Properties Of The Sendograph Metric Of Fuzzy Sets

In this thesis the linear properties of the fuzzy number spaces based on the sendograph metric are studied.The thesis is divided into the following parts:Part one is about the background and development of metrics and integrals of fuzzy valued functions, as well as the main purpose and content of this paper are introduced. Some preliminaries of fuzzy analysis related to the topics of this paper are introduced.Part two is the main part of the thesis. The attainability of sendograph metric is investigated. Therefore, the relative positions of the two points, at which the sendograph distance of two elements in E n is attained, are made clear. By using this result, we prove that sendograph metric is not translation invariant, which corrects a mistake in Metric space of fuzzy sets by Diamond and Koleden. Then by analyzing the attainable points locally, a lower bound for sendograph distance between elements in E 1 under translation is given, which is depending on the upper bound of Hausdorff distance between support and top of translation. When the translation is restricted to Lipschitzian fuzzy sets, the lower bound, which is shown to be optimal by a example, is given. The optimal bound of sendograph distance under scalar multiplication is also given. Additionally, it is shown that the convergence of sendograph distance is not affected by translation and the linear operations are continuous. The latter result implies that ( E 1, D )can be embedded as a positive cone into a topological vector space, thus making it possible for the theory of topological groups to the study of fuzzy numbers. Finally, we point out a mistake of the equivalence of two definition of measurability of fuzzy set valued mapping in the above mentioned book Metric space of fuzzy sets, and fix this error by switching the image of mapping from ( E n, d∞) to ( E n, D ). While we define fuzzy integrals by restricting the image of fuzzy set valued mapping to a separable space, this makes it possible to develop a systematic computable theory of fuzzy integral.Finally part three is the summary of the thesis and indicates what can be improved in further research.