Theory and Application of Hypersoft Set

2021·8.5 MB·

by  Smarandache, Florentin

2021
EDITION
AUTHOR
Prof. Dr. Florentin Smarandache Prof. Dr. Muhammad Saeed
Muhammad Saqlain Dr. Mohamed Abdel-Baset
Theory
and
Application of
Hypersoft
Set
Theory and Application of
Hypersoft Set

(Editors)

Prof. Dr. Florentin Smarandache Prof. Dr. Muhammad Saeed

Muhammad Saqlain Dr. Mohamed Abdel-Baset

ISBN 978-1-59973-699-0

Pons Publishing House
Brussels, 2021

Neutrosophic Science International Association (NSIA)
University of New Mexico 705 Gurley Ave., Gallup,
NM 87301, USA
Theory
and
Application
Of
Hypersoft
Set

ISBN 978-1-59973-699-0
Peer-Reviewers

Prof. Dr. Xiao Long Xin
School of Mathematics,
Northwest university,
Xian, China.

Associate Prof. Dr. Irfan Deli
Muallim Rıfat Faculty of Education,
7 Aralık University,
79000 Kilis, Turkey.

Assistant Prof. Dr. Muhammad Riaz
Department of Mathematics,
University of Punjab,
Lahore Pakistan.
Pons Publishing House / Pons asbl
Quai du Batelage, 5
1000 -Bruxelles
Belgium

DTP: George Lukacs

ISBN 978-1-59973-699-0

Neutrosophic Science International Association (NSIA)
University of New Mexico 705 Gurley Ave., Gallup,
NM 87301, USA
Table of Contents
Aims and Scope…………………………………………………………………………..I
Chapter 1
Muhammad Saeed, Atiqe Ur Rahman, Muhammad Ahsan, Florentin Smarandache
An Inclusive Study on Fundamentals of Hypersoft Set………………….…………………………..….1

Chapter 2
Irfan Deli
Hybrid set structures under uncertainly parameterized hypersoft sets: Theory and
applications…………………………………………………………………………………….………..….24

Chapter 3
Adem Yolcu, Taha Yasin Ozturk
Fuzzy Hypersoft Sets and It’s Application to Decision-Making……………………..…….…………50

Chapter 4
Muhammad Naveed Jafar, Muhammad Saeed, Muhammad Haseeb, Ahtasham Habib
Matrix Theory for Intuitionistic Fuzzy Hypersoft Sets and its application in Multi-Attributive
Decision-Making Problems……..…………………………………………………………………...……65

Chapter 5
Rana Muhammad Zulqarnain, Xiao Long Xin, Muhammad Saeed
A Development of Pythagorean fuzzy hypersoft set with basic operations and decision-making
approach based on the correlation coefficient…………………………………….………………...…..85

Chapter 6
Adeel Saleem, Muhammad Saqlain, Sana Moin
Development of TOPSIS using Similarity Measures and Generalized weighted distances for
Interval Valued Neutrosophic Hypersoft Matrices along with Application in MAGDM
Problems………...………………………………………………………………………………………...107

Chapter 7
Muhammad Umer Farooq, Muhammad Saqlain, Zaka-ur-Rehman
The Application of the Score Function of Neutrosophic Hypersoft Set in the Selection of SiC as
Gate Dielectric For MOSFET………………………………………………………………………….…138

Chapter 8
Mahrukh Irfan, Maryam Rani, Muhammad Saqlain, Muhammad Saeed
Tangent, Cosine, and Ye Similarity Measures of m-Polar Neutrosophic Hypersoft Sets………....155

Chapter 9
Muhammad Saeed, Muhammad Ahsan, Atiqe Ur Rahman
A Novel Approach to Mappings on Hypersoft Classes with Application…………………………175
Chapter 10
Atiqe Ur Rahman, Abida Hafeez, Muhammad Saeed, Muhammad Rayees Ahmad,
Ume-e-Farwa
Development of Rough Hypersoft Set with Application in Decision Making for the Best Choice of
Chemical Material…………………………………………………………………………………...……192

Chapter 11
Muhammad Saeed, Muhammad Khubab Siddique, Muhammad Ahsan,
Muhammad Rayees Ahmad, Atiqe Ur Rahman
A Novel Approach to the Rudiments of Hypersoft Graphs…………………………………………203

Chapter 12
Taha Yasin Ozturk and Adem Yolcu
On Neutrosophic Hypersoft Topological Spaces………………………...……………………………215
Aims and Scope

Florentin Smarandache generalize the soft set to the hypersoft set by
transforming the function 𝐹 into a multi-argument function. This extension
reveals that the hypersoft set with neutrosophic, intuitionistic, and fuzzy set
theory will be very helpful to construct a connection between alternatives
and attributes. Also, the hypersoft set will reduce the complexity of the case
study. The Book “Theory and Application of Hypersoft Set” focuses on
theories, methods, algorithms for decision making and also applications
involving neutrosophic, intuitionistic, and fuzzy information. Our goal is to
develop a strong relationship with the MCDM solving techniques and to
reduce the complexion in the methodologies. It is interesting that the
hypersoft theory can be applied on any decision-making problem without
the limitations of the selection of the values by the decision-makers. Some
topics having applications in the area: Multi-criteria decision making
(MCDM), Multi-criteria group decision making (MCGDM), shortest path
selection, employee selection, e-learning, graph theory, medical diagnosis,
probability theory, topology, and some more.
(Editors)
1
Chapter

An Inclusive Study on Fundamentals of Hypersoft Set
Muhammad Saeed1, Atiqe Ur Rahman2*, Muhammad Ahsan3, Florentin Smarandache4
1,2,3 University of Management and Technology, Lahore, Pakistan.
E-mail: [e-mail], E-mail: [e-mail], E-mail: [e-mail]
4 Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA. E-mail: [e-mail]

Abstract: Smarandache developed hypersoft set theory as an extension of soft set theory, to
adequate the existing concepts for multi-attribute function. In this study, essential elementary
properties e.g. not set, subset, absolute set, and aggregation operations e.g. union, intersection,
complement, AND, OR, restricted union, extended intersection, relevant complement, restricted
difference, restricted symmetric difference, are characterized under hypersoft set environment with
illustrated examples. New notions of relation, function and their basic properties are also discussed
for hypersoft sets. Moreover, matrix representation of hypersoft set is presented along with
different operations.

Keywords: Hypersoft Set, Hypersoft Relation, Hypersoft Function, Hypersoft Matrix.
1. Introduction
The theories like theory of probability, theory of fuzzy sets, and the interval mathematics, are
considered as mathematical means to tackle many intricate problems involving various
uncertainties in different fields of mathematical sciences. These theories have their own
complexities which restrain them to solve these problems successfully. The reason for these hurdles
is, possibly, the inadequacy of the parametrization tool. A mathematical tool is needed for dealing
with uncertainties which should be free of all such impediments. In 1999, Molodtsov [1]
introduced a mathematical tool called soft sets in literature as a new parameterized family of
subsets of the universe of discourse. In 2003, Maji et al. [2] extended the concept and introduced
some fundamental terminologies and operations like equality of two soft sets, subset and super set
of a soft set, complement of a soft set, null soft set, absolute soft set, AND, OR and also the
operations of union and intersection. They verified De Morgan's laws and a number of
other results. In 2005, Pei et al. [3] discussed the relationship between soft sets and
information systems. They showed the soft sets as a class of special information systems. In 2009,
Ali et al. [4] pointed out several assertions in previous work of Maji et al. and proposed new notions
such as the restricted intersection, the restricted union, the restricted difference and the extended
intersection of two soft sets. In 2010, 2011, Babitha et. al. [5,6] introduced
________________________________________________________________________
Theory and Application of Hypersoft Set 2
concept of soft set relation as a sub soft set of the Cartesian product of the soft sets and also
discussed many related concepts such as equivalent soft set relation, partition, composition
and function. In 2011, Sezgin et al. [7], Ge et al. [8], Fuli [9] gave some modifications in the
work of Maji et al. and also established some new results. In 2020, Saeed et al. [10] performed
an extensive inspection of the concept of soft elements and soft members in soft sets. Many
researchers [11–22] developed certain hybrids with soft sets to get more generalized results for
implementation in decision making and other related disciplines. In 2018, Smarandache [23]
introduced the concept of hypersoft set as a generalization of soft set.
Inthispaper,someessentialfundamentals(i.e. elementaryproperties,settheoreticoperations,
basic laws, set relations, set function and matrix representation) are conceptualized under hy-
persoft set environment. The rest of this article is structured as follows: Section 2 gives some
basic definitions and results on hyper soft sets. Section 3 presents elementary properties of
hypersoft sets. Section 4 describes set theoretic operations of hypersoft sets. Section 5 pro-
vides some basic properties, results and laws on hypersoft sets. Section 6 discusses hypersoft
relations and hypersoft functions. Section 7 presents the matrix representation of hypersoft
sets with some operations. Section 8 presents some hybrids of hypersoft sets and then last
section 9 concludes the paper.
2. Preliminaries
Here we recall some basic terminologies regarding soft set and hypersoft set. Throughout
the paper, U denotes the universe of discourse.
Definition 2.1. [1]
A pair (ζ ,Λ) is called a soft set over U, where ζ : Λ → P(U) and Λ be a set of attributes of
S S
U.
Definition 2.2. [2]
A soft set (ζ ,Λ ) is a soft subset of another soft set (ζ ,Λ ) if
S1 1 S2 2
(i) Λ ⊆ Λ , and
1 2
(ii) ζ (ω) ⊆ ζ (ω) for all ω ∈ Λ .
S1 S2 1
Definition 2.3. [2]
Union of two soft sets (ζ ,Λ ) and (ζ ,Λ ) is a soft set (ζ ,Λ ) with Λ = Λ ∪Λ and for
S1 1 S2 2 S3 3 3 1 2
ω ∈ Λ ,
3

ζ (ω) ω ∈ (Λ \Λ )
 S1 1 2
ζ (ω) = ζ (ω) ω ∈ (Λ \Λ )
S3 S2 2 1

 ζ (ω)∪ζ (ω) ω ∈ (Λ ∩Λ )
S1 S2 1 2