# An Algebraic Approach to Modular Inequalities Based on Interval-Valued Fuzzy Hypersoft Sets via Hypersoft Set-Inclusions

Authors: Atiqe Ur Rahman; Muhammad Saeed; Khuram Ali Khan; Ammara Nosheen; Rostin Matendo Mabela
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Interval-valued fuzzy hypersoft set is an emerging field of study which is projected to address the limitations of interval-valued fuzzy soft set for the entitlement of multiargument approximate function. This kind of function maps the subparametric tuples to power set of universe. It emphasizes on the partitioning of attributes into their respective subattribute values in the form of disjoint sets. These features make it a completely new mathematical tool for solving problems dealing with uncertainties. In this study, after characterization of essential properties, operations, and set-inclusions (L-inclusion and J-inclusion) of interval-valued fuzzy hypersoft set, some of its modular inequalities are discussed via set-inclusions. It is proved that all set-inclusion-based properties and inequalities are preserved when ordinary approximate function of interval-valued fuzzy soft set is replaced with multiargument approximate function of interval-valued fuzzy hypersoft set.


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## Body

## 1. Introduction

Molodtsov [1] initiated the concept of soft set (s-set) to equip fuzzy set-like models [2–4] with parameterization tool. This set employs the concept of approximate function which maps single set of parameters to initial set of alternatives. This function is also known as single-argument approximate function (SAAF) due to consideration of single set of parameters as its domain. Many researchers contributed towards the characterization of rudiments of s-sets but works of Maji et al. [5], Ali et al. [6], and Ge and Yang [7] for the investigation on set-theoretic operations and Babitha and Sunil [8, 9] for the introduction of relations and functions are more significant. Pei and Miao [10] introduced information system based on s-sets to handle the informational vagueness. Li [11] extended the previous work on soft operations and introduced some new operations. Feng and Li [12] investigated in detail the soft subset and soft product operations. Liu et al. [13] made discussion on generalized soft equal relations. Maji et al. [14] developed fuzzy soft set (fs-set) by combining fuzzy set (f-set) and s-set to deal uncertainties with parameterization tools. Yang et al. [15] hybridized interval-valued fuzzy set (ivf-set) [16] with s-set and developed interval-valued fuzzy soft set (ivfs-set) to tackle uncertain scenarios having interval nature of information and data. Jun and Yang [17] rectified some results on ivfs-sets presented by Yang et al. Chetia and Das [18] applied the notions of ivfs-sets in decision-making for medical diagnosis, Jiang et al. [19] calculated the entropy of ivfs-sets, and Feng et al. [20] characterized level soft sets based on ivfs-sets and applied them in decision-making. Liu et al. [21, 22] discussed some nonclassical properties of ivfs-sets and their modular inequalities based on soft J-inclusion.In many real-world decision-making scenarios, the classification of parameters into their respective subparametric-valued disjoint sets is considered necessary for having reliable and precise decisions. Soft set-like structures (hybridized structures of soft set) are inadequate to tackle such scenarios. Smarandache [23] conceptualized hypersoft set (hs-set) to address the limitations of soft set-like models. In hs-set, set of parameters is further partitioned into disjoint sets having subparametric values. It employs an approximate function which maps the cartesian product of attribute-valued nonoverlapping sets to collection of alternatives. In this way, this function is also called multiargument approximate function (MAAF). Saeed et al. [24] discussed some elementary properties and set-theoretic operations of hs-set with numerical examples. Abbas et al. [25] characterized the notions of hs-points and hs-function for their utilization in the development of hs-function spaces. Ihsan et al. [26] and Rahman et al. [27] developed hs-expert set and bijective hs-set respectively and discussed their applications in multiattribute decision-making (MADM). Rahman et al. [28] introduced a conceptual framework for classical convexity cum concavity under hs-set environment. The researchers Yolcu and Ozturk [29], Jafar and Saeed [30], and Debnath [31] discussed decision-making applications based on fuzzy hypersoft set (fhs-set) (a hybridized structure of f-set and hs-set). Rahman et al. [32] investigated the parameterization of hs-set under fuzzy setting and discussed its utilization in decision-making. The authors Saeed et al. [33] and Rahman et al. [34, 35] developed hybridized structures of fhs-set with complex set in order to tackle periodic nature of data.The existing literature on soft inclusions and modular inequalities for ivfs-sets is suitable for SAAF only but it is incapable to manage MAAF-settings. In other words, it can be viewed that the existing literature on fuzzy soft set is unable to provide a mathematical model which may tackle the following real-world situations collectively as a single model:(1)
The situation where uncertain nature of alternatives (entities in universal set) is required to be judged by assigning fuzzy membership grades to each entity corresponding to each parameter(2)
The scenarios where classification of parameters into their respective parametric valued subcollections is necessary to be considered(3)
The scenarios which has a big collection of interval-base information which is required to be tackled with the help of its interval-valued approximate settingTherefore, motivating from the above described shortcoming of literature, this study is aimed at developing a new structure ivfhs-sets which is more flexible as compared to existing models because it is capable to manage their limitations and is useful for having reliable and unbiased decisions due to deep focusing on parameters and their sub-parametric tuples. Some contributions of this research are (i) basic notions of ivfhs-set are characterized; (ii) the notions of soft inclusions discussed in [13, 17] are generalized for ivfhs-sets; and (iii) modular inequalities for ivfhs-sets based on hs-inclusions are explored by extending the concepts of Liu et al. [21, 22] and Jun and Yang [17]. The rest of the paper is structured as follows: Section 2 recalls some essential basic definitions, properties, and results relating to ivfs-set, hs-set, and fhs-set to support the main results. Section 3 presents the basic notions, properties, and inclusions of ivfhs-sets with discussion on some particular cases of ivfhs-sets. In Section 4, modular inequalities of ivfhs-sets via L-inclusion are discussed. In Section 5, modular inequalities of ivfhs-sets via J-inclusion are discussed. Section 6 summarizes the paper with some future directions.


## 2. Preliminaries

This section reviews few elementary terminologies and properties from literature for proper understanding of main results. Throughout the paper,X˘,ΓX˘,I, and P denote initial universe, power set of X˘, unit closed interval, and set of parameters, respectively.Definition 1 (see [16]).
Assume the setℤI=u^,v^:u^≤v^∀u^,v^∈I and the order relation ≤ℤI stated by u^1,v^1≤ℤIu^2,v^2 if and only if u^1≤u^2,v^1≤v^2 for all u^1,v^1,u^2,v^2∈ℤI, then ZI=ℤI,≤ℤI forms a complete lattice. An ivf-set Fiv over X˘ is characterized by mapping μ^:X˘⟶ℤI, where μ^ is called membership function of Fiv. The collection of all ivf-sets over X˘ is represented by Ωivfs.Definition 2 (see [1]).
LetX˘=x^1,x^2,⋯,x^n be an initial universe and P=p^1,p^2,⋯,p^n be a set of parameters then a SAAF is a mapping FS˘:Q⟶ΓX˘ and defined as FS˘p^1,p^2,⋯,p^k=Γx∧1,x∧2,⋯,x∧n, where ΓX˘ denotes the power set of X˘, Q⊆P with k≤n. The pair FS˘,Q is known as s-set and represented by S˘. The subsets FS˘p^i⊆X˘ are known as p^i-approximate sets having all p^i-approximate elements. The pair X˘,P is called soft-universe. The collection of s-sets is denoted by Ωss.Definition 3 (see [17]).
For any soft-universeX˘,P with Q⊆P, an ivfs-set FS˘iv=FS˘,Q is characterized by mapping FS˘:Q⟶Ωivfs, where Q is the same as stated in Definition 2, and FS˘ is known as SAAF of FS˘iv. The collection of ivfs-sets is denoted by Ωivfss.Definition 4 (see [15]).
LetFS˘iv=FS˘,Q1&GS˘iv=GS˘,Q2∈Ωivfss, then their soft product operations, i.e., ∧&∨ are given as:
(1)
The∧-product (AND-operation) of FS˘iv and GS˘iv is an ivfs-set defined by(1)FS˘iv∧GS˘iv=HS˘,Q1×Q2,such that
(2)HS˘p^1,p^2=FS˘p^1∩GS˘p^2,∀p^1,p^2∈Q1×Q2.(2)
The∨-product (OR-operation) of FS˘iv and GS˘iv is an ivfs-set defined by(3)FS˘iv∨GS˘iv=HS˘,Q1×Q2,such that
(4)HS˘p^1,p^2=FS˘p^1∪GS˘p^2,∀p^1,p^2∈Q1×Q2.
The following two soft-inclusions relations Jun’s inclusion⊆^J in [17] and Liu’s inclusion ⊆^L in [13] are prominent in literature for understanding the set-theoretic operations of ivfs-sets.Definition 5 (see [17]).
LetFS˘iv=FS˘,Q1&GS˘iv=GS˘,Q2∈Ωivfss, then
(1)
FS˘iv is said to be ivfs J-subset of GS˘iv, denoted by FS˘iv⊆^JGS˘iv, if for every p^1∈Q1∃p^2∈Q2 such that FS˘p^1⊆GS˘p^2(2)
FS˘iv and GS˘iv are said to be ivfs J-equal, denoted by FS˘iv=^JGS˘iv, if FS˘iv⊆^JGS˘iv and GS˘iv⊆^JFS˘ivLiu et al. [13] introduced the following soft inclusions by modifying the soft inclusion of Jun and Yang [17].Definition 6 (see [13]).
LetFS˘iv=FS˘,Q1&GS˘iv=GS˘,Q2∈Ωivfss, then
(1)
FS˘iv is said to be ivfs L-subset of GS˘iv, denoted by FS˘iv⊆^LGS˘iv, if for every p^1∈Q1∃p^2∈Q2 such that FS˘p^1=GS˘p^2(2)
FS˘iv and GS˘iv are said to be ivfs L-equal, denoted by FS˘iv=^LGS˘iv, if FS˘iv⊆^LGS˘iv and GS˘iv⊆^LFS˘iv
Note: both⊆^J and ⊆^L are termed as ivfs J-inclusion and ivfs L-inclusion respectively.Proposition 7.
IfFS˘iv⊆^LGS˘iv, then it implies FS˘iv⊆^JGS˘iv.Definition 8 (see [13]).
FS˘iv is said to be identical to GS˘iv, denoted by FS˘iv≡^GS˘iv, if Q1=Q2 and FS˘p^1=GS˘p^2 for every p^1∈Q1∃p^2∈Q2.Proposition 9.
IfFS˘iv≡^GS˘iv, then it implies FS˘iv⊆^LGS˘iv which further implies FS˘iv⊆^JGS˘iv.Propositions7 and 9 are not valid in general. Please refer to [12, 13] for detailed discussion regarding the generalization of these results.Definition 10 (see [23]).
LetX˘=x^1,x^2,⋯,x^n be an initial universe and P=p^1,p^2,⋯,p^n be a set of parameters. The respective attribute-valued nonoverlapping sets of each element of P are Q1=q^11,q^12,⋯,q^1n, Q2=q^21,q^22,⋯,q^2n, Q3=q^31,q^32,⋯,q^3n, …..,Qn=q^n1,q^n2,⋯,q^nn and Q=Q1×Q2×Q3×⋯.×Qn=q^1,q^2,q^3,⋯.,q^r, where each q^ii=1,2,⋯.,r is a n-tuple element of Q and r=∏i=1nQi, • denotes set cardinality, then a MAAF is a mapping FH˘:V⟶ΓX˘ and defined as FH˘q^1,q^2,⋯,q^k=Γx∧1,x∧2,⋯,x∧n, where ΓX˘ denotes the power set of X˘, V⊆Q with k≤r. The pair FH˘,V is known as hs-set and represented by H˘. The collection of all hs-sets is symbolized as Ωhss.Definition 11 (see [23]).
IfΓFX˘ be the collection of all fuzzy sets, then a hs-set H˘=FH˘,V is said to be fhs-set if FH˘:V⟶ΓFX˘, where V is same as discussed in Definition 10, and FH˘ν^ is an approximate element of fhs-set for ν^∈V.


## 3. Properties of ivfhs-Sets

In this section, novel notions of ivfhs-sets are characterized. During this characterization, focus is laid on those operations and properties which are essential to proceed further for the development of modular inequalities.Definition 12.
LetX˘=y^1,y^2,⋯,y^n be an initial universe and P=p^1,p^2,⋯,p^n be a set of parameters. The respective subparametric-valued disjoint sets are R1=r^11,r^12,⋯,r^1n, R2=r^21,r^22,⋯,r^2n, R3=r^31,r^32,⋯,r^3n,…..,Rn=r^n1,r^n2,⋯,r^nn,and R=R1×R2×R3×⋯.×Rn=r^1,r^2,r^3,⋯.,r^s, where each r^ii=1,2,⋯.,s is a n-tuple element of R and s=∏i=1nRi, • denotes set cardinality then the pair ΨFHS,W is known as ivfhs-set, where ΨFHS:W⟶Ωivfs and defined as ΨFHSr^1,r^2,⋯,r^k=Ωivfsy^1,y^2,⋯,y^n, and W⊆R with k≤s. The collection of all ivfhs-sets is symbolized as ℧ivfhssX˘,P.Example 1.
Suppose an organization plans to recruit a candidate to fill a vacant post of assistant manager. There are six candidates forming an initial universe of discourseX˘=C1,C2,C3,C4,C5,C6 and have been scrutinized by recruitment committee. The committee further requires evaluation to select one of these candidates. The evaluation indicators are qualification p^1, relevant experience in years p^2, and computer skill p^3. Their subparametric disjoint sets are R1=r^11=MBA, R2=r^21=5,r^22=7,r^23=10, and R3=r^31=MS−office respectively such that R=R1×R2×R3=r^1,r^2,r^3. Then, an ivfhs-set ΨFHS,R is structured as ΨFHS,R=ΨFHSr^1,r^1,ΨFHSr^2,r^2,ΨFHSr^3,r^3, where
(5)ΨFHSr^11,r^21,r^31=ΨFHSr^1=C10.2,0.4,C20.3,0.5,C30.4,0.6,C40.5,0.7,ΨFHSr^11,r^22,r^31=ΨFHSr^2=C10.3,0.6,C20.4,0.7,C30.5,0.8,C40.6,0.9,ΨFHSr^11,r^23,r^31=ΨFHSr^3=C10.1,0.5,C20.2,0.6,C30.3,0.7,C40.4,0.8.Hence,(6)ΨFHS,R=C10.2,0.4,C20.3,0.5,C30.4,0.6,C40.5,0.7,r^1,C10.3,0.6,C20.4,0.7,C30.5,0.8,C40.6,0.9,r^2,C10.1,0.5,C20.2,0.6,C30.3,0.7,C40.4,0.8,r^3.Its tabular representation is given in Table1.Definition 13.
The complement of an ivfhs-setΨFHS,R, denoted by ΨFHS,Rc, is defined by ΨFHS,Rc=ΨFHSc,~R where ΨFHSc:~R⟶Ωivfs and ~R stands for "not R.Example 2.
Considering data from Example1, we have
(7)ΨFHS,Rc=C10.6,0.8,C20.5,0.7,C30.4,0.6,C40.3,0.5,~r^1,C10.4,0.7,C20.3,0.6,C30.2,0.5,C40.1,0.4,~r^2,C10.5,0.9,C20.4,0.8,C30.3,0.7,C40.2,0.6,~r^3.
Its tabular representation is given in Table2.Definition 14.
LetG1=ΨFHS1,R1&G2=ΨFHS2,R2 be two ivfhs-sets then their hypersoft aggregation operations, i.e., ⊔~&⊓~ are given as:
(1)
Their union is an ivfhs-set defined byG1⊔~G2=ΨFHS3,R3 such that ΨFHS3p^=ΨFHS1p^∪ΨFHS2p^2∀p^∈R1∪R2 with maximum interval-valued fuzzy degrees respective to ΨFHS1p^ and ΨFHS2p^(2)
Their intersection is an ivfhs-set defined byG1⊓~G2=ΨFHS4,R4 such that ΨFHS4p^=ΨFHS1p^∩ΨFHS2p^2∀p^∈R1∩R2 with minimum interval-valued fuzzy degrees respective to ΨFHS1p^ and ΨFHS2p^Example 3.
Considering Example1, we have following two ivfhs-sets G1=ΨFHS1,R1&G2=ΨFHS2,R2.
(8)G1=ΨFHS1,R1=C10.2,0.4,C20.3,0.5,C30.4,0.6,C40.5,0.7,r^1,C10.3,0.6,C20.4,0.7,C30.5,0.8,C40.6,0.9,r^2,C10.1,0.5,C20.2,0.6,C30.3,0.7,C40.4,0.8,r^3,G2=ΨFHS2,R2=C10.3,0.4,C20.4,0.5,C30.5,0.6,C40.6,0.7,r^1,C10.4,0.6,C20.5,0.7,C30.6,0.8,C40.7,0.9,r^2,C10.2,0.5,C20.3,0.6,C30.4,0.7,C40.5,0.8,r^3,then
(9)G1⊔~G2=ΨFHS3,R3=C10.3,0.4,C20.4,0.5,C30.5,0.6,C40.6,0.7,r^1,C10.4,0.6,C20.5,0.7,C30.6,0.8,C40.7,0.9,r^2,C10.2,0.5,C20.3,0.6,C30.4,0.7,C40.5,0.8,r^3,and
(10)G1⊓~G2=ΨFHS4,R4=C10.2,0.4,C20.3,0.5,C30.4,0.6,C40.5,0.7,r^1,C10.3,0.6,C20.4,0.7,C30.5,0.8,C40.6,0.9,r^2,C10.1,0.5,C20.2,0.6,C30.3,0.7,C40.4,0.8,r^3.Definition 15.
LetG1=ΨFHS1,R1&G2=ΨFHS2,R2 be two ivfhs-sets, then their hypersoft product operations, i.e., ⊗~&⊕~ are given as:
(1)
The⊗~-product (AND-operation) is an ivfhs-set defined by G1⊗~G2=ΨFHS3,R1×R2 such that ΨFHS3p^1,p^2=ΨFHS1p^1∩ΨFHS2p^2∀p^1,p^2∈R1×R2(2)
The⊕~-product (OR-operation) is an ivfhs-set defined by G1⊕~G2=ΨFHS4,R1×R2 such that ΨFHS4p^1,p^2=ΨFHS1p^1∪ΨFHS2p^2∀p^1,p^2∈R1×R2Example 4.
Considering the values of two ivfhs-setsG1=ΨFHS1,R1&G2=ΨFHS2,R2 from Example 3, then
(11)G1⊗~G2=ΨFHS3,R1×R2=C10.2,0.4,C20.3,0.5,C30.4,0.6,C40.5,0.7,r^1,r^1,C10.3,0.6,C20.4,0.7,C30.5,0.8,C40.6,0.9,r^1,r^2,C10.1,0.5,C20.2,0.6,C30.3,0.7,C40.4,0.8,r^1,r^3,C10.3,0.4,C20.4,0.5,C30.5,0.6,C40.6,0.7,r^2,r^1,C10.3,0.6,C20.4,0.7,C30.5,0.8,C40.6,0.9,r^2,r^2,C10.2,0.5,C20.3,0.6,C30.4,0.7,C40.5,0.8,r^2,r^3,C10.1,0.4,C20.2,0.5,C30.3,0.6,C40.4,0.7,r^3,r^1,C10.1,0.5,C20.2,0.6,C30.3,0.7,C40.4,0.8,r^3,r^2,C10.1,0.5,C20.2,0.6,C30.3,0.7,C40.4,0.8,r^3,r^3,and
(12)G1⊕~G2=ΨFHS4,R1×R2=C10.3,0.4,C20.4,0.5,C30.5,0.6,C40.6,0.7,r^1,r^1,C10.4,0.6,C20.5,0.7,C30.6,0.8,C40.7,0.9,r^1,r^2,C10.2,0.5,C20.3,0.6,C30.4,0.7,C40.5,0.8,r^1,r^3,C10.3,0.6,C20.4,0.7,C30.5,0.8,C40.6,0.9,r^2,r^1,C10.4,0.6,C20.5,0.7,C30.6,0.8,C40.7,0.9,r^2,r^2,C10.2,0.5,C20.3,0.6,C30.4,0.7,C40.5,0.8,r^2,r^3,C10.3,0.5,C20.4,0.6,C30.5,0.7,C40.6,0.8,r^3,r^1,C10.4,0.6,C20.5,0.7,C30.6,0.8,C40.7,0.9,r^3,r^2,C10.2,0.5,C20.3,0.6,C30.4,0.7,C40.5,0.8,r^3,r^3.
Their tabular representations are presented in Tables3 and 4.Table 1
Interval-valued fuzzy hypersoft setΨFHS,R.
R↓\X˘⟶C1C2C3C4r^1[0.2,0.4][0.3,0.5][0.4,0.6][0.5,0.7]r^2[0.3,0.6][0.4,0.7][0.5,0.8][0.6,0.9]r^3[0.1,0.5][0.2,0.6][0.3,0.7][0.4,0.8]Table 2
Complement of ivfhs-setΨFHS,R.
~R↓\X˘⟶C1C2C3C4~r^1[0.6,0.8][0.5,0.7][0.4,0.6][0.3,0.5]~r^2[0.4,0.7][0.3,0.6][0.2,0.5][0.1,0.4]~r^3[0.5,0.9][0.4,0.8][0.3,0.7][0.2,0.6]Table 3
Tabular representation ofG1⊗~G2.
R1×R2↓\X˘⟶C1C2C3C4r^1,r^1[0.2,0.4][0.3,0.5][0.4,0.6][0.5,0.7]r^1,r^2[0.3,0.6][0.4,0.7][0.5,0.8][0.6,0.9]r^1,r^3[0.1,0.5][0.2,0.6][0.3,0.7][0.4,0.8]r^2,r^1[0.3,0.4][0.4,0.5][0.5,0.6][0.6,0.7]r^2,r^2[0.3,0.6][0.4,0.7][0.5,0.8][0.6,0.9]r^2,r^3[0.2,0.5][0.3,0.6][0.4,0.7][0.5,0.8]r^3,r^1[0.1,0.4][0.2,0.5][0.3,0.6][0.4,0.7]r^3,r^2[0.1,0.5][0.2,0.6][0.3,0.7][0.4,0.8]r^3,r^3[0.1,0.5][0.2,0.6][0.3,0.7][0.4,0.8]Table 4
Tabular representation ofG1⊕~G2.
R1×R2↓\X˘⟶C1C2C3C4r^1,r^1[0.3,0.4][0.4,0.5][0.5,0.6][0.6,0.7]r^1,r^2[0.4,0.6][0.5,0.7][0.6,0.8][0.7,0.9]r^1,r^3[0.2,0.5][0.3,0.6][0.4,0.7][0.5,0.8]r^2,r^1[0.3,0.6][0.4,0.7][0.5,0.8][0.6,0.9]r^2,r^2[0.4,0.6][0.5,0.7][0.6,0.8][0.7,0.9]r^2,r^3[0.2,0.5][0.3,0.6][0.4,0.7][0.5,0.8]r^3,r^1[0.3,0.5][0.4,0.6][0.5,0.7][0.6,0.8]r^3,r^2[0.4,0.6][0.5,0.7][0.6,0.8][0.7,0.9]r^3,r^3[0.2,0.5][0.3,0.6][0.4,0.7][0.5,0.8]Now, we present the generalized version ofJ-inclusion and L-inclusion for ivfhs-sets with entitlement of multiargument approximate functions.Definition 16.
LetG1=ΨFHS1,R1&G2=ΨFHS2,R2 be two ivfhs-sets then
(1)
G1 is said to be ivfhs J-subset of G2, denoted by A⊆^JB, if for every r^1∈R1∃r^2∈R2 such that ΨFHS1r^1⊆ΨFHS2r^2(2)
G1 and G2 are said to be ivfhs J-equal, denoted by G1=^JG2, if G1⊆^JG2 and G2⊆^JG1Definition 17.
LetG1=ΨFHS1,R1&G2=ΨFHS2,R2 be two ivfhs-sets, then
(1)
G1 is said to be ivfs L-subset of G2, denoted by G1⊆^LG2, if for every r^1∈R1∃r^2∈R2 such that ΨFHS1r^1=ΨFHS2r^2(2)
G1 and G2 are said to be ivfs L-equal, denoted by G1=^LG2, if G1=^LG2 and G2=^LG1
Note: both⊆^J and ⊆^L are named as ivfhs J-inclusion and ivfhs L-inclusion, respectively.Proposition 18.
IfG1⊆^LG2, then it implies G1⊆^JG2.Definition 19.
LetG1=ΨFHS1,R1&G2=ΨFHS2,R2 be two ivfhs-sets then G1 is said to be identical to G2, denoted by G1≡^G2, if R1=R2 and ΨFHS1r^1=ΨFHS2r^2 for every r^1∈R1∃r^2∈R2.Proposition 20.
IfG1≡^G2, then it implies G1⊆^LG2 which further implies G1⊆^JG2.Proof.
LetG1≡^G2, then by Definition 19, we have R1=R2 and ΨFHS1r^1=ΨFHS2r^2 for every r^1∈R1∃r^2∈R2. By Definition 17 part (1), we get G1⊆^LG2. Now, applying Proposition 18, we obtain G1⊆^JG2.It is pertinent to mention here that the results presented in Propositions18 and 20 are not legitimate in general. Both ⊆^J and ⊆^L are preorder for ℧ivfhssX˘,P.Proposition 21.
The relations=^J and =^L satisfy the properties of equivalence relation on ℧ivfhssX˘,P.Proof.
Applying the concept stated in Definitions16 and Definitions 17, it is clear that both =^J and =^L satisfy reflexive property as G1=^JG1 and G1=^LG1. Their symmetric and transitive nature can also be deduced from these mentioned definitions. These properties collectively conclude that both =^J and =^L are equivalence relations.
We know from classical set theory, for a setD≠∅ with a preorder ≤, an upward directed set is a set D,≤ in which every pair of elements in D has an upper bound, i.e., for d1,d2∈D, there exists d3 such that d1≤d3andd2≤d3. The following definition is the generalized set theoretic version of upward directed set under hypersoft set environment.Definition 22.
An ivfhs-setΨFHS,R with R≠∅ is called an upward directed ivfhs-set (UD-ivfhss) if for r^1,r^2∈R, there exists r^3∈R such that
(13)ΨFHSr^1∪ΨFHSr^2⊆ΨFHSr^3.Example 5.
Considering data from Example1, we have ivfhs-set G=ΨFHS,R as given below
(14)ΨFHS,R=C10.2,0.4,C20.3,0.5,C30.4,0.6,C40.5,0.7,r^1,C10.1,0.5,C20.2,0.6,C30.3,0.7,C40.4,0.8,r^2,C10.3,0.6,C20.4,0.7,C30.5,0.8,C40.6,0.9,r^3,with tabular representation as given below in Table 5. It can easily be seen that ΨFHSr^1∪ΨFHSr^2⊆ΨFHSr^3 which shows that ΨFHS,R is an UD-ivfhss.Proposition 23.
An ivfhs-setG=ΨFHS,R with WG=ΨFHSr^:r^∈R is an UD-ivfhss if and only if WG,⊆ is an UD-ivfhss.Proof.
LetG=ΨFHS,R be an UD-ivfhss then by definition of UD-ivfhss, then following conditions hold:
(i)
R≠∅ and WG=ΨFHSr^:r^∈R⊆Ωivfs;WG≠∅(ii)
forr^1,r^2∈R∃r^3∈R such that ΨFHSr^1∪ΨFHSr^2⊆ΨFHSr^3
The second condition implies thatΨFHSr^1⊆ΨFHSr^3 and ΨFHSr^2⊆ΨFHSr^3 which proves that WG,⊆ is an UD-ivfhss.
Conversely, letWG,⊆ is an UD-ivfhss, then the below given clauses hold due to definition of UD-ivfhss:
(i)
bothR and WG are nonempty sets(ii)
forr^1,r^2∈R, there exists an upper bound for ΨFHSr^1 and ΨFHSr^2 in WG which means ∃r^3∈R such that ΨFHSr^1⊆ΨFHSr^3 and ΨFHSr^2⊆ΨFHSr^3
The clause (ii) further impliesΨFHSr^1∪ΨFHSr^2⊆ΨFHSr^3 which shows that G=ΨFHS,R is an UD-ivfhss.Proposition 24.
An ivfhs-setG=ΨFHS,R with R≠∅ is an UD-ivfhss if and only if G=^JG⊕~G.Proof.
LetG⊕~G=YFHS,R×R with G=ΨFHS,R be an UD-ivfhss. Therefore, there exists r^3∈R corresponding to pair r^1,r^2∈R×R s.t.
(15)YFHSr^1,r^2=ΨFHSr^1∪ΨFHSr^2,and
(16)ΨFHSr^1∪ΨFHSr^2⊆ΨFHSr^3.
Combining above equations, we obtainYFHSr^1,r^2⊆ΨFHSr^3 which shows that G⊕~G⊆^JG but we know that G⊆^JG⊕~G; hence, G=^JG⊕~G.
Conversely, letG=^JG⊕~G then G⊕~G⊆^JG implies YFHS,R×R⊆^JG, that is, there exists r^3∈R corresponding to pair r^1,r^2∈R×R s.t.
(17)YFHSr^1,r^2=ΨFHSr^1∪ΨFHSr^2⊆ΨFHSr^3,which proves that G=ΨFHS,R be an UD-ivfhss.Corollary 25.
LetG=ΨFHS,R be an ivfhs-set with R≠∅, then the given below statements are equivalent:
(1)
G is an UD-ivfhss over X˘(2)
WG=ΨFHSr^:r^∈R is an UD-ivfhss w.r.t ⊆(18)G=^JG⊕~G,G⊕~G⊆^JG.Proof.
These can easily be verified by considering the consequences of Proposition23 and Proposition 24.Table 5
An upward directed interval-valued fuzzy hypersoft setΨFHS,R.
R↓\X˘⟶C1C2C3C4r^1[0.2,0.4][0.3,0.5][0.4,0.6][0.5,0.7]r^2[0.1,0.5][0.2,0.6][0.3,0.7][0.4,0.8]r^3[0.3,0.6][0.4,0.7][0.5,0.8][0.6,0.9]


## 4. Modular Inequalities of ivfhs-Sets viaL-Inclusion

Liu et al. [22] discussed some modular inequalities for ivfs-sets which employs approximate function that is unable to tackle multiargument settings (i.e., cartesian product of sub-parametric valued disjoint sets); therefore, in this section, such modular inequalities are generalized to manage such kind of settings.LetX˘=o^1,o^2,⋯,o^n be an initial universe and C=c^1,c^2,⋯,c^n be a set of parameters. The respective subparametric-valued disjoint sets are K1=k^11,k^12,⋯,k^1n, K2=k^21,k^22,⋯,k^2n, K3=k^31,k^32,⋯,k^3n,....., Kn=k^n1,k^n2,⋯,k^nn,and K=K1×K2×K3×.⋯×Kn=k^1,k^2,k^3,.⋯,k^α, where each k^ii=1,2,.⋯,α is a n-tuple element of K, and α=∏i=1nKi, • denotes set cardinality, then the pair ψFHS,K is known as ivfhs-set, where ψFHS:K⟶Ωivfs and defined as ψFHSk^1,k^2,⋯,k^p=Ωivfso^1,o^2,⋯,o^n, and K⊆C with p≤α.Theorem 26.
LetΘ^1=ψFHS1,K1&Θ^2=ψFHS2,K2 be two ivfhs-sets, then
(19)Θ^1⊆^LΘ^1⊕~Θ^2,Θ^2⊆^LΘ^1⊕~Θ^2,Θ^1⊗~Θ^2⊆^LΘ^1,Θ^1⊗~Θ^2⊆^LΘ^1.Theorem 27 (Generalized commutativity of ivfhs-sets).
LetΘ^1=ψFHS1,K1&Θ^2=ψFHS2,K2 be two ivfhs-sets, then
(20)Θ^1⊗~Θ^2⊆^LΘ^2⊗~Θ^1,Θ^1⊕~Θ^2⊆^LΘ^2⊕~Θ^1.Theorem 28.
LetΘ^1=ψFHS1,K1,Θ^2=ψFHS2,K2&Θ^3=ψFHS3,K3 be three ivfhs-sets with Θ^1⊆^LΘ^2, then
(21)Θ^3⊕~Θ^1⊆^LΘ^3⊕~Θ^2,Θ^3⊕~Θ^1⊆^LΘ^2⊕~Θ^3,Θ^1⊕~Θ^3⊆^LΘ^2⊕~Θ^3,Θ^1⊕~Θ^3⊆^LΘ^3⊕~Θ^2.Proof.
(1)
LetΘ^3⊕~Θ^1=ψFHS3,K3⊕~ψFHS1,K1=ζFHS1,K3×K1 and Θ^3⊕~Θ^2=ψFHS3,K3⊕~ψFHS2,K2=ζFHS2,K3×K2. Since given that Θ^1⊆^LΘ^2 which implies that there exists k^2∈K2 for every k^1∈K1 such that(22)ψFHS1k^1=ψFHS2k^2.
Letk^3,k^1∈K3×K1, then by definition of ⊕~, we have
(23)ζFHS1k^3,k^1=ψFHS3k^3∪ψFHS1k^1.
By combining Equation (22) and Equation (23), we get
(24)ζFHS1k^3,k^1=ψFHS3k^3∪ψFHS2k^2.
Similarly fork^3,k^2∈K3×K2, we get
(25)ζFHS2k^3,k^2=ψFHS3k^3∪ψFHS2k^2.
From Equation (24) and Equation (25), we get ζFHS1k^3,k^1=ζFHS2k^3,k^2, which shows that Θ^3⊕~Θ^1⊆^LΘ^3⊕~Θ^2.
(2)
According to second part of Theorem27, we have Θ^2⊕~Θ^3⊆^LΘ^3⊕~Θ^2 which implies that(26)ψFHS2k^2∪ψFHS3k^3=ψFHS3k^3∪ψFHS2k^2.
Therefore, from Equations (24), (25), and (26), it is vivid that ζFHS1k^3,k^1=ζFHS2k^2,k^3 which shows that Θ^3⊕~Θ^1⊆^LΘ^2⊕~Θ^3.
Part 3 and part 4 can easily be verified with the help of Theorem27(2) and above results.Theorem 29.
LetψFHSi,Ki&ΦFHSi,Si be two ivfhs-sets with ψFHS1,K1⊆^LΦFHS1,S1 and ψFHS2,K2⊆^LΦFHS2,S2, then
(27)ψFHS1,K1⊕~ψFHS2,K2⊆^LΦFHS1,S1⊕~ΦFHS2,S2.Theorem 30.
LetΘ^1=ψFHS1,K1,Θ^2=ψFHS2,K2&Θ^3=ψFHS3,K3 be three ivfhs-sets with Θ^1⊆^LΘ^2 then
(28)Θ^3⊗~Θ^1⊆^LΘ^3⊗~Θ^2,Θ^3⊗~Θ^1⊆^LΘ^2⊗~Θ^3,Θ^1⊗~Θ^3⊆^LΘ^2⊗~Θ^3,Θ^1⊗~Θ^3⊆^LΘ^3⊗~Θ^2.Proof.
(1)
LetΘ^3⊗~Θ^1=ψFHS3,K3⊗~ψFHS1,K1=ξFHS1,K3×K1 and Θ^3⊗~Θ^2=ψFHS3,K3⊗~ψFHS2,K2=ξFHS2,K3×K2. Since given that Θ^1⊆^LΘ^2 which implies that there exists k^2∈K2 for every k^1∈K1 such that(29)ψFHS1k^1=ψFHS2k^2.
Letk^3,k^1∈K3×K1, then by definition of ⊗~, we have
(30)ξFHS1k^3,k^1=ψFHS3k^3∩ψFHS1k^1.
By combining Equation (29) and Equation (30), we get
(31)ξFHS1k^3,k^1=ψFHS3k^3∩ψFHS2k^2.
Similarly, fork^3,k^2∈K3×K2, we get
(32)ξFHS2k^3,k^2=ψFHS3k^3∩ψFHS2k^2.
From Equation (31) and Equation (32), we get ξFHS1k^3,k^1=ξFHS2k^3,k^2 which shows that Θ^3⊗~Θ^1⊆^LΘ^3⊗~Θ^2.
(2)
According to Theorem27, we have Θ^2⊗~Θ^3⊆^LΘ^3⊗~Θ^2 which implies that(33)ψFHS2k^2∩ψFHS3k^3=ψFHS3k^3∩ψFHS2k^2.
Therefore from Equations (31), (32), and (33), it is vivid that ξFHS1k^3,k^1=ξFHS2k^2,k^3 which shows that Θ^3⊗~Θ^1⊆^LΘ^2⊗~Θ^3.
Part 3 and part 4 can easily be verified with the help of Theorem27(2) and above results.Theorem 31.
LetψFHSi,Ki&ΦFHSi,Si be two ivfhs-sets with ψFHS1,K1⊆^LΦFHS1,S1 and ψFHS2,K2⊆^LΦFHS2,S2, then ψFHS1,K1⊗~ψFHS2,K2⊆^LΦFHS1,S1⊗~ΦFHS2,S2.Theorem 32 (Generalized distributive inequalities of ivfhs-sets).
LetΘ^1=ψFHS1,K1,Θ^2=ψFHS2,K2&Θ^3=ψFHS3,K3 be three ivfhs-sets then
(34)Θ^1⊕~Θ^2⊗~Θ^3⊆^LΘ^1⊗~Θ^3⊕~Θ^2⊗~Θ^3,Θ^1⊗~Θ^2⊕~Θ^3⊆^LΘ^1⊕~Θ^3⊗~Θ^2⊕~Θ^3,Θ^1⊗~Θ^2⊕~Θ^3⊆^LΘ^1⊗~Θ^2⊕~Θ^1⊗~Θ^3,Θ^1⊕~Θ^2⊗~Θ^3⊆^LΘ^1⊕~Θ^2⊗~Θ^1⊕~Θ^3.


## 5. Modular Inequalities of ivfhs-Sets viaJ-Inclusion

Jun and Yang [17] discussed some modular inequalities for ivfs-sets by extending the concept presented by Liu et al. [22], and this concept too shows inadequacy regarding multiargument approximate settings (i.e., cartesian product of subparametric valued disjoint sets); therefore, in this section, such modular inequalities are generalized to manage such kind of settings.LetX˘=u^1,u^2,⋯,u^n be an initial universe and V=v^1,v^2,⋯,v^n be a set of parameters. The respective subparametric-valued disjoint sets are D1=d^11,d^12,⋯,d^1n, D2=d^21,d^22,⋯,d^2n, D3=d^31,d^32,⋯,d^3n,....., Dn=d^n1,d^n2,⋯,d^nn,and D=D1×D2×D3×.⋯×Dn=d^1,d^2,d^3,.⋯,d^s, where each d^ii=1,2,⋯.,s is an n-tuple element of D and s=∏i=1nDi, • denotes set cardinality, then the pair ΨFHS,D is known as ivfhs-set where ΨFHS:D⟶Ωivfs and defined as ΨFHSd^1,d^2,⋯,d^k=Ωivfsu^1,u^2,⋯,u^n, and D⊆V with k≤s.Theorem 33.
LetW1=ΨFHS1,D1&W2=ΨFHS2,D2 be two ivfhs-sets, then
(35)W1⊆^JW1⊕~W2,W2⊆^JW1⊕~W2,W1⊗~W2⊆^JW1,W1⊗~W2⊆^JW1.Theorem 34.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets with W1⊆^JW2, then
(36)W3⊕~W1⊆^JW3⊕~W2,W3⊕~W1⊆^JW2⊕~W3,W1⊕~W3⊆^JW2⊕~W3,W1⊕~W3⊆^JW3⊕~W2.Theorem 35.
LetΨFHSi,Di&ΦFHSi,Si be two ivfhs-sets with ΨFHS1,D1⊆^JΦFHS1,S1 and ΨFHS2,D2⊆^JΦFHS2,S2, then
(37)ΨFHS1,D1⊕~ΨFHS2,D2⊆^JΦFHS1,S1⊕~ΦFHS2,S2.Theorem 36.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets with W1⊆^JW2, then
(38)W3⊗~W1⊆^JW3⊗~W2,W3⊗~W1⊆^JW2⊗~W3,W1⊗~W3⊆^JW2⊗~W3,W1⊗~W3⊆^JW3⊗~W2.Theorem 37.
LetΨFHSi,Ri&ΦFHSi,Si be two ivfhs-sets with
(39)ΨFHS1,R1⊆^JΦFHS1,S1,and
(40)ΨFHS2,R2⊆^JΦFHS2,S2,then
(41)ΨFHS1,R1⊗~ΨFHS2,R2⊆^JΦFHS1,S1⊗~ΦFHS2,S2.Theorem 38.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets, then W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3.Proof.
From Theorem32(1), we have
(42)W1⊕~W2⊗~W3⊆^LW1⊗~W3⊕~W2⊗~W3,and then, after applying Proposition 18, we get
(43)W1⊕~W2⊗~W3⊆^JW1⊗~W3⊕~W2⊗~W3,and by Theorem 33, we obtain W1⊗~W3⊆^JW1; therefore,
(44)W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3,implies
(45)W1⊗~W3⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3,which leads to following final result due to transitivity of ⊆^J(46)W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3.Corollary 39.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets then
(47)W2⊕~W1⊗~W3⊆^JW1⊕~W2⊗~W3,W1⊕~W2⊗~W3⊆^JW1⊕~W3⊗~W2,W2⊕~W1⊗~W3⊆^JW1⊕~W3⊗~W2.Proof.
From Theorem27(2), we have
(48)W2⊕~W1=^LW1⊕~W2,and then, after applying Proposition 18, we get
(49)W2⊕~W1=^JW1⊕~W2,implies
(50)W2⊕~W1⊆^JW1⊕~W2.
Taking⊗~ on both sides of above inequality with W3, we have
(51)W2⊕~W1⊗~W3⊆^JW1⊕~W2⊗~W3,but by Theorem 38(52)W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3,which leads to following final result due to transitivity of ⊆^J(53)W2⊕~W1⊗~W3⊆^JW1⊕~W2⊗~W3.
Other parts can easily be validated in the similar manner.Corollary 40.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets then
(54)W3⊗~W2⊕~W1⊆^JW1⊕~W2⊗~W3,W3⊗~W1⊕~W2⊆^JW1⊕~W3⊗~W2,W3⊗~W2⊕~W1⊆^JW1⊕~W3⊗~W2,W3⊗~W1⊕~W2⊆^JW1⊕~W2⊗~W3.Corollary 41.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets, then
(55)W3⊗~W2⊕~W1⊆^JW2⊗~W3⊕~W1,W3⊗~W1⊕~W2⊆^JW3⊗~W2⊕~W1,W3⊗~W2⊕~W1⊆^JW3⊗~W2⊕~W1,W3⊗~W1⊕~W2⊆^JW2⊗~W3⊕~W1.Corollary 42.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets, then
(56)W2⊕~W1⊗~W3⊆^JW2⊗~W3⊕~W1,W1⊕~W2⊗~W3⊆^JW3⊗~W2⊕~W1,W2⊕~W1⊗~W3⊆^JW3⊗~W2⊕~W1,W1⊕~W2⊗~W3⊆^JW2⊗~W3⊕~W1.Theorem 43.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets. If W1⊆^JW3, then
(57)W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3⊕~W3.Proof.
From Theorem32(2), we have
(58)W1⊕~W2⊗~W3⊆^LW1⊕~W2⊗~W1⊕~W3,and then, after applying Proposition 18, we get
(59)W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W1⊕~W3.since given that,
(60)W1⊆^JW3.
Taking⊕~ on both sides of above inequality with W3, we have
(61)W1⊕~W3⊆^JW3⊕~W3,implies
(62)W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3⊕~W3,such that
(63)W1⊕~W2⊗~W1⊕~W3⊆^JW1⊕~W2⊗~W3⊕~W3,which leads to following final result due to transitivity of ⊆^J(64)W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3⊕~W3.Example 6.
LetX˘=u^1,u^2,u^3,u^4,u^5 be an initial universe and V=v^1,v^2,v^3,v^4,v^5,v^6 be a set of attributes. The respective attribute-valued disjoint sets are D1=d^11, D2=d^21,d^22, D3=d^31,d^32, D4=d^41,d^42D5=d^51D6=d^61, and D=D1×D2×D3×.⋯×Dn=d^1,d^2,d^3,d^4,d^5,d^6,d^7,d^8. Let us take E1=d^1,d^2,d^3,E2=d^4,d^5, and E3=d^6,d^7,d^8 as subsets of D, then we have three ivfhs-sets W1=ΨFHS1,E1,W2=ΨFHS2,E2&W3=ΨFHS3,E3 with
(65)ΨFHS1d^1=u^10.2,0.7,u^20.3,0.8,u^30.4,0.8,u^40.5,0.6,u^50.6,0.7,ΨFHS1d^2=u^10.3,0.7,u^20.4,0.8,u^30.5,0.8,u^40.6,0.7,u^50.7,0.8,ΨFHS1d^3=u^10.4,0.7,u^20.5,0.8,u^30.6,0.8,u^40.7,0.8,u^50.5,0.8,ΨFHS2d^4=u^10.5,0.8,u^20.6,0.8,u^30.7,0.8,u^40.8,0.9,u^50.6,0.8,ΨFHS2d^5=u^10.6,0.8,u^20.7,0.8,u^30.8,0.9,u^40.8,0.9,u^50.7,0.9,ΨFHS3d^6=u^10.3,0.8,u^20.4,0.9,u^30.5,0.9,u^40.6,0.7,u^50.7,0.8,ΨFHS3d^7=u^10.4,0.8,u^20.5,0.9,u^30.6,0.9,u^40.7,0.8,u^50.8,0.9,ΨFHS3d^8=u^10.5,0.8,u^20.6,0.9,u^30.7,0.9,u^40.8,0.9,u^50.6,0.9.
It is clear thatΨFHS1d^1⊆ΨFHS3d^6, ΨFHS1d^2⊆ΨFHS3d^7, and ΨFHS1d^3⊆ΨFHS3d^8; therefore, W1⊆^JW3.
ConsiderW4=W2⊗~W2 or ΦFHS,E2×E3=ΨFHS2,E2⊗~ΨFHS3,E3 with
(66)ΦFHSd^4,d^6=u^10.3,0.8,u^20.4,0.8,u^30.5,0.8,u^40.6,0.7,u^50.6,0.8,ΦFHSd^4,d^7=u^10.4,0.8,u^20.5,0.8,u^30.6,0.8,u^40.7,0.8,u^50.6,0.8,ΦFHSd^4,d^8=u^10.5,0.8,u^20.6,0.8,u^30.7,0.8,u^40.8,0.9,u^50.6,0.8,ΦFHSd^5,d^6=u^10.3,0.8,u^20.4,0.8,u^30.5,0.9,u^40.6,0.7,u^50.7,0.8,ΦFHSd^5,d^7=u^10.4,0.8,u^20.5,0.8,u^30.6,0.9,u^40.7,0.8,u^50.7,0.9,ΦFHSd^5,d^8=u^10.5,0.8,u^20.6,0.8,u^30.7,0.9,u^40.8,0.9,u^50.6,0.9.
Now, letW5=W1⊕~W2⊗~W2 or
(67)YFHS,E1×E2×E3=ΨFHS1,E1⊕~ΨFHS2,E2⊗~ΨFHS3,E3,with
(68)YFHSd^1,d^4,d^6=u^10.3,0.8,u^20.4,0.8,u^30.5,0.8,u^40.6,0.7,u^50.6,0.8,YFHSd^1,d^4,d^7=u^10.4,0.8,u^20.5,0.8,u^30.6,0.8,u^40.7,0.8,u^50.6,0.8,YFHSd^1,d^4,d^8=u^10.5,0.8,u^20.6,0.8,u^30.7,0.8,u^40.8,0.9,u^50.6,0.8,YFHSd^1,d^5,d^6=u^10.3,0.8,u^20.4,0.8,u^30.5,0.9,u^40.6,0.7,u^50.7,0.8,YFHSd^1,d^5,d^7=u^10.4,0.8,u^20.5,0.8,u^30.6,0.9,u^40.7,0.8,u^50.7,0.9,YFHSd^1,d^5,d^8=u^10.5,0.8,u^20.6,0.8,u^30.7,0.9,u^40.8,0.9,u^50.6,0.9,YFHSd^2,d^4,d^6=u^10.3,0.8,u^20.4,0.8,u^30.5,0.8,u^40.6,0.7,u^50.7,0.8,YFHSd^2,d^4,d^7=u^10.4,0.8,u^20.5,0.8,u^30.6,0.8,u^40.7,0.8,u^50.7,0.8,YFHSd^2,d^4,d^8=u^10.5,0.8,u^20.6,0.8,u^30.7,0.8,u^40.8,0.9,u^50.7,0.8,YFHSd^2,d^5,d^6=u^10.3,0.8,u^20.4,0.8,u^30.5,0.9,u^40.6,0.7,u^50.7,0.8,YFHSd^2,d^5,d^7=u^10.4,0.8,u^20.5,0.8,u^30.6,0.9,u^40.7,0.8,u^50.7,0.9,YFHSd^2,d^5,d^8=u^10.5,0.8,u^20.6,0.8,u^30.7,0.9,u^40.8,0.9,u^50.7,0.9,YFHSd^3,d^4,d^6=u^10.4,0.8,u^20.5,0.8,u^30.6,0.8,u^40.7,0.8,u^50.7,0.8,YFHSd^3,d^4,d^7=u^10.4,0.8,u^20.5,0.8,u^30.6,0.8,u^40.7,0.8,u^50.6,0.8,YFHSd^3,d^4,d^8=u^10.5,0.8,u^20.6,0.8,u^30.7,0.8,u^40.8,0.9,u^50.6,0.8,YFHSd^3,d^5,d^6=u^10.4,0.8,u^20.5,0.8,u^30.6,0.9,u^40.7,0.8,u^50.7,0.8,YFHSd^3,d^5,d^7=u^10.4,0.8,u^20.5,0.8,u^30.6,0.9,u^40.7,0.8,u^50.7,0.9,YFHSd^3,d^5,d^8=u^10.5,0.8,u^20.6,0.8,u^30.7,0.9,u^40.8,0.9,u^50.6,0.9.
Now, we findW6=W1⊕~W2=ΨFHS1,E1⊕~ΨFHS2,E2=ΦFHS1,E1×E2 with
(69)ΦFHS1d^1,d^4=u^10.5,0.8,u^20.6,0.8,u^30.7,0.8,u^40.8,0.9,u^50.6,0.8,ΦFHS1d^1,d^5=u^10.6,0.8,u^20.7,0.8,u^30.8,0.9,u^40.8,0.9,u^50.7,0.9,ΦFHS1d^2,d^4=u^10.5,0.8,u^20.6,0.8,u^30.7,0.8,u^40.8,0.9,u^50.7,0.8,ΦFHS1d^2,d^5=u^10.6,0.8,u^20.7,0.8,u^30.8,0.9,u^40.8,0.9,u^50.7,0.9,ΦFHS1d^3,d^4=u^10.5,0.8,u^20.6,0.8,u^30.7,0.8,u^40.8,0.9,u^50.6,0.8,ΦFHS1d^3,d^5=u^10.6,0.8,u^20.7,0.8,u^30.8,0.9,u^40.8,0.9,u^50.7,0.9.
Similarly,W7=W3⊕~W3=ΨFHS3,E3⊕~ΨFHS3,E3=ΦFHS2,E3×E3 with
(70)ΦFHS2d^6,d^6=u^10.3,0.8,u^20.4,0.9,u^30.5,0.9,u^40.6,0.7,u^50.7,0.8,ΦFHS2d^6,d^7=u^10.4,0.8,u^20.5,0.9,u^30.6,0.9,u^40.7,0.8,u^50.8,0.9,ΦFHS2d^6,d^8=u^10.5,0.8,u^20.6,0.9,u^30.7,0.9,u^40.8,0.9,u^50.7,0.9,ΦFHS2d^7,d^6=u^10.4,0.8,u^20.5,0.9,u^30.6,0.9,u^40.7,0.8,u^50.8,0.9,ΦFHS2d^7,d^7=u^10.4,0.8,u^20.5,0.9,u^30.6,0.9,u^40.7,0.8,u^50.8,0.9,ΦFHS2d^7,d^8=u^10.5,0.8,u^20.6,0.9,u^30.7,0.9,u^40.8,0.9,u^50.8,0.9,ΦFHS2d^8,d^6=u^10.5,0.8,u^20.6,0.9,u^30.7,0.9,u^40.8,0.9,u^50.7,0.9,ΦFHS2d^8,d^7=u^10.5,0.8,u^20.6,0.9,u^30.7,0.9,u^40.8,0.9,u^50.8,0.9,ΦFHS2d^8,d^8=u^10.5,0.8,u^20.6,0.9,u^30.7,0.9,u^40.8,0.9,u^50.6,0.9.
Lastly,W8=W6⊗~W7=W1⊕~W2⊗~W3⊕~W3=ΦFHS1,E1×E2⊗~ΦFHS2,E3×E3=ΦFHS3,E1×E2×E3×E3 with
(71)ΦFHS3d^1,d^4,d^6,d^6=u^10.3,0.8,u^20.4,0.8,u^30.5,0.8,u^40.6,0.7,u^50.6,0.8,ΦFHS3d^1,d^4,d^6,d^7=u^10.4,0.8,u^20.5,0.8,u^30.6,0.8,u^40.7,0.8,u^50.6,0.8,ΦFHS3d^1,d^4,d^6,d^8=u^10.5,0.8,u^20.6,0.8,u^30.7,0.8,u^40.8,0.9,u^50.6,0.8,⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯ΦFHS3d^3,d^5,d^8,d^8=u^10.5,0.8,u^20.6,0.8,u^30.7,0.9,u^40.8,0.9,u^50.6,0.9.
It can be seen that(72)W5⊆^JW8=W1⊕~W2⊗~W3⊆^J,
W1⊕~W2⊗~W3⊕~W3 is not valid in general.Corollary 44.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets. If W1⊆^JW3, then
(73)W1⊕~W3⊗~W2⊆^JW1⊕~W2⊗~W3⊕~W3,W1⊕~W3⊗~W2⊆^JW2⊕~W1⊗~W3⊕~W3,W1⊕~W2⊗~W3⊆^JW2⊕~W1⊗~W3⊕~W3.Proof.
From Theorem27(2), we have
(74)W2⊗~W3=^LW3⊗~W2,and then, after applying Proposition 18, we get
(75)W2⊗~W3=^JW3⊗~W2,implies
(76)W3⊗~W2⊆^JW2⊗~W3.
Taking⊕~ on both sides of above inequality with W1, we have
(77)W1⊕~W3⊗~W2⊆^JW1⊕~W2⊗~W3,but by Theorem 43, we get
(78)W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3⊕~W3,which leads to following final result due to transitivity of ⊆^J(79)W1⊕~W3⊗~W2⊆^JW1⊕~W2⊗~W3⊕~W3.
Other parts can easily be validated in the similar manner.Corollary 45.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets. If W1⊆^JW3, then
(80)W1⊕~W3⊗~W2⊆^JW3⊕~W3⊗~W2⊕~W1,W1⊕~W2⊗~W3⊆^JW3⊕~W3⊗~W2⊕~W1,W1⊕~W2⊗~W3⊆^JW3⊕~W3⊗~W1⊕~W2,W1⊕~W3⊗~W2⊆^JW3⊕~W3⊗~W1⊕~W2.Corollary 46.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets. If W1⊆^JW3, then
(81)W3⊗~W2⊕~W1⊆^JW3⊕~W3⊗~W2⊕~W1,W2⊗~W3⊕~W1⊆^JW3⊕~W3⊗~W2⊕~W1,W2⊗~W3⊕~W1⊆^JW3⊕~W3⊗~W1⊕~W2,W3⊗~W2⊕~W1⊆^JW3⊕~W3⊗~W1⊕~W2.Corollary 47.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets. If W1⊆^JW3, then
(82)W3⊗~W2⊕~W1⊆^JW2⊕~W1⊗~W3⊕~W3,W2⊗~W3⊕~W1⊆^JW2⊕~W1⊗~W3⊕~W3,W2⊗~W3⊕~W1⊆^JW1⊕~W2⊗~W3⊕~W3,W3⊗~W2⊕~W1⊆^JW1⊕~W2⊗~W3⊕~W3.Theorem 48.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets. If W1⊆^JW3 and W3 is an UD-ivfhss, then we have
(83)W1⊕~W2⊗~W3=^JW1⊕~W2⊗~W3.Proof.
Since we know from Theorem43 that
(84)W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3⊕~W3,
As given thatW3 is an UD-ivfhss, therefore, W3⊕~W3=^JW3 which implies W3⊕~W3⊆^JW3 such that
(85)W1⊕~W2⊗~W3⊆^JW1⊕~W2⊗~W3,implies
(86)W1⊕~W2⊗~W3⊕~W3⊆^JW1⊕~W2⊗~W3,which leads to following final result due to transitivity of ⊆^J(87)W1⊕~W2⊕~W3⊆^JW1⊕~W2⊗~W3.Corollary 49.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets. If W1⊆^JW3 and W3 is an UD-ivfhss, then
(88)W1⊕~W3⊗~W2=^JW1⊕~W2⊗~W3,W1⊕~W3⊗~W2=^JW2⊕~W1⊗~W3,W1⊕~W2⊗~W3=^JW2⊕~W1⊗~W3.Proof.
Since we know from Theorem27 that W2⊗~W3=^LW3⊗~W2 which further implies that W2⊗~W3=^JW3⊗~W2, i.e.,
(89)W2⊗~W3⊆^JW3⊗~W2,and
(90)W3⊗~W2⊆^JW2⊗~W3.
By applying Theorem36, we have
(91)W1⊕~W2⊗~W3⊆^JW1⊕~W3⊗~W2,and
(92)W1⊕~W3⊗~W2⊆^JW1⊕~W2⊗~W3,so
(93)W1⊕~W3⊗~W2=^JW1⊕~W2⊗~W3,since by Theorem 48, we have
(94)W1⊕~W2⊗~W3=^JW1⊕~W2⊗~W3.
Hence,(95)W1⊕~W3⊗~W2=^JW1⊕~W2⊗~W3.Corollary 50.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets. If W1⊆^JW3 and W3 is an UD-ivfhss, then
(96)W3⊗~W2⊕~W1=^JW1⊕~W2⊗~W3,W3⊗~W2⊕~W1=^JW2⊕~W1⊗~W3,W2⊗~W3⊕~W1=^JW2⊕~W1⊗~W3,W2⊗~W3⊕~W1=^JW1⊕~W2⊗~W3.Corollary 51.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets. If W1⊆^JW3 and W3 is an UD-ivfhss, then
(97)W3⊗~W2⊕~W1=^JW3⊗~W1⊗~W2,W3⊗~W2⊕~W1=^JW3⊗~W2⊗~W1,W2⊗~W3⊕~W1=^JW3⊗~W2⊗~W1,W2⊗~W3⊕~W1=^JW3⊗~W1⊗~W2.Corollary 52.
LetW1=ΨFHS1,D1,W2=ΨFHS2,D2&W3=ΨFHS3,D3 be three ivfhs-sets. If W1⊆^JW3 and W3 is an UD-ivfhss, then
(98)W1⊕~W3⊗~W2=^JW3⊗~W1⊕~W2,W1⊕~W3⊗~W2=^JW3⊗~W2⊕~W1,W1⊕~W2⊗~W3=^JW3⊗~W2⊕~W1,W1⊕~W2⊗~W3=^JW3⊗~W1⊕~W2.


### 5.1. Discussion

Now, we prove the flexibility of our presented model ivfhs-set through structural comparison based on some important evaluating features like DoM (degree of membership), SAAF (single-argument approximate function), MAAF (multiargument approximate function), DFPT (deep focus on parametric tuples), and IVTD (interval-valued type data). The Table6 presents this comparison with some relevant existing studies. Some of the advantages of the proposed model are as under:
(1)
It is capable to manage the uncertain nature of alternatives (entities in universal set) by assigning fuzzy membership grades to each entity corresponding to each parameter(2)
It has ability to tackle the scenarios where classification of parameters into their respective parametric-valued subcollections is necessary to be considered(3)
It is useful to manage big collection of interval-base information with the help of its interval-valued approximate settingTable 6
Structural comparison of proposed structure with existing relevant models.
ReferencesStructuresDoMSAAFMAAFDFPTIVTDMolodtsov[1]f-set×✓×××Zadeh [2]f-set✓××××Maji et al. [14]fs-set✓✓×××Yang et al. [15]ivfs-set✓✓××✓Zadeh [16]ivf-set✓×××✓Proposed modelivfhs-set✓✓✓✓✓In short, the ivfhs-set tackle all the above three situations collectively in one model.


## 5.1. Discussion

Now, we prove the flexibility of our presented model ivfhs-set through structural comparison based on some important evaluating features like DoM (degree of membership), SAAF (single-argument approximate function), MAAF (multiargument approximate function), DFPT (deep focus on parametric tuples), and IVTD (interval-valued type data). The Table6 presents this comparison with some relevant existing studies. Some of the advantages of the proposed model are as under:
(1)
It is capable to manage the uncertain nature of alternatives (entities in universal set) by assigning fuzzy membership grades to each entity corresponding to each parameter(2)
It has ability to tackle the scenarios where classification of parameters into their respective parametric-valued subcollections is necessary to be considered(3)
It is useful to manage big collection of interval-base information with the help of its interval-valued approximate settingTable 6
Structural comparison of proposed structure with existing relevant models.
ReferencesStructuresDoMSAAFMAAFDFPTIVTDMolodtsov[1]f-set×✓×××Zadeh [2]f-set✓××××Maji et al. [14]fs-set✓✓×××Yang et al. [15]ivfs-set✓✓××✓Zadeh [16]ivf-set✓×××✓Proposed modelivfhs-set✓✓✓✓✓In short, the ivfhs-set tackle all the above three situations collectively in one model.


## 6. Conclusion

In this research, some essential elementary rudiments (i.e., properties, set-theoretic operations, and set-inclusions) of ivfhs-set are conceptualized, and then, some modular inequalities of ivfhs-set are established by employing the concept ofL-inclusion and J-inclusion. It is observed that the transformation of approximate function from ivfs-set to ivfhs-set preserve all set-inclusion-based properties and inequalities. As this paper focuses on the fuzzy membership with interval setting under hs-set environment, so it is inadequate for the scenarios where the consideration of falsity degree and indeterminacy degree is mandatory. Therefore, the future work may include the extension of this study to tackle above said scenarios. This can also be extended to the development of algebraic structures based on fuzzy hypersoft set with interval-valued setting.



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*Source: 1384541-2022-05-10.xml*