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Although there are many researches on the similarity measure for intuitionistic fuzzy sets, some of them have some defects. To overcome these shortcomings, based on the Euclidean distance between intuitionistic fuzzy set and three special intuitionistic fuzzy sets, a new similarity measure for intuitionistic fuzzy sets is proposed in this paper. The proposed similarity measure is compared with the existing similarity measures in intuitionistic fuzzy environment by numerical examples. Moreover, we apply the proposed similarity measure to pattern recognition problems in intuitionistic fuzzy environment. The experimental results show that the proposed similarity measure for intuitionistic fuzzy sets is reasonable and effective. [ABSTRACT FROM AUTHOR]

As a direct extension of intuitionistic fuzzy sets, picture fuzzy sets have a powerful ability to deal with vague, uncertain and inconsistent information. The distance between picture fuzzy sets is an effective mathematical tool to distinguish the difference between objects. Although various distances have been proposed, there are some distances that do not meet the axiomatic definition of distance or some distances that induce counter-intuitive results. To address the above defects, this paper gives a new distance between picture fuzzy sets by aggregating three-dimensional divergence, due to the capability of divergence to distinguish information. This new distance not only satisfies the axiomatic definition, but also overcomes the counter-intuitive defects. Furthermore, the proposed distance is applied to solve the multi-attribute decision-making problem. Through comparison and analysis of decision results, this distance not only provides reasonable decision results, but also has high degree of confidence. [ABSTRACT FROM AUTHOR]

Similarity measure is an important tool to evaluate the degree of similarity between different objects. Therefore, a large number of similarity measures on picture fuzzy sets have been studied. However, some existing similarity measures have not been unified into a framework. The aim of this paper is to study a construction method of similarity measure on picture fuzzy sets, based on this, we can unify some existing similarity measures into a framework, and then propose some new similarity measures. Firstly, the concept of picture fuzzy equivalence is introduced. Some construction methods for picture fuzzy equivalence are given. Secondly, a construction method of similarity measure on picture fuzzy sets is proposed. Some new similarity measures on picture fuzzy sets based on picture fuzzy equivalences are given and some existing similarity measures are unified into a framework. Finally, the proposed similarity measures are applied to pattern recognition. By comparing the degrees of confidence, we find that our degrees of confidence are higher than some similarity measures. The experimental results show that the proposed similarity measures can effectively distinguish some similar picture fuzzy sets. [ABSTRACT FROM AUTHOR]

Among the most classic measures in single valued neutrosophic sets (SVNS) theory, the distance is an important tool to compare and calculate degree of difference between SVNS. Although there exist some types of distance for single valued neutrosophic sets, most of them lack of strictly axiomatic definition and exist counter-intuitive cases. In this paper, a novel distance between single valued neutrosophic sets based on matrix norm is given. Then we proved that the new distance satisfies the axiomatic definition of the metric. Finally, the distance is applicated to pattern recognition and medical diagnoses. [ABSTRACT FROM AUTHOR]