Equivalent Structures of Interval Sets and Fuzzy Interval Sets

Abstract

Ideas of interval sets come from the lower and upper approximations of rough sets to study a unified structure of rough sets and their generalizations. Starting from the interval sets and their operations, this paper summarizes and analyzes other sets that have similarities with the interval sets or fuzzy interval sets. Our conclusions are that interval sets are mathematically equivalent to shadowed sets and flou sets, respectively, and fuzzy interval sets are mathematically equivalent to interval-valued fuzzy sets and intuitionistic fuzzy sets, respectively. C 2017 Wiley Periodicals, Inc.

Interval‐valued equivalence measures respecting uncertainty in image processing

“…The presented results are mainly theoretical but accompanied with an example of application: an algorithm for image thresholding, which obtains comparable results with recent image thresholding algorithms in the literature, 49 illustrating the applicability of the proposed theoretical concepts and justifying their introduction. Furthermore, we believe that the presented results have high potential in applications by considering uncertainty of information which is inevitable in real life problems beyond image thresholding, and, in future works, we would like to apply the obtained results in decision making problems (cf 52‐54 …”

Entropy‐based shadowed set approximation of intuitionistic fuzzy sets

“…29 This latter connection is of particular interest as orthopairs can be characterized as nonfuzzy IFSs, 30,31 that is, IFSs in which only lack of knowledge is present. Despite this connection, the problem of approximating IFSs into Shadowed Sets has scarcely been addressed: El-Hawy et al showed how a Shadowed Set could be obtained starting from intuitionistic fuzzy numbers 32,33 ; Hu et al 34 studied the relationships among interval sets (which are equivalent to Shadowed Sets) and IFSs, while Gonzalez et al 35 and Ibrahim and William-West 36 studied Shadowed Sets in the more general context of type-2 Fuzzy Sets. None of these works addressed the general problem of approximating an IFS with a shadowed set.…”