of a set of students and the efficiency of the same set of students working in a group, it will be seen that the efficiencies are not equal. This is due to the reasons that the efficiency of the students working in a team is not the addition of the efficiency of each students working on their own. The concept of fuzzy measure does not require additivity. It requires monotonicity.

4.2 Definition of Fuzzy Measure

Let X = {x1, x1, x3, … , xn} be a finite set of reference. A fuzzy measure on X is a mapping μ : P(X) ⟶ [0, 1], where P(X) is a power set of X (2X) that ...

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