

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6483
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fermatean Double-Valued Neutrosophic Soft Topological
Spaces

Raed Hatamleh1, Naser Odat1, Hamza Ali Abujabal2, Faria Khan3, Arif Mehmood3,∗,
Alaa M. Abd El-latif4, Husham M. Attaalfadeel4, Abdelhalim Hasnaoui4

1 Department of Mathematics, Faculty of Science, Jadara University, P.O. Box 733, Irbid 21110,
Jordan
2 Department of Mathematics, King Abdulaziz University, P.O. Box 80003, Jeddah 21580,
Saudi Arabia
3 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail
Khan 29050, KPK, Pakistan
4 Department of Mathematics, College of Science, Northern Border University, Arar 91431,
Saudi Arabia

Abstract. This paper introduces the concept of Fermatean Double Valued Neutrosophic Soft Set (FD-
VNSS), an advanced mathematical framework for modeling uncertainty, vagueness, and indeterminacy
in complex systems. A FDVNSS extends traditional neutrosophic and soft set theories by incorporating
two layers of truth and falsity degrees: absolute and relative. This multi-dimensional representation al-
lows for more nuanced decision-making, particularly in domains characterized by conflicting or imprecise
information, such as medical diagnostics. The formal definition of FDVNSS is established, including the
conditions under which membership degrees satisfy the Fermatean constraint, i.e., the sum of the cubes
of the degrees lies within a defined range. Key operations such as subset, null and absolute FDVNSS,
complement, union, and intersection are rigorously defined and exemplified. Furthermore, the algebraic
properties of FDVNSSs under these operations are explored, and foundational propositions-analogous to
classical set theory- are proven to hold. Examples drawn from health care contexts, such as evaluating
diseases, symptoms, or medical responses, are provided to illustrate the applicability of FDVNSS in real-
world problems. Overall, this framework provides a more expressive and flexible tool for reasoning under
high levels of uncertainty and offers significant potential for application in intelligent decision support
systems. The notion of a Fermatean Double Valued Neutrosophic Soft Topological Space (FDVNSTS) is
introduced. The concepts of semi-open (s-open), pre-open (p-open), and b-open sets are defined within
the context of FDVNSTS. Among these generalized open sets, the pre-open set is selected for further
exploration, and several fundamental topological notions are developed based on this definition. These
include the closure, exterior, boundary, and interior in FDVNSTS.

2020 Mathematics Subject Classifications: 54A05

Key Words and Phrases: Fuzzy set, Fermatean fuzzy soft set, neutrosophic soft set, Fermatean double
valued neutrosophic soft set, Fermatean double valued neutrosophic soft union, Fermatean double valued
neutrosophic soft intersection, AND operator, OR operator, Fermatean double valued neutrosophic soft
topological space

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6483

Email addresses: (raed@jadara.edu.jo) (R. Hatamleh) , (nodat@jadara.edu.jo) (N. Odat),
(haabujabal@kau.edu.sa) (H. A. Abujabal), (fariak474@gmail.com) (F. Khan),
(mehdaniyal@gmail.com (A. Mehmood), (alaa.ali@nbu.edu.sa) (Alaa. M. Abd El-latif),
(husham.alhassan@nbu.edu.sa) ( H. M. Attaalfadeel), (Abdllhalim.hasanawa@nbu.edu.sa) (A. Hasnaoui)

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 (3) (2025), 6483 2 of 25

1. Introduction

In the 1870s, set theory emerged through the groundbreaking work of Georg Cantor and
Richard Dedekind [1], revealing its profound significance across a wide range of real-world appli-
cations. Classical set theory, based on the concept of crisp sets, defines membership in absolute
terms an element either belongs to a set or it does not. However, this rigid framework was
unable to capture the nuances of partial membership. To address this limitation, Lotfi Zadeh
introduced fuzzy set theory in 1965, allowing elements to have varying degrees of membership
[2]. Fuzzy sets were later generalized by Pawlak as rough sets in 1982 [3], and by Molodtsov
as soft sets in 1999 [4]. These extensions have demonstrated significant utility in managing
uncertainty across a wide range of real-world applications, including engineering, economics,
social sciences, environmental sciences, and medical sciences [5, 8].
Fuzzy soft sets [7], intuitionistic fuzzy sets [9], intuitionistic fuzzy soft sets [10], hesitant fuzzy
sets [11], hesitant fuzzy soft sets [12], picture fuzzy sets [13], picture fuzzy soft sets [14], hyper
soft sets [16], neutrosophic soft sets [16], and neutrosophic hyper soft sets [17] are several notable
variants developed through the generalization of truth (membership), falsity (non-membership),
and hesitancy (indeterminacy). In this context, we highlight key scholarly contributions related
to soft sets, neutrosophic sets, and Fermatean sets. In 2003, Maji introduced the foundational
concepts of soft set theory, including basic elements and operators [18], and in 2009, Ali et al.
proposed modified operators [19]. Subsequently, the theory evolved as Cagman and Enginoglu
introduced the concept of the soft matrix [20]; Babitha and Sunil defined relations and functions
on soft sets along with their properties [21, 22]; and Yang and Guo explored closure and kernel
concepts within soft relations and soft mappings [23]. Further contributions to soft set theory
have been made by various mathematicians [26, 27]. As the theory continued to evolve through
the aforementioned developments, researchers began exploring its connections with algebraic
structures. Aktas¸ and Cagman introduced the concept of soft groups [28], Acer defined soft
rings [5], and Aslam and Qurashi investigated sub algebraic structures associated with soft
groups [29]. Drawing inspiration from philosophical logic?particularly notions of relative and
absolute truth and falsity?as well as real-world contexts such as game outcomes (win, loss,
tie), voting results (in favor, against, abstention), numerical classifications (positive, negative,
neutral), binary responses (yes, no, not applicable), and decision-making scenarios (acceptance,
rejection, hesitation, pending decisions), Smarandache introduced neutrosophic sets. These
tri-component sets, rooted in the concept of ’neutral wisdom,’ account for membership, non-
membership, and indeterminacy simultaneously [30, 31]. Neutrosophic set theory has continued
to evolve through numerous significant contributions. Wang et al. proposed single-valued and
interval-valued neutrosophic sets [32, 33], while Salama and Alblowi introduced neutrosophic
topological spaces [34]. Georgiou developed the concept of soft topological spaces [35], and Bera
and Mahapatra extended this to neutrosophic soft topological spaces [36]. Various mathemat-
ical constructs have been formulated in relation to neutrosophic sets, including measures and
integrals [37], lattices [38], vector spaces [39], continuous functions [44], entropy [41], groups
and subgroups [42], and algebraic structures such as soft rings and soft fields [43].
Mathematicians have also explored a wide range of applications for neutrosophic techniques,
including image processing [44], medical diagnosis [45, 46], and multi-criteria decision-making
[47, 49], often employing similarity measures, neutrosophic logic, and hyper soft graphs [50].
To address limitations in expressing degrees of membership and non-membership in intuition-
istic and Pythagorean fuzzy sets, Senapati and Yager introduced the Fermatean fuzzy set [51].
This development opened new avenues for research: Broumi et al. applied complex Fermatean
neutrosophic graphs to decision-making [52]; Bilgin et al. introduced Fermatean neutrosophic
topological spaces [53]; and Salsabeela and John explored the use of the TOPSIS technique on
Fermatean fuzzy soft sets [54]. Further in-depth studies on neutrosophic theory and its gener-
alizations can be found in [[63]-[70]]. The foundational work on supra soft topological spaces
by El-Sheikh and Abd El-Latif [71] introduced decompositions and notions of soft continuity,


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 3 of 25

setting a precedent for analyzing generalized topologies under uncertainty. This was further de-
veloped through the introduction of soft supra compactness [72], providing compactness criteria
applicable to soft topological frameworks. Additionally, Abd El-Latif and Hosny [73] explored
supra open soft sets and related separation axioms, deepening the theoretical basis for soft
separation in such topologies. These contributions collectively offer a conceptual and structural
foundation for extending topological properties to Fermatean Double Valued Neutrosophic Soft
Sets (FDVNSS), where uncertainty is characterized by both degrees and interrelated member-
ship values, enabling more robust modeling in complex decision and AI-driven environments.
Structure of the Paper: Section 2 presents the basic definitions required for understanding
the subsequent sections. Section 3 introduces key concepts related to FDVNSS. This includes
definitions and examples of the FDVNSS itself, its subset, null set, absolute set, complement,
union, and intersection. Additionally, several propositions are discussed, along with the FD-
VNSS ?AND? and ?OR? operators, each accompanied by relevant examples to ensure a clear
understanding of these concepts.

2. Preliminaries

This section is devoted to the basic definitions necessary for the subsequent sections.

Definition 1. [2] Let X be a non-empty set. A fuzzy set A in X is characterized by its
membership function µA : X → [0, 1] and µA(X) is interpreted as the degree of membership of
element X in fuzzy set A ,for each x ∈ X. It is clear that A is completely determined by the
set of tuples

A = {(x, µA(x)) : x ∈ X}.

Definition 2. [10] The intuitionistic fuzzy sets defined on a non?empty set X as objects having
the form A = {(x, FA(x), GA(x)) : x ∈ X}, where the functions FA(x) : X → [0, 1] and
GA(x) : X → [0, 1], denote the degree of membership and the degree of non-membership of each
element x ∈ X to the set A respectively, and the

0 ≤ FA(x) +GA(x) ≤ 1, forall x ∈ X.

Clearly, when GA(x) = 1− FA(x), for every x ∈ X, the set A becomes a fuzzy set.

Definition 3. [51]Fermatean fuzzy set F in the universe set X is an object with the type
F = {(u, F (x), G(x)) : x ∈ X} where F : U → [0, 1] and G : X → [0, 1] with the condition

0 ≤ (F (x))3 + (G(x))c ≤ 1∀ x ∈ X

Definition 4. [32]A single valued neutrosophic set A on the universe of discourse X is defined
as A = (x : TA(x), IA(x), FA(x)) : x ∈ X where T : I : X → [0, 1] and 0 ≤ TA(x) + IA(x) +
FA(x) ≤ 3.

Definition 5. [4]Let X be the universal set and E be the attributes with respect to X. Let
P (X) be the power set of X and A ⊆ E. Pair (F,A) is called a soft set over U and its mapping
is given as F : A → P (X). It is defined as, (F,A) = {F (e)P (X) : e ∈ E, F (e) = ϕ if e /∈ A}.

Definition 6. [54] Let E be any set of deferent parameters, and let X be the universe, A ⊆ E.
A Fermatean fuzzy soft set (FFSS) and X is defined as the pair (F,A) where F is mapping
given by F : A → FFS(X), Where FFS(X) is the set of all Fermatean fuzzy sets over X. Here
for any parameters e ∈ A, F (e) is the Fermatean fuzzy set over X, defined as:

F (e) = {(x, αF (e)(x), βF (e) : x ∈ X}.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 4 of 25

Where, αF (e)(x) is the degree of membership of x with respect to the parameter e and βF (e) is
the degree of non-membership of x with respect to the parameter e. These degrees satisfy the
condition:

0 ≤ (αF (e)(x))
3 + (βF (e))

3 ≤ 1.

Thus the Fermatean fuzzy soft set (F,E) can be expressed as:

(F,E) = {(e, {x, (αF (e)(x), βF (e))}) : e ∈ A, x ∈ X}.

Definition 7. [59] Let X be an initial universe set and E be a set of parameters . Consider
A ⊆ E. Let P (X) denotes the set of all neutrosophic sets of X. The collection (F,A) is termed
to be the soft neutrosophic set over X, where F is a mapping given by F : A → P (X).

Definition 8. [61] Consider X and E as a universe set and a set of parameters, respectively.
Let P (X) denotes the set Intuitionistic fuzzy set of X. Let A ⊆ E. A pair (F,A) is an
intuitionistic fuzzy soft set over X. Where F is a mapping given by F : A → P (X).

Definition 9. [62] Let É be the set of parameters and X be the key set. Let P (X) represent the
power set of X. Then, a quadri-partitioned neutrosophic soft set (F̃ , É) over X is a mapping
F̃ : É → P (X), where F̃ is the function of (F̃ , É). Symbolically,

(F̃ , É) =
{(

è, ⟨s,AbTF̃ (è)(s),ReTF̃ (è)(s),ReFF̃ (è)(s),AbFF̃ (è)(s) : s ∈ X⟩
)
: è ∈ É

}
.

Where, AbTF̃ (è)(s), ReTF̃ (è)(s), ReFF̃ (è)(s), and AbFF̃ (è)(s) ∈ [0, 1] respectively, AbTF̃ (è)(s),

ReTF̃ (è)(s), ReFF̃ (è)(s), AbFF̃ (è)(s) are called the absolute true-membership, relative true-membership,unknown
membership, confusion-membership, ignorance-membership, relative false-membership, and ab-
solute false-membership function of F̃ (è). Since the supremum of each function is 1 and the
infimum is 0 so the inequality;

0 ≤ AbTF̃ (è)(s) + ReTF̃ (è)(s) + ReFF̃ (è)(s) +AbFF̃ (è)(s) ≤ 4.

Definition 10. [62] Let (F̃ , É) be a QPNSS over the key set X. Then, the complement of
(F̃ , É) is denoted by (F̃ , É)c and is defined as follows:

(F̃ , É)c =
{(

è, ⟨s,AbFF̃ (è)(s),ReFF̃ (è)(s),ReTF̃ (è)(s),AbTF̃ (è)(s)⟩ : s ∈ X
)
: è ∈ É

}
.

So,
(
(F̃ , É)c

)c
= (F̃ , É).

Definition 11. [62] Let (F̃ , É) and (G̃, É) be two QPNSSs over the key set X. Then, (F̃ , É)⊆̃(G̃, É)
if

AbTF̃ (è)(s) ⪯ AbTG̃(è)(s),

ReTF̃ (è)(s) ⪯ ReTG̃(è)(s),

ReFF̃ (è)(s) ⪰ ReFG̃(è)(s),

AbFF̃ (è)(s) ⪰ AbFG̃(è)(s),

for all è ∈ É and s ∈ X. If (F̃ , É)⊆̃(G̃, É) and (F̃ , É)⊇̃(G̃, É), then (F̃ , É) = (G̃, É).

Definition 12. [62] Let (F̃ , É) and (G̃, É) be two QPNSSs over the key set X such that (F̃ , É) ̸=
(G̃, É). Then their union is denoted by (F̃ , É)⋓̃(G̃, É) = (H̃, É) and is defined as:

(H̃, É) =

{(
è, ⟨s,AbTH̃(è)(s),ReTH̃(è)(s),ReFH̃(è)(s),AbFH̃(è)(s)⟩ : s ∈ X

)
: è ∈ É

}
.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 5 of 25

where

AbTH̃(è)(s) = max
{
AbTF̃ (è)(s),AbTG̃(è)(s)

}
,

ReTH̃(è)(s) = max
{
ReTF̃ (è)(s),ReTG̃(è)(s)

}
,

ReFH̃(è)(s) = min
{
ReFF̃ (è)(s),ReFG̃(è)(s)

}
,

AbFH̃(è)(s) = min
{
AbFF̃ (è)(s),AbFG̃(è)(s)

}
.

Definition 13. [62] Let (F̃ , É) and (G̃, É) be two QPNSSs over the key set X such that (F̃ , É) ̸=
(G̃, É). Then their intersection is denoted by (F̃ , É)⋒̃(G̃, É) = (H̃, É) and is defined as:

(H̃, É) =
{(

è, ⟨s,AbTH̃(è)(s),ReTH̃(è)(s),ReFH̃(è)(s),AbFH̃(è)(s)⟩ : s ∈ X
)
: è ∈ É

}
.

where

AbTH̃(è)(s) = min
{
AbTF̃ (è)(s),AbTG̃(è)(s)

}
,

ReTH̃(è)(s) = min
{
ReTF̃ (è)(s),ReTG̃(è)(s)

}
,

ReFH̃(è)(s) = max
{
ReFF̃ (è)(s),ReFG̃(è)(s)

}
,

AbFH̃(è)(s) = max
{
AbFF̃ (è)(s),AbFG̃(è)(s)

}
.

Definition 14. [62] Let (F̃ , É) and (G̃, É) be two QPNSSs over the key set X such that (F̃ , É) ̸=
(G̃, É). Then, their difference is (H̃, É) = (F̃ , É) \ (G̃, É) and is defined as:

(H̃, É) = (F̃ , É) ⋒ (G̃, É)c,

such that

(H̃, É) =
{(

è, ⟨s,AbTH̃(è)(s),ReTH̃(è)(s),ReFH̃(è)(s),AbFH̃(è)(s)⟩
)
: s ∈ X, è ∈ É

}
.

where

AbTH̃(è)(s) = min
{
AbTF̃ (è)(s),AbTG̃(è)(s)

}
,

ReTH̃(è)(s) = min
{
ReTF̃ (è)(s),ReTG̃(è)(s)

}
,

ReFH̃(è)(s) = max
{
ReFF̃ (è)(s),ReFG̃(è)(s)

}
,

AbFH̃(è)(s) = max
{
AbFF̃ (è)(s),AbFG̃(è)(s)

}
.

Definition 15. [62] Let (F̃ , É) and (G̃, É) be two QPNSSs over the key set X. Then, the
”AND” operation on them is denoted by (F̃ , É) ∧ (G̃, É) = (H̃, É × É) and is defined as:

(H̃, É × É) =

{(
(è1, è2), ⟨s,AbTH̃(è1, è2)(s),ReTH̃(è1, è2)(s),

ReFH̃(è1, è2)(s),AbFH̃(è1, è2)(s) : s ∈ X⟩
)
: (è1, è2) ∈ É × É

}
.

where

AbTH̃(è)(s) = min
{
AbTF̃ (è)(s),AbTG̃(è)(s)

}
,


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 6 of 25

ReTH̃(è)(s) = min
{
ReTF̃ (è)(s),ReTG̃(è)(s)

}
,

ReFH̃(è)(s) = max
{
ReFF̃ (è)(s),ReFG̃(è)(s)

}
,

AbFH̃(è)(s) = max
{
AbFF̃ (è)(s),AbFG̃(è)(s)

}
.

Definition 16. [62]Let (F̃ , É) and (G̃, É) be two QPNSSs over the key set X. Then, the ”OR”
operation on them is denoted by (F̃ , É) ∨ (G̃, É) = (H̃, É × É) and is defined as:

(H̃, É × É) =

{(
(è1, è2), ⟨s,AbTH̃(è1,è2)

(s),ReTH̃(è1,è2)
(s),

ReFH̃(è1,è2)
(s),AbFH̃(è1,è2)

(s) : s ∈ X⟩
)
: (è1, è2) ∈ É × É

}
.

where

AbTH̃(è)(s) = max
{
AbTF̃ (è)(s),AbTG̃(è)(s)

}
,

ReTH̃(è)(s) = max
{
ReTF̃ (è)(s),ReTG̃(è)(s)

}
,

ReFH̃(è)(s) = min
{
ReFF̃ (è)(s),ReFG̃(è)(s)

}
,

AbFH̃(è)(s) = min
{
AbFF̃ (è)(s),AbFG̃(è)(s)

}
.

3. Fundamental of Fermatean Double Valued Neutrosophic Soft Sets

This section is devoted to new of Fermatean double valued neutrosophic soft sets. The
concept indeterminacy refers to the inherent uncertainty present in all aspects of life. Recog-
nizing and incorporating this uncertainty makes research and scientific inquiry more realistic
and sensitive by acknowledging the complex, often ambiguous nature of reality. In real-world
scenarios, indeterminacy can sometimes exhibit a tendency toward truth?it holds more truth
value than falsehood?yet it cannot be definitively classified as true. Conversely, it may lean
more toward falsehood than truth, without being conclusively false. These meticulous cases
require a more exact categorization to improve understanding and interpretation. To address
this, indeterminacy can be classified into two distinct types:
1. Indeterminacy leaning toward truth (IT ) or ReT (x): When the indeterminate state
has a greater affinity with truth than falsehood but still falls short of being classified as true.
2. Indeterminacy leaning toward falsehood (IF ) or ReF (x): When the indeterminate
state is closer to falsehood than truth but cannot be conclusively labeled as false. This refined
classification IT or ReT (x) and IF or ReF (x)allows for a more accurate and detailed represen-
tation of uncertainty. It provides deeper insight into the spectrum of indeterminacy, enabling
researchers and theorists to engage with complex phenomena more precisely and thoughtfully.
In this section, the Fermatean double valued neutrosophic soft set, Fermatean double valued
neutrosophic soft subset, FDVNSS null set and its example, FDVNSS absolute set and its
example, FDVNSS complement and its example, FDVNSS union and its example, FDVNSS in-
tersection and its example, several propositions, the FDVNSS ?AND? operator and its example,
and the FDVNSS ?OR? operator and its examples are presented for a clearer understanding of
these concepts.

Definition 17. Let X be a non-empty set (universal). A Fermatean double valued neutrosophic
soft set (DVNSS) is defined as

{x, T (x), ReT (x), ReF (x), F (x) : x ∈ X},


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 7 of 25

Where T (x), ReT (x), ReF (x), F (x) ∈ [0, 1], 0 ≤ (T (x))3+(ReT (x))3+(ReF (x))3+(F (x))3 ≤ 2.
Then 0 ≤ (T (x))3 + (ReF (x))3 + (ReF (x))3 + (F (x))3 ≤ 2, for all x ∈ X. T (x) is the degree
of true membership, ReT (x) is the degree of relative truth membership, ReT (x is the degree of
relative false membership and F (x) is the degree of false membership. Here T (x) and F (x) are
dependent components and ReT (x) and ReF (x) is an independent component.

Example 1. Let X = {x1, x2, x3} to be set of three diseases. Consider a set of charac-
teristics or symptoms E = {e1, e2, e3}, where e1 = contagious, e2 = severity and e3 =
treatment availability. We defined a function F : E → FDV NSS(X) as follows:

(F,E) =

{e1 = {⟨x1, 0.2, 0.4, 0.2, 0.1⟩, ⟨x2, 0.4, 0.1, 0.1, 0.6⟩, ⟨x3, 0.3, 0.03, 0.07, 0.7⟩} ,
e2 = {⟨x1, 0.3, 0.2, 0.2, 0.5⟩, ⟨x2, 0.2, 0.2, 0.1, 0.4⟩, ⟨x3, 0.7, 0.4, 0.4, 0.2⟩} ,
e3 = {⟨x1, 0.6, 0.1, 0.1, 0.1⟩, ⟨x2, 0.3, 0.3, 0.3, 0.9⟩, ⟨x2, 0.1, 0.1, 0.1, 0.3⟩}

}

F (e1) = {⟨x1, 0.2, 0.4, 0.2, 0.1⟩, ⟨x2, 0.4, 0.1, 0.1, 0.6⟩, ⟨x3, 0.3, 0.03, 0.07, 0.7⟩} ;

F (e2) = {⟨x1, 0.3, 0.2, 0.2, 0.5⟩, ⟨x2, 0.2, 0.2, 0.1, 0.4⟩, ⟨x3, 0.7, 0.4, 0.4, 0.2⟩} ;

F (e3) = {⟨x1, 0.6, 0.1, 0.1, 0.1⟩, ⟨x2, 0.3, 0.3, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} .

Then the FDVNSS over (X,E) in matrix form can be expressed as:

(F,E) =

 (⟨0.2, 0.4, 0.2, 0.1⟩), (⟨0.4, 0.1, 0.1, 0.6⟩), (⟨0.3, 0.03, 0.07, 0.7⟩)
(⟨0.3, 0.2, 0.2, 0.5⟩), (⟨0.2, 0.2, 0.1, 0.4⟩), (⟨0.7, 0.4, 0.4, 0.2⟩)
(⟨0.6, 0.1, 0.1, 0.1⟩), (⟨0.3, 0.3, 0.3, 0.9⟩), (⟨0.1, 0.1, 0.1, 0.3⟩)

 .

Where ith row vector represents F (ei), the ith column vector represent xi. Its clearly that
(0.2)3+(0.1)3 ≤ 1 and (0.2)3+(0.4)3+(0.2)3+(0.1)3 ≤ 2 which satisfy the FDVNSS condition.

Definition 18. Let (F,E) and (G,E) be two FDVNSS on (X,E). Then (F,E)is called to be a
Fermatean double valued neutrosophic soft subset of (G,E) if F (e) ⊆ G(e), ∀ e ∈ E , here we
say that (F,E) ⊆ (G,E)).

Example 2. Let X = {x1, x2, x3} to be set of three types of cancer diseases. Let E = {e1, e2, e3}
be a set of medical parameters, where e1 = early − stage diagnosis, e2 = high mortality rate
and e3 = responsive to treatment. Let (F,E) is FDVNSS over (X,E) defined as follows:

F (e1) = {⟨x1, 0.2, 0.2, 0.1, 0.2⟩, ⟨x2, 0.4, 0.3, 0.3, 0.4⟩, ⟨x3, 0.1, 0.1, 0.1, 0.9⟩} ;

F (e2) = {⟨x1, 0.5, 0.1, 0.2, 0.8⟩, ⟨x2, 0.7, 0.1, 0.2, 0.6⟩, ⟨x3, 0.1, 0.1, 0.1, 0.6⟩} ;

F (e3) = {⟨x1, 0.2, 0.2, 0.4, 0.3⟩, ⟨x2, 0.3, 0.1, 0.4, 0.8⟩, ⟨x3, 0.4, 0.1, 0.2, 0.7⟩} .

Let (G,E) is FDVNSS over (X,E) defined as follows:

G(e1) = {⟨x1, 0.3, 0.2, 0.2, 0.1⟩, ⟨x2, 0.6, 0.2, 0.5, 0.2⟩, ⟨x3, 0.7, 0.1, 0.1, 0.8⟩} ;

G(e2) = {⟨x1, 0.6, 0.2, 0.2, 0.7⟩, ⟨x2, 0.9, 0.2, 0.2, 0.4⟩, ⟨x3, 0.3, 0.4, 0.1, 0.3⟩} ;

G(e3) = {⟨x1, 0.4, 0.2, 0.5, 0.1⟩, ⟨x2, 0.6, 0.3, 0.3, 0.7⟩, ⟨x3, 0.5, 0.2, 0.2, 0.5⟩} .

It follows that (F,E) ⊆ (G,E).


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Definition 19. A FDVNSS is said to be a null FDVNSS, denoted by Fϕ such that

Eϕ = {(e, {x, 0, 0, 0, 1}) : e ∈ E, c ∈ X}.

Example 3. Let X = {x1, x2, x3} to be set of three COVID-19 variants and E = {e1, e2, e3}
be a set of medical characteristics, where e1 = high transmission rate, e2 = severe symptoms
and e3 = resistant to treatment. We defined a function F : E → FDV NSS(X) which is a
FDVNSS over (X,E) defined as follows:

F (e1) = {⟨x1, 0, 0, 0, 1⟩, ⟨x2, 0, 0, 0, 1⟩, ⟨x3, 0, 0, 0, 1⟩} ;

F (e2) = {⟨x1, 0, 0, 0, 1⟩, ⟨x2, 0, 0, 0, 1⟩, ⟨x3, 0, 0, 0, 1⟩} ;

F (e3) = {⟨x1, 0, 0, 0, 1⟩, ⟨x2, 0, 0, 0, 1⟩, ⟨x3, 0, 0, 0, 1⟩} .

(F,E) =

 (⟨0, 0, 0, 1⟩), (⟨0, 0, 0, 1⟩), (⟨0, 0, 0, 1⟩)
(⟨0, 0, 0, 1⟩), (⟨0, 0, 0, 1⟩), (⟨0, 0, 0, 1⟩)
(⟨0, 0, 0, 1⟩), (⟨0, 0, 0, 1⟩), (⟨0, 0, 0, 1⟩)

 .

This means (F,E) is null FDVNSS.

Definition 20. A FDVNSS is said to be an absolute FDVNSS , denoted by XE such that

XE = {(e, {x, 1, 0, 0, 0}) : e ∈ E, x ∈ X}.

Example 4. Let X = {x1, x2, x3} to be set of three AIDS case. Let E = {e1, e2, e3} be a set of
medical parameters, where e1 = low immunity level, e2 = presence of opportunistic infections
and e3 = response to antiretroviral therapy. Let (F,E) be FDVNSS over (X,E) defined as
follows:

F (e1) = {⟨x1, 1, 0, 0, 0⟩, ⟨x2, 1, 0, 0, 0⟩, ⟨x3, 1, 0, 0, 0⟩} ;

F (e2) = {⟨x1, 1, 0, 0, 0⟩, ⟨x2, 1, 0, 0, 0⟩, ⟨x3, 1, 0, 0, 0⟩} ;

F (e3) = {⟨x1, 1, 0, 0, 0⟩, ⟨x2, 1, 0, 0, 0⟩, ⟨x3, 1, 0, 0, 0⟩} .

(F,E) =

 (⟨1, 0, 0, 0⟩), (⟨1, 0, 0, 0⟩), (⟨1, 0, 0, 0⟩)
(⟨1, 0, 0, 0⟩), (⟨1, 0, 0, 0⟩), (⟨1, 0, 0, 0⟩)
(⟨1, 0, 0, 0⟩), (⟨1, 0, 0, 0⟩), (⟨1, 0, 0, 0⟩)

 .

This means (F,E) is absolute FDVNSS.

Definition 21. Let (F,E) be a FDVNSS over (X,E). Then the complement of (F,E), denoted
by (F,E)c is defined by F c(e) = D(e), ∀ e ∈ E where D(e) denoted the Fermatean Double
Valued Neutrosophic complement, where

(F̃ , E)c = {⟨x, FF̃ (e)(x), ReFF̃ (e)(x), ReTF̃ (e)(x), TF̃ (e)(x)⟩ : x ∈ X}.

Obvious that ((F̃ , E)c)c = (F̃ , E).


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 9 of 25

Example 5. Refer to the matrix form illustrated in Example 1.

(F,E) =

{e1 = {⟨x1, 0.1, 0.2, 0.4, 0.2⟩, ⟨x2, 0.6, 0.1, 0.1, 0.4⟩, ⟨x3, 0.7, 0.07, 0.03, 0.3⟩} ,
e2 = {⟨x1, 0.5, 0.2, 0.2, 0.3⟩, ⟨x2, 0.4, 0.1, 0.2, 0.2⟩, ⟨x3, 0.2, 0.4, 0.4, 0.7⟩} ,
e3 = {⟨x1, 0.1, 0.1, 0.1, 0.6⟩, ⟨x2, 0.9, 0.3, 0.3, 0.3⟩, ⟨x3, 0.3, 0.1, 0.1, 0.1⟩}

}

By using the FDVNSS complement , we have (F,E)c = (G,E), where (G,E) is:

(G,E) =

 (⟨0.2, 0.4, 0.2, 0.1⟩), (⟨0.4, 0.1, 0.1, 0.6⟩), (⟨0.3, 0.03, 0.07, 0.7⟩)
(⟨0.3, 0.2, 0.2, 0.5⟩), (⟨0.2, 0.2, 0.1, 0.4⟩), (⟨0.7, 0.4, 0.4, 0.2⟩)
(⟨0.6, 0.1, 0.1, 0.1⟩), (⟨0.3, 0.3, 0.3, 0.9⟩), (⟨0.1, 0.1, 0.1, 0.3⟩)

 .

Definition 22. The union of two FDVNSSs (F,E) and (G,E), denoted by (F,E)∪̃(G,E), is
a FDV NSS(H,E) such that H : E → FDV NSS(X) defined by H(e) = (F (e)∪̃G(e)) is a
Fermatean double valued neutrosophic soft union, where:

(F,E)∪̃(G,E) =
{
x, (maxTF̃ (e1)

(x),maxReTF̃ (e2)
(x),minReFF̃ (e1)

(x),minFF̃ (e2)
(x))

}
.

Example 6. A universal set of objects: X = {x1, x2, x3} possibly individuals whose blood
pressure is being analyzed. A set of parameters: E = {e1, e2, e3} likely representing different
blood pressure characteristics, such as e1 : Systolic pressure, e2 : Diastolic pressure, e3 :
Pulse pressure or any relevant metric. Then the FDVNSS (F,E) is defined as follows:

F (e1) = {⟨x1, 0.2, 0.4, 0.2, 0.1⟩), (⟨x2, 0.4, 0.1, 0.1, 0.6⟩, ⟨x3, 0.3, 0.03, 0.07, 0.7⟩} ;

F (e2) = {⟨x1, 0.3, 0.2, 0.2, 0.5⟩, ⟨x2, 0.2, 0.2, 0.1, 0.4⟩, ⟨x3, 0.7, 0.4, 0.4, 0.2⟩} ;

F (e3) = {⟨x1, 0.6, 0.1, 0.1, 0.1⟩, ⟨x2, 0.3, 0.3, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} .

Let us define another FDVNSS (G,E) as follows:

G(e1) = {⟨x1, 0.3, 0.1, 0.1, 0.1⟩, ⟨x2, 0.2, 0.2, 0.2, 0.3⟩, ⟨x3, 0.3, 0.1, 0.1, 0.2⟩} ;

G(e2) = {⟨x1, 0.2, 0.2, 0.1, 0.3⟩, ⟨x2, 0.1, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} ;

G(e3) = {⟨x1, 0.4, 0.2, 0.1, 0.4⟩, ⟨x2, 0.1, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.5, 0.4, 0.3⟩} .

Then their FDVNSS union is given as follows:

(F,E)∪̃(G,E) =

{e1 = {⟨x1, 0.3, 0.4, 0.1, 0.1⟩, ⟨x2, 0.4, 0.2, 0.1, 0.3⟩, ⟨x3, 0.3, 0.1, 0.07, 0.2⟩} ,
e2 = {⟨x1, 0.3, 0.2, 0.1, 0.3⟩, ⟨x2, 0.2, 0.4, 0.1, 0.4⟩, ⟨x3, 0.7, 0.4, 0.1, 0.2⟩} ,
e3 = {⟨x1, 0.6, 0.2, 0.1, 0.1⟩, ⟨x2, 0.3, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.5, 0.1, 0.3⟩}

}

In matrix notation, we write it as follows:

(H,E) =

 (⟨0.3, 0.4, 0.1, 0.1⟩), (⟨0.4, 0.2, 0.1, 0.3⟩), (⟨0.3, 0.1, 0.07, 0.2⟩)
(⟨0.3, 0.2, 0.1, 0.3⟩), (⟨0.2, 0.4, 0.1, 0.4⟩), (⟨0.7, 0.4, 0.1, 0.2⟩)
(⟨0.6, 0.2, 0.1, 0.1⟩), (⟨0.3, 0.4, 0.3, 0.4⟩), (⟨0.1, 0.5, 0.1, 0.3⟩)

 .

Definition 23. The intersection of two FDVNSSs (F,E) and (G,E), denoted by (F,E)∩̃(G,E),
is a FDV NSS(H,E) such that H : E → FDV NSS(X) defined by H(e) = (F (e)∩̃G(e)) is a
Fermatean double valued neutrosophic soft intersection, where:

(F,E)∩̃(G,E) =
{
x, (minTF̃ (e1)

(x),minReTF̃ (e2)
(x),maxReFF̃ (e1)

(x),maxFF̃ (e2)
(x))

}
.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 10 of 25

Example 7. Let us consider Example 1.

F (e1) = {⟨x1, 0.2, 0.4, 0.2, 0.1⟩, ⟨x2, 0.4, 0.1, 0.1, 0.6⟩, ⟨x3, 0.3, 0.03, 0.07, 0.7⟩} ;

F (e2) = {⟨x1, 0.3, 0.2, 0.2, 0.5⟩, ⟨x2, 0.2, 0.2, 0.1, 0.4⟩, ⟨x3, 0.7, 0.4, 0.4, 0.2⟩} ;

F (e3) = {⟨x1, 0.6, 0.1, 0.1, 0.1⟩, ⟨x2, 0.3, 0.3, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} .

Let us define another FDVNSS (G,E) as follows:

G(e1) = {⟨x1, 0.3, 0.1, 0.1, 0.1⟩, ⟨x2, 0.2, 0.2, 0.2, 0.3⟩, ⟨x3, 0.3, 0.1, 0.1, 0.2⟩} ;

G(e2) = {⟨x1, 0.2, 0.2, 0.1, 0.3⟩, ⟨x2, 0.1, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} ;

G(e3) = {⟨x1, 0.4, 0.2, 0.1, 0.4⟩, ⟨x2, 0.1, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.5, 0.4, 0.3⟩} .

Then their FDVNSS intersection is given as follows:

(F,E)∩̃(G,E) =

{e1 = {⟨x1, 0.2, 0.1, 0.2, 0.1⟩, ⟨x2, 0.2, 0.1, 0.2, 0.6⟩, ⟨x3, 0.3, 0.03, 0.1, 0.7⟩} ,
e2 = {⟨x1, 0.2, 0.2, 0.2, 0.5⟩, ⟨x2, 0.1, 0.2, 0.3, 0.4⟩, ⟨x3, 0.1, 0.1, 0.4, 0.3⟩} ,
e3 = {⟨x1, 0.4, 0.1, 0.1, 0.4⟩, ⟨x2, 0.1, 0.3, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.4, 0.3⟩}

}

In matrix notation, we write it as follows:

(H,E) =

 (⟨0.2, 0.1, 0.2, 0.1⟩), (⟨0.2, 0.1, 0.2, 0.6⟩), (⟨0.3, 0.03, 0.1, 0.7⟩)
(⟨0.2, 0.2, 0.2, 0.5⟩), (⟨0.1, 0.2, 0.3, 0.4⟩), (⟨0.1, 0.1, 0.4, 0.3⟩)
(⟨0.4, 0.1, 0.1, 0.4⟩), (⟨0.1, 0.3, 0.3, 0.9⟩), (⟨0.1, 0.1, 0.4, 0.3⟩)

 .

Proposition 1. Let(F,E), (G,E) and (H,E) are three FDVNSSs over (X,E) then the follow-
ing holds: 1.

(F,E)∪̃(G,E) = (G,E)∪̃(F,E)

2.
(F,E)∩̃(G,E) = (G,E)∩̃(F,E)

3.
(F,E)∪̃((G,E)∪̃(H,E)) = ((F,E)∪̃(G,E))∪̃(H,E)

4.
(F,E)∩̃((G,E)∩̃(H,E)) = ((F,E)∩̃(G,E))∩̃(H,E)

Proof. Obvious.

Proposition 2. Let(F,E) and (G,E) are two FDVNSSs over (X,E) then the following holds:
1.

((F,E)∪̃(G,E))c = ((F,E))c∪̃((G,E))c

2.
((F,E)∩̃(G,E))c = ((F,E))c∩̃((G,E))c

Proof. 1. ((F,E)∪̃(G,E))c = ((F,E))c∪̃((G,E))c.
2. ((F,E)∩̃(G,E))c = ((F,E))c∩̃((G,E))c.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 11 of 25

Definition 24. (F,A) and (G,B) are two FDVNSSs then ”(F,A)AND(G,B)”, defined by
(F,A) ∧ (G,B) is denoted by (F,A) ∧ (G,B) = (H,A × B). Where H(α, β) = (H(e1, e2)(x)),
for all (e1, e2) ∈ A×B, such that

H(e1, e2) = (F (e1)∩̃G(e2)), ∀ (e1, e2) ∈ A×B,

where ∩̃ is Fermatean Double Valued neutrosophic soft intersection.

Example 8. Assume the universe consists of three malaria patients, that is, X = {x1, x2, x2}
and there are three parameters, E = {e1, e2, e3}, which describe their health conditions based on
specific malaria-related factors (e.g., fever level, parasite count, and response to medication).
Suppose Dr. X wants to evaluate a patient based on only one of these parameters. Let there be
two observations, (F,A) and (G,A) made by two experts, defined as follows:

(F,E) ∧ (G,E) =
{
x, (minTF̃ (e1)

(x),minReTF̃ (e2)
(x),maxReFF̃ (e1)

(x),maxFF̃ (e2)
(x))

}
.

We have

F (e1) = {⟨x1, 0.2, 0.4, 0.2, 0.1⟩, ⟨x2, 0.4, 0.1, 0.1, 0.6⟩, ⟨x3, 0.3, 0.03, 0.07, 0.7⟩} ;

F (e2) = {⟨x1, 0.3, 0.2, 0.2, 0.5⟩, ⟨x2, 0.2, 0.2, 0.1, 0.4⟩, ⟨x3, 0.7, 0.4, 0.4, 0.2⟩} ;

F (e3) = {⟨x1, 0.6, 0.1, 0.1, 0.1⟩, ⟨x2, 0.3, 0.3, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} .
and

G(e1) = {⟨x1, 0.3, 0.1, 0.1, 0.1⟩, ⟨x2, 0.2, 0.2, 0.2, 0.3⟩, ⟨x3, 0.3, 0.1, 0.1, 0.2⟩} ;

G(e2) = {⟨x1, 0.2, 0.2, 0.1, 0.3⟩, ⟨x2, 0.1, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} ;

G(e3) = {⟨x1, 0.4, 0.2, 0.1, 0.4⟩, ⟨x2, 0.1, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.5, 0.4, 0.3⟩} .
So,

(F,E)∧(G,E) =

{(e1, e1) = {⟨x1, 0.2, 0.1, 0.2, 0.1⟩, ⟨x2, 0.2, 0.1, 0.2, 0.6⟩, ⟨x3, 0.3, 0.03, 0.1, 0.7⟩} ,
(e1, e2) = {⟨x1, 0.2, 0.2, 0.2, 0.3⟩, ⟨x2, 0.1, 0.03, 0.1, 0.6⟩, ⟨x3, 0.1, 0.03, 0.1, 0.7⟩} ,
(e1, e3) = {⟨x1, 0.2, 0.2, 0.2, 0.4⟩, ⟨x2, 0.1, 0.1, 0.3, 0.6⟩, ⟨x3, 0.1, 0.03, 0.4, 0.7⟩}

}

=

{(e2, e1) = {⟨x1, 0.3, 0.1, 0.2, 0.5⟩, ⟨x2, 0.2, 0.2, 0.2, 0.4⟩, ⟨x3, 0.3, 0.1, 0.2, 0.4⟩} ,
(e2, e2) = {⟨x1, 0.2, 0.2, 0.2, 0.5⟩, ⟨x2, 0.1, 0.2, 0.3, 0.4⟩, ⟨x3, 0.1, 0.1, 0.4, 0.3⟩} ,
(e2, e3) = {⟨x1, 0.3, 0.2, 0.2, 0.5⟩, ⟨x2, 0.1, 0.2, 0.3, 0.4⟩, ⟨x3, 0.2, 0.2, 0.2, 0.5⟩}

}

=

{(e3, e1) = {⟨x1, 0.3, 0.1, 0.1, 0.1⟩, ⟨x2, 0.2, 0.2, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} ,
(e3, e2) = {⟨x1, 0.2, 0.1, 0.1, 0.3⟩, ⟨x2, 0.1, 0.03, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} ,
(e3, e3) = {⟨x1, 0.4, 0.1, 0.1, 0.4⟩, ⟨x2, 0.1, 0.3, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.4, 0.3⟩}

}

In matrix notation, we write it as follows:

(H,E) =



(⟨0.2, 0.1, 0.2, 0.1⟩), (⟨0.2, 0.1, 0.2, 0.6⟩), (⟨0.3, 0.03, 0.1, 0.7⟩)
(⟨0.2, 0.2, 0.2, 0.3⟩), (⟨0.1, 0.03, 0.1, 0.6⟩), (⟨0.1, 0.03, 0.1, 0.7⟩)
(⟨0.2, 0.2, 0.2, 0.4⟩), (⟨0.1, 0.1, 0.3, 0.6⟩), (⟨0.1, 0.03, 0.4, 0.7⟩)
(⟨0.3, 0.1, 0.2, 0.5⟩), (⟨0.2, 0.2, 0.2, 0.4⟩), (⟨0.3, 0.1, 0.2, 0.4⟩)
(⟨0.2, 0.2, 0.2, 0.5⟩), (⟨0.1, 0.2, 0.3, 0.4⟩), (⟨0.1, 0.1, 0.4, 0.3⟩)
(⟨0.3, 0.2, 0.2, 0.5⟩), (⟨0.1, 0.2, 0.3, 0.4⟩), (⟨0.2, 0.2, 0.2, 0.5⟩)
(⟨0.3, 0.1, 0.1, 0.1⟩), (⟨0.2, 0.2, 0.3, 0.9⟩), (⟨0.1, 0.1, 0.1, 0.3⟩)
(⟨0.2, 0.1, 0.1, 0.3⟩), (⟨0.1, 0.03, 0.3, 0.9⟩), (⟨0.1, 0.1, 0.1, 0.3⟩)
(⟨0.4, 0.1, 0.1, 0.4⟩), (⟨0.1, 0.3, 0.3, 0.9⟩), (⟨0.1, 0.1, 0.4, 0.3⟩)


.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 12 of 25

Definition 25. (F,A) and (G,B) are two FDVNSSs then ”(F,A)OR(G,B)”, defined by (F,A)∨
(G,B) is denoted by (F,A) ∨ (G,B) = (H,A × B). Where H(α, β) = (H(e1, e2)(x)), for all
(e1, e2) ∈ A×B, such that

H(e1, e2) = (F (e1)∪̃G(e2)), ∀ (e1, e2) ∈ A×B,

Example 9. Let us consider Example 1.

F (e1) = {⟨x1, 0.2, 0.4, 0.2, 0.1⟩, ⟨x2, 0.4, 0.1, 0.1, 0.6⟩, ⟨x3, 0.3, 0.03, 0.07, 0.7⟩} ;

F (e2) = {⟨x1, 0.3, 0.2, 0.2, 0.5⟩, ⟨x2, 0.2, 0.2, 0.1, 0.4⟩, ⟨x3, 0.7, 0.4, 0.4, 0.2⟩} ;

F (e3) = {⟨x1, 0.6, 0.1, 0.1, 0.1⟩, ⟨x2, 0.3, 0.3, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} .

and
G(e1) = {⟨x1, 0.3, 0.1, 0.1, 0.1⟩, ⟨x2, 0.2, 0.2, 0.2, 0.3⟩, ⟨x3, 0.3, 0.1, 0.1, 0.2⟩} ;

G(e2) = {⟨x1, 0.2, 0.2, 0.1, 0.3⟩, ⟨x2, 0.1, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} ;

G(e3) = {⟨x1, 0.4, 0.2, 0.1, 0.4⟩, ⟨x2, 0.1, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.5, 0.4, 0.3⟩} .

So,

(F,E)∨(G,E) =

{(e1, e1) = {⟨x1, 0.3, 0.4, 0.1, 0.1⟩, ⟨x2, 0.4, 0.2, 0.1, 0.3⟩, ⟨x3, 0.3, 0.1, 0.07, 0.2⟩} ,
(e1, e2) = {⟨x1, 0.2, 0.4, 0.1, 0.1⟩, ⟨x2, 0.4, 0.1, 0.07, 0.2⟩, ⟨x3, 0.3, 0.1, 0.07, 0.3⟩} ,
(e1, e3) = {⟨x1, 0.4, 0.4, 0.1, 0.1⟩, ⟨x2, 0.4, 0.4, 0.1, 0.4⟩, ⟨x3, 0.3, 0.5, 0.07, 0.3⟩}

}

=

{(e2, e1) = {⟨x1, 0.3, 0.2, 0.1, 0.1⟩, ⟨x2, 0.2, 0.2, 0.1, 0.3⟩, ⟨x3, 0.7, 0.4, 0.1, 0.2⟩} ,
(e2, e2) = {⟨x1, 0.3, 0.2, 0.1, 0.3⟩, ⟨x2, 0.2, 0.2, 0.07, 0.2⟩, ⟨x3, 0.3, 0.1, 0.07, 0.3⟩} ,
(e2, e3) = {⟨x1, 0.4, 0.2, 0.1, 0.4⟩, ⟨x2, 0.2, 0.4, 0.1, 0.4⟩, ⟨x3, 0.7, 0.5, 0.4, 0.2⟩}

}

=

{(e3, e1) = {⟨x1, 0.6, 0.1, 0.1, 0.1⟩, ⟨x2, 0.3, 0.3, 0.2, 0.3⟩, ⟨x3, 0.3, 0.1, 0.1, 0.2⟩} ,
(e3, e2) = {⟨x1, 0.6, 0.2, 0.1, 0.1⟩, ⟨x2, 0.3, 0.3, 0.07, 0.2⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} ,
(e3, e3) = {⟨x1, 0.6, 0.2, 0.1, 0.1⟩, ⟨x2, 0.3, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.5, 0.1, 0.3⟩}

}

In matrix notation, we write it as follows:

(H,E) =



(⟨0.3, 0.4, 0.1, 0.1⟩), (⟨0.4, 0.2, 0.1, 0.3⟩), (⟨0.3, 0.03, 0.1, 0.7⟩)
(⟨0.2, 0.4, 0.1, 0.1⟩), (⟨0.14, 0.1, 0.07, 0.2⟩), (⟨0.3, 0.1, 0.07, 0.3⟩)
(⟨0.4, 0.4, 0.1, 0.1⟩), (⟨0.4, 0.4, 0.1, 0.4⟩), (⟨0.3, 0.5, 0.07, 0.3⟩)
(⟨0.3, 0.2, 0.1, 0.1⟩), (⟨0.2, 0.2, 0.1, 0.3⟩), (⟨0.7, 0.4, 0.1, 0.2⟩)

(⟨0.3, 0.2, 0.1, 0.3⟩), (⟨0.2, 0.2, 0.07, 0.2⟩), (⟨0.3, 0.1, 0.07, 0.3⟩)
(⟨0.4, 0.2, 0.1, 0.4⟩), (⟨0.2, 0.4, 0.1, 0.4⟩), (⟨0.7, 0.4, 0.4, 0.2⟩)
(⟨0.6, 0.1, 0.1, 0.1⟩), (⟨0.3, 0.3, 0.2, 0.3⟩), (⟨0.3, 0.1, 0.1, 0.2⟩)
(⟨0.6, 0.2, 0.1, 0.1⟩), (⟨0.3, 0.3, 0.07, 0.2⟩), (⟨0.1, 0.1, 0.1, 0.3⟩)
(⟨0.6, 0.2, 0.1, 0.1⟩), (⟨0.3, 0.4, 0.3, 0.4⟩), (⟨0.1, 0.5, 0.1, 0.3⟩)


.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 13 of 25

Example 10. Assume the universe consists of three tuberculosis (TB) patients, that is, X =
{x1, x2, x2} and there are three parameters, E = {e1, e2, e3} is a set of parameters representing
medical indicators of TB severity, such as: e1 : Chest X−ray findings, e2 : Sputum smear results,
e3 : Clinical symptoms (e.g. fever weight loss cough). Suppose Dr. X wants to evaluate a pa-
tient based on only one of these parameters. Let there be two observations, (F,A) and (G,A)
made by two experts, defined as follows:

F (e1) = {⟨x1, 0.2, 0.4, 0.2, 0.1⟩, ⟨x2, 0.4, 0.1, 0.1, 0.6⟩, ⟨x3, 0.3, 0.03, 0.07, 0.7⟩} ;

F (e2) = {⟨x1, 0.3, 0.2, 0.2, 0.5⟩, ⟨x2, 0.2, 0.2, 0.1, 0.4⟩, ⟨x3, 0.7, 0.4, 0.4, 0.2⟩} ;

F (e3) = {⟨x1, 0.6, 0.1, 0.1, 0.1⟩, ⟨x2, 0.3, 0.3, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} .

and
G(e1) = {⟨x1, 0.3, 0.1, 0.1, 0.1⟩, ⟨x2, 0.2, 0.2, 0.2, 0.3⟩, ⟨x3, 0.3, 0.1, 0.1, 0.2⟩} ;

G(e2) = {⟨x1, 0.2, 0.2, 0.1, 0.3⟩, ⟨x2, 0.1, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} ;

G(e3) = {⟨x1, 0.4, 0.2, 0.1, 0.4⟩, ⟨x2, 0.1, 0.4, 0.3, 0.4⟩, ⟨x3, 0.1, 0.5, 0.4, 0.3⟩} .

So,

(F,E)∧(G,E) =

{(e1, e1) = {⟨x1, 0.2, 0.1, 0.2, 0.1⟩, ⟨x2, 0.2, 0.1, 0.2, 0.6⟩, ⟨x3, 0.3, 0.03, 0.1, 0.7⟩} ,
(e1, e2) = {⟨x1, 0.2, 0.2, 0.2, 0.3⟩, ⟨x2, 0.1, 0.03, 0.1, 0.6⟩, ⟨x3, 0.1, 0.03, 0.1, 0.7⟩} ,
(e1, e3) = {⟨x1, 0.2, 0.2, 0.2, 0.4⟩, ⟨x2, 0.1, 0.1, 0.3, 0.6⟩, ⟨x3, 0.1, 0.03, 0.4, 0.7⟩}

}

=

{(e2, e1) = {⟨x1, 0.3, 0.1, 0.2, 0.5⟩, ⟨x2, 0.2, 0.2, 0.2, 0.4⟩, ⟨x3, 0.3, 0.1, 0.2, 0.4⟩} ,
(e2, e2) = {⟨x1, 0.2, 0.2, 0.2, 0.5⟩, ⟨x2, 0.1, 0.2, 0.3, 0.4⟩, ⟨x3, 0.1, 0.1, 0.4, 0.3⟩} ,
(e2, e3) = {⟨x1, 0.3, 0.2, 0.2, 0.5⟩, ⟨x2, 0.1, 0.2, 0.3, 0.4⟩, ⟨x3, 0.2, 0.2, 0.2, 0.5⟩}

}

=

{(e3, e1) = {⟨x1, 0.3, 0.1, 0.1, 0.1⟩, ⟨x2, 0.2, 0.2, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} ,
(e3, e2) = {⟨x1, 0.2, 0.1, 0.1, 0.3⟩, ⟨x2, 0.1, 0.03, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.1, 0.3⟩} ,
(e3, e3) = {⟨x1, 0.4, 0.1, 0.1, 0.4⟩, ⟨x2, 0.1, 0.3, 0.3, 0.9⟩, ⟨x3, 0.1, 0.1, 0.4, 0.3⟩}

}

In matrix notation, we write it as follows:

(H,E) =



(⟨0.2, 0.1, 0.2, 0.1⟩), (⟨0.2, 0.1, 0.2, 0.6⟩), (⟨0.3, 0.03, 0.1, 0.7⟩)
(⟨0.2, 0.2, 0.2, 0.3⟩), (⟨0.1, 0.03, 0.1, 0.6⟩), (⟨0.1, 0.03, 0.1, 0.7⟩)
(⟨0.2, 0.2, 0.2, 0.4⟩), (⟨0.1, 0.1, 0.3, 0.6⟩), (⟨0.1, 0.03, 0.4, 0.7⟩)
(⟨0.3, 0.1, 0.2, 0.5⟩), (⟨0.2, 0.2, 0.2, 0.4⟩), (⟨0.3, 0.1, 0.2, 0.4⟩)
(⟨0.2, 0.2, 0.2, 0.5⟩), (⟨0.1, 0.2, 0.3, 0.4⟩), (⟨0.1, 0.1, 0.4, 0.3⟩)
(⟨0.3, 0.2, 0.2, 0.5⟩), (⟨0.1, 0.2, 0.3, 0.4⟩), (⟨0.2, 0.2, 0.2, 0.5⟩)
(⟨0.3, 0.1, 0.1, 0.1⟩), (⟨0.2, 0.2, 0.3, 0.9⟩), (⟨0.1, 0.1, 0.1, 0.3⟩)
(⟨0.2, 0.1, 0.1, 0.3⟩), (⟨0.1, 0.03, 0.3, 0.9⟩), (⟨0.1, 0.1, 0.1, 0.3⟩)
(⟨0.4, 0.1, 0.1, 0.4⟩), (⟨0.1, 0.3, 0.3, 0.9⟩), (⟨0.1, 0.1, 0.4, 0.3⟩)


.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 14 of 25

4. Fermatean Double Valued Neutrosophic Soft Set Topological Space

The notion of a Fermatean Double Valued Neutrosophic Soft Topological Space (FDVNSTS)
is introduced in this section. The concepts of semi-open (s-open), pre-open (p-open), and b-
open sets are defined within the context of FDVNSTS. Among these generalized open sets, the
pre-open set is selected for further exploration, and several fundamental topological notions are
developed based on this definition. These include the closure, exterior, boundary, and interior
in FDVNSTS.

Definition 26. Let FDVNSS (X̃, É) be the family of all FDVNSSs, and let τ ⊂ FDV NSS(X̃, É),
then τ is a Fermatean double-valued neutrosophic soft topology (FDVNST) on X̃ if:

(i) 0(X̃,É), 1(X̃,É) ∈ τ ,

(ii) The union of any number of FDVNSSs in τ belongs to τ ,

(iii) The intersection of a finite number of FDVNSSs in τ belongs to τ .

Then, (X̃, τ, É) is said to be a Fermatean double-valued neutrosophic soft topological space
(FDVNSTS) over X̃.

Example 11. A universal set of objects: X = {x1, x2, x3} possibly individuals whose blood
pressure is being analyzed. A set of parameters: E = {e1, e2, e3} likely representing different
blood pressure characteristics, such as e1 : Systolic pressure, e2 : Diastolic pressure, e3 :
Pulse pressure or any relevant metric. Then

τFDV NSS =
{
0(X,É), 1(X,É), (F,E), (G,E), (H,E)

}
defines Fermatean double valued neutrosophic soft topology over X. Here, the Fermatean double
valued neutrosophic soft sets (F,E), (G,E), (H,E) over X are defined as following: Then their
FDVNSS is

(F,E) =

{
e1 = {⟨x1, 0.6, 0.7, 0.3, 0.2⟩, ⟨x2, 0.6, 0.7, 0.8, 0.8⟩, ⟨x3, 0.6, 0.7, 0.8, 0.1⟩} ,
e2 = {⟨x1, 0.7, 0.8, 0.8, 0.8⟩, ⟨x2, 0.7, 0.8, 0.8, 0.9⟩, ⟨x3, 0.7, 0.8, 0.8, 0.8⟩}

}

(G,E) =

{
e1 = {⟨x1, 0.5, 0.5, 0.8, 0.9⟩, ⟨x2, 0.4, 0.4, 0.9, 0.9⟩, ⟨x3, 0.2, 0.3, 0.9, 0.1⟩} ,
e2 = {⟨x1, 0.4, 0.5, 0.9, 0.8⟩, ⟨x2, 0.6, 0.5, 0.9, 0.9⟩, ⟨x3, 0.2, 0.7, 0.9, 0.8⟩}

}

(H,E) =

{
e1 = {⟨x1, 0.2, 0.3, 0.9, 0.9⟩, ⟨x2, 0.0, 0.0, 0.9, 0.9⟩, ⟨x3, 0.0, 0.0, 0.9, 0.9⟩} ,
e2 = {⟨x1, 0.4, 0.5, 0.9, 0.8⟩, ⟨x2, 0.6, 0.4, 0.9, 0.9⟩, ⟨x3, 0.2, 0.7, 0.9, 0.8⟩}

}

Let us define FDVNSS (F,E) as follows:

F (e1) = {⟨x1, 0.6, 0.7, 0.3, 0.2⟩, ⟨x2, 0.6, 0.7, 0.8, 0.8⟩, ⟨x3, 0.6, 0.7, 0.8, 0.1⟩} ;

F (e2) = {⟨x1, 0.7, 0.8, 0.8, 0.8⟩, ⟨x2, 0.7, 0.8, 0.8, 0.9⟩, ⟨x3, 0.7, 0.8, 0.8, 0.8⟩} ;

Let us define FDVNSS (G,E) as follows:

G(e1) = {⟨x1, 0.5, 0.5, 0.8, 0.9⟩, ⟨x2, 0.4, 0.4, 0.9, 0.9⟩, ⟨x3, 0.2, 0.3, 0.9, 0.1⟩} ;

G(e2) = {⟨x1, 0.4, 0.5, 0.9, 0.8⟩, ⟨x2, 0.6, 0.5, 0.9, 0.9⟩, ⟨x3, 0.2, 0.7, 0.9, 0.8⟩} ;


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 15 of 25

Let us define another FDVNSS (H,E) as follows:

H(e1) = {⟨x1, 0.2, 0.3, 0.9, 0.9⟩, ⟨x2, 0.0, 0.0, 0.9, 0.9⟩, ⟨x3, 0.0, 0.0, 0.9, 0.9⟩} ;

H(e2) = {⟨x1, 0.4, 0.5, 0.9, 0.8⟩, ⟨x2, 0.6, 0.4, 0.9, 0.9⟩, ⟨x3, 0.2, 0.7, 0.9, 0.8⟩} ;

Then their FDVNSS union is given as follows:

(F,E)∪̃(G,E) = (F,E) =

{
e1 = {⟨x1, 0.6, 0.7, 0.3, 0.2⟩, ⟨x2, 0.6, 0.7, 0.8, 0.8⟩, ⟨x3, 0.6, 0.7, 0.8, 0.1⟩} ,
e2 = {⟨x1, 0.7, 0.8, 0.8, 0.8⟩, ⟨x2, 0.7, 0.8, 0.8, 0.9⟩, ⟨x3, 0.7, 0.8, 0.8, 0.8⟩}

}

(F,E)∪̃(H,E) = (F,E) =

{
e1 = {⟨x1, 0.6, 0.7, 0.3, 0.2⟩, ⟨x2, 0.6, 0.7, 0.8, 0.8⟩, ⟨x3, 0.6, 0.7, 0.8, 0.1⟩} ,
e2 = {⟨x1, 0.7, 0.8, 0.8, 0.8⟩, ⟨x2, 0.7, 0.8, 0.8, 0.9⟩, ⟨x3, 0.7, 0.8, 0.8, 0.8⟩}

}

(H,E)∪̃(G,E) = (G,E) =

{
e1 = {⟨x1, 0.5, 0.5, 0.8, 0.9⟩, ⟨x2, 0.4, 0.4, 0.9, 0.9⟩, ⟨x3, 0.2, 0.3, 0.9, 0.1⟩} ,
e2 = {⟨x1, 0.4, 0.5, 0.9, 0.8⟩, ⟨x2, 0.6, 0.5, 0.9, 0.9⟩, ⟨x3, 0.2, 0.7, 0.9, 0.8⟩}

}
Then their FDVNSS intersection is given as follows:

(F,E)∩̃(G,E) = (G,E) =

{
e1 = {⟨x1, 0.5, 0.5, 0.8, 0.9⟩, ⟨x2, 0.4, 0.4, 0.9, 0.9⟩, ⟨x3, 0.2, 0.3, 0.9, 0.1⟩} ,
e2 = {⟨x1, 0.4, 0.5, 0.9, 0.8⟩, ⟨x2, 0.6, 0.5, 0.9, 0.9⟩, ⟨x3, 0.2, 0.7, 0.9, 0.8⟩}

}

(F,E)∩̃(H,E) = (H,E) =

{
e1 = {⟨x1, 0.2, 0.3, 0.9, 0.9⟩, ⟨x2, 0.0, 0.0, 0.9, 0.9⟩, ⟨x3, 0.0, 0.0, 0.9, 0.9⟩} ,
e2 = {⟨x1, 0.4, 0.5, 0.9, 0.8⟩, ⟨x2, 0.6, 0.4, 0.9, 0.9⟩, ⟨x3, 0.2, 0.7, 0.9, 0.8⟩}

}

(G,E)∩̃(H,E) = (H,E) =

{
e1 = {⟨x1, 0.2, 0.3, 0.9, 0.9⟩, ⟨x2, 0.0, 0.0, 0.9, 0.9⟩, ⟨x3, 0.0, 0.0, 0.9, 0.9⟩} ,
e2 = {⟨x1, 0.4, 0.5, 0.9, 0.8⟩, ⟨x2, 0.6, 0.4, 0.9, 0.9⟩, ⟨x3, 0.2, 0.7, 0.9, 0.8⟩}

}

Theorem 1. Let (X, τ1, É) and (X, τ2, É) be two FDVNSSTS. Then τ1∩̃τ2 is a FDVNSSTS
on X.

Proof. The first and third requirements are clear, and we move forward as follows for the
second condition. Let {(£̃i, É); i ∈ I} ∈ τ1∩̃τ2 then (£̃i, É) ∈ τ1, (£̃i, É) ∈ τ1 as τ1 and τ2 are
FDVNSTSs on X, then ∪̃i(£̃i, É) ∈ τ1 and ∪̃i(£̃i, É) ∈ τ2. So ∪̃i(£̃i, É) ∈ τ1∩̃τ2.

Remark 1. Let (X, τ1, É) and (X, τ2, É) be two FDVNSSTS. Then τ1∩̃τ2 is not necessarily to
be FDVNSSTS on X.

Definition 27. Let (X, τ1, É) be a FDVNSBTS over X, and (F̃ , É) be a FDVNSS then:

(i) (F̃ , É) is FDVNS semi-open if (F̃ , É) ⊆ NScl(NSint(F̃ , É)).

(ii) (F̃ , É) is FDVNS pre-open (p-open) if (F̃ , É) ⊆ NSint(NScl(F̃ , É)).

(iii) (F̃ , É) is FDVNS ∗b open if

(F̃ , É) ⊆ NScl(NSint(F̃ , É)) ∪NSint(NScl(F̃ , É)),

and FDVNS ∗b close if

(F̃ , É) ⊇ NScl(NSint(F̃ , É)) ∩NSint(NScl(F̃ , É)).


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 16 of 25

Definition 28. Let (X, τ1, É) be a FDVNSBTS over X, and (F̃ , É) be a FDVNS, then interior
of (F̃ , É), denoted by (F̃ , É)◦, is the union of all FDVNS p-open sets of (F̃ , É). Clearly, (F̃ , É)◦

is the largest FDVNS p-open set contained in (F̃ , É).

Definition 29. Let (X, τ1, É) be a FDVNSBTS, and (F̃ , É) be a FDVNS, the frontier of (F̃ , É)
denoted by Fr((F̃ , É)), is a FDVNS point s1

⋋
⟨φ1,φ2,φ3,φ4⟩ is frontier of (F̃ , É) if every FDVNS

p-open set comprising s1
⋋
⟨φ1,φ2,φ3,φ4⟩ comprises at least one point of (F̃ , É) and at least one

FDVNS point of (F̃ , É)c.

Definition 30. If (X, τ1, É) is a FDVNSBTS and (F̃ , É) is a FDVNS, then the exterior of
(F̃ , É), denoted by Ext((F̃ , É)), is a FDVNS point s1

⋋
⟨φ1,φ2,φ3,φ4⟩ is called exterior of (F̃ , É) if

s1
⋋
⟨φ1,φ2,φ3,φ4⟩ is in the interior of (F̃ , É)c, that is FDVNS p-open set (G̃, É) such that

s1
⋋
⟨φ1,φ2,φ3,φ4⟩ ∈ (G̃, É) ⊆ (F̃ , É)c.

Definition 31. If (X, τ1, É) and ( ˜⟨Y ⟩,F1, É) are FDVNSBTSs, and (f, ϕ) : (X, τ1, É) →
( ˜⟨Y ⟩,F1, É) is a FDVNS mapping, then (f, ϕ) is said to be a FDVNS p-close mapping if the
image (f, ϕ)(F̃ , É) of each FDVNS p-closed set (F̃ , É) over X̃ is a FDVNS p-closed set in ˜⟨Y ⟩.

Theorem 2. Let (X, τ1, É) be a FDVNSBTSs over X and (F̃ , É) is FDVNS subset. Then,
(F̃ , É) is a FDVNS p-open set if and only if (F̃ , É) = (F̃ , É)◦.

Proof. Let (F̃ , É) be a FDVNS p-open set. Then, the biggest FDVNS p-open set surrounded
by (F̃ , É) is equal to (F̃ , É). Hence, (F̃ , É) = (F̃ , É)◦.

Contrariwise, it is known that (F̃ , É)◦ is a HPNS p-open set, and if (F̃ , É) = (F̃ , É)◦, then
(F̃ , É) is a FDVNS p-open set.

Theorem 3. Let (X, τ1, É) be a FDVNSBTS over X, and let (F̃ , É) and (G̃, ∂) be FDVNS
subsets then

(i) [(F̃ , É)◦]◦ = (F̃ , É)◦,

(ii) (0(⟨X̃⟩,É))
◦ = 0(⟨X̃⟩,É) and (1(⟨X̃⟩,É))

◦ = 1(⟨X̃⟩,É),

(iii) (F̃ , É) ⊆ (G̃, É) ⇒ (F̃ , É)◦ ⊆ (G̃, É)◦,

(iv) [(F̃ , É) ∩ (G̃, É)]◦ = (F̃ , É)◦ ∩ (G̃, É)◦,

(v) (F̃ , É)◦ ∪ (G̃, É)◦ ⊆ [(F̃ , É) ∪ (G̃, É)]◦.

Proof.

(i) (F̃ , É)◦ = (G̃, É) then (G̃, ∂) ∈ τ̃ iff (G̃, É) = (F̃ , É)◦. So
[(F̃ , É)◦]◦ = (F̃ , É)◦

(ii) Since 0(⟨X̃⟩,É) and 1(⟨X̃⟩,É) are always FDVNS p-open sets, so

(0(⟨X̃⟩,É))
◦ = 0(⟨X̃⟩,É), and (1(⟨X̃⟩,É))

◦ = 1(⟨X̃⟩,É).

(iii) It is known that (F̃ , É)◦ ⊆ (F̃ , É) ⊆ (G̃, É) and (G̃, É)◦ ⊆ (G̃, É). Since (G̃, É)◦ is the
biggest FDVNS p-open set enclosed in (G̃, É) and so, (F̃ , É)◦ ⊆ (G̃, É)◦.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 17 of 25

(iv) Since (F̃ , É) ∩ (G̃, É) ⊆ (F̃ , É) and (F̃ , É) ∩ (G̃, É) ⊆ (G̃, É), then

[(F̃ , É) ∩ (G̃, É)]◦ ⊆ (F̃ , É)◦and[(F̃ , É) ∩ (G̃, É)]◦ ⊆ (G̃, É)◦.

so,
[(F̃ , É) ∩ (G̃, É)]◦ ⊆ (F̃ , É)◦ ∩ (G̃, É)◦.

On other way, since (F̃ , É)◦ ⊆ (F̃ , É) and (G̃, É)◦ ⊆ (G̃, É), then

(F̃ , É)◦ ∩ (G̃, É)◦ ⊆ (F̃ , É) ∩ (G̃, É).

also [(F̃ , É) ∩ (G̃, É)]◦ ⊆ (F̃ , É) ∩ (G̃, É) is the biggest FDVNS p-open set.
⇒ (F̃ , É)◦ ∩ (G̃, É)◦ ⊆ [(F̃ , É) ∩ (G̃, É)]◦

Thus,
[(F̃ , É) ∩ (G̃, É)]◦ = (F̃ , É)◦ ∩ (G̃, É)◦.

(v) Since (F̃ , É) ⊆ (F̃ , É) ∪ (G̃, É) and (G̃, É) ⊆ (F̃ , É) ∪ (G̃, É), then

(F̃ , É)◦ ⊆ [(F̃ , É) ∪ (G̃, É)]◦, and (G̃, É)◦ ⊆ [(G̃, É) ∪ (G̃, É)]◦.

⇒ (F̃ , É)◦ ∪ (G̃, É)◦ ⊆ [(F̃ , É) ∪ (G̃, É)]◦.

Theorem 4. Let (X, τ1, É) be a FDVNSBTS over X, and if (F̃ , É) is a FDVNS subset, then

(F̃ , É) is a FDVNS p-closed set if and only if (F̃ , É) = (F̃ , É).

Proof. Let (F̃ , É) is a FDVNS p-closer set, then:

(F̃ , É)d = (F̃ , É)

this implies that
(F̃ , É) ∪ (F̃ , É)d ∼= (F̃ , É)

⇒
(F̃ , É) ∼= (F̃ , É)

and conversely let (F̃ , É) ∼= (F̃ , É), this implies that

(F̃ , É) ∪ (F̃ , É)d ∼= (F̃ , É)

⇒
(F̃ , É)d = (F̃ , É)

this implies, (F̃ , É) is a FDVNS p-closed set.

Theorem 5. Let (X, τ1, É) be a FDVNSBTS over X, and let (F̃ , É) and (G̃, É) be HPNS
subsets then

(i) (F̃ , É) = (F̃ , É),

(ii) (0(⟨X̃⟩,É)) = 0(X̃,É) and (1(⟨X̃⟩,É)) = (1(⟨X̃⟩,É)),

(iii) (F̃ , É) ⊆ ⟨(G̃, É)⟩ ⇒ (F̃ , É) ⊆ (G̃, É),

(iv) [(F̃ , É) ∪ (G̃, É)] = (F̃ , É) ∪ (G̃, É),

(v) [(F̃ , É) ∩ (G̃, É)] ⊆ (F̃ , É) ∩ (G̃, É).


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 18 of 25

Proof.

(i) If (F̃ , É) = (G̃, É) then (G̃, É) is a FDVNS p-closed set.Hence, if (G̃, É) = (G̃, É). There-
fore

[(F̃ , É)] = (F̃ , É)

(ii) Since 0(⟨X̃⟩,É) and 1(⟨X̃⟩,É) are always FDVNS p-closure set so by the above result

(0(⟨X̃⟩,É)) = 0(⟨X̃⟩,É), and (1(⟨X̃⟩,É)) = 1(⟨X̃⟩,É).

(iii) Since (F̃ , É) ⊆ (F̃ , É) and (G̃, É) ⊆ (G̃, É), so (F̃ , É) ⊆ (G̃, É) ⊆ (G̃, É) .Since (G̃, É) is

the smallest FDVNS p-closure set covering in (F̃ , É) then (F̃ , É) ⊆ (G̃, É).

(iv) Since (F̃ , É) ⊆ (F̃ , É) ∪ (G̃, É) and (G̃, É) ⊆ 0(⟨X̃⟩,É) ∪ (G̃, É) then

(F̃ , É) ⊆ [(F̃ , É) ∪ (G̃, É)]and(G̃, É) ⊆ [(F̃ , É) ∪ (G̃, É)].

so,

(F̃ , É) ∪ (G̃, É) ⊆ [(F̃ , É) ∪ (G̃, É)].

Conversely, since (F̃ , É) ⊆ (F̃ , É) and (G̃, É) ⊆ (G̃, É), then

(F̃ , É) ∪ (G̃, É) ⊆ (F̃ , É) ∪ (G̃, É).

Besides, [(F̃ , É) ∪ (G̃, É)] is the smallest FDVNS p-closed set that enclosing (F̃ , É)∪(G̃, É)

therefore, [(F̃ , É) ∪ (G̃, É)] ⊆ (F̃ , É) ∪ (G̃, É).

Thus, [(F̃ , É) ∪ (G̃, É)] = (F̃ , É) ∪ (G̃, É).

(v) Since ⟨(0(⟨X̃⟩,É))⟩ ∩X ⊆ (F̃ , É) ∩X and [(F̃ , É) ∩X] is the smallest FDVNS p-closed

set that enclosing (F̃ , É) ∩X, then [(F̃ , É) ∩X] ⊆ (F̃ , É) ∩X.

Theorem 6. Let (X, τ1, É) be a FDVNSBTSs over X, and let (F̃ , É) be a FDVNSS then,

(i) [(F̃ , É)]c = [(F̃ , É)c]◦,

(ii) [(F̃ , É)◦]c = [(F̃ , É)c].

Proof.

(i)

(F̃ , É) = ∩{(G̃, É) ∈ (X, τ1, É)c : (G̃, É) ⊇ (F̃ , É)}

⇒ [(F̃ , É)]c =
[
∩{(G̃, É) ∈ (X, τ1, É)c : (G̃, É) ⊇ (F̃ , É)}

]c
= ∪{(G̃, É)c ∈ (X, τ1, É) : (G̃, É)c ⊆ (F̃ , É)c}
= [(F̃ , É)c]◦.

(ii)

(F̃ , É)◦ = ∪{(G̃, É) ∈ (X, τ1, É) : (G̃, É) ⊆ (F̃ , É)}

⇒ [(F̃ , É)◦]c =
[
∩{(G̃, É) ∈ (X, τ1, É) : (G̃, É) ⊆ (F̃ , É)}

]c
= ∩{(G̃, É)c ∈ (X, τ1, É)c : (G̃, É)c ⊇ (F̃ , É)c}

= [(F̃ , É)c].


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 19 of 25

Theorem 7. Let (X, τ1, É) be a FDVNSBTS over X. If (F̃ , É) and (G̃, É) are FDVNS subsets,
then:

(i) Ext((F̃ , É) ∪ (G̃, É)) = Ext((F̃ , É)) ∪ Ext((G̃, É)).

(ii) Ext((F̃ , É) ∩ (G̃, É)) ⊇ Ext((F̃ , É)) ∪ Ext((G̃, É)).

(iii) Fr((F̃ , É) ∪ (G̃, É)) ⊆ Fr(F̃ , É) ∪ Fr(G̃, É).

(iv) Fr((F̃ , É) ∩ (G̃, É)) ⊆ Fr(F̃ , É) ∪ Fr(G̃, É).

Proof.

(i) Since
Ext((F̃ , É) ∪ (G̃, É)) = (((F̃ , É) ∪ (ℑ̃, ∂))c)◦

= ((F̃ , É)c ∩ (G̃, É)c)◦

= ((F̃ , É)c)◦ ∩ ((G̃, É)c)◦

= Ext((F̃ , É)) ∩ Ext((G̃, É)).

(ii)
Ext((F̃ , É) ∩ (G̃, É)) = (((F̃ , É) ∩ (G̃, É))c)◦

= (((F̃ , É)c ∪ (G̃, É)c))◦

⊇ ((F̃ , É)c)◦ ∪ ((G̃, É)c)◦

= Ext((F̃ , É)) ∪ Ext((G̃, É)).

(iii)

Fr((F̃ , É) ∪ (G̃, É)) = (F̃ , É) ∪ (G̃, É) ∩ ((F̃ , É) ∪ (G̃, É))c

= ((F̃ , É) ∪ (G̃, É)) ∩ (F̃ , É)c ∩ (G̃, É)c ⊆ ((F̃ , É)) ∪ ((G̃, É)) ∩ (F̃ , É)c ∩ (G̃, É)c)

= {((F̃ , É)) ∪ ((G̃, É)) ∩ (F̃ , É)c ∩ (G̃, É)c}

= {((F̃ , É) ∩ (F̃ , É)c) ∪ (G̃, É) ∩ (F̃ , É)c} ∪ {((G̃, É)c) ∩ ((F̃ , É)c ∩ (G̃, É)c)}

= {Fr((F̃ , É)) ∩ ((G̃, É)c)} ∪ {Fr((G̃, É)) ∩ (F̃ , É)c}
⊆ Fr((F̃ , É)) ∪ Fr((G̃, É)).

(iv)

Fr((F̃ , É) ∩ (G̃, É)) = (F̃ , É) ∩ (G̃, É) ∩ ((F̃ , É) ∩ (G̃, É))c

⊆ (F̃ , É) ∩ ((F̃ , É)c) ∪ (G̃, É)c)

= {((F̃ , É)) ∪ ((G̃, É)) ∩ (F̃ , É)c} ∪ {((F̃ , É)) ∪ ((G̃, É)) ∩ (G̃, É)c}

= {Fr((F̃ , É)) ∩ ((G̃, É))} ∪ {((F̃ , É)) ∩ Fr((G̃, É))}
⊆ Fr((F̃ , É)) ∪ Fr((G̃, É)).


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 20 of 25

5. Conclusion and Future Work

In this paper, we introduced Fermatean Double-Valued Neutrosophic Soft Sets (FDVNSS),
an advanced mathematical framework that significantly enhances the modeling of uncertainty,
vagueness, and indeterminacy in complex environments. By incorporating both absolute and
relative degrees of truth and falsity, FDVNSS provides a more nuanced and expressive struc-
ture compared to traditional neutrosophic and soft set approaches. The formal definition of
FDVNSS was established under the Fermatean constraint, ensuring that the sum of the cubes
of membership values lies within a bounded range. We also defined key operations?such as
subset, null and absolute FDVNSS, complement, union, and intersection?and explored their al-
gebraic properties, demonstrating alignment with classical set-theoretic behavior. Applications
in medical decision-making were provided to illustrate the practical relevance of FDVNSS in
handling imprecise, incomplete, or conflicting information, highlighting its potential in intelli-
gent decision support systems. The notion of a Fermatean Double Valued Neutrosophic Soft
Topological Space (FDVNSTS) is introduced. The concepts of semi-open (s-open), pre-open
(p-open), and b-open sets are defined within the context of FDVNSTS. Among these gener-
alized open sets, the pre-open set is selected for further exploration, and several fundamental
topological notions are developed based on this definition. These include the closure, exterior,
boundary, and interior in FDVNSTS.
Future work will focus on extending the FDVNSS framework to dynamic and real-time sys-
tems, integrating it with machine learning and data-driven inference mechanisms. Additional
research will also explore optimization techniques, ranking algorithms, and hybrid models that
combine FDVNSS with other uncertainty handling tools, such as fuzzy cognitive maps and
rough sets. Moreover, domain-specific applications, particularly in healthcare, finance, and en-
vironmental risk assessment, will be pursued to further validate and refine the effectiveness of
this framework in real-world scenarios. Future research could explore the integration of Double-
Valued Neutrosophic Soft Topological Spaces (DV-NSTS) with machine learning techniques to
enhance uncertainty handling in complex data analysis. The Epanechnikov-Pareto distribution
[74] could be incorporated into DV-NSTS to develop a Neutrosophic Principal Component Anal-
ysis (N-PCA) that effectively manages indeterminacy during feature extraction, while concepts
from lacunary statistical convergence [77] could provide robust criteria for analyzing neutro-
sophic eigenvalues in dimensionality reduction. The ranked set sampling approach [79] might
be adapted for improved clustering of imbalanced data within DV-NSTS frameworks, potentially
leading to advanced neutrosophic versions of k-means or fuzzy c-means algorithms that account
for membership, indeterminacy, and non-membership as soft topological properties. Building
on the stability analysis of Volterra integral equations [75], new neutrosophic regression models
could be formulated where coefficients are represented as Double-Valued Neutrosophic Soft Sets
(DV-NSS) and topological continuity ensures model stability, with potential applications in fi-
nancial forecasting where market trends exhibit inherent vagueness. The reliability estimation
methods using Pareto and Benktander distributions [78, 82] could be extended to create novel
neutrosophic entropy measures for feature selection in machine learning pipelines, particularly
valuable in IoT networks with varying data reliability. The modified Midzuno scheme [76] might
be employed to optimize feature subset selection processes in high-dimensional neutrosophic
datasets. Furthermore, the ratio estimator modifications [80] could inspire the development
of specialized loss functions for Double-Valued Neutrosophic Neural Networks, while the frac-
tional differential equations framework [81] might enable more sophisticated time-series analysis
within DV-NSTS. Future work could also focus on implementing practical DV-NSS libraries in
programming environments like Python or R to facilitate real-world applications in healthcare
diagnostics, autonomous vehicle perception systems, and financial market prediction, where
conventional methods struggle with incomplete or contradictory information. These directions
would benefit from both theoretical advancements in defining topological convergence properties
for DV-NSTS and empirical validation through case studies across various domains.


R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6483 21 of 25

Future studies will investigate the integration of symbolic n-plithogenic intervals [83] with
dimensionality reduction methods such as PCA and t-SNE, as well as clustering algorithms like
K-means, to enhance pattern recognition and classification in complex fuzzy and neutrosophic
environments. In the same way, the algebraic framework built on CCIFS in bisemirings [84]
will be broadened with Fermatean double-valued neutrosophic sets. Merging this with PCA,
t-SNE, and K-means could improve visualization and analysis in situations involving uncertain,
high-dimensional data.

Acknowledgements

We extend our sincere gratitude to Prof. Dr. Arif Mehmood, principal investigator of our
collaborative project and leader of a 176-member research team. His exceptional leadership,
vision, and unwavering support have been instrumental to our success. His guidance not only
shaped our work but also inspired us to maintain professionalism and perseverance throughout
this journey. We are truly thankful for his invaluable mentorship.

The authors extend their appreciation to the Deanship of Scientific Research at Northern
Border University, Arar, KSA for funding this research work through the project number ”NBU-
FFR-2025-1153-05”.

Authors’ Contributions: All authors read and approved the final manuscript.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report
regarding the present study.

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