Some Results on Hausdorff Metric for Intuitionistic Fuzzy Sets

Abstract

Fuzzy set theory was frstly introduced by L. A. Zadeh in 1965 [1]. In fuzzy sets, every element in the set is accompanied with a function μ(x): X → [0, 1], called membership function. The membership function may have uncertainty in some applications because of the subjectivity of the expert or the missing information in the model. Hence some extensions of fuzzy set theory were proposed [2-4]. One of these extensions is Atanassov’s intuitionistic fuzzy set (IFS) theory [2]. In 1986, Atanassov [2] introduced the concept of intuitionistic fuzzy sets and carried out rigorous researches to develop the theory. In this set concept, he introduced a new degree ν(x) : X → [0, 1], called non-membership function, such that the sum μ+ν is less than or equal to 1. Hence the difference 1 −(μ+ν) is regarded as degree of hesitation. Since intuitionistic fuzzy set theory contains membership function, non-membership function and the degree of hesitation, it can be regarded as a tool which is more flexible and closer to human reasoning in handling uncertainty due to imprecise knowledge or data. The Hausdorff metric distance between the alpha cuts of fuzzy numbers is one of the most used metric on fuzzy set theory. It measures how far fuzzy numbers are [5]. In this work we will give some results based on the maximum metric involving Hausdorff metric to measure the distance between intuitionistic fuzzy numbers. We will show that intuitionistic fuzzy numbers are complete under the maximum metric based on Hausdorff metric. As a result we will prove that that space of continuous intuitionistic fuzzy number valued functions are complete under this metric.