

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5523
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

AI-Assisted Wearable Devices for Promoting Human
Health and Strength Using Complex Interval-Valued

Picture Fuzzy Soft Relations

Raed Hatamleh1, Abdullah Al-Husban2, Sulima Ahmed Mohammed Zubair3,∗,
Mawahib Elamin4, Maha Mohammed Saeed5, Eisa Abdolmaleki6, Takaaki Fujita7,
Giorgio Nordo8, Arif Mehmood9,∗

1 Department of Mathematics, Faculty of Science , Jadara University, Irbid 21110, Jordan
2 Department of Mathematics, Faculty of Science and Technology, Irbid National University,
P.O. Box: 2600 Irbid, Jordan.
Jadara Research Center, Jadara University, Irbid 21110, Jordan
3 Department of Mathematics, College of Sciences, Qassim University, Buraydah 51452,
Saudi Arabia
4 Department of Mathematics, College of Science, Qassim University, Buraydah, 51452,
Saudi Arabia
5 Department of Mathematics, Faculty of Sciences, King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, 15 Saudi, Arabia
6 Department of Mathematics, Tonekabon Branch, Islamic Azad University,
1477893780 Tonekabon, Iran
7 Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan
8 MIFT Department of Mathematical and Computer Science, Physical Sciences and Earth Sciences
- University of Messina, 98166 Sant’Agata, Messina, Italy
9 Department of Mathematics, Institute of Numerical Sciences, Gomal University,
Dera Ismail Khan 29050, KPK, Pakistan.

Abstract. Wearable technology allows users to monitor and track various fitness-related activities and
data, including calories burned, distance traveled, heart rate, and sleep patterns. To choose best app
for fitness, here introduced a new innovation in fuzzy algebra named as complex interval valued picture
fuzzy soft relations (CIVPFSR) by defining the cartesian product (CP) of two complex interval valued
picture fuzzy soft sets (CIVPFSS). Additionally, the many kinds of CIVPFS relations are described, and
their outcomes are also mentioned in this article. The CIVPFSS has a complex structure that includes
multidimensional variables for membership, abstinence, and non-membership degrees. As a result, this
research offers modeling techniques for wearable technology that are based on CIVPFSR. The score
function for best human decision making is also formulated in this process. Lastly, a comparison of current
structures is done to demonstrate the viability of the suggested work with the conclusion highlighting
its potential to significantly advance wearable technology. Additionally, a topological space is developed
for CIVPFSR, providing a geometric interpretation of these relations and offering insights into their
behavior in multi-dimensional decision spaces. The basic operations, operators, related theorems, and
results are discussed. Results concerning the interiors and closures are also addressed.

2020 Mathematics Subject Classifications: 54A05
Key Words and Phrases: Complex interval valued picture fuzzy soft relations, Complex interval
valued picture fuzzy soft sets, Score function, Fitness tracker app, Wearable technology

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5523

Email addresses: (raed@jadara.edu.jo) (R. Hatamleh),(dralhosban@inu.edu.jo) (A. Al-Husban),
(s.zebir@qu.edu.sa) (S. A. M. Zubair), (ma.elhag@qu.edu.sa) (M. Elamin), (mmmohammed@kau.edu.sa)
(M. M. Saeed), (eabdolmaleki@aihe.ac.ir) (E. Abdolmaleki), (t171d603@gunma-u.ac.jp) (T. Fujita),
(giorgio.nordo@unime.it) (G. Nordo), (mehdaniyal@gmail.com) (A. Mehmood)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 2 of 38

1. Introduction

Many daily decisions are characterized by uncertainty. When choosing a course of action or
decision that could have multiple alternative outcomes, an ambiguous situation results. Human
decisions are typically ambiguous and imprecise. It is a necessary component of life. Meanwhile,
Zadeh [44] introduced the theory of fuzzy sets (FSs) in 1965, which deals with the ambiguity
and uncertainty of human decision-making and successfully identifies, processes, and resolves
uncertainty in mathematics. A membership function is assigned by an FS to every element
whose values fall within the interval [0, 1]. Fuzzy sets are used to quickly and effectively answer
ambiguous and uncertain real-world problems. The development of fuzzy set theory allowed for
the handling and modeling of uncertainty in novel ways.

According to Klir [20], crisp sets are only defined as yes or no. On the other hand, uncertainty
is incompatible with the crisp theory, which can only handle precise information. Mendal et al.
[26] introduced the idea of fuzzy relations (FRs), which specify the degree, grade, and strength
of positive interactions between any pair of FSs. A closer membership number to 1 indicates a
positive relationship. A membership value that is closer to 0 indicates a weak relationship. To
lessen ambiguity in decision-making, Zadeh [45] proposed the interval-valued fuzzy sets (IVFSs)
in 1975. These sets substitute an interval of membership for a single value of membership. The
interval’s extremes are sub-intervals of [0, 1]. An extension of fuzzy relations (FRs) named
interval-valued fuzzy relations (IVFRs) was introduced by Bustince and Burillo [11].

The concept of intuitionistic fuzzy sets (IFSs), which address the membership and non-
membership of any element, was first put forth by Atanassov [7]. The term IFSs encompasses a
broader scope than FSs. The idea of intuitionistic fuzzy relations (IFRs), which simultaneously
explain a relation’s effectiveness and ineffectiveness, was first presented by Burillo et al. [10].
It’s a continuation of FRs.

An expansion of FSs and IFSs known as picture fuzzy sets (PFSs) is a new concept developed
by Cuong et al. [22]. Three stages of an element are covered in this structure: the first is the
membership level, the second is the indeterminacy, and the third is the non-membership level.
Additionally, the sum of the membership, indeterminacy, and non-membership belongs to [0, 1].
They also presented the idea of picture fuzzy relations (PFRs).

Following the definition of FSs, a new development in fuzzy sets called the concept of com-
plex fuzzy sets (CFSs) was first introduced by Ramot et al. [32]. CFS represents membership
grades as a complex number, i.e., rϋs(ϋ)2πi, where r is the amplitude term and s(ϋ) is the phase
term. A CFS can model an issue with periodicity. It reduces the possibility of mistakes and mis-
understandings. Ramot et al.[32] also provided a definition of complex fuzzy relations (CFRs).
Rani et al. [33] established complex intuitionistic fuzzy sets (CIFSs), where membership and
non-membership are represented by complex numbers. The complex intuitionistic fuzzy rela-
tions (CIFRs) were introduced by Jan et al. [18] through an investigation of cybersecurity and
cybercrimes in the oil and gas industry.

1.1. Research Gap

Despite the growing use of wearable technology to track fitness data, there is a significant
research gap in optimizing the selection process for the best fitness app or device. While fitness
trackers provide useful data on calories burned, heart rate, distance, and sleep patterns, current
methods of choosing the right app often do not fully account for the complex relationships
between various factors, such as user preferences and multidimensional fitness data. Existing
studies have not explored advanced mathematical models, like Complex Interval Valued Picture
Fuzzy Soft Relations (CIVPFSR), that can incorporate multiple variables such as membership,
abstinence, and non-membership in a user’s decision-making process. A topological space can
also be defined on this structure, providing a formal framework for analyzing its properties.
In addition, both basic and advanced results related to topology can be integrated into this


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 3 of 38

structure, allowing for a deeper exploration of its underlying characteristics and behaviors This
research aims to fill this gap by applying CIVPFSR to develop a more refined model for selecting
the most suitable fitness app or device, thereby enhancing the decision-making process and
advancing the field of wearable technology.

1.2. Motivation

The research presented in reference by Jan et al. [17] explores Generative Adversarial Net-
works (GANs), powerful models that generate data samples based on statistical distributions.
However, processing visual data often leads to uncertainties and misperceptions. To tackle this
challenge, the authors introduce a groundbreaking mathematical framework called complex pic-
ture fuzzy soft relations (CPFSRs), which merges concepts from picture fuzzy sets and soft sets.
This innovative framework provides a strong solution for managing inconsistent information and
enhances decision-making by simplifying complex data structures. The study not only leverages
CPFSRs to evaluate and select the optimal GAN but also conducts a comprehensive compara-
tive analysis to demonstrate the superiority of the proposed approach. In a novel contribution,
the research extends this work by incorporating R-interval valued data with complex picture
fuzzy soft information, offering fresh insights and addressing intriguing applications with clear
and insightful explanations.

1.3. Significance of the Study

By enabling users to watch vital parameters like heart rate, distance walked, calories burned,
and sleep patterns, wearable technology has completely changed how we track and maximize
our fitness journeys. These gadgets provide unmatched flexibility, enabling users to customize
their workout and health regimens, whether they are independent or built into smartwatches.
Fitness trackers offer a user-friendly platform for goal-setting, progress monitoring, and obtain-
ing comprehensive insights into one’s fitness journey when paired with companion smartphone
apps. This work presents complex interval-valued picture fuzzy soft relations (CIVPFSR), a
revolutionary development in fuzzy algebra that goes one step further in choosing the best
fitness software. This novel idea provides a new way to assess fitness technology since it is
defined by the Cartesian product (CP) of two complex interval-valued picture fuzzy soft sets
(CIVPFSS). In order to demonstrate the complex, multifaceted character of CIVPFSS-which
encompasses membership, abstention, and non-membership degrees-the paper examines a va-
riety of CIVPFS interactions and their results. Using this sophisticated framework, the study
offers state-of-the-art wearable technology modeling methods along with a recently developed
score function for better decision-making. A comparison analysis with current models is shown
to illustrate the practicality of this technique and highlight how this work has the potential to
greatly improve wearable technology and fitness tracking in the future.

1.4. Literature Review

Complex picture fuzzy sets (CPFSs) are a novel idea developed by Akram et al. [3]. The
given concept is an expansion of CIFSs and includes an additional degree of indeterminacy
to convey the neutral effect of each element. Each of the three phases is represented by a
complex number formed by combining the interval’s left and right extremes, which belongs
to [0, 1]. Complex interval-valued fuzzy sets (CIVFSs) were introduced by Greenfield et al.
[14]. Complex interval-valued fuzzy relations (CIVFRs) are utilized to examine the interactions
between two or more CIVFSs. Nasir et al. [29] proposed an application of CIVFRs to medical
diagnosis in 2021. Complex interv al-valued intuitionistic fuzzy sets (CIVIFSs), which consist
of the membership and non-membership degrees of complex values, were introduced by Garg
et al. [13]. Even with all these advancements in decision-making, choosing the right course of
action is still a challenge for people.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 4 of 38

Molodtsov [28] developed the ideas of soft sets (SS) in 1999, which help in decision-making
in unexpected ways. SSs use specific criteria to select the best options. Alkhazaleh et al. [6]
defined the soft multiset theory, and Yang et al. [42] suggested a generalization of SSs. Maji
et al. [25] presented an application of SSs in decision-making difficulties. Babitha and Sunil [8]
established the idea of soft relations (SRs) and the study of soft sets. Park et al. [30] explored
a few of the effects of equivalence SRs. Maji et al. [24] introduced the idea of the fuzzy soft
set (FSS) by combining the FSs and SSs. It reduces uncertainty, which promotes better daily
decision-making. Ali et al. [5] made a comment on SSs, harsh SSs, and FSSs. For FSS decision-
making, Feng et al. [12] developed a flexible method. Yao et al. [43] clarified the differences
between soft sets and FSSs.

FRs and SRs are combined to create fuzzy soft relations (FSRs), a novel idea created by
Borah et al. [9]. While Sut et al. [37] recommended using FSRs in decision-making, Mockor and
Hurtik [27] approximated FSSs using FSRs in conjunction with image processing. The concept
of complex fuzzy soft sets (CFSSs) was proposed by Thirunavukarasu et al. [39]. Tamir et al.
[38] provided an overview of the theory and applications of complex fuzzy logic. The idea of
a sophisticated multi-fuzzy soft expert set and its applications was created by Al-Qudah and
Hassan [4]. Yang et al. [41] developed the interval-valued fuzzy soft set (IVFSS), an example
of an uncertainty model that is more realistic than the FS. Tripathy et al. [40] discussed the
use of IVFSSs in group decision-making processes. Selvachandran et al. [34] introduced the
innovative idea of CIVFSS through an application. The concept of intuitionistic fuzzy soft
sets (IFSSs), an extension of fuzzy soft sets (FSSs), was introduced by Maji et al. [23]. The
notion of a complex intuitionistic fuzzy soft set (CIFSS) was first presented by Kumar et al.
[21]. With complex values, it is an IFSS generalized form. It handles multifaceted issues
and improves the accuracy of human decision-making. Kumam et al. [19] introduced a novel
advancement in fuzzy theory called the picture fuzzy soft set (PFSS). A new notion of complex
picture fuzzy soft set (CPFSS) was introduced by Jan et al. [17]. Introduces by Shihadeh et al.
[35] two-fold fuzzy n refined neutrosophic rings for n ≥ 3, providing a framework for handling
uncertainty in algebraic structures. Relevant for wearable devices by supporting robust data
processing. Presents two-fold fuzzy algebras based on neutrosophic real numbers, offering tools
for modeling imprecise data by Shihadeh et al. [36]. Enhances wearable device performance
in uncertain decision-making. Rajalakshmi et al. [31] explores neutrosophic ideals in ordered
ternary semigroups. Crucial for structured data analysis, such as health monitoring in wearables.
Abubaker et al. [2] is an Solves neutrosophic singular boundary value problems using (LPM)
polynomials. Assists wearable devices in managing variable health thresholds. Investigates
threshold conversion numbers for graph products and neutrosophic graphs. Supports threshold-
based decision-making in wearables by Abubaker et al. [1]. Hatamleh et al. [16] uses a complex
tangent trigonometric approach with rung fuzzy sets, improving multi-sensor data fusion in
wearables. Hatamleh et al. [15] by explores generalized weighted operators in trigonometric
rung interval-valued fuzzy settings, aiding adaptive algorithms for health monitoring. This
literature review aims to give a thorough overview of the state of CIVPFSS research today and
to highlight the key findings, applications, and advancements in the field.
The concept of Complex Interval-Valued Picture Fuzzy Soft Sets (CIVPFSSs) represents a
significant advancement in the area of multi-dimensional decision-making under uncertainty.
This framework extends old fuzzy and soft set theories by incorporating a complex-valued
membership function with interval and picture fuzzy dimensions. The presence of interval-
valued grades allows the representation of uncertainty in a range rather than a fixed value,
while picture fuzzy components enable the modeling of hesitancy alongside positive, neutral,
and negative membership degrees. Together, these features make CIVPFSSs highly suitable for
tackling problems that involve intricate and ambiguous data structures.

CIVPFSSs are particularly effective in applications requiring multi-faceted evaluation crite-
ria. For instance, in wearable technology, selecting the most efficient device involves balancing


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 5 of 38

multiple parameters such as accuracy, comfort, durability, cost, and connectivity. CIVPFSSs
enable a holistic approach by incorporating interval-valued grades to accommodate varying lev-
els of certainty in the performance of these parameters while addressing hesitancy using picture
fuzzy characteristics. The amplitude term of the membership grade captures the strength or
reliability of a parameter, while the phase term reflects its periodic variations over time.

This approach also supports the modeling of relations such as reflexivity, symmetry, transi-
tivity, and equivalence, making it a versatile tool for analyzing interconnected datasets. Through
its complex interval-valued representation, CIVPFSSs can effectively manage incomplete, in-
consistent, and ambiguous data, outperforming traditional methodologies in both accuracy
and interpretability. These features make CIVPFSSs an indispensable framework for modern
decision-making problems, particularly in fields like wearable health monitoring, environmental
modeling, and intelligent systems design. The score function in CIVPFSSs allows for a system-
atic assessment of parameters by integrating interval-valued grades to accommodate uncertainty
and picture fuzzy characteristics to address hesitancy in decision-making.

The structure of the paper is outlined as follows:

2. Preliminaries / Basic Concepts

This section explains some predefined concepts related to fuzzy algebra, including fuzzy
sets, complex fuzzy sets, fuzzy soft sets, complex fuzzy soft sets, complex interval-valued fuzzy
soft sets, complex interval-valued intuitionistic fuzzy soft sets, picture fuzzy soft sets, complex
picture fuzzy soft sets, and interval-valued fuzzy sets.

Definition 1. [44] A fuzzy set FS A on a universal set X with a mapping µ(ϋ) : X → [0, 1] is
defined as:

A = {(ϋ, µ(ϋ)) | ϋ ∈ X}

where µ is the membership function of ϋ.

Definition 2. [32] Let X be a universal set; then, a CFS A with mapping r, s : X → [0, 1] is
expressed as:

A = {(ϋ, r(ϋ)e(s(ϋ))2πi) : ϋ ∈ X}

where r and s are the amplitude term and phase term of the membership, respectively.

Definition 3. [7] Let X be a universal set; then, an IFS A is expressed as:

A = {(ϋ, µ(ϋ), ν(ϋ)) : ϋ ∈ X}

where 0 ≤ µ(ϋ) + ν(ϋ) ≤ 1. Further, the functions µ : X → [0, 1] and ν : X → [0, 1] represent
the degree of membership and non-membership of the element ϋ ∈ X.

Definition 4. [22] A PFS A on a universal set X is defined as:

A = {(ϋ, µ(ϋ), ϕ(ϋ), ν(ϋ)) : ϋ ∈ X}

where µ(ϋ), ϕ(ϋ), ν(ϋ) ∈ [0, 1] denote the degree of membership, neutral, and non-membership,
respectively. Also, 0 ≤ µ(ϋ) + ϕ(ϋ) + ν(ϋ) ≤ 1.

Definition 5. [13] An CIVIFS A on a universal set X is defined as:

A = {ϋ, [µ−(ϋ), µ+(ϋ)], [ν−(ϋ), ν+(ϋ)] : ϋ ∈ X}

where µ−(ϋ), µ+(ϋ) ∈ [0, 1] and ν−(ϋ), ν+(ϋ) ∈ [0, 1], which satisfy 0 ≤ µ+(ϋ) + ν+(ϋ) ≤ 1 and
0 ≤ µ−(ϋ) + ν−(ϋ) ≤ 1.

The values of µ−(ϋ), ν−(ϋ) and µ+(ϋ), ν+(ϋ) denote the lower and upper terms of the de-
gree of membership and non-membership, respectively. Also, µ−(ϋ), µ+(ϋ), ν−(ϋ), ν+(ϋ) have
complex values, including amplitude and phase terms.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 6 of 38

Definition 6. [30] Let X be a universal set and E be a set of parameters. Let P (X) denote the
set of all possible subsets of X. Then, a FSS (A, T ) with T ⊆ E and a mapping A : E → P (X)
is represented as:

A = {(ϋ, µ(ϋ)) : ϋ ∈ E,µ(ϋ) ∈ P (X)}

Example 1. Let E be a set of parameters and let X be a set of refrigerator companies. Assume
that each membership degree assigned by experts and the refrigerator attribute as a function of
some parameter are represented by a fuzzy soft set (F,E).

X = {x1, x2, x3, x4}, i.e., x1 = LG, x2 = Samsung, x3 = Orient, x4 = Daewoo.

E = {ϋ1, ϋ2, ϋ3}, i.e., ϋ1 = Design, ϋ2 = Beautiful, ϋ3 = Digital.

F (ϋ1) = {x1 = 0.40, x2 = 0.51, x3 = 0.72, x4 = 0.33},

F (ϋ2) = {x1 = 0.34, x2 = 0.85, x3 = 0.66, x4 = 0.67},

F (ϋ3) = {x1 = 0.68, x2 = 0.49, x3 = 0.30, x4 = 0.71}.

The fuzzy soft set (F,E) is a parameterized family of {F (ϋi), i = 1, 2, 3}.

Definition 7. [39] Let X be a universal set and E be a set of parameters. Let CP (X) denote
the set of all possible complex fuzzy subsets of X. Then, a CFSS (A, T ) with T ⊆ E and a
mapping A : E → CP (X) is represented as:

A = {(ϋ, µc(ϋ)) : ϋ ∈ E,µ(ϋ) ∈ CP (X)} = {(ϋ, rµ(ϋ)esµ(ϋ)2πi) : ϋ ∈ E},

where
µc(ϋ) = rµ(ϋ)e

sµ(ϋ)2πi,

and rµ(ϋ), sµ(ϋ) : E → [0, 1], where rµ and sµ are the amplitude term and phase term of the
degree of membership.

Definition 8. [23] Let X be a universal set and E be a set of parameters. Let IP (X) denote
the set of all possible intuitionistic fuzzy subsets of X. Then, an IFSS (A, T ) with T ⊆ E and
a mapping A : E → IP (X) is represented as:

A = {(ϋ, µ(ϋ), ν(ϋ)) : ∀ϋ ∈ E,µ(ϋ), ν(ϋ) ∈ IP (X)},

where µ(ϋ) and ν(ϋ) are the membership and non-membership degrees, respectively.

Example 2. From Example 1, assume an intuitionistic fuzzy soft set (F,E) describing the
characteristics of refrigerators with respect to some parameters, where each membership and
non-membership degree is assigned by experts.

F (ϋ1) = {x1 = (0.41, 0.53), x2 = (0.50, 0.63), x3 = (0.31, 0.62), x4 = (0.33, 0.44)},
F (ϋ2) = {x1 = (0.25, 0.36), x2 = (0.17, 0.58), x3 = (0.39, 0.50), x4 = (0.41, 0.50)},
F (ϋ3) = {x1 = (0.43, 0.54), x2 = (0.35, 0.56), x3 = (0.23, 0.46), x4 = (0.30, 0.42)}.

Then, the intuitionistic fuzzy soft set (F,E) is a parameterized family, i.e., {F (ϋi), i =
1, 2, 3}.


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Definition 9. [21] Let X be a universal set and E be a set of parameters. Let CIP (X) denote
the set of all possible complex intuitionistic fuzzy subsets of X. Then, a CIFSS (A, T ) with
T ⊆ E and a mapping A : E → CIP (X) is represented as:

A = {(ϋ, µc(ϋ), νc(ϋ)) : ∀ϋ ∈ E,µc(ϋ), νc(ϋ) ∈ CIP (X)} = {(ϋ, rµ(ϋ)esµ(ϋ)2πi, rν(ϋ)esν(ϋ)2πi) : ϋ ∈ E},

where rµ, rν are the amplitude terms of the membership and non-membership degrees, respec-
tively, and sµ, sν are the phase terms of the membership and non-membership degrees, respec-
tively.

Definition 10. [19] Let X be a universal set and E be a set of parameters. Let PP (X) denote
the set of all possible picture fuzzy subsets of X. Then, a PFSS (A, T ) with T ⊆ E and a
mapping A : E → PP (X) is represented as:

A = {(ϋ, µ(ϋ), ϕ(ϋ), ν(ϋ)) : ϋ ∈ E,µ(ϋ), ϕ(ϋ), ν(ϋ) ∈ PP (X)},

where µ(ϋ), ϕ(ϋ), ν(ϋ) are the membership, neutral, and non-membership degrees, respectively.

Example 3. From Example 1, assume a PFSS (F,E) describing the characteristics of the refrig-
erator with respect to some parameters, where each membership, neutral, and non-membership
degree is assigned by experts.
F (ϋ1) = {x1 = (0.10, 0.35, 0.50), x2 = (0.20, 0.31, 0.45), x3 = (0.31, 0.21, 0.50), x4 = (0.23, 0.44, 0.12)},
F (ϋ2) = {x1 = (0.25, 0.36, 0.20), x2 = (0.17, 0.38, 0.40), x3 = (0.39, 0.41, 0.31), x4 = (0.11, 0.30, 0.43)},
F (ϋ3) = {x1 = (0.13, 0.34, 0.23), x2 = (0.25, 0.46, 0.12), x3 = (0.23, 0.16, 0.43), x4 = (0.30, 0.12, 0.50)}.

Then, the picture fuzzy soft set (F,E) is a parameterized family, i.e., {F (ϋi), i = 1, 2, 3}.

Definition 11. [17] Let X be a universal set and E be a set of parameters. Let CPP (X)
denote the set of all possible complex picture fuzzy subsets of X. Then, a CPFSS (A, T ) and
T ⊆ E with a mapping A : E → CPP (X) is represented as:

A = {(ϋ, µc(ϋ), φ(ϋ), ψc(ϋ)) : ∀µ(ϋ), φ(ϋ), ψ(ϋ) ∈ CPP (X)}
= {(ϋ, rµ(ϋ)esµ(ϋ)2πi, ϋ, rφ(ϋ)esφ(ϋ)2πi, ϋ, rψ(ϋ)esψ(ϋ)2πi) : ϋ ∈ E}

where rµ, rφ, rψ are the amplitude terms of membership, neutral, and non-membership degree,
respectively, and sµ, sφ, sψ are the phase terms of membership, neutral, and non-membership
degree, respectively.

3. Main Result

We clarify the idea of the CP of two complex interval-valued picture fuzzy soft sets, complex
interval-valued picture fuzzy soft relations, and their types. Furthermore, we describe their
examples and useful results.

Definition 12. Let X be a universal set and E be a set of parameters. Let CIV P (X) denote
the set of all possible complex interval-valued picture fuzzy subsets of X. Then, a CIVPFSS
(A, T ) and T ⊆ E with a mapping A : E → PPP (X) is represented as:

F =

ϋ;


[µ−(ϋ), µ+(ϋ)][eϋ2πi, eϋ
+2πi],

[φ−(ϋ), φ+(ϋ)][eφ
−2πi, eφ

+2πi],

[∂−(ϋ), ∂+(ϋ)][ef
−(ϋ)2πi, ef

+(ϋ)2πi]

 : ϋ ∈ X


Where µ+(ϋ), µ−(ϋ), φ+(ϋ), φ−(ϋ), ψ+(ϋ), ψ−(ϋ) show the amplitude terms of the degree of the
membership, neutral, and non-membership degree, respectively. ϋ+(ϋ), ϋ−(ϋ), j+(ϋ), j−(ϋ), f+(ϋ), f−(ϋ)
show the phase terms of the degree of membership, neutral, and non-membership, respectively.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 8 of 38

Example 4. Suppose the universal set X = {X1, X2, X3} consists of three car companies, i.e.,
X1 = Mercedes, X2 = Porsche, X3 = McLaren, and there are three parameters E = {ϋ1, ϋ2, ϋ3},
i.e., ϋ1 = Price, ϋ2 = Engine, ϋ3 = Model. Then,

F =



ϋ1 =

([0.22, 0.26]e2πi[0.12,0.21]), ([0.13, 0.25]e2πi[0.28,0.34]), ([0.12, 0.31]e2πi[0.32,0.42])

([0.41, 0.42]e2πi[0.14,0.22]), ([0.32, 0.45]e2πi[0.23,0.42]), ([0.22, 0.45]e2πi[0.23,0.42])

([0.31, 0.42]e2πi[0.14,0.22]), ([0.34, 0.44]e2πi[0.34,0.43]), ([0.21, 0.30]e2πi[0.34,0.44])



ϋ2 =

([0.23, 0.34]e2πi[0.32,0.43]), ([0.17, 0.42]e2πi[0.22,0.52]), ([0.43, 0.52]e2πi[0.23,0.43])

([0.21, 0.41]e2πi[0.23,0.42]), ([0.14, 0.24]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.42])

([0.15, 0.31]e2πi[0.12,0.42]), ([0.12, 0.24]e2πi[0.26,0.43]), ([0.13, 0.22]e2πi[0.14,0.44])



ϋ3 =

([0.34, 0.53]e2πi[0.33,0.43]), ([0.04, 0.90]e2πi[0.38,0.47]), ([0.25, 0.35]e2πi[0.34,0.43])

([0.13, 0.22]e2πi[0.26,0.45]), ([0.12, 0.21]e2πi[0.26,0.45]), ([0.15, 0.25]e2πi[0.26,0.45])

([0.12, 0.23]e2πi[0.23,0.44]), ([0.01, 0.05]e2πi[0.25,0.42]), ([0.12, 0.21]e2πi[0.26,0.45])




In the above observation, each value shows the membership, neutral, and non-membership

degree of each company. Here, X1, X2, X3 represent the first, second, and third values of the
universal set. The terms in ϋ1 show the parameters of the universal set, where the first term
shows the price, the second term shows the engine, and the third term shows the model. The
same applies for ϋ2 and ϋ3.

Definition 13. Suppose that (F,A) and (G,B) are two complex interval-valued picture fuzzy
soft sets on universal sets X, Y , and A,B ⊆ E. Let (F,A) × (G,B) = (H,C). Then, the CP
of CIVPFSSs is denoted by:

F =

ϋ;

[µ−(ϋ), µ+(ϋ)][eϋ

−(ϋ)2πi, eϋ
+(ϋ)2πi],

[φ−(ϋ), φ+(ϋ)][ej
−(ϋ)2πi, ej

+(ϋ)2πi],

[∂−(ϋ), ∂+(ϋ)][ef
−(ϋ)2πi, ef

+(ϋ)2πi]

 : ϋ ∈ X


G =

ώ;

[µ−(ώ), µ+(ώ)][eϋ

−(ώ)2πi, eϋ
+(ώ)2πi],

[φ−(ώ), φ+(ώ)][ej
−(ώ)2πi, ej

+(ώ)2πi],

[∂−(ώ), ∂+(ώ)][ef
−(ώ)2πi, ef

+(ώ)2πi]

 : ώ ∈ Y


Define:

F ×G = H =

(ϋ, ώ);


[µ−(ϋ, ώ), µ+(ϋ, ώ)][eϋ

−(ϋ,ώ)2πi, eϋ
+(ϋ,ώ)2πi],

[φ−(ϋ, ώ), φ+(ϋ, ώ)][ej
−(ϋ,ώ)2πi, ej

+(ϋ,ώ)2πi],

[∂−(ϋ, ώ), ∂+(ϋ, ώ)][ef
−(ϋ,ώ)2πi, ef

+(ϋ,ώ)2πi]

 : ϋ ∈ X, ώ ∈ Y


Where:

µ+(ϋ, ώ) = min[µ+(ϋ), µ+(ώ)], µ−(ϋ, ώ) = min[µ−(ϋ), µ−(ώ)]

φ+(ϋ, ώ) = min[φ+(ϋ), φ+(ώ)], φ−(ϋ, ώ) = min[φ−(ϋ), φ−(ώ)]

∂+(ϋ, ώ) = max[∂+(ϋ), ∂+(ώ)], ∂−(ϋ, ώ) = max[∂−(ϋ), ∂−(ώ)]

ϋ+(ϋ, ώ) = min[ϋ+(ϋ), ϋ+(ώ)], ϋ−(ϋ, ώ) = min[ϋ−(ϋ), ϋ−(ώ)]


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 9 of 38

j+(ϋ, ώ) = min[j+(ϋ), j+(ώ)], j−(ϋ, ώ) = min[j−(ϋ), j−(ώ)]

f+(ϋ, ώ) = max[f+(ϋ), f+(ώ)], f−(ϋ, ώ) = max[f−(ϋ), f−(ώ)]

Example 5. Suppose the universal set X = {X1, X2, X3} consists of three car companies,
i.e., X1 = Mercedes, X2 = Porsche, X3 = McLaren, and there are three parameters (E =
{ϋ1, ϋ2, ϋ3}), i.e., ϋ1 = Price, ϋ2 = Engine, ϋ3 = Model. Let (F,A) = F and (G,B) = G be two
CIVPFSS on X, as shown below:

F =



ϋ1 =

([0.22, 0.26]e2πi[0.12,0.21]), ([0.41, 0.42]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.14,0.22])([0.13, 0.25]e2πi[0.28,0.34]), ([0.32, 0.45]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.42])

([0.12, 0.31]e2πi[0.32,0.42]), ([0.22, 0.45]e2πi[0.23,0.42]), ([0.21, 0.30]e2πi[0.34,0.43])


ϋ2 =

([0.23, 0.34]e2πi[0.32,0.43]), ([0.21, 0.41]e2πi[0.23,0.42]), ([0.15, 0.31]e2πi[0.12,0.42])([0.17, 0.42]e2πi[0.22,0.52]), ([0.14, 0.24]e2πi[0.23,0.42]), ([0.12, 0.24]e2πi[0.26,0.34])

([0.43, 0.52]e2πi[0.23,0.43]), ([0.34, 0.44]e2πi[0.23,0.42]), ([0.13, 0.22]e2πi[0.14,0.44])


ϋ3 =

([0.34, 0.53]e2πi[0.33,0.43]), ([0.13, 0.22]e2πi[0.26,0.45]), ([0.12, 0.23]e2πi[0.23,0.44])([0.04, 0.90]e2πi[0.38,0.47]), ([0.12, 0.21]e2πi[0.26,0.45]), ([0.01, 0.05]e2πi[0.25,0.42])

([0.25, 0.35]e2πi[0.34,0.43]), ([0.15, 0.25]e2πi[0.26,0.45]), ([0.12, 0.21]e2πi[0.26,0.45])




And

G =



ϋ1 =

([0.12, 0.26]e2πi[0.12,0.31]), ([0.32, 0.41]e2πi[0.26,0.45]), ([0.21, 0.42]e2πi[0.14,0.25])([0.23, 0.25]e2πi[0.18,0.34]), ([0.11, 0.21]e2πi[0.26,0.45]), ([0.24, 0.44]e2πi[0.13,0.42])

([0.22, 0.31]e2πi[0.22,0.42]), ([0.22, 0.33]e2πi[0.26,0.45]), ([0.11, 0.30]e2πi[0.24,0.43])


ϋ2 =

([0.13, 0.34]e2πi[0.22,0.43]), ([0.32, 0.41]e2πi[0.26,0.45]), ([0.25, 0.31]e2πi[0.32,0.42])([0.27, 0.42]e2πi[0.12,0.52]), ([0.22, 0.27]e2πi[0.26,0.45]), ([0.22, 0.24]e2πi[0.16,0.34])

([0.33, 0.52]e2πi[0.13,0.43]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.13, 0.22]e2πi[0.24,0.44])


ϋ3 =

([0.24, 0.53]e2πi[0.23,0.43]), ([0.18, 0.26]e2πi[0.26,0.45]), ([0.12, 0.24]e2πi[0.33,0.44])([0.05, 0.90]e2πi[0.28,0.47]), ([0.32, 0.41]e2πi[0.26,0.45]), ([0.03, 0.05]e2πi[0.15,0.42])

([0.15, 0.35]e2πi[0.24,0.43]), ([0.32, 0.51]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])





.

In the above observation, each value shows the membership, neutral, and non-membership degree
of each company. Each row represents the parametric observations.

Then, their CP of (F,A) and (G,B) is defined as: The parametric observation H is given
as:


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 10 of 38

H =



(ϋ1, ϋ1) :
(
([0.12, 0.26]e2πi[0.12,0.21], [0.32, 0.41]e2πi[0.14,0.22], [0.31, 0.42]e2πi[0.14,0.25]),

([0.13, 0.25]e2πi[0.18,0.34], [0.11, 0.21]e2πi[0.23,0.42], [0.34, 0.44]e2πi[0.23,0.42]),

([0.12, 0.31]e2πi[0.22,0.42], [0.22, 0.33]e2πi[0.26,0.45], [0.21, 0.30]e2πi[0.34,0.43])
)
,

(ϋ1, ϋ2) :
(
([0.13, 0.26]e2πi[0.12,0.21], [0.32, 0.41]e2πi[0.14,0.22], [0.31, 0.42]e2πi[0.32,0.42]),

([0.13, 0.25]e2πi[0.12,0.34], [0.22, 0.27]e2πi[0.23,0.42], [0.34, 0.44]e2πi[0.23,0.34]),

([0.12, 0.31]e2πi[0.13,0.42], [0.16, 0.29]e2πi[0.26,0.45], [0.21, 0.30]e2πi[0.34,0.44])
)
,

(ϋ1, ϋ3) :
(
([0.22, 0.26]e2πi[0.12,0.21], [0.18, 0.26]e2πi[0.14,0.22], [0.31, 0.42]e2πi[0.33,0.44]),

([0.05, 0.25]e2πi[0.28,0.34], [0.32, 0.41]e2πi[0.26,0.45], [0.34, 0.44]e2πi[0.23,0.42]),

([0.12, 0.31]e2πi[0.24,0.42], [0.22, 0.45]e2πi[0.26,0.45], [0.22, 0.31]e2πi[0.36,0.45])
)
,

(ϋ2, ϋ1) :
(
([0.13, 0.26]e2πi[0.12,0.21], [0.32, 0.41]e2πi[0.14,0.22], [0.31, 0.42]e2πi[0.32,0.42]),

([0.13, 0.25]e2πi[0.12,0.34], [0.22, 0.27]e2πi[0.23,0.42], [0.34, 0.44]e2πi[0.23,0.34]),

([0.12, 0.31]e2πi[0.13,0.42], [0.16, 0.29]e2πi[0.26,0.45], [0.21, 0.30]e2πi[0.34,0.44])
)
,

(ϋ3, ϋ3) :
(
([0.53, 0.24]e2πi[0.23,0.43], [0.13, 0.41]e2πi[0.26,0.45], [0.24, 0.12]e2πi[0.33,0.44]),

([0.90, 0.04]e2πi[0.28,0.47], [0.12, 0.21]e2πi[0.26,0.45], [0.05, 0.03]e2πi[0.25,0.42]),

([0.35, 0.15]e2πi[0.34,0.43], [0.15, 0.25]e2πi[0.26,0.45], [0.31, 0.22]e2πi[0.36,0.45])
)
.



.

Definition 14. The complex interval-valued picture fuzzy soft relation R is a subset of CP of
two CIVPFSSs.

Example 6. From Example 5, take a subset of H which is CIVPFSR:


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 11 of 38

R =



((ϋ1, ϋ2),
(
[0.13, 0.26]e2πi[0.12,0.21], [0.32, 0.41]e2πi[0.14,0.22], [0.31, 0.42]e2πi[0.32,0.42]

)
,(

[0.13, 0.25]e2πi[0.12,0.34], [0.22, 0.27]e2πi[0.23,0.42], [0.34, 0.44]e2πi[0.23,0.34]
)
,(

[0.12, 0.31]e2πi[0.13,0.42], [0.16, 0.29]e2πi[0.26,0.45], [0.21, 0.30]e2πi[0.34,0.44]
)
,

((ϋ1, ϋ3),
(
[0.22, 0.26]e2πi[0.12,0.21], [0.18, 0.26]e2πi[0.14,0.22], [0.31, 0.42]e2πi[0.33,0.44]

)
,(

[0.05, 0.25]e2πi[0.28,0.34], [0.32, 0.41]e2πi[0.26,0.45], [0.34, 0.44]e2πi[0.23,0.42]
)
,(

[0.12, 0.31]e2πi[0.24,0.42], [0.22, 0.45]e2πi[0.26,0.45], [0.22, 0.31]e2πi[0.36,0.45]
)
,

((ϋ2, ϋ1),
(
[0.13, 0.26]e2πi[0.12,0.21], [0.32, 0.41]e2πi[0.14,0.22], [0.31, 0.42]e2πi[0.32,0.42]

)
,(

[0.13, 0.25]e2πi[0.12,0.34], [0.22, 0.27]e2πi[0.23,0.42], [0.34, 0.44]e2πi[0.23,0.34]
)
,(

[0.12, 0.31]e2πi[0.13,0.42], [0.16, 0.29]e2πi[0.26,0.45], [0.21, 0.30]e2πi[0.34,0.44]
)
,

((ϋ3, ϋ3),
(
[0.53, 0.24]e2πi[0.23,0.43], [0.13, 0.41]e2πi[0.26,0.45], [0.24, 0.12]e2πi[0.33,0.44]

)
,(

[0.90, 0.04]e2πi[0.28,0.47], [0.12, 0.21]e2πi[0.26,0.45], [0.05, 0.03]e2πi[0.25,0.42]
)
,(

[0.35, 0.15]e2πi[0.34,0.43], [0.15, 0.25]e2πi[0.26,0.45], [0.31, 0.22]e2πi[0.36,0.45]
)



.

Definition 15. Suppose that (F,A) is a CIVPFSS on universal set X and the CIVPFSR is
defined as:

R =

(ϋ, ώ)


[µ−(ϋ, ώ), µ+(ϋ, ώ)]

[
eϋ

−(ϋ,ώ)2πi, eϋ
+(ϋ,ώ)2πi

]
[φ−(ϋ, ώ), φ+(ϋ, ώ)]

[
ej

−(ϋ,ώ)2πi, ej
+(ϋ,ώ)2πi

]
[∂−(ϋ, ώ), ∂+(ϋ, ώ)]

[
ef

−(ϋ,ώ)2πi, ef
+(ϋ,ώ)2πi

]
 : (ϋ, ώ) ∈ R


Then the inverse of the CIVPFSS relation is denoted by R−1 and is defined as:

R−1 =

(ώ, ϋ)


[µ−(ώ, ϋ), µ+(ώ, ϋ)]

[
eϋ

−(ώ,ϋ)2πi, eϋ
+(ώ,ϋ)2πi

]
[φ−(ώ, ϋ), φ+(ώ, ϋ)]

[
ej

−(ώ,ϋ)2πi, ej
+(ώ,ϋ)2πi

]
[∂−(ώ, ϋ), ∂+(ώ, ϋ)]

[
em

−(ώ,ϋ)2πi, em
+(ώ,ϋ)2πi

]
 : (ώ, ϋ) ∈ R−1


Example 7. Consider the CIVPFSS relation R from Example 6. Then, its inverse is calculated
as:

R−1 =


(ϋ2, ϋ1),

(
[0.13, 0.26] e2πi[0.12,0.21], [0.32, 0.41] e2πi[0.14,0.22], [0.31, 0.42] e2πi[0.32,0.42]

)
,(

[0.13, 0.25] e2πi[0.12,0.34], [0.22, 0.27] e2πi[0.23,0.42], [0.34, 0.44] e2πi[0.23,0.34]
)
,(

[0.12, 0.31] e2πi[0.13,0.42], [0.16, 0.29] e2πi[0.26,0.45], [0.21, 0.30] e2πi[0.34,0.44]
)
,

 .

R−1 =


(ϋ3, ϋ1),

(
[0.22, 0.26] e2πi[0.12,0.21], [0.18, 0.26] e2πi[0.14,0.22], [0.31, 0.42] e2πi[0.333,0.44]

)
,(

[0.05, 0.25] e2πi[0.28,0.34], [0.32, 0.41] e2πi[0.26,0.45], [0.34, 0.44] e2πi[0.23,0.42]
)
,(

[0.12, 0.31] e2πi[0.24,0.42], [0.22, 0.45] e2πi[0.26,0.45], [0.22, 0.31] e2πi[0.36,0.45]
)
,

 .


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 12 of 38

R−1 =


(ϋ1, ϋ2),

(
[0.13, 0.26] e2πi[0.12,0.21], [0.32, 0.41] e2πi[0.14,0.22], [0.31, 0.42] e2πi[0.32,0.42]

)
,(

[0.13, 0.25] e2πi[0.12,0.34], [0.22, 0.27] e2πi[0.23,0.42], [0.34, 0.44] e2πi[0.23,0.34]
)
,(

[0.12, 0.31] e2πi[0.13,0.42], [0.16, 0.29] e2πi[0.26,0.45], [0.21, 0.30] e2πi[0.34,0.44]
)
,

 .

R−1 =


(ϋ3, ϋ3),

(
[0.53, 0.24] e2πi[0.23,0.43], [0.13, 0.41] e2πi[0.26,0.45], [0.24, 0.12] e2πi[0.33,0.44]

)
,(

[0.90, 0.04] e2πi[0.28,0.47], [0.12, 0.21] e2πi[0.26,0.45], [0.05, 0.03] e2πi[0.25,0.42]
)
,(

[0.35, 0.15] e2πi[0.34,0.43], [0.15, 0.25] e2πi[0.26,0.45], [0.31, 0.22] e2πi[0.36,0.45]
)
,

 .

Definition 16. For a universal set X, let a CIVPFSS (F,A) and R1 be a CIVPFSR on (F,A).
Then:

• R1 is said to be a CIVPFS reflexive relation on (F,A) if (ϋ, ϋ) ∈ R1, for all ϋ ∈ (F,A).

• R1 is said to be a CIVPFS irreflexive relation on (F,A) if (ϋ, ϋ) /∈ R1, for all ϋ ∈ (F,A).

• R1 is said to be a CIVPFS antisymmetric relation on (F,A) if for all ϋ, ώ ∈ (F,A),
(ϋ, ώ) ∈ R1 and (ώ, ϋ) ∈ R1; then ϋ = ώ.

• R1 is said to be a CIVPFS complete relation on (F,A) if (ϋ, ώ) ∈ R1 or (ώ, ϋ) ∈ R1, for
all ϋ, ώ ∈ (F,A).

• R1 is said to be a CIVPFS transitive relation on (F,A) if (ϋ, ώ) ∈ R1 and (ώ, γ) ∈ R1;
then, (ϋ, γ) ∈ R1, for all ϋ, ώ, γ ∈ (F,A).

• R1 is said to be a CIVPFS equivalence relation on (F,A) if R1 is a CIVPFS reflexive
relation, CIVPFS symmetric relation, and CIVPFS transitive relation on (F,A).

• R1 is said to be a CIVPFS preorder relation on (F,A) if R1 is a CIVPFS reflexive relation
and CIVPFS transitive relation on (F,A).

• R1 is said to be a CIVPFS strict-order relation on (F,A) if R1 is a CIVPFS irreflexive
relation and CIVPFS transitive relation on (F,A).

• R1 is said to be a CIVPFS partial-order relation on (F,A) if R1 is a CIVPFS preorder
relation and CIVPFS antisymmetric relation on (F,A).

• R1 is said to be a CIVPFS linear-order relation on (F,A) if R1 is a CIVPFS partial-order
relation and CIVPFS complete relation on (F,A).

Example 8. From Example 5, we define the following relations:
i. The CIVPFS reflexive relation R1 is

R1 =


(ϋ1, ϋ1),

(
([0.12, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.14,0.25])

)
,(

([0.13, 0.25]e2πi[0.18,0.34]), ([0.11, 0.21]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.42])
)
,(

([0.12, 0.31]e2πi[0.22,0.42]), ([0.22, 0.33]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.43])
)




R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 13 of 38

R1 =


(ϋ2, ϋ2),

(
([0.13, 0.34]e2πi[0.22,0.43]), ([0.21, 0.41]e2πi[0.26,0.45]), ([0.25, 0.31]e2πi[0.12,0.42])

)
,(

([0.17, 0.42]e2πi[0.12,0.52]), ([0.14, 0.24]e2πi[0.26,0.45]), ([0.22, 0.24]e2πi[0.26,0.34])
)
,(

([0.33, 0.52]e2πi[0.13,0.43]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.13, 0.22]e2πi[0.24,0.44])
)



R1 =


(ϋ3, ϋ3),

(
([0.53, 0.24]e2πi[0.23,0.43]), ([0.13, 0.41]e2πi[0.26,0.45]), ([0.24, 0.12]e2πi[0.33,0.44])

)
,(

([0.90, 0.04]e2πi[0.28,0.47]), ([0.12, 0.21]e2πi[0.26,0.45]), ([0.05, 0.03]e2πi[0.25,0.42])
)
,(

([0.35, 0.15]e2πi[0.34,0.43]), ([0.15, 0.25]e2πi[0.26,0.45]), ([0.31, 0.22]e2πi[0.36,0.45])
)


ii. The CIVPFS irreflexive relation R2 is

R2 =


(ϋ1, ϋ2),

(
([0.13, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.32,0.42])

)
,(

([0.13, 0.25]e2πi[0.12,0.34]), ([0.22, 0.27]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.34])
)
,(

([0.12, 0.31]e2πi[0.13,0.42]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.44])
)



R2 =


(ϋ2, ϋ3),

(
([0.23, 0.34]e2πi[0.23,0.43]), ([0.18, 0.26]e2πi[0.26,0.45]), ([0.15, 0.31]e2πi[0.33,0.44])

)
,(

([0.05, 0.42]e2πi[0.28,0.47]), ([0.14, 0.24]e2πi[0.26,0.45]), ([0.12, 0.024]e2πi[0.26,0.42])
)
,(

([0.15, 0.35]e2πi[0.23,0.43]), ([0.32, 0.44]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])
)



R2 =


(ϋ3, ϋ1),

(
([0.22, 0.26]e2πi[0.12,0.21]), ([0.18, 0.26]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.33,0.44])

)
,(

([0.05, 0.25]e2πi[0.28,0.34]), ([0.32, 0.41]e2πi[0.26,0.45]), ([0.34, 0.44]e2πi[0.23,0.42])
)
,(

([0.12, 0.31]e2πi[0.24,0.42]), ([0.22, 0.45]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])
)



R2 =


(ϋ3, ϋ2),

(
([0.23, 0.34]e2πi[0.23,0.43]), ([0.18, 0.26]e2πi[0.26,0.45]), ([0.15, 0.31]e2πi[0.33,0.44])

)
,(

([0.05, 0.42]e2πi[0.28,0.47]), ([0.14, 0.24]e2πi[0.26,0.45]), ([0.12, 0.024]e2πi[0.26,0.42])
)
,(

([0.15, 0.35]e2πi[0.23,0.43]), ([0.32, 0.44]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])
)


Example 9. From example 5, we define the following relations:

i. The CIVPFS symmetric relation R1 is:


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 14 of 38

R1 =



(ϋ1, ϋ2),
(
([0.13, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.32,0.42])

)
,(

([0.13, 0.25]e2πi[0.12,0.34]), ([0.22, 0.27]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.34])
)
,(

([0.12, 0.31]e2πi[0.13,0.42]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.44])
)
,

(ϋ2, ϋ1),
(
([0.13, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.32,0.42])

)
,(

([0.13, 0.25]e2πi[0.12,0.34]), ([0.22, 0.27]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.34])
)
,(

([0.12, 0.31]e2πi[0.13,0.42]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.44])
)
,

(ϋ2, ϋ3),
(
([0.23, 0.34]e2πi[0.23,0.43]), ([0.18, 0.26]e2πi[0.26,0.45]), ([0.15, 0.31]e2πi[0.33,0.44])

)
,(

([0.05, 0.42]e2πi[0.28,0.47]), ([0.14, 0.24]e2πi[0.26,0.45]), ([0.12, 0.024]e2πi[0.26,0.42])
)
,(

([0.15, 0.35]e2πi[0.23,0.43]), ([0.32, 0.44]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])
)
,

(ϋ3, ϋ2),
(
([0.23, 0.34]e2πi[0.23,0.43]), ([0.18, 0.26]e2πi[0.26,0.45]), ([0.15, 0.31]e2πi[0.33,0.44])

)
,(

([0.05, 0.42]e2πi[0.28,0.47]), ([0.14, 0.24]e2πi[0.26,0.45]), ([0.12, 0.024]e2πi[0.26,0.42])
)
,(

([0.15, 0.35]e2πi[0.23,0.43]), ([0.32, 0.44]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])
)


ii. The CIVPFS antisymmetric relation R2 is:

R2 =



(ϋ1, ϋ1),
(
([0.12, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.14,0.25])

)
,(

([0.13, 0.25]e2πi[0.18,0.34]), ([0.11, 0.21]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.42])
)
,(

([0.12, 0.31]e2πi[0.22,0.42]), ([0.22, 0.33]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.43])
)
,

(ϋ2, ϋ2),
(
([0.13, 0.34]e2πi[0.22,0.43]), ([0.21, 0.41]e2πi[0.26,0.45]), ([0.25, 0.31]e2πi[0.12,0.42])

)
,(

([0.17, 0.42]e2πi[0.12,0.52]), ([0.14, 0.24]e2πi[0.26,0.45]), ([0.22, 0.24]e2πi[0.26,0.34])
)
,(

([0.33, 0.52]e2πi[0.13,0.43]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.13, 0.22]e2πi[0.24,0.44])
)
,

(ϋ3, ϋ3),
(
([0.53, 0.24]e2πi[0.23,0.43]), ([0.13, 0.41]e2πi[0.26,0.45]), ([0.24, 0.12]e2πi[0.33,0.44])

)
,(

([0.90, 0.04]e2πi[0.28,0.47]), ([0.12, 0.21]e2πi[0.26,0.45]), ([0.05, 0.03]e2πi[0.25,0.42])
)
,(

([0.35, 0.15]e2πi[0.34,0.43]), ([0.15, 0.25]e2πi[0.26,0.45]), ([0.31, 0.22]e2πi[0.36,0.45])
)


iii. The CIVPFS transitive relation R3 is:


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 15 of 38

R3 =



(ϋ2, ϋ1),
(
([0.13, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.32,0.42])

)
,(

([0.13, 0.25]e2πi[0.12,0.34]), ([0.22, 0.27]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.34])
)
,(

([0.12, 0.31]e2πi[0.13,0.42]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.44])
)
,

(ϋ2, ϋ2),
(
([0.13, 0.34]e2πi[0.22,0.43]), ([0.21, 0.41]e2πi[0.26,0.45]), ([0.25, 0.31]e2πi[0.12,0.42])

)
,(

([0.17, 0.42]e2πi[0.12,0.52]), ([0.14, 0.24]e2πi[0.26,0.45]), ([0.22, 0.24]e2πi[0.26,0.34])
)
,(

([0.33, 0.52]e2πi[0.13,0.43]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.13, 0.22]e2πi[0.24,0.44])
)
,

(ϋ2, ϋ3),
(
([0.23, 0.34]e2πi[0.23,0.43]), ([0.18, 0.26]e2πi[0.26,0.45]), ([0.15, 0.31]e2πi[0.33,0.44])

)
,(

([0.05, 0.42]e2πi[0.28,0.47]), ([0.14, 0.24]e2πi[0.26,0.45]), ([0.12, 0.024]e2πi[0.26,0.42])
)
,(

([0.15, 0.35]e2πi[0.23,0.43]), ([0.32, 0.44]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])
)
,

(ϋ3, ϋ1),
(
([0.22, 0.26]e2πi[0.12,0.21]), ([0.18, 0.26]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.33,0.44])

)
,(

([0.05, 0.25]e2πi[0.28,0.34]), ([0.32, 0.41]e2πi[0.26,0.45]), ([0.34, 0.44]e2πi[0.23,0.42])
)
,(

([0.12, 0.31]e2πi[0.24,0.42]), ([0.22, 0.45]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])
)
,

(ϋ3, ϋ2),
(
([0.23, 0.34]e2πi[0.23,0.43]), ([0.18, 0.26]e2πi[0.26,0.45]), ([0.15, 0.31]e2πi[0.33,0.44])

)
,(

([0.05, 0.42]e2πi[0.28,0.47]), ([0.14, 0.24]e2πi[0.26,0.45]), ([0.12, 0.024]e2πi[0.26,0.42])
)
,(

([0.15, 0.35]e2πi[0.23,0.43]), ([0.32, 0.44]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])
)
,

(ϋ3, ϋ3),
(
([0.53, 0.24]e2πi[0.23,0.43]), ([0.13, 0.41]e2πi[0.26,0.45]), ([0.24, 0.12]e2πi[0.33,0.44])

)
,(

([0.90, 0.04]e2πi[0.28,0.47]), ([0.12, 0.21]e2πi[0.26,0.45]), ([0.05, 0.03]e2πi[0.25,0.42])
)
,(

([0.35, 0.15]e2πi[0.34,0.43]), ([0.15, 0.25]e2πi[0.26,0.45]), ([0.31, 0.22]e2πi[0.36,0.45])
)




R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 16 of 38

iv. The CIVPFS equivalence relation R4 is:

R4 =



(ϋ1, ϋ1),
((

[0.12, 0.26]e2πi[0.12,0.21], [0.32, 0.41]e2πi[0.14,0.22], [0.31, 0.42]e2πi[0.14,0.25]
)
,(

[0.13, 0.25]e2πi[0.18,0.34], [0.11, 0.21]e2πi[0.23,0.42], [0.34, 0.44]e2πi[0.23,0.42]
)
,(

[0.12, 0.31]e2πi[0.22,0.42], [0.22, 0.33]e2πi[0.26,0.45], [0.21, 0.30]e2πi[0.34,0.43]
))

,

(ϋ2, ϋ2),
((

[0.13, 0.34]e2πi[0.22,0.43], [0.21, 0.41]e2πi[0.26,0.45], [0.25, 0.31]e2πi[0.12,0.42]
)
,(

[0.17, 0.42]e2πi[0.12,0.52], [0.14, 0.24]e2πi[0.26,0.45], [0.22, 0.24]e2πi[0.26,0.34]
)
,(

[0.33, 0.52]e2πi[0.13,0.43], [0.16, 0.29]e2πi[0.26,0.45], [0.13, 0.22]e2πi[0.24,0.44]
))

,

(ϋ3, ϋ2),
((

[0.23, 0.34]e2πi[0.23,0.43], [0.18, 0.26]e2πi[0.26,0.45], [0.15, 0.31]e2πi[0.33,0.44]
)
,(

[0.05, 0.42]e2πi[0.28,0.47], [0.14, 0.24]e2πi[0.26,0.45], [0.12, 0.24]e2πi[0.26,0.42]
)
,(

[0.15, 0.35]e2πi[0.23,0.43], [0.32, 0.44]e2πi[0.26,0.45], [0.22, 0.31]e2πi[0.36,0.45]
))

,

(ϋ3, ϋ3),
((

[0.53, 0.24]e2πi[0.23,0.43], [0.13, 0.41]e2πi[0.26,0.45], [0.24, 0.12]e2πi[0.33,0.44]
)
,(

[0.90, 0.04]e2πi[0.28,0.47], [0.12, 0.21]e2πi[0.26,0.45], [0.05, 0.03]e2πi[0.25,0.42]
)
,(

[0.35, 0.15]e2πi[0.34,0.43], [0.15, 0.25]e2πi[0.26,0.45], [0.31, 0.22]e2πi[0.36,0.45]
))


v. The CIVPFS complete relation R5 is:

R5 =



(ϋ1, ϋ2),
((

[0.13, 0.26]e2πi[0.12,0.21], [0.32, 0.41]e2πi[0.14,0.22], [0.31, 0.42]e2πi[0.32,0.42]
)
,(

[0.13, 0.25]e2πi[0.12,0.34], [0.22, 0.27]e2πi[0.23,0.42], [0.34, 0.44]e2πi[0.23,0.34]
)
,(

[0.12, 0.31]e2πi[0.13,0.42], [0.16, 0.29]e2πi[0.26,0.45], [0.21, 0.30]e2πi[0.34,0.44]
))

,

(ϋ2, ϋ3),
((

[0.23, 0.34]e2πi[0.23,0.43], [0.18, 0.26]e2πi[0.26,0.45], [0.15, 0.31]e2πi[0.33,0.44]
)
,(

[0.05, 0.42]e2πi[0.28,0.47], [0.14, 0.24]e2πi[0.26,0.45], [0.12, 0.24]e2πi[0.26,0.42]
)
,(

[0.15, 0.35]e2πi[0.23,0.43], [0.32, 0.44]e2πi[0.26,0.45], [0.22, 0.31]e2πi[0.36,0.45]
))


Example 10. Consider the CP from Example 5, we have;
i. The CIVPFS preorder relation is expressed as:


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 17 of 38

R1 =



((ϋ1, ϋ1),
(
([0.12, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.14,0.25])

)
,(

([0.13, 0.25]e2πi[0.18,0.34]), ([0.11, 0.21]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.42])
)
,(

([0.12, 0.31]e2πi[0.22,0.42]), ([0.22, 0.33]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.43])
)
),

((ϋ1, ϋ2),
(
([0.13, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.32,0.42])

)
,(

([0.13, 0.25]e2πi[0.12,0.34]), ([0.22, 0.27]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.34])
)
,(

([0.12, 0.31]e2πi[0.13,0.42]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.44])
)
),

((ϋ2, ϋ2),
(
([0.13, 0.34]e2πi[0.22,0.43]), ([0.21, 0.41]e2πi[0.26,0.45]), ([0.25, 0.31]e2πi[0.12,0.42])

)
,(

([0.17, 0.42]e2πi[0.12,0.52]), ([0.14, 0.241]e2πi[0.26,0.45]), ([0.22, 0.24]e2πi[0.26,0.34])
)
,(

([0.33, 0.52]e2πi[0.13,0.43]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.13, 0.22]e2πi[0.24,0.44])
)
),

((ϋ3, ϋ3),
(
([0.53, 0.24]e2πi[0.23,0.43]), ([0.13, 0.41]e2πi[0.26,0.45]), ([0.24, 0.12]e2πi[0.33,0.44])

)
,(

([0.90, 0.04]e2πi[0.28,0.47]), ([0.12, 0.21]e2πi[0.26,0.45]), ([0.05, 0.03]e2πi[0.25,0.42])
)
,(

([0.35, 0.15]e2πi[0.34,0.43]), ([0.15, 0.25]e2πi[0.26,0.45]), ([0.31, 0.22]e2πi[0.36,0.45])
)
)



.

ii. The CIVPFS strict-order relation is expressed as:

R2 =



((ϋ2, ϋ1),
(
([0.13, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.32,0.42])

)
,(

([0.13, 0.25]e2πi[0.12,0.34]), ([0.22, 0.27]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.34])
)
,(

([0.12, 0.31]e2πi[0.13,0.42]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.44])
)
),

((ϋ3, ϋ1),
(
([0.22, 0.26]e2πi[0.12,0.21]), ([0.18, 0.26]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.333,0.44])

)
,(

([0.05, 0.25]e2πi[0.28,0.34]), ([0.32, 0.41]e2πi[0.26,0.45]), ([0.34, 0.44]e2πi[0.23,0.42])
)
,(

([0.12, 0.31]e2πi[0.24,0.42]), ([0.22, 0.45]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])
)
),

((ϋ3, ϋ2),
(
([0.23, 0.34]e2πi[0.23,0.43]), ([0.18, 0.26]e2πi[0.26,0.45]), ([0.15, 0.31]e2πi[0.33,0.44])

)
,(

([0.05, 0.42]e2πi[0.28,0.47]), ([0.14, 0.24]e2πi[0.26,0.45]), ([0.12, 0.024]e2πi[0.26,0.42])
)
,(

([0.15, 0.35]e2πi[0.23,0.43]), ([0.32, 0.44]e2πi[0.26,0.45]), ([0.22, 0.31]e2πi[0.36,0.45])
)
)



.

The CIVPFS partial-order relation is expressed as:

R̃3 =



((ϋ1, ϋ1),
(
([0.12, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.14,0.25])

)
,(

([0.13, 0.25]e2πi[0.18,0.34]), ([0.11, 0.21]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.42])
)
,(

([0.12, 0.31]e2πi[0.22,0.42]), ([0.22, 0.33]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.43])
)
),

((ϋ2, ϋ1),
(
([0.13, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.32,0.42])

)
,(

([0.13, 0.25]e2πi[0.12,0.34]), ([0.22, 0.27]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.34])
)
,(

([0.12, 0.31]e2πi[0.13,0.42]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.44])
)
),

((ϋ2, ϋ2),
(
([0.13, 0.34]e2πi[0.22,0.43]), ([0.21, 0.41]e2πi[0.26,0.45]), ([0.25, 0.31]e2πi[0.12,0.42])

)
,(

([0.17, 0.42]e2πi[0.12,0.52]), ([0.14, 0.241]e2πi[0.26,0.45]), ([0.22, 0.24]e2πi[0.26,0.34])
)
,(

([0.33, 0.52]e2πi[0.13,0.43]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.13, 0.22]e2πi[0.24,0.44])
)
),

((ϋ3, ϋ3),
(
([0.53, 0.24]e2πi[0.23,0.43]), ([0.13, 0.41]e2πi[0.26,0.45]), ([0.24, 0.12]e2πi[0.33,0.44])

)
,(

([0.90, 0.04]e2πi[0.28,0.47]), ([0.12, 0.21]e2πi[0.26,0.45]), ([0.05, 0.03]e2πi[0.25,0.42])
)
,(

([0.35, 0.15]e2πi[0.34,0.43]), ([0.15, 0.25]e2πi[0.26,0.45]), ([0.31, 0.22]e2πi[0.36,0.45])
)
)



.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 18 of 38

iii. The CIVPFS partial-order relation is expressed as:

R3 =



((ϋ1, ϋ1),
(
([0.12, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.14,0.25])

)
,(

([0.13, 0.25]e2πi[0.18,0.34]), ([0.11, 0.21]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.42])
)
,(

([0.12, 0.31]e2πi[0.22,0.42]), ([0.22, 0.33]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.43])
)
),

((ϋ2, ϋ1),
(
([0.13, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.32,0.42])

)
,(

([0.13, 0.25]e2πi[0.12,0.34]), ([0.22, 0.27]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.34])
)
,(

([0.12, 0.31]e2πi[0.13,0.42]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.44])
)
),

((ϋ2, ϋ2),
(
([0.13, 0.34]e2πi[0.22,0.43]), ([0.21, 0.41]e2πi[0.26,0.45]), ([0.25, 0.31]e2πi[0.12,0.42])

)
,(

([0.17, 0.42]e2πi[0.12,0.52]), ([0.14, 0.241]e2πi[0.26,0.45]), ([0.22, 0.24]e2πi[0.26,0.34])
)
,(

([0.33, 0.52]e2πi[0.13,0.43]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.13, 0.22]e2πi[0.24,0.44])
)
),

((ϋ3, ϋ3),
(
([0.53, 0.24]e2πi[0.23,0.43]), ([0.13, 0.41]e2πi[0.26,0.45]), ([0.24, 0.12]e2πi[0.33,0.44])

)
,(

([0.90, 0.04]e2πi[0.28,0.47]), ([0.12, 0.21]e2πi[0.26,0.45]), ([0.05, 0.03]e2πi[0.25,0.42])
)
,(

([0.35, 0.15]e2πi[0.34,0.43]), ([0.15, 0.25]e2πi[0.26,0.45]), ([0.31, 0.22]e2πi[0.36,0.45])
)
)



.

iv. The CIVPFS linear-order relation is expressed as:

R4 =



((ϋ1, ϋ1),
(
([0.12, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.14,0.25])

)
,(

([0.13, 0.25]e2πi[0.18,0.34]), ([0.11, 0.21]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.42])
)
,(

([0.12, 0.31]e2πi[0.22,0.42]), ([0.22, 0.33]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.43])
)
),

((ϋ2, ϋ1),
(
([0.13, 0.26]e2πi[0.12,0.21]), ([0.32, 0.41]e2πi[0.14,0.22]), ([0.31, 0.42]e2πi[0.32,0.42])

)
,(

([0.13, 0.25]e2πi[0.12,0.34]), ([0.22, 0.27]e2πi[0.23,0.42]), ([0.34, 0.44]e2πi[0.23,0.34])
)
,(

([0.12, 0.31]e2πi[0.13,0.42]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.21, 0.30]e2πi[0.34,0.44])
)
),

((ϋ2, ϋ2),
(
([0.13, 0.34]e2πi[0.22,0.43]), ([0.21, 0.41]e2πi[0.26,0.45]), ([0.25, 0.31]e2πi[0.12,0.42])

)
,(

([0.17, 0.42]e2πi[0.12,0.52]), ([0.14, 0.241]e2πi[0.26,0.45]), ([0.22, 0.24]e2πi[0.26,0.34])
)
,(

([0.33, 0.52]e2πi[0.13,0.43]), ([0.16, 0.29]e2πi[0.26,0.45]), ([0.13, 0.22]e2πi[0.24,0.44])
)
),

((ϋ3, ϋ3),
(
([0.53, 0.24]e2πi[0.23,0.43]), ([0.13, 0.41]e2πi[0.26,0.45]), ([0.24, 0.12]e2πi[0.33,0.44])

)
,(

([0.90, 0.04]e2πi[0.28,0.47]), ([0.12, 0.21]e2πi[0.26,0.45]), ([0.05, 0.03]e2πi[0.25,0.42])
)
,(

([0.35, 0.15]e2πi[0.34,0.43]), ([0.15, 0.25]e2πi[0.26,0.45]), ([0.31, 0.22]e2πi[0.36,0.45])
)
)



.

Theorem 1. A CIVPFS relation R is a CIVPFS symmetric relation on a CIVPFS set F if
and only if R = R−1.

Proof: Suppose that R = R−1. Then,

(ϋ, ώ) ∈ R =⇒ (ώ, ϋ) ∈ R−1 =⇒ (ώ, ϋ) ∈ R.

Thus, R is a CIVPFS symmetric relation on a CIVPFS set F .
Conversely, assume that R is a CIVPFS symmetric relation on a CIVPFS set.

(ϋ, ώ) ∈ R =⇒ (ώ, ϋ) ∈ R.

However,
(ώ, ϋ) ∈ R =⇒ (ώ, ϋ) ∈ R−1 =⇒ R = R−1.

Hence, proved.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 19 of 38

Theorem 2. A CIVPFS relation R is a CIVPFS transitive relation on a CIVPFS set F if and
only if R ◦R ⊆ R.

Proof: Suppose that R is a CIVPFS transitive relation on a CIVPFS set F . Let (ϋ, X) ∈
R ◦ R. Then, according to the definition of a CIVPFS transitive relation,

(ϋ, ώ) ∈ R and (ώ,X) ∈ R =⇒ (ϋ, X) ∈ R.

Thus, R ◦R ⊆ R.
Conversely, assume that R ◦R ⊆ R. Then, For (ϋ, ώ) ∈ R and (ώ,X) ∈ R,

(ϋ, X) ∈ R ◦ R ⊆ R =⇒ (ϋ, X) ∈ R.

Thus, R is a CIVPFS transitive relation on the CIVPFS set F .

4. Application:

This section explores an application of the proposed framework to identify the best fitness
tracking app. The method evaluates multiple apps based on predefined criteria using the newly
introduced concepts. The results demonstrate the practicality and effectiveness of the proposed
approach in decision-making scenarios.

4.1. Fitness tracking app

Wearable technology, called fitness trackers, enables users to keep tabs on their fitness-
related activities and data, like calories burned and distance traveled. Fitness trackers can be
integrated into smartwatches or be used as standalone devices. Typically, a mobile app that
lets users manage information and utilize app features is connected to the device.

In the middle of the 1960s, the first functional fitness tracker was created. However, wearable
technology and fitness trackers really took off a decade ago. In 2012, the first Fitbit was released,
and in 2014, the first Apple Watch was introduced. Since then, wearable technology and fitness
trackers have grown in popularity.

According to a recent Pew Research Center study, 21% of Americans wear smartwatches
or fitness trackers on a regular basis. Similarly, a 2019 Gallup poll found that over 40% of
Americans track their fitness-related activities using an app or gadget.

Fitness trackers are used for a variety of purposes, but the primary goal is to improve or
preserve one’s health. The tool or app keeps users informed, helps them stay motivated, and lets
them monitor their progress. Many people have started pursuing healthier lifestyles in recent
years, and they have embraced technologies that encourage them to monitor their progress.
Millennials are the generation that has embraced this trend the most; some have even dubbed
them “the wellness generation.”

The procedure of this application is explained in Figure (1).
First of all, we defined a universal set, which includes some fundamental fitness tracking

apps. The universal set is defined as:

X = {X1, X2, X3, X4}

This universal set consists of four types of fitness tracking apps, where:

• X1 = Mobile application,

• X2 = Web application,

• X3 = Wearable integration, and

• X4 = Social networking integration.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 20 of 38

Figure 1: . Fitness tracking app application procedure.

Figure 2: Summary of different fitness tracking apps.

The types of fitness tracking apps are discussed in Figure (2).
The summary of different apps is as follows:

⋇ Mobile Application:
A mobile application is designed to help users monitor and manage their physical activity, health
metrics, and wellness goals. It typically includes features like:

• Tracking workouts,

• Counting steps,

• Monitoring heart rate,

• Providing nutrition information, and


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 21 of 38

• Setting fitness goals.

⋇ Web Application:
The online version of a fitness tracking app, referred to as a web application, is accessible
through a web browser. Without having to download the app to your device, you can:

• Monitor your fitness activities,

• Set objectives,

• View your progress, and

• Use other features.

It functions as an online version of the app rather than a downloadable application for your
computer or phone.
⋇ Wearable Integration:
The term “wearable integration” describes an app’s capacity to link and synchronize with
wearable gadgets, such as:

• Smartwatches,

• Fitness trackers, or

• Other wearable technology.

The app collects information from the wearable device, including:

• Heart rate,

• Steps taken,

• Distance traveled, and

• Sleep patterns.

This synchronization provides a more thorough and accurate picture of fitness activities and
progress.
⋇ Social Networking Integration:
Social networking integration in fitness tracking apps allows users to connect and share their
fitness activities and progress with friends or followers on social media platforms. This feature
enables users to:

• Share accomplishments, challenges, and updates on platforms like Facebook, Instagram,
and Twitter,

• Participate in friendly competitions with their social network,

• Discuss fitness experiences, and

• Stay motivated.

⋇ Fitness Tracking App Parameters:
The parameters of a fitness tracking app are defined as:

E = {ϋ1, ϋ2, ϋ3, ϋ4}

where:

• ϋ1 = Users’ profiles,


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 22 of 38

• ϋ2 = Activity tracking,

• ϋ3 = Workout plans, and

• ϋ4 = Progress monitoring.

A summary of fitness tracking app parameters is shown in Figure (3).

Figure 3: Summary of fitness tracking app parameters.

The summary of different parameters is as follows:
⋇ User Profiles:
Users create profiles with personal information, goals, and preferences, which allow the app to
customize workouts and meal plans according to this input. Goal setting enables users to set
specific goals, track progress, and adjust their daily routine accordingly.

The parameters related to user profiles are defined as:

ϋ1 =



(((
[0.27, 0.36]e2πi[0.12,0.21], [0.40, 0.42]e2πi[0.14,0.22], [0.31, 0.42]e2πi[0.14,0.22]

)
,(

[0.13, 0.35]e2πi[0.28,0.34], [0.32, 0.40]e2πi[0.23,0.42], [0.34, 0.43]e2πi[0.23,0.42]
)
,(

[0.12, 0.31]e2πi[0.32,0.42], [0.21, 0.45]e2πi[0.23,0.42], [0.21, 0.30]e2πi[0.34,0.43]
)
,(

[0.12, 0.33]e2πi[0.32,0.42], [0.21, 0.40]e2πi[0.23,0.42], [0.21, 0.31]e2πi[0.34,0.43]
)))


⋇ Activity Tracking:
Activity tracking involves the practice of measuring and collecting data on an individual’s
physical and psychological activity to keep track of and maintain documentation regarding
their health and wellness.

The parameters related to activity tracking are defined as:

ϋ2 =



(((
[0.23, 0.38]e2πi[0.32,0.43], [0.21, 0.40]e2πi[0.23,0.42], [0.15, 0.35]e2πi[0.12,0.42]

)
,(

[0.17, 0.40]e2πi[0.22,0.52], [0.14, 0.24]e2πi[0.23,0.42], [0.12, 0.24]e2πi[0.26,0.34]
)
,(

[0.40, 0.42]e2πi[0.23,0.43], [0.34, 0.40]e2πi[0.23,0.42], [0.13, 0.28]e2πi[0.14,0.44]
)
,(

[0.12, 0.36]e2πi[0.32,0.42], [0.22, 0.45]e2πi[0.23,0.42], [0.21, 0.30]e2πi[0.34,0.43]
)))




R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 23 of 38

⋇ Workout Plan:
A workout plan includes the type of exercise you need to do and how long you need to do them.
This plan helps you maintain a consistent routine, so you can achieve your fitness goals. It gives
you an organized structure of how your exercise should go. It can also help you make working
out more creative and will help create more progress. Many fitness experts recommend creating
a plan, so you can see the progress you’ve made and gradually achieve your goals.

The parameters related to the workout plan are defined as:

ϋ3 =



(((
[0.34, 0.53]e2πi[0.33,0.43], [0.13, 0.20]e2πi[0.26,0.45], [0.12, 0.20]e2πi[0.23,0.44]

)
,(

[0.04, 0.80]e2πi[0.38,0.47], [0.12, 0.22]e2πi[0.26,0.45], [0.01, 0.06]e2πi[0.25,0.42]
)
,(

[0.25, 0.34]e2πi[0.34,0.43], [0.15, 0.24]e2πi[0.26,0.45], [0.12, 0.24]e2πi[0.26,0.45]
)
,(

[0.12, 0.37]e2πi[0.32,0.42], [0.20, 0.45]e2πi[0.23,0.42], [0.21, 0.36]e2πi[0.34,0.43]
)))


⋇ Progress Monitoring:
Users love to see what they have achieved through their workout sessions. When a wearable
device collects the information and sends it to your fitness tracking application, users can
efficiently monitor it. From how many miles they have covered during running sessions to how
much time was taken, it allows them to track their progress efficiently.

The parameters related to progress monitoring are defined as:

ϋ4 =



(((
[0.34, 0.43]e2πi[0.33,0.43], [0.13, 0.20]e2πi[0.26,0.45], [0.12, 0.20]e2πi[0.23,0.44]

)
,(

[0.06, 0.80]e2πi[0.38,0.47], [0.12, 0.25]e2πi[0.26,0.45], [0.01, 0.02]e2πi[0.25,0.42]
)
,(

[0.25, 0.35]e2πi[0.34,0.43], [0.24, 0.25]e2πi[0.26,0.45], [0.12, 0.22]e2πi[0.26,0.45]
)
,(

[0.12, 0.41]e2πi[0.32,0.42], [0.22, 0.44]e2πi[0.23,0.42], [0.21, 0.32]e2πi[0.34,0.43]
)))


Experts examine the fitness tracking app in which all the parameters are considered. Let
(F,A) = F represent observations by an expert, who assigns values of membership, neutral,
and non-membership based on the parameters.

Suppose that the corresponding membership, neutral, and non-membership matrices are as
follows:

F =



ϋ1 =
(
[0.27, 0.36] e2πi[0.12,0.21], [0.40, 0.42] e2πi[0.14,0.22], [0.31, 0.42] e2πi[0.14,0.22]

)
,(

[0.13, 0.35] e2πi[0.28,0.34], [0.32, 0.40] e2πi[0.23,0.42], [0.34, 0.43] e2πi[0.23,0.42]
)
,(

[0.12, 0.31] e2πi[0.32,0.42], [0.21, 0.45] e2πi[0.23,0.42], [0.21, 0.30] e2πi[0.34,0.43]
)
,(

[0.12, 0.33] e2πi[0.32,0.42], [0.21, 0.40] e2πi[0.23,0.42], [0.21, 0.31] e2πi[0.34,0.43]
)



F =



ϋ2 =
(
[0.23, 0.38] e2πi[0.32,0.43], [0.21, 0.40] e2πi[0.23,0.42], [0.15, 0.35] e2πi[0.12,0.42]

)
,(

[0.17, 0.40] e2πi[0.22,0.52], [0.14, 0.24] e2πi[0.23,0.42], [0.12, 0.24] e2πi[0.26,0.34]
)
,(

[0.40, 0.42] e2πi[0.23,0.43], [0.34, 0.40] e2πi[0.23,0.42], [0.13, 0.28] e2πi[0.14,0.44]
)
,(

[0.12, 0.36] e2πi[0.32,0.42], [0.22, 0.45] e2πi[0.23,0.42], [0.21, 0.30] e2πi[0.34,0.43]
)




R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 24 of 38

F =



ϋ3 =
(
[0.34, 0.53] e2πi[0.33,0.43], [0.13, 0.20] e2πi[0.26,0.45], [0.12, 0.20] e2πi[0.23,0.44]

)
,(

[0.04, 0.80] e2πi[0.38,0.47], [0.12, 0.22] e2πi[0.26,0.45], [0.01, 0.06] e2πi[0.25,0.42]
)
,(

[0.25, 0.34] e2πi[0.34,0.43], [0.15, 0.24] e2πi[0.26,0.45], [0.12, 0.24] e2πi[0.26,0.45]
)
,(

[0.12, 0.37] e2πi[0.32,0.42], [0.20, 0.45] e2πi[0.23,0.42], [0.21, 0.36] e2πi[0.34,0.43]
)



F =



ϋ4 =
(
[0.34, 0.43] e2πi[0.33,0.43], [0.13, 0.20] e2πi[0.26,0.45], [0.12, 0.20] e2πi[0.23,0.44]

)
,(

[0.06, 0.80] e2πi[0.38,0.47], [0.12, 0.25] e2πi[0.26,0.45], [0.01, 0.02] e2πi[0.25,0.42]
)
,(

[0.25, 0.35] e2πi[0.34,0.43], [0.24, 0.25] e2πi[0.26,0.45], [0.12, 0.22] e2πi[0.26,0.45]
)
,(

[0.12, 0.41] e2πi[0.32,0.42], [0.22, 0.44] e2πi[0.23,0.42], [0.21, 0.32] e2πi[0.34,0.43]
)


Assume that the first three values of matrices ϋ1, ϋ2, ϋ3 of each parameter correspond to the
values of membership, neutral, and non-membership assigned by an expert. The value (λ) of
each parameter corresponds to the value of membership, neutral, and non-membership assigned
by an expert, and the fourth indicates the general belongingness value in the fitness tracking
app. Then, the self-CP of (F,A) is expressed CP of fitness tracking app as shown in Figures
(4)-(7).

Figure 4: CP of fitness tracking app for ϋ1

4.2. Score Function

The CP of two CIVPFSSs are shown in the above table. The complex values are converted
into real values to find the score values. We use the following formula to compute the score:[
µ+(ϋ)2 + ϋ+(ϋ)2 − µ−(ϋ)2 − ϋ−(ϋ)2

]
+
[
φ+(ϋ)2 + j+(ϋ)2 − φ−(ϋ)2 − j−(ϋ)2

]
−
[
ϕ+(ϋ)2 + f+(ϋ)2 − ϕ−(ϋ)2 − f−(ϋ)2

]
where: µ+(ϋ), µ−(ϋ) represent the membership values, ϋ+(ϋ), ϋ−(ϋ) are the neutral values,

φ+(ϋ), φ−(ϋ) represent the non-membership values, j+(ϋ), j−(ϋ) are the neutral values for
non-membership, ϕ+(ϋ), ϕ−(ϋ) represent the fuzzy sets for membership values, f+(ϋ), f−(ϋ)
are the non-membership values for fuzzy sets.

The values that we obtain from the formula are given below in the Table 1.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 25 of 38

Figure 5: CP of fitness tracking app for ϋ2

Figure 6: CP of fitness tracking app for ϋ3

Figure 7: CP of fitness tracking app for ϋ4

To find the best fitness tracking app, we must first determine the highest numerical degree
in each row while ignoring the last column. The intended value of λ is multiplied by the product
of these numerical degrees to get each fitness tracking app score. We cannot compare any app
with itself. So, in each same ordered pair like (ϋ3, ϋ2), we write a cross (×). The fitness tracking
app with the highest score is the finest. Since it is not a novel attempt to compare, we do not
look at the numerical degree of the fitness tracking app of the identical parametric ordered pair.

The grade values are calculated in Table 2. Now, calculate the score function as:

S(X1) = (0.426× 0.322) + (0.426× 0.322) = 0.274


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 26 of 38

Ordered Pair X1 X2 X3 λ

(ϋ1, ϋ1) 0.224 0.023 0.131 0.259
(ϋ1, ϋ2) 0.287 0.014 0.156 0.321
(ϋ1, ϋ3) 0.257 -0.083 0.132 0.182
(ϋ1, ϋ4) 0.280 -0.083 0.132 0.165
(ϋ2, ϋ1) 0.287 0.014 0.156 0.321
(ϋ2, ϋ2) 0.312 -0.066 0.423 0.081
(ϋ2, ϋ3) 0.320 0.080 0.336 0.147
(ϋ2, ϋ4) 0.322 0.426 0.334 0.228
(ϋ3, ϋ1) 0.012 -0.008 0.133 0.182
(ϋ3, ϋ2) 0.320 0.080 0.336 0.147
(ϋ3, ϋ3) 0.328 0.230 0.760 0.122
(ϋ3, ϋ4) 0.318 0.137 0.761 0.122
(ϋ4, ϋ1) 0.280 -0.083 0.032 0.165
(ϋ4, ϋ2) 0.322 0.426 0.334 0.228
(ϋ4, ϋ3) 0.318 0.137 0.761 0.122
(ϋ4, ϋ4) 0.367 0.147 0.413 0.124

Table 1: Score values

Table 2: Grade Table

R (ϋ1, ϋ1) (ϋ1, ϋ2) (ϋ1, ϋ3) (ϋ1, ϋ4) (ϋ2, ϋ1) (ϋ2, ϋ2) (ϋ2, ϋ3) (ϋ2, ϋ4)

hi h3 h3 h3 h3 h3 h2 h2 h1

Highest Degree × 0.321 0.182 0.165 0.321 × 0.336 0.426

λ × 0.287 0.257 0.280 0.287 × 0.320 0.322

R (ϋ3, ϋ1) (ϋ3, ϋ2) (ϋ3, ϋ3) (ϋ3, ϋ4) (ϋ4, ϋ1) (ϋ4, ϋ2) (ϋ4, ϋ3) (ϋ4, ϋ4)

hi h3 h2 h2 h2 h3 h1 h2 h2

Highest Degree 0.182 0.336 × 0.761 0.165 0.426 0.761 ×

λ 0.012 0.320 × 0.318 0.280 0.322 0.318 ×

S(X2) = (0.336× 0.320) + (0.336× 0.320) + (0.761× 0.318) + (0.761× 0.318) = 0.699

S(X3) = (0.321× 0.287) + (0.182× 0.257) + (0.165× 0.280) + (0.321× 0.287)

+ (0.182× 0.012) + (0.165× 0.280) = 0.326

Hence, X2 is a good fitness tracking app and works effectively.

5. Comparative Analysis

In this section, the concept of CIVPFSS is compared with some preceding structures in the
theory of FS, IFS, PFS, CFS, CIFS, CPFS, IVFS, IVIFS, IVPFS, CIVFS, CIVIFS, CIVPFS,
CFSS, CIFSS, CPFSS, CIVFSS, and CIVIFSS.

5.1. CIVPFSS Comparison with FS, IFS, and PFS

FS has a single membership degree, IFS adds a non-membership degree, and PFS includes
membership, neutral, and non-membership degrees. CIVPFSS combines complex intervals and
picture fuzzy sets to model uncertainty in a multi-dimensional and complex way, making it


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suitable for complex and dynamic systems. In contrast, FS, IFS, and PFS are more suitable
for simpler applications, with FS being the most basic and PFS being more advanced but still
limited to three membership degrees.

5.2. CIVPFSS Comparison with CFS, CIFS, and CPFS

CFS is the simplest, with a single complex membership degree. CIFS adds a non-membership
degree, increasing the uncertainty dimension. CPFS adds a neutral membership degree, further
increasing the uncertainty dimension. CIVPFSS combines complex intervals and picture fuzzy
sets, allowing for the most flexible and nuanced uncertainty modeling. CIVPFSS is more suitable
for complex and dynamic systems, while CFS, CIFS, and CPFS are more suitable for simpler
applications.

5.3. CIVPFSS Comparison with IVFS, IVIFS, and IVPFS

IVFS is the simplest, with a single interval membership degree. IVIFS adds a non-membership
degree, increasing the uncertainty dimension. IVPFS adds a neutral membership degree, further
increasing the uncertainty dimension. CIVPFSS associates complex intervals and picture fuzzy
sets, allowing for the most flexible and nuanced uncertainty modeling, and is more suitable for
complex and dynamic systems.

5.4. CIVPFSS Comparison with CIVFS, CIVIFS, and CIVPFS

CIVFS is the simplest, with a single complex interval membership degree. CIVIFS adds a
non-membership degree, increasing the uncertainty dimension. CIVPFS adds a neutral member-
ship degree, further increasing the uncertainty dimension. CIVPFSS combines complex intervals
and picture fuzzy sets, allowing for the most flexible and nuanced uncertainty modeling, and
is more suitable for complex and dynamic systems. The main difference between CIVPFSS
and CIVPFS is that CIVPFSS uses soft sets, which allow for more flexibility in defining the
membership degrees, whereas CIVPFS uses a more traditional fuzzy set approach.

5.5. CIVPFSS Comparison with CFSS, CIFSS, and CPFSS

CFSS is the simplest, with a single complex membership degree. CIFSS adds a non-
membership degree, increasing the uncertainty dimension. CPFSS adds a neutral membership
degree, further increasing the uncertainty dimension. CIVPFSS combines complex intervals and
picture fuzzy sets, allowing for the most flexible and nuanced uncertainty modeling, and is more
suitable for complex and dynamic systems. The main difference between CIVPFSS and CPFSS
is that CIVPFSS uses interval-valued membership degrees, which allow for more flexibility in
defining the membership degrees, whereas CPFSS uses a more traditional fuzzy set approach.

5.6. CIVPFSS Comparison with CIVFSS and CIVIFSS

CIVFSS is the simplest, with a single complex interval membership degree. CIVIFSS adds
a non-membership degree, increasing the uncertainty dimension. CIVPFSS combines complex
intervals and picture fuzzy sets, allowing for the most flexible and nuanced uncertainty modeling,
and is more suitable for complex and dynamic systems. The main difference between CIVPFSS
and CIVIFSS is that CIVPFSS uses picture fuzzy sets, which allow for more nuanced and
complex uncertainty modeling, whereas CIVIFSS uses intuitionistic fuzzy sets, which are more
suitable for simple and static systems. CIVPFSS is a more general and flexible framework that
can handle complex and dynamic systems, while CIVFSS and CIVIFSS are more specific and
suitable for simpler applications.


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Table 3: Comparison Table

Structure Membership Neutral Non-membership Interval values Multidimensional

CFS Yes No No No Yes

CIFS Yes No Yes No Yes

CPFS Yes Yes Yes No Yes

CIVIFSS Yes No Yes Yes Yes

CIVPFSS Yes Yes Yes Yes Yes

6. Comprehensive Discussion on Complex Interval Valued Picture Fuzzy
Soft Topological Spaces

In this section, the most important space is introduced, known as the complex interval
valued picture fuzzy soft topological space. The related results and theorems are discussed it
full detailed. The connection between interior and closure is also reflected.

Definition 17. Let τ be the collection of complex interval valued picture fuzzy soft sets over
X, then τ is said to be a complex interval valued picture fuzzy soft topology (TST) on X if:

(i) ∅̈, Ẍ ∈ τ ,

(ii) The union of any number of members of complex interval valued picture fuzzy soft sets in
τ belongs to τ ,

(iii) The intersection of any two complex interval valued picture fuzzy soft sets in τ belongs to
τ .

Then (X, τ,E) is called a complex interval valued picture fuzzy soft topological space over X.

Definition 18. Let (X, τ,E) be a complex interval valued picture fuzzy soft topological space
over X. Then the members of τ are said to be complex interval valued picture fuzzy soft open
sets in X.

Definition 19. Let (X, τ,E) be a complex interval valued picture fuzzy soft topological space
over X. Then the members of τ are said to be complex interval valued picture fuzzy soft closed
sets in X if their relative complements (F,E)′ belong to τ .

Definition 20. Let X be initial universe sets, E be a set of parameters, and τ = {∅̈, Ẍ}. Then
τ is called the complex interval valued picture fuzzy soft indiscrete topology on X, and (X, τ,E)
is said to be a complex interval valued picture fuzzy soft indiscrete space over X.

Definition 21. Let X be initial universe sets, E be a set of parameters, and let τ be the
collection of all complex interval valued picture fuzzy soft sets which can be defined over X. The
τ is called the complex interval valued picture fuzzy soft discrete topology on X, and (X, τ,E)
is said to be a complex interval valued picture fuzzy soft discrete space over X.

Theorem 3. Let (X, τ,E) and (X, τ ′, E) be two complex interval valued picture fuzzy soft
topological spaces over the common initial sets X. Then (X, τ ∩̈ τ ′ , E) is a complex interval
valued picture fuzzy soft topological space over X.
Proof.

(i) ∅̈, Ẍ, belongs to τ ∩̈τ ′.

(ii) Let {Gi, E/i ∈ Î} be a family of complex interval valued picture fuzzy soft sets in τ ∩̈τ ′.
Then (Gi, E) ∈ τ and (Gi, E) ∈ τ ′ for all i ∈ Î, so ∪̈i∈Î(Gi, E) ∈ τ and ∪̈i∈Î(Gi, E) ∈ τ ′.
Thus ∪̈i∈Î(Gi, E) ∈ τ ∩ τ ′.


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(iii) Let the two complex interval valued picture fuzzy soft sets (H,E), (Î , E) ∈ τ ∩̈ τ ′ . Then
(H,E), (Î , E) ∈ τ and (H,E), (Î , E) ∈ τ ′. Since (H,E)∩̈(G,E) ∈ τ and (H,E)∩̈(G,E) ∈
τ ′, so (H,E)∩̈(G,E) ∈ τ ∩̈ τ ′ .

Definition 22. Let (X, τ,E) be a complex interval valued picture fuzzy soft topological space
over X and (G,E) be the complex interval valued picture fuzzy soft set over common universal

sets X. Then the complex interval valued picture fuzzy soft closure of (G,E), denoted by (G,E),
is the intersection of all complex interval valued picture fuzzy soft closed sets of (G,E). Thus,

(G,E) is the smallest complex interval valued picture fuzzy soft closed set over X which contains
(G,E).

Theorem 4. Let (X, τ,E) be a complex interval valued picture fuzzy soft topological space over
X and let (H,E), (Î , E) be complex interval valued picture fuzzy soft sets over X. Then:

(i) ∅̈ = ∅̈ and Ẍ = Ẍ.

(ii) (H,E)⊆̈(H,E) implies (H,E) is a complex interval valued picture fuzzy soft closed set

and (H,E) contains (H,E).

(iii) (H,E) is a complex interval valued picture fuzzy soft closed set if and only if (H,E)≈̈(H,E).

(iv) (H,E) = (H,E).

(v) (H,E)⊆̈(Î , E) implies (H,E)⊆̈(Î , E).

(vi) (H,E)∪̈(Î , E) = (H,E)∪̈(Î , E).

(vii) (H,E)∩̈(Î , E)⊆̈(H,E)∩̈(Î , E).

Proof.

(i) This is obvious.

(ii) Let {(Hi, E) | i ∈ Î} be the family of all the complex interval valued picture fuzzy closed
sets containing (H,E). Then, by definition, we know that:

(H,E)∩̈ ∈ Î(Hi, E) → (1).

Now, since {(Hi, E) | i ∈ Î} is a complex interval valued picture fuzzy soft closed set
∀ i ∈ I ⇒ ∩̈i ∈ Î(Hi, E) is also a complex interval valued picture fuzzy soft closed set.
Since an arbitrary intersection of complex interval valued picture fuzzy soft closed sets is

a complex interval valued picture fuzzy soft closed set, (H,E) is a complex interval valued
picture fuzzy soft closed set (from(1)).

⇒ Thus, (H,E) is a complex interval valued picture fuzzy soft closed set. Now, we prove

that (H,E)⊇̈(H,E). We know that ∀ i ∈ I, {(Hi, E) | i ∈ Î}⊇̈(H,E).

⇒ (H,E)⊆̈ ∩̈i ∈ Î (Hi, E) =⇒ (H,E) ⊆̈ (H,E) [using(1)]

⇒ (H,E)⊆̈(H,E) Thus (H,E) contains (H,E). Hence, (H,E) is a complex interval

valued picture fuzzy soft closed set and (H,E) contains (H,E).


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(iii) Let (H,E) be a complex interval valued picture fuzzy soft set. To prove (H,E) = (H,E),
suppose (H,E) is a complex interval valued picture fuzzy soft closed set. Now, we have
(H,E)⊇̈(H,E), so (H,E) is a << (T, S) >> closed set containing (H,E) → (1). But

(H,E) is the smallest complex interval valued picture fuzzy soft closed set containing

(H,E) → (2). Therefore from (1) and (2), it follows that (H,E) is smaller then (H,E)

that is (H,E)⊆̈(H,E). But from from (ii) of this theorem, we have (H,E)⊆̈(H,E) is

always true. Therefore we have (H,E)⊆̈(H,E). Thus (H,E) = (H,E). Consequently , if

(H,E) is a complex interval valued picture fuzzy soft closed set then (H,E) = (H,E).

(iv) Since (H,E) is a complex interval valued picture fuzzy soft closed set, therefore by (iii),

we have (H,E = (H,E).

(v) If (H,E)⊆̈(Î , E), then (H,E)⊆̈(Î , E). Suppose (H,E)⊆̈(Î , E). We know that (Î , E)⊆̈(Î , E),
and we have

(H,E)⊆̈(Î , E)⊆̈(Î , E).

Therefore, (H,E)⊆̈(Î , E). Therefore, (Î , E) is a complex interval valued picture fuzzy soft

closed set containing (Î , E) → (1). But (H,E) is the smallest complex interval valued

picture fuzzy soft closed set containing (H,E) → (2) it follows that (H,E) is smaller than

(Î , E), that is (H,E)⊆̈(Î , E). Thus if (H,E)⊆̈(Î , E). Then (H,E)⊆̈(Î , E).

(vi) We know (H,E)⊆̈(H,E)∪̈(Î , E) and (Î , E)⊆̈(H,E)∪̈(Î , E). Therefore:

(H,E)⊆̈(H,E)∪̈(Î , E) and (Î , E)⊆̈(H,E)∪̈(Î , E).

Since (H,E)⊆̈(Î , E) implies (H,E)⊆̈(Î , E)

⇒ {(H,E)∪̈(Î , E)}⊆̈{(H,E)∪̈(Î , E)} ∪̈{(H,E)∪̈(Î , E)}.

(H,E)∪̈(Î , E)⊆̈(H,E)∪̈(Î , E) → (1).

Also, from the complex interval valued picture fuzzy soft closure property, we have (H,E)⊆̈(H,E) and (Î , E)⊆̈(Î , E).

Thus(H,E)∪̈(Î , E)⊆̈(H,E)∪̈(Î , E) =⇒ (H,E)∪̈(Î , E)
is the complex interval valued picture fuzzy soft closed set containing (H,E)∪̈(Î , E). But
(H,E)∪̈(Î , E) is the smallest complex interval valued picture fuzzy soft closed set contain-
ing (H,E)∪̈(Î , E) → (2)

Comparing (1) and (2), we have (H,E)∪̈(Î , E) is smaller than (H,E)∪̈(Î , E). Thus, from

(1) and (2), we have (H,E)∪̈(Î , E) = (H,E)∪̈(Î , E).

(vii) Since (H,E)∩̈(Î , E)⊆̈(H,E),, so by part (v), (H,E)∩̈(Î , E)⊆̈(H,E) and (H,E)∩̈(Î , E)⊆̈(Î , E).

Thus, (H,E)∩̈(Î , E)⊆̈(H,E)∩̈(Î , E).

Definition 23. Let (X, τ,A) be a complex interval valued picture fuzzy soft topological space
over X and (H,A) be a complex interval valued picture fuzzy soft set. Then we associate point
wise complex interval valued picture fuzzy soft closure of (F,E) over X, which is denoted by

(H,A) and defined as (H,A)α̈ = (H,A)α̈ where (H,A)α̈ is the complex interval valued picture
fuzzy soft closure of (H,A)α̈ in (X, τ,A) for each α̈ ∈ A.


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Theorem 5. Let (X, τ,A) be a complex interval valued picture fuzzy soft topological space over

X and (H,A) be a complex interval valued picture fuzzy soft set. Then (H,A)⊆̈(H,A).

Proof. For any parameter α̈ ∈ E, (H,A)α̈ is the smallest complex interval valued picture fuzzy

soft closed set in (X, τ,A) which contains (H,A)α̈. Moreover, if (H,A)α̈ = ( L, A), then ( L, A)
is also a complex interval valued picture fuzzy soft closed set in (X, τ,A) containing (H,A)α̈.

This implies that (H,A)α̈ =, (H,A)α̈⊆̈( L, A). Thus, (H,A)⊆̈(H,A). □

Theorem 6. Let (X, τ,A) be the complex interval valued picture fuzzy soft topological space

and (F,A) be the complex interval valued picture fuzzy soft set over (X). Then, (F ,A)⊆̈(F,A).
Proof. Let (X, τ,A) be the complex interval valued picture fuzzy soft topological space over X.

If (F ,A)⊆̈(F,A), then (F ,A) is a complex interval valued picture fuzzy soft closed set and so

(F ,A)ć ∈ τ .

Conversely, if (F ,A)ć ∈ τ , then complex interval valued picture fuzzy soft closed set contain-

ing (F,A). By the above theorem, (F ,A)⊆̈(F,A), and by the definition of the complex interval
valued picture fuzzy soft closure of (F,A), any complex interval valued picture fuzzy closed set

over X that contains (F,A) will contain (F ,A). Thus, (F,A)⊆̈(F ,A), hence (F ,A) = (F,A).
□

Definition 24. Let (X, τ,A) be a complex interval valued picture fuzzy soft topological space
over X and (H,A) be a complex interval valued picture fuzzy soft set. Let ex ∈ E. Then ex is
said to be a complex interval valued picture fuzzy soft interior point of (H,A) if there exists a
complex interval valued picture fuzzy soft open set (K, A) such that ex ∈ (H,A)⊆̈(K, A).

Definition 25. Let (X, τ,A) be a complex interval valued picture fuzzy soft topological space
over X and (F,A) be a complex interval valued picture fuzzy soft set. Let ex ∈ A. Then (F,A)
is said to be a complex interval valued picture fuzzy soft neighborhood of ex if there exists a
complex interval valued picture fuzzy soft open set (K, A) such that ex ∈ (F,A)⊆̈(K, A).

Theorem 7. Let (X, τ,A) be a complex interval valued picture fuzzy soft topological space over
X. Let (F,E) be a complex interval valued picture fuzzy soft set over X, and let ex ∈ E. If
ex is a complex interval valued picture fuzzy soft interior point of (F,E), then ex is a complex
interval valued picture fuzzy soft interior point of (F,E)α̈ in (X, τ,A) for each α̈ ∈ E.

The above theorem is not true in general.

Theorem 8. Let (X, τ,A) be a complex interval valued picture fuzzy soft topological space over
the initial parameter (X). Then:

(i) Each ex ∈ E has a complex interval valued picture fuzzy soft neighborhood.

(ii) The intersection of any two complex interval valued picture fuzzy soft neighborhoods of
a complex interval valued picture fuzzy soft point ex is again a complex interval valued
picture fuzzy soft neighborhood.

(iii) Every complex interval valued picture fuzzy soft superset of a complex interval valued
picture fuzzy soft neighborhood of a point ex is again a complex interval valued picture
fuzzy soft neighborhood of the point ex.

Proof. (i). For any ex ∈ Ẍ, we have ex ∈ Ẍ⊆̈Ẍ. Thus, Ẍ is a complex interval valued
picture fuzzy soft neighborhood of ex.

(ii). Let (X, τ,A) be a complex interval valued picture fuzzy soft topological space, and let
ex ∈ E be any complex interval valued picture fuzzy soft point. Let (F,E) and (G,E) be any
two complex interval valued picture fuzzy soft neighborhoods of ex. To prove that (F,E)∩̈(G,E)
is also a complex interval valued picture fuzzy soft neighborhood of ex, we note that (F,E) being


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a complex interval valued picture fuzzy soft neighborhood of ex implies there exists a complex
interval valued picture fuzzy soft open set (K, E) such that ex ∈ (K, E)⊆̈(F,E). Similarly,
(G,E) is a complex interval valued picture fuzzy soft neighborhood of ex, implying there exists
a complex interval valued picture fuzzy soft open set ( L, E) such that ex ∈ ( L, E)⊆̈(G,E).

Now, (K, E)∩̈( L, E) is a complex interval valued picture fuzzy soft open set, and from the pre-
vious conditions, we have ex ∈ [(K, E)∩̈( L, E)]⊆̈[(F,E)∩̈(G,E)]. Thus, there exists a complex
interval valued picture fuzzy soft open set [(K, E)∩̈( L, E)] such that ex ∈ [(K, E)∩̈( L, E)]⊆̈[(F,E)∩̈(G,E)].
From the definition of a complex interval valued picture fuzzy soft neighborhood, it follows that
(F,E)∩̈(G,E) is a complex interval valued picture fuzzy soft neighborhood of ex. Hence, the in-
tersection of any two complex interval valued picture fuzzy soft neighborhoods is again a complex
interval valued picture fuzzy soft neighborhood.

(iii). Let (X, τ,E) be a complex interval valued picture fuzzy soft topological space, and let
ex ∈ E be any complex interval valued picture fuzzy soft point. Let (F,E) be a complex interval
valued picture fuzzy soft neighborhood of ex, and let (G,E) be any complex interval valued picture
fuzzy soft superset of (F,E). Since (G,E) is also a complex interval valued picture fuzzy soft
neighborhood of ex, there exists a complex interval valued picture fuzzy soft open set (H,E) such
that ex ∈ (H,E)⊆̈(F,E).

Now, (F,E) being a complex interval valued picture fuzzy soft subset of (G,E) implies
(G,E)⊇̈(F,E), which gives (F,E)⊆̈(F,E). From the previous results, we have ex ∈ (H,E)⊆̈(F,E)⊆̈(G,E),
which implies ex ∈ (H,E)⊆̈(G,E). Therefore, there exists a complex interval valued picture
fuzzy soft open set (H,E) such that ex ∈ (H,E)⊆̈(G,E). Hence, (G,E) is a complex interval
valued picture fuzzy soft neighborhood of ex. Thus, every complex interval valued picture fuzzy
soft superset of a complex interval valued picture fuzzy soft neighborhood is again a complex
interval valued picture fuzzy soft neighborhood of that point.

Theorem 9. Let (X, τ,E) be a complex interval valued picture fuzzy soft topological space. Let
(F,E) be any complex interval valued picture fuzzy soft subset over X. Then the following hold
true:

(i) (F,E)◦ is a complex interval valued picture fuzzy soft open set contained in (F,E), i.e.
(F,E)◦ is a complex interval valued picture fuzzy soft open set and (F,E)◦⊆̈(F,E).

(ii) (F,E)◦ is the largest complex interval valued picture fuzzy soft open set contained in (F,E).

(iii) (F,E) is complex interval valued picture fuzzy soft open if and only if (F,E)=̈(F,E)◦.

Proof. By the definition of complex interval valued picture fuzzy soft interior, we have
(F,E)◦ = ¨∪λ∈A(H,E)λ, where {(H,E)λ} : λ ∈ A is the family of all complex interval val-
ued picture fuzzy soft open sets contained in (F,E). (H,E)λ⊆̈(F,E) ∀ λ ∈ Λ implies the union
of all complex interval valued picture fuzzy soft open sets implies that open sets by the definition
of complex interval valued picture fuzzy soft topological space. Also, we have (H,E)λ⊆̈(F,E) ∀
λ ∈ Λ ⇒ ¨∪λ∈A(H,E)λ⊆̈(F,E)
⇒ (F,E)◦⊆̈(F,E). Hence, (F,E)◦ is a complex interval valued picture fuzzy soft open set and
(F,E)◦⊆̈(F,E).

(ii) from (i), we have that (F,E)◦ is a complex interval valued picture fuzzy soft open set
contained in (F,E). Let (H,E) be any complex interval valued picture fuzzy soft open set
contained in (F,E). Let (H,E) be any complex interval valued picture fuzzy soft open set
contained in (F,E). This implies that the family {(H,E)λ : λ ∈ Λ} = , the family of all complex
interval valued picture fuzzy soft open sets contained in (F,E) implies (H,E)⊆̈ ¨∪λ∈A(H,E)λ ⇒
(H,E)⊆̈(F,E)◦. ⇒ (F,E)◦⊇̈(H,E). ⇒ (F,E)◦ is larger than every complex interval valued
picture fuzzy soft open set contained in (F,E). Thus, (F,E)◦ is the largest complex interval
valued picture fuzzy soft open set contained in (F,E).

(ii). Suppose (F,E) is complex interval valued picture fuzzy soft open. Therefore, (F,E) is
a complex interval valued picture fuzzy soft open set contained in (F,E), (i.e. (F,E)⊆̈(F,E))


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→ (1). But (F,E)◦ is the largest complex interval valued picture fuzzy soft open set contained
in (F,E) → (2). Therefore, from (1) and (2), it follows that (F,E)◦ must be larger than (F,E),
that is, (F,E)◦⊇̈(F,E) or (F,E) is smaller than (F,E)◦, that is (F,E)⊆̈(F,E)◦ → (3). But
(F,E)◦⊇̈(F,E) is always true → (4). From (3) and (4), we have (F,E)=̈(F,E)◦. Note that
the right-hand side result, i.e., (F,E)◦ is a complex interval valued picture fuzzy soft open set,
implies that the left-hand side, i.e., (F,E), must also be a complex interval valued picture fuzzy
soft open set. Consequently, (F,E) is a complex interval valued picture fuzzy soft open set. If
(F,E)=̈(F,E)◦, then (F,E) is a complex interval valued picture fuzzy soft open set. Hence,
(F,E) is complex interval valued picture fuzzy soft open if and only if (F,E)=̈(F,E)◦.

Theorem 10. Let (X, τ,E) be the complex interval valued picture fuzzy soft topological space.
Let (F,E) and (G,E) be any two complex interval valued picture fuzzy soft subsets over X.
Then the following properties hold true:

(i) Ẍ◦=̈Ẍ.

(ii) ∅̈◦ = ∅̈.

(iii) If (F,E)⊆̈(G,E), then (F,E)◦⊆̈(G,E)◦.

(iv) [(F,E)∩̈(G,E)]◦=̈(F,E)◦∩̈(G,E)◦.

(v) [(F,E)◦]◦ = (F,E)◦.

(vi) (F,E)◦∪̈(G,E)◦⊆̈[(F,E)∪̈(G,E)]◦.

Proof.

(i) We know that Ẍ is a complex interval valued picture fuzzy soft open set. This implies
Ẍ◦=̈Ẍ. (Since (F,E) is open if and only if (F,E)=̈(F,E)◦). Therefore, Ẍ◦=̈Ẍ.

(ii) The result follows from (i).

(iii) Suppose (F,E)⊆̈(G,E). Then we know that (F,E)◦⊆̈(F,E) and (F,E)⊆̈(G,E), therefore
(F,E)◦⊆̈(G,E). Therefore, (F,E)◦ is a ≪ (T, S) ≫ complex interval valued picture fuzzy
soft open set contained in (G,E) → (1). But (G,E)◦ is the largest complex interval
valued picture fuzzy soft open set contained in (G,E) → (2). From (1) and (2), we have
that (G,E)◦ is larger than (F,E)◦, which implies (F,E)◦⊆̈(G,E)◦. Thus, (F,E)⊆̈(G,E)
implies (F,E)◦⊆̈(G,E)◦.

(iv) Let (X, τ,E) be the complex interval valued picture fuzzy soft topological space. To prove

[(F,E)∩̈(G,E)]◦=̈(F,E)◦∩̈(G,E)◦,

we know that (F,E)∩̈(G,E) ̂̂⊆(F,E) and (F,E)∩̈(G,E)⊆̈(G,E), which implies

[(F,E)∩̈(G,E)]◦⊆̈(F,E)◦ and [(F,E)∩̈(G,E)]◦⊆̈(G,E)◦(by(iii)).

This implies
[(F,E)∩̈(G,E)]◦⊆̈(F,E)◦∩̈(G,E)◦.

Also, we have (F,E)◦⊆̈(F,E) and (G,E)◦⊆̈(G,E), which implies

(F,E)◦∩̈(G,E)◦⊆̈(F,E)∩̈(G,E),

which implies that (F,E)◦∩̈(G,E)◦ is a complex interval valued picture fuzzy soft open set
contained in (F,E)∩̈(G,E) ⇒ (2). But [(F,E)∩̈(G,E)]◦ is the largest complex interval val-
ued picture fuzzy soft open set contained in (F,E)∩̈(G,E) ⇒ (3). Therefore, from (2) and
(3), it follows that [(F,E)∩̈(G,E)]◦ is larger than (F,E)◦∩̈(G,E)◦, that is (F,E)◦∩̈(G,E)◦

is smaller than [(F,E)∩̈(G,E)]◦ this leads to (F,E)◦∩̈(G,E)◦⊆̈ [(F,E)∩̈(G,E)]◦ → (4).
From (1) and (4) it follows that [(F,E)∩̈(G,E)]◦ =̈(F,E)◦∩̈(G,E)◦.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 34 of 38

(v) We know that [(F,E)◦]◦ is a complex interval valued picture fuzzy soft open set. Let us
assume (F,E)◦=̈(H,E). Therefore, (H,E) is a complex interval valued picture fuzzy soft
open set, which implies (H,E)=̈(H,E)◦. Therefore, (F,E)◦=̈[(F,E)◦]◦, hence the result.

(vi) Since (F,E)⊆̈(F,E)∪̈(G,E) and (G,E)⊆̈(F,E)∪̈(G,E), by (iii), we have

(F,E)◦⊆̈[(F,E)∪̈(G,E)]◦ and (F,E)◦⊆̈[(F,E)∪̈(G,E)]◦.

So that
(F,E)◦∪̈(G,E)◦⊆̈[(F,E)∪̈(G,E)]◦,

since (F,E)◦∪̈(G,E)◦ is a complex interval valued picture fuzzy soft open set.

Remark 1. Let (X, τ,E) be the complex interval valued picture fuzzy soft topological space over
X, and let (F,E) be any complex interval valued picture fuzzy soft open set. Then we always
have

(H,E)◦⊆̈(F,E)⊆̈(F,E).

Remark 2. (H,E)=̈[Ẍ − (H,E)] and bd (H,E) = bd [Ẍ − (H,E)].

To prove;

bd(H,E) = bd(H,E) ∩̈ [Ẍ − (H,E)] ⇒ (1)

bd [Ẍ − (H,E)] = [Ẍ − (H,E)]∩̈ [Ẍ − (Ẍ − (H,E))].

Hence,

[Ẍ − (H,E)]∩̈ (H,E) ⇒ (H,E)∩̈ [Ẍ − (H,E)]∩̈ = bd(H,E).

Thus,
bd (H,E) = bd [Ẍ − (H,E)].

Theorem 11. Let (X, τ,E) be the complex interval valued picture fuzzy soft topological space.
Let (H,E) be any complex interval valued picture fuzzy soft subset of Ẍ. Then the following
properties are true:

(i) bd(H,E) = (H,E)− (H,E)◦.

(ii) (H,E)◦ = (H,E)− bd(H,E).

(iii) Ẍ=̈(H,E)◦∩̈bd(H,E)∪̈[Ẍ − (H,E)◦].

Proof.

(i) We know that bd(H,E) = (H.E) =̈ (H,E) ∩̈ (H,E)′

⇒ (H,E) = (H,E)− [Ẍ − (Ẍ − (H,E))]◦ = (H,E)− [Ẍ(Ẍ − (H,E))]◦

.
Thus, = (H,E)− (H,E)◦ = bd (H,E) = (H,E)− (H,E)◦.

(ii) Consider

(H,E)− bd (H,E) = (H,E)− [(H,E)− (Ẍ − (H,E))]

= (H,E) ∩̈ [Ẍ − (H,E) ∩̈ (Ẍ − (H,E))]

= (H,E) ∩̈ [(Ẍ − (H,E)) ∪̈ (Ẍ − [Ẍ − (H,E)])]


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 35 of 38

.
This simplifies to

⇒ (H,E) ∩̈ [(Ẍ − (H,E)) ∪̈ (Ẍ − [(H,E)◦])]

= (H,E) ∩̈ [(Ẍ − (H,E)) ∪̈ [(H,E) ∩̈ (H,E)◦]]

. Therefore, (H,E)◦ = (H,E)− bd (H,E).

(iii) Consider the right-hand side

(H,E)◦∪̈bd (H,E)∪̈ [Ẍ − (H,E)◦]

= (H,E)◦ ∪̈ bd (H,E) ∪̈ (Ẍ − (H,E))

= (H,E) ∪̈ (Ẍ − (H,E)) = Ẍ.

Further, we know that bd (H,E) = (H,E) − (H,E)◦, which implies that bd(H,E) and
(H,E)◦ are disjoint → (1). Replacing (H,E) with (Ẍ−(H,E)◦), are disjoint sets. Hence,

Ẍ = (H,E) ∪̈ (Ẍ − (H,E)), which is a complex interval valued picture fuzzy soft disjoint
union.

7. Conclusion and Future Work

This research introduces the CP of two CIVPFSSs, a novel framework aimed at enhancing
the applicability of fuzzy soft set theory in addressing multi-criteria decision-making problems.
The study presents groundbreaking concepts such as CIVPFSS symmetric relation, CIVPFSS
antisymmetric relation, CIVPFSS asymmetric relation, CIVPFSS complete relation, CIVPFSS
transitive relation, CIVPFSS equivalence relation, CIVPFSS partial-order relation, CIVPFSS
strict-order relation, CIVPFSS preorder relation, and CIVPFSS equivalence classes. A practical
application of the proposed CIVPFSS framework is demonstrated in evaluating fitness tracking
applications. This application incorporates diverse criteria and attributes, leveraging expert
opinions expressed through neutral, membership, and non-membership values. The innovative
score function associated with CIVPFSS relations was utilized during the decision-making pro-
cess, ensuring an objective and comprehensive evaluation. The findings reveal that CIVPFSS
relations exhibit superior performance compared to existing fuzzy soft set structures, particu-
larly in addressing periodicity challenges inherent in decision-making scenarios. The robustness
and flexibility of CIVPFSS relations make them a promising tool for modeling complex and
uncertain systems. This study opens new avenues for extending the generalization of fuzzy soft
sets, paving the way for the development of advanced mathematical models with wide-ranging
applications in domains such as artificial intelligence, data science, and operations research.
Future research could focus on refining these concepts to create even more sophisticated struc-
tures capable of addressing emerging challenges across various scientific and industrial fields.
The creation of a topological space for CIVPFSR builds a geometrical framework which clarifies
how decision factors connect while examining choice behaviors operating in multi-dimensional
domains. This work identifies the key operations and operators with their relevant theorems
and major results. The text explores results that pertain to both interiors and closures among
specific concepts then details their theoretical applications. Future developments in wearable
technology might concentrate on improving personalization through the use of real-time AI-
driven data processing for instant insights, linking wearables with larger health management
systems, and incorporating a variety of data kinds. The CIVPFSR model may become more
scalable with more development, allowing it to be used to bigger datasets and more intricate


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5523 36 of 38

health variables. The capabilities of wearable technology could also be further increased, mak-
ing them more essential to total health management, by investigating cross-domain applications
and long-term health tracking, as well as by creating sophisticated user interfaces for improved
accessibility.

Acknowledgements

We are thankful to Sulima Ahmed Mohammed Zubair of the Department of Mathematics,
College of Sciences, Qassim University, Buraydah 51452, Saudi Arabia, for supporting this work.

Author contributions: All authors Equally Contributed.

Conflicts of Interest: The author(s) declare(s) that there are no conflicts of interest
regarding the publication of this paper.

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