To measure the fuzziness of fuzzy sets, this paper introduces a distance-based and a fuzzy entropybased measurements. Then these measurements are generalized to measure the fuzziness of fuzzy partition, namely partition fuzziness. According to the relationship between the validity of fuzzy partition and its partition fuzziness, a family of cluster validity functions is proposed based on the mod...

In this article, we would like to revisit and comment on the widely used definition of cardinality of fuzzy sets. For this purpose we have given a brief description of the history of development of fuzzy cardinality. In the process, we can find that the existing definition fails to give a proper cardinality while dealing with complementation of fuzzy sets. So there arises the need of defining t...

Two measures that quantify distinguishability of fuzzy sets are addressed in this paper: similarity, which exhibits sound theoretical properties but it is usually computationally intensive, and possibility, whose calculation can be very efficient but does not exhibit the same properties of similarity. It is shown that under mild conditions – usually met in interpretable fuzzy modelling – possib...

This paper describes a method for fuzzy classification and recognition of two-dimensional (2-D) shapes, such as handwritten characters, image contours, etc. A fuzzy model is derived for each considered shape from a fuzzy description of a set of instances of this shape. A fuzzy description of a shape instance, in its turn, exploits appropriate fuzzy partitions of the two dimensions of the shape....

Groundwater is an important source of water. Since the control and removal of pollution are expensive, it is essential to identify the possible sources of contamination and to correctly classify groundwater on the basis of its intrinsic and integrated vulnerability. In the last years, many fuzzy models have been developed. There is a lack, however, of studies concerning the application of fuzzy...

We studied the Cauchy problem of fuzzy di erential equations on the basis of introducing the concept of di erentiation which is a generalization of the de nition of H-di erentiability due to Puri and Ralescu (1983), for fuzzy set-valued mappings of real variables whose values are in the fuzzy number space (E; D). The existence and uniqueness theorem is obtained for the Cauchy problem x′=f(t; x)...

The main purpose of this article is to highlight the fact that there are some drawbacks in the existing definition of complementation of fuzzy sets and hence the geometrical representation of fuzzy sets on the basis of such definition which itself is defective would have no meaning. As a result the theorems or formulas which were rooted in the geometrical representation would become unacceptabl...

This chapter takes advantage of using fuzzy classifier rules to capture the correlations between genes. The main motivation to conduct this study is that a fuzzy classifier rule is essentially an “if-then” rule that contains linguistic terms to represent the feature values. This representation of a rule that demonstrates the correlations among the genes is very simple to understand and interpre...

In this paper it is pointed out that a classification is always made taking into account all the available classes, i.e., by means of a classification system. The approach presented in this paper generalizes the classical definition of fuzzy partition as defined by Ruspini, which is now conceived as a quite often desirable objective that can be usually obtained only after a long learning proces...

A considerable amount of fuzzy spatial data emerged in various applications leads to investigation of fuzzy spatial data and their fuzzy relations. Because of complex requirements, it is challenging to propose a general fuzzy spatial relationship representation and a general algorithm for determining all fuzzy spatial relations. This paper, presents a general characterization of representing fu...