Abstract
Hypersoft set is a novel area of study which is established as an extension of soft set to handle its limitations. It employs a new approximate mapping called multi-argument approximate function which considers the Cartesian product of attribute-valued disjoint sets as its domain and the power set of universe as its co-domain. The domain of this function is broader as compared to the domain of soft approximate function. It is capable of handling the scenario where sub-attribute-valued sets are considered more significant than taking merely single set of attributes. In this study, notions of set inclusion, convex (concave) sets, strongly convex (concave) sets, strictly convex (concave) sets, convex hull, and convex cone are conceptualized for the multi-argument approximate function. Based on these characterized notions, some set-theoretic inequalities are established with generalized properties and results. The set-theoretic version of classical Jensen’s type inequalities is also discussed with the help of proposed notions.
Highlights
In order to provide the parameterization tool to fuzzy setlike models [1–3] for dealing with uncertain data, in 1999, Molodtsov [4] employed the concept of soft approximate function for defining a novel model soft set. is set employs sa-function which maps single set of parameters to power set of initial universe
Abbas et al [13] investigated the notions of soft points and discussed their limitations, comparisons, and challenges. e hybridized structures of s-set with fuzzy set, intuitionistic fuzzy set, and neutrosophic set were developed by the authors [14–16]
In numerous real-life situations, parameters are to be further classified into their respective parametric-valued disjoint sets. e existing s-set model is deficient for dealing with such sort of partitioning of parameters
Summary
In order to provide the parameterization tool to fuzzy setlike models [1–3] for dealing with uncertain data, in 1999, Molodtsov [4] employed the concept of soft approximate function (sa-function) for defining a novel model soft set (sset). is set employs sa-function which maps single set of parameters to power set of initial universe. Many scholars discussed rudiments of s-sets but the contributions of Ali et al [5], Babitha and Sunil [6, 7], Ge and Yang [8], Li [9], Maji et al [10], Pei and Miao [11], and Sezgin and Atagun [12] are considered prominent regarding the characterization of elementary properties and set-theoretic operations of s-sets. In numerous real-life situations, parameters are to be further classified into their respective parametric-valued disjoint sets. Hypersoft set (hs-set) [17] is developed to generalize s-set to manage real-life scenarios with subparametric disjoint sets. In order to utilize hs-set in various areas of knowledge, Abbas et al [18] and Saeed et al [19] investigated the elementary properties and operations of hs-set. Musa and Asaad [27] discussed the bipolarity of hs-set and investigated some of its properties and aggregation operations
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