The Pythagorean triples of integers satisfying x + y = z have been studied and enumerated since Babylonian times. Since Diophantus, it has been known that this set of triples is related to the standard rational parameterization of the unit circle, ( t 2−1 t2+1 , 2t t2+1 ). The Pythagorean triple solutions, which are relatively prime, have the elementary and beautiful characterization as integer...

Fuzzy models are present everywhere from natural to artificial structures, embodying the dynamic processes in physical, biological, and social systems. As real-life problems often uncertain on account of inconsistent indeterminate information, it seems very demanding for an expert solve those using a fuzzy model. In this regard, we develop hybrid new model m-polar Diophantine neutrosophic N-sof...

The aim of this manuscript is to present a new concept of hesitant q-rung orthopair fuzzy sets (Hq-ROFSs) by combining the concept of the q-ROFSs as well as Hesitant fuzzy sets. The proposed concept is the generalization of the fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and Pythagorean fuzzy sets as well as intuitionistic hesitant fuzzy sets (IHFSs) and hesitant Pythagorean fuz...

The aim of this paper is to introduce the concept a generalized Pythagorean fuzzy soft set (GPFSS), which combination sets and sets. Several important operations GPFSS including complement, restricted union, extended intersection are discussed. basic properties presented. Further, an algorithm GPFSSs given solve decision-making. Finally, comparative analysis between approach some existing appro...

Abstract The complex q-rung orthopair fuzzy set (Cq-ROFS) is the extension of Pythagorean (CPFS) in which sum q-power real part (imaginary part) support for and against limited by one; however, it difficult to express hesitant information. In this study, conception (Cq-ROHFS) combining Cq-ROFS (HFS) proposed, its properties are discussed, obviously, Cq-ROHFS can reflect uncertainties structure ...

Abstract This article contributes to the advancement and evolution of outranking decision-making methodologies, with a novel essay on ELimination Choice Translating REality (ELECTRE) family methods. Its primary target is unfold constituents expound implementation ELECTRE II method for group decision making in complex Pythagorean fuzzy framework. results method. By inception, it intrinsically su...

Scientific progression has allowed researchers to develop novel and innovative ways deal with uncertainty in data, allowing for the development of more precise accurate data representation models. This paper aims extend an already reported concept Cubic Pythagorean fuzzy set Fuzzy Soft Set (CPFSs) as it incorporates both interval-valued sets (IVPFS) (PFS) at same time, providing a targeted appr...