Fermatean fuzzy sets and their variants
Vishnu Narayan Mishra; Tarun Kumar; Mukesh Kumar Sharma; Laxmi Rathour
Abstract
This paper aims to study Pythagorean and Fermatean Fuzzy Subgroups (FFSG) in the context of -norm and -conorm functions. The paper examines the extensions of fuzzy subgroups, specifically "Pythagorean Fuzzy Subgroups (PFSG)" and "FFSG", along with their properties. In the existing literature ...
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This paper aims to study Pythagorean and Fermatean Fuzzy Subgroups (FFSG) in the context of -norm and -conorm functions. The paper examines the extensions of fuzzy subgroups, specifically "Pythagorean Fuzzy Subgroups (PFSG)" and "FFSG", along with their properties. In the existing literature on Pythagorean and FFSG, the standard properties for membership and non-membership functions are based on the "min" and "max" operations, respectively. However, in this work, we develop a theory that utilizes the -norm for "min" and the -conorm for "max", providing definitions of Pythagorean and FFSG with these functions, along with relevant examples. By incorporating this approach, we introduce multiple options for selecting the minimum and maximum values. Additionally, we prove several results related to Pythagorean and FFSG using the -norm and -conorm, and discuss important properties associated with them.
Fermatean fuzzy sets and their variants
Paul Augustine Ejegwa; Doonen Zuakwagh
Abstract
Fermatean Fuzzy Sets (FFSs) provide an effective way to handle uncertainty and vagueness by expanding the scope of membership and Non-Membership Degrees (NMDs) of Intuitionistic Fuzzy Set (IFS) and Pythagorean Fuzzy Set (PFS), respectively. FFS handles uncertain information more easily in the process ...
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Fermatean Fuzzy Sets (FFSs) provide an effective way to handle uncertainty and vagueness by expanding the scope of membership and Non-Membership Degrees (NMDs) of Intuitionistic Fuzzy Set (IFS) and Pythagorean Fuzzy Set (PFS), respectively. FFS handles uncertain information more easily in the process of decision making. The concept of composite relation is an operational information measure for decision making. This study establishes Fermatean fuzzy composite relation based on max-average rule to enhance the viability of FFSs in machine learning via soft computing approach. Some numerical illustrations are provided to show the merit of the proposed max-average approach over existing the max-min-max computational process. To demonstrate the application of the approach, we discuss some pattern recognition problems of building materials and mineral fields with the aid of the Fermatean fuzzy modified composite relation and Fermatean fuzzy max-min-max approach to underscore comparative analyses. In recap, the objectives of the paper include: 1) discussion of FFS and its composite relations, 2) numerical demonstration of Fermatean fuzzy composite relations, 3) establishment of a decision application framework under FFS in pattern recognition cases, and 4) comparative analyses to showcase the merit of the new approach of Fermatean fuzzy composite relation. In future, this Fermatean fuzzy modified composite relation could be studied in different environments like picture fuzzy sets, spherical fuzzy sets, and so on.
