n Power Root Fuzzy Sets and Its Topology

Tareq M. Al-shami; Hariwan Z. Ibrahim; Abdelwaheb Mhemdi; Radwan Abu-Gdairi

doi:10.5391/IJFIS.2022.22.4.350

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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(4): 350-365

Published online December 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.4.350

n Power Root Fuzzy Sets and Its Topology

Tareq M. Al-shami¹, Hariwan Z. Ibrahim², Abdelwaheb Mhemdi³, and Radwan Abu-Gdairi⁴

¹Department of Mathematics, Sana’a University, Sana’a, Yemen Future University, Egypt
²Department of Mathematics, Faculty of Education, University of Zakho, Zakho, Kurdistan Region-Iraq
³Department of Mathematics, College of Sciences and Humanities in Aflaj, Prince Sattam bin Abdulaziz University, Riyadh, Saudi Arabia
⁴Department of Mathematics, Faculty of Science, Zarqa University, P.O. Box 13110 Zarqa, Jordan

Correspondence to :
Abdelwaheb Mhemdi (mhemdiabd@gmail.com)

Received: October 15, 2022; Revised: November 23, 2022; Accepted: December 12, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

One of the most useful expansions of fuzzy sets for coping with information uncertainties is the q-rung orthopair fuzzy sets. In such circumstances, in this article, we define a novel extension of fuzzy sets called n^th power root fuzzy set (briefly, nPR-fuzzy set) and elucidate their relationship with intuitionistic fuzzy sets, SR-fuzzy sets, CR-fuzzy sets and q-rung orthopair fuzzy sets. Then, we provide the necessary set of operations for nPR-fuzzy sets as well as study their various features. Furthermore, we familiarize the concept of nPR-fuzzy topology and investigate the basic aspects of this topology. In addition, we define separated nPR-fuzzy sets and then present the concept of disconnected nPR-fuzzy sets. Moreover, we study and characterize nPR-fuzzy continuous maps in great depth. Finally, we establish 𝕿₀ and 𝕿₁ in nPR-fuzzy topologies and discover the links between them.

Keywords: nPR-fuzzy sets, Operations, nPR-fuzzy topology, Separated nPR-FSs, Connected nPR-FS, nPR-fuzzy continuous maps, 𝕿₀, 𝕿₁

1. Introduction

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Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

Zadeh [39] established the concept of fuzzy sets to deal with imprecise data, and various studies on generalizations of the concept of fuzzy set were undertaken after that. From health sciences to computer science, from physical sciences to arts, and from engineering and humanities to life sciences, it is seen to have numerous applications connected to fuzzy set theory in both theoretical and practical investigations. As examples of theoretical studies, Kirisci [28] discussed triangular fuzzy numbers, and Zarasiz studied algebraic fuzzy structure [40] and fuzzy matrices [41]. Pawlak [32] was the first one who proposed the concept of rough sets, and Molodtsov [31] introduced the concept of soft sets as a generic mathematical tool for dealing with ambiguous objects. These two types of non-crisp sets contribute to coping with uncertain environments in different fields; see, for example [5, 6, 8–10, 12–15]. The merging between fuzzy sets and some doubtfulness techniques such as rough sets and soft sets have been studied; in this regard, Ahmad and Kharal [1] combined fuzzy sets with soft sets, then Cağman et al. [19] showed how fuzzy soft sets applied to optimal choices. Alcantud [2] explored the relationships between the topological structures in fuzzy and soft settings. Al-shami et al. [11] presented generalized umbral for orthopair fuzzy soft sets called (a, b)-Fuzzy soft sets. Investigation of covering-based rough set from fuzzy view was the main goal of some manuscripts; see, for example, [18, 37, 38]. In fuzzy algebra, Gulzar with his coauthors debated some types of fuzzy subgroups [25] as well as displayed interesting applications of complex IFSs in group theory [26]. New hybridizations of fuzzy set theory with some types of soft sets were established in [3, 33].

Atanassov [17] defined intuitionistic fuzzy sets as one of the intriguing generalizations of fuzzy sets with excellent application. Applications of intuitionistic fuzzy sets can be found in a variety of domains, including optimization issues, decisionmaking, and medical diagnostics [22–24]. However, in many circumstances, the decision maker may assign degrees of membership and non-membership to a given attribute such that their aggregate is larger than one. As a result, Yager [30] introduced the Pythagorean fuzzy set notion, which is a generalization of intuitionistic fuzzy sets and a more effective tool for solving uncertain issues. Pythagorean fuzzy sets are a type of fuzzy set that can be used to characterize uncertain data more effectively and precisely than intuitionistic fuzzy sets. Senapati et al. [35] explored Fermatean fuzzy sets and introduced fundamental procedures on them. Another form of generalized Pythagorean fuzzy set named (3,2)-Fuzzy sets and SR-Fuzzy sets was defined by [27] and [4], respectively. Yager [29] developed the concept of the q-rung orthopair fuzzy sets and because of their greater range of depicting membership grades, they are more likely to be used in uncertain situations than other types of fuzzy sets. Al-shami et al. [4] presented another kind of fuzzy set called SR-Fuzzy set and discussed their properties in details. Saliha et al. [34] proposed CR-fuzzy sets and found the fundamental set of operations for them. Recently, Al-shami [7] has defined a new class of IFSs called “(2,1)-Fuzzy sets” and investigated main properties. He also has generated new weighted aggregated operators using this class and applied to multi-criteria decision-making problems.

One of the interesting themes is studying topology with respect to fuzzy sets. This line of research began by Chang [20] in 1968. He explored the main properties of some topological concepts and ideas via fuzzy set environment. In 1997, Çoker [21] defined the concept of an intuitionistic fuzzy topological space. He defined the main topological concepts via this space such as continuity, compactness, connectedness, and separation axioms. Recently, Turkarslan et al. [36] have familiarized a q-rung orthopair fuzzy topological space and studied some its properties. Recently, Ameen et al. [16] have initiated the concept of infra-fuzzy topological structures and explored main features.

In this paper, we propose a new environment of fuzzy sets so-called “n^th power root fuzzy set” and compare it with the other types of fuzzy sets. The fundamental benefit of n^th power root fuzzy sets is that they can be used in a wide range of decision-making situations. Then, we propose the set of operations for the n^th power root fuzzy set and explore their main features. Also, the concept of topology for n^th power root fuzzy sets is investigated. Thereafter, we describe separated and disconnected n^th power root fuzzy sets under this topology, and also we study n^th power root fuzzy continuous maps in details. Ultimately, we study two types of separation axioms in n^th power root fuzzy topology.

2. nPR-Fuzzy Sets

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Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

In this section, we study the notion of n^th power root fuzzy sets, and establish main properties. For computations, we use only three decimal places in whole paper.

Definition 2.1

Let ℕ be the set of natural numbers and A be a universal set. Then, the n^th power root fuzzy set (briefly, nPR-FS) ξ, which is a set of ordered pairs over A, is defined as following:

ξ={〈w,ϖξ(w),ωξ(w)〉:w∈A},

where ϖξ:A→[0,1] is the degree of membership and ωξ:A→[0,1] is the degree of non-membership functions such that

0≤(ϖξ(w))n+ωξ(w)n≤1,

for each w∈A and n ∈ ℕ \ {1}.

Then, there is a degree of indeterminacy of w∈A to ξ defined by

πξ(w)=1-[(ϖξ(w))n+ωξ(w)n].

It is clear that (ϖξ(w))n+ωξ(w)n+πξ(w)=1. Otherwise, πξ(w)=0 whenever (ϖξ(w))n+ωξ(w)n=1.

For the sake of simplicity, we shall mention the symbol ξ = (ω̄_ξ, ω_ξ) for the nPR-FS ξ={〈w,ϖξ(w),ωξ(w)〉:w∈A}. We noticed that 2PR-fuzzy is called SR-FS in [?], and 3PR-fuzzy is called CR-FS in [34]. If 0≤ϖξ(w)+ωξ(w)≤1 (resp. 0≤ϖξ(w)n+ωξ(w)n≤1), then ξ = (ω̄_ξ, ω_ξ) is called intuitionistic fuzzy set (IFS) [17](resp. n-rung orthopair fuzzy set (n-ROFS) [29]).

The spaces of some kinds of nPR-fuzzy membership grades are displayed in Figure 1.

Remark 2.2

From Figure 2, we get that

the space of 4-rung orthopair fuzzy membership grades is larger than the space of 4PR-fuzzy membership grades.
ξ = (ω̄_ξ ≈ 0.724, ω_ξ ≈ 0.276) is a point of intersection between 4PR-fuzzy and intuitionistic fuzzy sets.
for ω̄_ξ ∈ (0, 0.724) and ω_ξ ∈ (0.276, 1) the space of 4PR-fuzzy membership grades starts to be larger than the space of intuitionistic membership grades.
for ω̄_ξ ∈ (0.724, 1) and ω_ξ ∈ (0, 0.276) the space of 4PR-fuzzy membership grades starts to be smaller than the space of intuitionistic membership grades.
ξ=(ϖξ=5-12,ωξ=7-352) is a point of intersection between 4PR-fuzzy and SR-fuzzy.
for ϖξ∈(0,5-12) and ωξ∈(7-352,1) the space of 4PR-fuzzy membership grades starts to be larger than the space of SR-fuzzy membership grades.
for ϖξ∈(5-12,1) and ωξ∈(0,7-352) the space of 4PR-fuzzy membership grades starts to be smaller than the space of SR-fuzzy membership grades.
ξ = (ω̄_ξ ≈ 0.819, ω_ξ ≈ 0.091) is a point of intersection between 4PR-fuzzy and CR-fuzzy sets.
for ω̄_ξ ∈ (0, 0.819) and ω_ξ ∈ (0.091, 1) the space of 4PR-fuzzy membership grades starts to be larger than the space of CR-fuzzy membership grades.
for ω̄_ξ ∈ (0.819, 1) and ω_ξ ∈ (0, 0.091) the space of 4PR-fuzzy membership grades starts to be smaller than the space of CR-fuzzy membership grades.

Remark 2.3

It is clear that for any nPR-FS ξ = (ω̄_ξ, ω_ξ) we have 0≤ϖξn+ωξn≤ϖξn+ωξn≤1, then ξ is an n-rung orthopair fuzzy set. Therefore, every nPR-fuzzy set is an n-rung orthopair fuzzy set.

Definition 2.4

Let ξ = (ω̄_ξ, ω_ξ), ξ₁ = (ω̄_ξ_₁, ω_ξ_₁) and ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) be three nPR-fuzzy sets (nPR-FSs), then

ξ1∩ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2}).
ξ1∪ξ2=(max{ϖξ1,ϖξ2},min{ωξ1,ωξ2}).
ξc=(ωξn2,(ϖξ)n2).

Example 2.5

Suppose that ξ₁ = (0.6, 0.5) and ξ₂ = (0.5, 0.6) are both 4PR-FSs for A={w}. Then,

ξ1∩ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})=(min{0.6,0.5},max{0.5,0.6})=(0.5,0.6).
ξ1∪ξ2=(max{ϖξ1,ϖξ2},min{ωξ1,ωξ2})=(max{0.6,0.5},min{0.5,0.6})=(0.6,0.5).
ξ1c≈(0.958,0.000).

Theorem 2.6

If ξ = (ω̄_ξ, ω_ξ) is a nPR-FS, then ξc is also a nPR-FS and (ξc)c=ξ.

Proof

Since 0≤ϖξn≤1,0≤ωξn≤1 and 0≤(ϖξ)n+ωξn≤1, then

0≤(ωξn2)n+(ϖξ)n2n=(ϖξ)n+ωξn≤1,

and hence

0≤(ωξn2)n+(ϖξ)n2n≤1.

Thus, ξc is a nPR-FS and it is obvious that (ξc)c=(ωξn2,(ϖξ)n2)c=(ϖξ,ωξ).

Theorem 2.7

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁) and ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) be two nPR-FSs, then

ξ₁ ∩ ξ₂ = ξ₂ ∩ ξ₁.
ξ₁ ∪ ξ₂ = ξ₂ ∪ ξ₁.

Proof

From Definition 2.4, we can obtain:

ξ1∩ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})=(min{ϖξ2,ϖξ1},max{ωξ2,ωξ1})=ξ2∩ξ1.
The proof is similar to (1).

Theorem 2.8

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁) and ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) be two nPR-FSs, then

(ξ₁ ∩ ξ₂) ∪ ξ₂ = ξ₂.
(ξ₁ ∪ ξ₂) ∩ ξ₂ = ξ₂.

Proof

From Definition 2.4, we can obtain:

(ξ1∩ξ2)∪ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})∪(ϖξ2,ωξ2)=(max{min{ϖξ1,ϖξ2},ϖξ2},min{max{ωξ1,ωξ2},ωξ2})=(ϖξ2,ωξ2)=ξ2.
(ξ1∪ξ2)∩ξ2=(max{ϖξ1,ϖξ2},min{ωξ1,ωξ2})∩(ϖξ2,ωξ2)=(min{max{ϖξ1,ϖξ2},ϖξ2},max{min{ωξ1,ωξ2},ωξ2})=(ϖξ2,ωξ2)=ξ2.

Theorem 2.9

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁), ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) and ξ₃ = (ω̄_ξ_₃, ω_ξ_₃) be three nPR-FSs, then

ξ₁ ∩ (ξ₂ ∩ ξ₃) = (ξ₁ ∩ ξ₂) ∩ ξ₃.
ξ₁ ∪ (ξ₂ ∪ ξ₃) = (ξ₁ ∪ ξ₂) ∪ ξ₃.

Proof

For the three nPR-FSs ξ₁, ξ₂ and ξ₃, according to Definition 2.4, we can obtain

1. ξ1∩(ξ2∩ξ3)=(ϖξ1,ωξ1)∩(min{ϖξ2,ϖξ3},max{ωξ2,ωξ3})=(min{ϖξ1,min{ϖξ2,ϖξ3}},max{ωξ1,max{ωξ2,ωξ3}})=(min{min{ϖξ1,ϖξ2},ϖξ3},max{max{ωξ1,ωξ2},ωξ3})=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})∩(ϖξ3,ωξ3)=(ξ1∩ξ2)∩ξ3.
2. The proof is similar to (1).

Theorem 2.10

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁), ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) and ξ₃ = (ω̄_ξ_₃, ω_ξ_₃) be three nPR-FSs, then

(ξ₁ ∪ ξ₂) ∩ ξ₃ = (ξ₁ ∩ ξ₃) ∪ (ξ₂ ∩ ξ₃).
(ξ₁ ∩ ξ₂) ∪ ξ₃ = (ξ₁ ∪ ξ₃) ∩ (ξ₂ ∪ ξ₃).

Proof

It can be proved following similar arguments given to the proof of its counterpart in [7].

Theorem 2.11

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁) and ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) be two nPR-FSs, then

(ξ1∩ξ2)c=ξ1c∪ξ2c.
(ξ1∪ξ2)c=ξ1c∩ξ2c.

Proof

For the two nPR-FSs ξ₁ and ξ₂, according to Definition 2.4, we can obtain:

1. (ξ1∩ξ2)c=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})c=(max{ωξ1n2,ωξ2n2},min{(ϖξ1)n2,(ϖξ2)n2})=(ωξ1n2,(ϖξ1)n2)∪(ωξ2n2,(ϖξ2)n2)=ξ1c∪ξ2c.
2. The proof is similar to (1).

Definition 2.12

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁) and ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) be two nPR-FSs, then

ξ₁ = ξ₂ if and only if ω̄_ξ_₁ = ω̄_ξ_₂ and ω_ξ_₁ = ω_ξ_₂.
ξ₁ ≥ ξ₂ if and only if ω̄_ξ_₁ ≥ ω̄_ξ_₂ and ω_ξ_₁ ≤ ω_ξ_₂.
ξ₂ ⊂ ξ₁ or ξ₁ ⊃ ξ₂ if ξ₁ ≥ ξ₂.

Example 2.13

Let A={w1,w2}, then

ξ₁ = ξ₂ for two 4PR-FSs ξ1={〈w1,0.62,0.51〉,〈w2,0.45,0.76〉} and ξ2={〈w1,0.62,0.51〉,〈w2,0.45,0.76〉}.
ξ₂ ≤ ξ₁ and ξ₂ ⊂ ξ₁ for two 4PR-FSs ξ1={〈w1,0.62,0.51〉,〈w2,0.45,0.76〉} and ξ2={〈w1,0.61,0.52〉,〈w2,0.44,0.77〉}.

3. Some Topological Concepts via nPR-Fuzzy Topology

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Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

In this section, we define a new class of fuzzy topology using nPR-fuzzy subsets. We initiate the main concepts via this class such as connectedness, continuity and separation axioms. We describe these concepts and illustrate the relations between them with aid of counterexamples.

3.1 nPR-Fuzzy Topology

Definition 3.1

Let ℑ be a family of nPR-fuzzy subsets of a non-empty set A. If

1A,0A∈ℑ where 1A=(1,0) and0A=(0,1),
ξ1∩ξ2∈ℑ, for any ξ₁, ξ2∈ℑ,
∪i∈Iξi∈ℑ, for any {ξi}i∈N⊂ℑ,

then ℑ is called an nPR-fuzzy topology (briefly, nPR-FT) on A and (A,ℑ) is an nPR-fuzzy topological space (briefly, nPR-FTS). Each member of ℑ is called an open nPR-FS. We call ξ a closed nPR-FS if its complement is an open nPR-FS.

Now, we give an example of 7PR-FTS.

Example 3.2

Let A={w1,w2} and ξ_i be 7PR-fuzzy subsets of A such that ω̄_ξ_{_i} and ω_ξ_{_i} are corresponding membership and non-membership functions of ξ_i for each i = 1, 2, 3, 4, 5. If

ϖξ1(w1)=0.52, ωξ1(w1)=0.72,ϖξ1(w2)=0.61, ωξ1(w2)=0.72,ϖξ2(w1)=0.55, ωξ2(w1)=0.68,ϖξ2(w2)=0.62, ωξ2(w2)=0.73,ϖξ3(w1)=0.51, ωξ3(w1)=0.73,ϖξ3(w2)=0.59, ωξ3(w2)=0.74,ϖξ4(w1)=0.55, ωξ4(w1)=0.68,ϖξ4(w2)=0.62, ωξ4(w2)=0.72,ϖξ5(w1)=0.52, ωξ5(w1)=0.72,ϖξ5(w2)=0.61, ωξ5(w2)=0.73,

then the family of 7PR-fuzzy sets ℑ={1A,0A,ξ1,ξ2,ξ3,ξ4,ξ5} is 7PR-FT on A.

Remark 3.3

Every nPR-FT is an n-rung orthopair fuzzy topology because every nPR-fuzzy subset of a set can be considered as an n-rung orthopair fuzzy subset. The next example elaborates that n-rung orthopair fuzzy topological space need not be an nPR-FTS.

Example 3.4

Let A={w1,w2} and ξ_i be 2-rung orthopair fuzzy subsets of A such that ω̄_ξ_{_i} and ω_ξ_{_i} are corresponding membership and non-membership functions of ξ_i for each i = 1, 2. If

ϖξ1(w1)=0.96, ωξ1(w1)=0.25,ϖξ1(w2)=0.92, ωξ1(w2)=0.35,ϖξ2(w1)=0.95, ωξ2(w1)=0.25,ϖξ2(w2)=0.91, ωξ2(w2)=0.35,

then ℑ={1A,0A,ξ1,ξ2} is 2-rung orthopair fuzzy topology but ℑ is not 2PR-FT.

Remark 3.5

We call ℑ={1A,0A} the indiscrete nPR-FT on A.
If ℑ contains all nPR-fuzzy subsets, then we call ℑ the discrete nPR-FT on A.

Definition 3.6

Let ξ₁ and ξ₂ be two nPR-fuzzy subsets in an nPR-FTS (A,ℑ). Then, ξ₂ is called a neighbourhood of ξ₁ if there exists an open nPR-fuzzy subset ξ₃ such that ξ₁ ⊂ ξ₃ ⊂ ξ₂.

Theorem 3.7

An nPR-fuzzy set ξ₁ is open in an nPR-FTS (A,ℑ) if and only if it contains a neighbourhood of its each subset.

Proof

Let ξ₁ be an open nPR-fuzzy set and ξ₂ be an nPR-fuzzy set such that ξ₂ ⊂ ξ₁. Since ξ₂ ⊂ ξ₁ ⊂ ξ₁ and ξ₁ is an open nPR-fuzzy set, then ξ₁ is a neighborhood of ξ₂. On the other hand, assume that ξ₁ is a neighborhood of its each subset. From Definition 3.6, for arbitrary ξ₂ ⊂ ξ₁ there exists an open nPR-fuzzy set ξ_ξ_₂ such that ξ₂ ⊂ ξ_ξ_₂ ⊂ ξ₁. Thus, we have ξ₁ ⊂ ∪_ξ_₂⊂_ξ_₁ξ_ξ_₂ and since for all ξ₂ ⊂ ξ₁ and ξ_ξ_₂ ⊂ ξ₁, we get ∪_ξ_₂⊂_ξ_₁ξ_ξ_₂ ⊂ ξ₁. Consequently, we obtain ξ₁ = ∪_ξ_₂⊂_ξ_₁ξ_ξ_₂ which implies ξ₁ is an open nPR-fuzzy set.

3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators

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Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

Definition 3.8

Let (A,ℑ) be an nPR-FTS and ξ = (ω̄_ξ, ω_ξ) be an nPR-FS in A. Then, the nPR-fuzzy interior and nPR-fuzzy closure of ξ are, respectively, defined by

int(ξ)=∪{ξ2:ξ2 is open nPR-FS in A and ξ2⊂ξ}.
cl(ξ)=∩{ξ1:ξ1 is closed nPR-FS in A and ξ⊂ξ1}.

Remark 3.9

Let ( A,ℑ) be an nPR-FTS and ξ be any nPR-FS in A. Then,

int(ξ) is an open nPR-FS.
cl(ξ) is a closed nPR-FS.
int(1A)=cl(1A)=1A and int(0A)=cl(0A)=0A.

Example 3.10

Consider the 7PR-FTS (A,ℑ) in Example 3.2. If ξ={〈w1,0.53,0.73〉,〈w2,0.63,0.74〉}, then int(ξ)=ξ3 and cl(ξ)=1A.

Theorem 3.11

Let (A,ℑ) be an nPR-FTS and ξ₁, ξ₂ be two nPR-FSs in A. Then,

int(ξ1)⊂ξ1⊂cl(ξ1).
int(ξ1)⊂int(ξ2) and cl(ξ1)⊂cl(ξ2) if ξ₁ ⊂ ξ₂.
ξ₁ is an open nPR-FS if and only if ξ1=int(ξ1).
ξ₁ is a closed nPR-FS if and only if ξ1=cl(ξ1).

Proof

(1) and (2) are obvious.

(3) and (4) follow from Definition 3.8.

Theorem 3.12

Let (A,ℑ) be an nPR-FTS and ξ₁, ξ₂ be two nPR-FSs in A. Then,

int(ξ1)∪int(ξ2)⊂int(ξ1∪ξ2).
cl(ξ1∩ξ2)⊂cl(ξ1)∩cl(ξ2).
int(ξ1∩ξ2)=int(ξ1)∩int(ξ2).
cl(ξ1)∪cl(ξ2)=cl(ξ1∪ξ2).

Proof

Since ξ₁ ⊂ ξ₁ ∪ ξ₂ and ξ₂ ⊂ ξ₁ ∪ ξ₂, then by Theorem 3.11 (2), we have int(ξ1)⊂int(ξ1∪ξ2) and int(ξ2)⊂int(ξ1∪ξ2). Therefore, int(ξ1)∪int(ξ2)⊂int(ξ1∪ξ2)∪int(ξ1∪ξ2)=int(ξ1∪ξ2).
Since ξ₁∩ξ₂ ⊂ ξ₁ and ξ₁∩ξ₂ ⊂ ξ₂, then by Theorem 3.11 (2), we have cl(ξ1∩ξ2)⊂cl(ξ1) and cl(ξ1∩ξ2)⊂cl(ξ2). Therefore, cl(ξ1∩ξ2)=cl(ξ1∩ξ2)∩cl(ξ1∩ξ2)⊂cl(ξ1)∩cl(ξ2).
Since int(ξ1∩ξ2)⊂int(ξ1) and int(ξ1∩ξ2)⊂int(ξ2), we obtain int(ξ1∩ξ2)⊂int(ξ1)∩int(ξ2). On the other hand, from the facts int(ξ1)⊂ξ1 and int(ξ2)⊂ξ2 we have int(ξ1)∩int(ξ2)⊂ξ1∩ξ2 and int(ξ1)∩int(ξ2)∈ℑ; we see that int(ξ1)∩int(ξ2)⊂int(ξ1∩ξ2), and hence int(ξ1∩ξ2)=int(ξ1)∩int(ξ2).
Can be proved similar to (3).

Theorem 3.13

Let (A,ℑ) be an nPR-FTS and ξ be nPR-FS in A. Then,

cl(ξc)=int(ξ)c.
cl(ξc)c=int(ξ).
int(ξc)c=cl(ξ).
int(ξc)=cl(ξ)c.

Proof

Let ξ={〈w,ϖξ(w),ωξ(w)〉:w∈A} and suppose that the family of open nPR-fuzzy sets contained in ξ are indexed by the family {〈w,ϖξi(w),ωξi(w)〉:i∈J}. Then, int(ξ)={〈w,⋁ϖξi(w),⋀ωξi(w)〉} and hence

int(ξ)c={〈w,⋀ωξi(w)n2,(⋁ϖξi(w))n2〉}. Since ξc={〈w,ωξ(w)n2,(ϖξ(w))n2〉} and ω̄_ξ_{_i} ≤ ω̄_ξ, ω_ξ_{_i} ≥ ω_ξ for each i ∈ J, so we obtain that 〈w,ωξi(w)n2,(ϖξi(w))n2〉 is the family of closed nPR-fuzzy sets containing ξc, that is,
cl(ξc)={〈w,⋀ωξi(w)n2,(⋁ϖξi(w))n2〉}.
Thus, cl(ξc)=int(ξ)c.
{〈w,ωξi(w)n2,(ϖξi(w))n2〉} is the family of closed nPR-fuzzy sets containing ξc, so,
cl(ξc)={〈m,⋀ωξi(m)n2,(⋁ϖξi(w))n2〉}
and
cl(ξc)c={〈m, (⋁ϖξi(m))n2n2,(⋀ωξi(m)n2)n2〉}={〈m,⋁ϖξi(m),⋀ωξi(m)〉}=int(ξ).
from (1), we can getcl(ξ)=int(ξc)c.
from (2), we can get cl(ξ)c=int(ξc).

3.3 nPR-Fuzzy Connectedness

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Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

Definition 3.14

Two nPR-FSs ξ₁ and ξ₂ of an nPR-FTS (A,ℑ) are called separated nPR-FSs if cl(ξ1)⋀ξ2=0A and ξ1⋁cl(ξ2)=0A.

If ξ₁ and ξ₂ are separated nPR-FSs, then ξ1⋀ξ2=0A. The converse may not be true in general, as it is shown in the following example.

Example 3.15

Consider the 7PR-FTS (A,ℑ) in Example 3.2. If ξ1={〈w1,0,1〉,〈w2,0.64,0.71〉} and ξ2={〈w1,0.52,0.87〉,〈w2,0,1〉}, then ξ1∩ξ2=0A but ξ₁ and ξ₂ are not separated nPR-FSs.

Corollary 3.16

If ξ₁ and ξ₂ are two nPR-FSs in an nPR-FTS (A,ℑ) such that ξ1∩ξ2=0A, then ξ1⊂ξ2c.

Proof

Fromξ1∩ξ2=0A, we get min{ϖξ1,ϖξ2}=0 and max{ωξ1,ωξ2}=1. If the inequality ϖξ1≤ωξ2n2 were not true, then there would exist wo∈A such that ϖξ1(wo)>ωξ2(wo)n2, then, since ϖξ1(wo)>0 and hence, ωξ1(wo)<1, we obtain ωξ2(wo)=1 which is an obvious contradiction. Similarly, if the inequality ω_ξ_₁ ≥ (ω̄_ξ_₂)^n² were not true, then there would exist woo∈A such that ωξ1(woo)<(ϖξ2(woo))n2, then, since ϖξ2(woo)>0 and hence, ωξ2(woo)<1, we obtain ωξ1(woo)=1which is an obvious contradiction. So, we get ω_ξ_₁ ≥ (ω̄_ξ_₂ )^n², and thus ξ1⊂ξ2c.

Theorem 3.17

If ξ1≠0A and ξ2≠0A are any two nPR-FSs in an nPR-FTS (A,ℑ), then the following statements are true:

If ξ₁ and ξ₂ are separated nPR-FSs, ξ₁₁ ⊂ ξ₁ and ξ₂₁ ⊂ ξ₂, then ξ₁₁ and ξ₂₁ are also separated nPR-FSs.
If ξ1∩ξ2=0A such that ξ₁ and ξ₂ are both closed nPR-FSs, then ξ₁ and ξ₂ are separated nPR-FSs.

Proof

Since ξ₁₁ ⊂ ξ₁, then cl(ξ11)⊂cl(ξ1). Then, ξ2∩cl(ξ1)=0A implies ξ21∩cl(ξ1)=0A and so ξ21∩cl(ξ11)=0A. Similarly ξ11∩cl(ξ21)=0A. Hence, ξ₁₁ and ξ₂₁ are separated nPR-FSs.
Since ξ1=cl(ξ1),ξ2=cl(ξ2) and ξ1∩ξ2=0A, then cl(ξ1)∩ξ2=0A and cl(ξ2)∩ξ1=0A. Hence, ξ₁ and ξ₂ are separated nPR-FSs.

Remark 3.18

If ξ1∩ξ2=0A such that ξ₁ and ξ₂ are both open nPR-FSs in an nPR-FTS (A,ℑ), then ξ₁ and ξ₂ are not separated nPR-FSs in general.

Example 3.19

Let A={w1,w2} and ξ_i be 4PR-fuzzy subsets of A such that ω̄_ξ_{_i} and ω_ξ_{_i} are corresponding membership and non-membership functions of ξ_i for each i = 1, 2, 3. If

ϖξ1(w1)=0, ωξ1(w1)=1,ϖξ1(w2)=0.25, ωξ1(w2)=0.86,ϖξ2(w1)=0.26, ωξ2(w1)=0.85,ϖξ2(w2)=0, ωξ2(w2)=1,ϖξ3(w1)=0.26, ωξ3(w1)=0.85,ϖξ3(w2)=0.25, ωξ3(w2)=0.86,

then ℑ={1A,0A,ξ1,ξ2,ξ3} is 4PR-FT and ξ1∩ξ2=0A, but ξ₁ and ξ₂ are not separated 4PR-FSs.

Corollary 3.20

If ξ₁ and ξ₂ are separated nPR-FSs in an nPR-FTS (A,ℑ), then there exist two open nPR-FSs ξ₃ and ξ₄ in (A,ℑ) such that ξ₁ ⊂ ξ₃ and ξ₂ ⊂ ξ₄.

Proof

Let ξ₁ and ξ₂ be separated nPR-FSs. Set ξ4=cl(ξ1)c and ξ3=cl(ξ2)c, then ξ₃ and ξ₄ are open nPR-FSs such that ξ₁ ⊂ ξ₃ and ξ₂ ⊂ ξ₄.

Theorem 3.21

Let the following statements in an nPR-FTS (A,ℑ),

0A and 1A are the only open nPR-FSs and closed nPR-FSs in (A,ℑ).
there are no two open nPR-FSs ξ₁ and ξ₂ such that ξ1≠0A,ξ2≠0A,ξ1∩ξ2=0A and 1A=ξ1∪ξ2.
there are no two closed nPR-FSs ξ₁ and ξ₂ such that ξ1≠0A,ξ2≠0A,ξ1∩ξ2=0A and 1A=ξ1∪ξ2.
there are no two separated nPR-FSs ξ₁ and ξ₂ such that ξ1≠0A,ξ2≠0A and 1A≠ξ1∪ξ2.

Then, (1) ⇒ (2) ⇔ (3) ⇔ (4) hold.

Proof

(1) ⇒ (2): Suppose (2) is false and that 1A=ξ1∪ξ2, where ξ₁ and ξ₂ are two open nPR-FSs such that ξ1≠0A,ξ2≠0A and ξ1∩ξ2=0A. By Theorems 2.6, 2.11 (2) and Corollary 3.16, we have ξ1c=ξ2 and hence ξ₂ is open nPR-FS and closed nPR-FS in (A,ℑ) such that 1A≠ξ2≠0A, which shows that (1) is false.

(2) ⇔ (3): This is clear.

(3) ⇒ (4): If (4) is false, then 1A=ξ1∪ξ2, where ξ₁ and ξ₂ are two separated nPR-FSs such that ξ1≠0A and ξ2≠0A. By Theorems 2.11 (2), we have 0A=1Ac=(ξ1∪ξ2)c=ξ1c∩ξ2c. By Corollary 3.16 and Theorem 2.6, we get ξ1c⊂(ξ2c)c=ξ2. Since cl(ξ2)∩ξ1=0A, we conclude that cl(ξ2)⊂ξ2 so ξ₂ is closed nPR-FS. Similarly, ξ₁ must be closed nPR-FS. Therefore, (3) is false.

(4) ⇒ (3): If (3) is false, then 1A=ξ1∪ξ2, where ξ₁ and ξ₂ are two closed nPR-FSs such that ξ1≠0A,ξ2≠0A and ξ1∩ξ2=0A. Then, ξ₁ and ξ₂ are separated nPR-FSs by Theorem 3.11 (4). Therefore, (4) is false.

Definition 3.22

Let (A,ℑ) be an nPR-FTS and ξ be an nPR-FS in A, then ξ is called disconnected nPR-FS if there are two separated nPR-FSs ξ₁ and ξ₂ in A such that ξ1≠0A,ξ2≠0A and ξ = ξ₁ ∪ ξ₂. If ξ is not disconnected nPR-FS, then ξ is called connected nPR-FS.

Example 3.23

Let A={w1,w2} and ξ_i be nPR-fuzzy subsets of A such that ω̄_ξ_{_i} and ω_ξ_{_i} are corresponding membership and non-membership functions of ξ_i for each i = 1, 2. If

ϖξ1(w1)=0, ωξ1(w1)=1, ϖξ1(w2)=1, ωξ1(w2)=0,ϖξ2(w1)=1, ωξ2(w1)=0, ϖξ2(w2)=0, ωξ2(w2)=1,

then ℑ={1A,0A,ξ1,ξ2} is nPR-FT and hence 1A is disconnected nPR-FS because there are two nPR-FSs ξ₁ and ξ₂ such that ξ1≠0A,ξ2≠0A,cl(ξ1)∩ξ2=0A,ξ1∩cl(ξ2)=0A and 1A=ξ1∪ξ2.

Remark 3.24

An nPR-FS 1A

is connected in every indiscrete nPR-FTS (A,ℑ).

Remark 3.25

An nPR-FS 1A is connected if any (therefore all) of the conditions (2) – (4) in Theorem 3.21 hold.

Remark 3.26

According to the Definition 3.22 and Remark 3.25, 1A is disconnected nPR-FS if we can write 1A=ξ1∪ξ2, where the following (equivalent) statements are true:

ξ₁ and ξ₂ are open nPR-FSs such that ξ1≠0A,ξ2≠0A and ξ1∩ξ2=0A.
ξ₁ and ξ₂ are closed nPR-FSs such that ξ1≠0A,ξ2≠0A and ξ1∩ξ2=0A.
ξ₁ and ξ₂ are separated nPR-FSs such that ξ1≠0A and ξ2≠0A.

Corollary 3.27

Let (A,ℑ) be an nPR-FTS. Then, 1A is connected nPR-FS if 0A and 1A are the only open nPR-FSs and closed nPR-FSs in (A,ℑ).

Proof

Follows from Theorem 3.21 and Remark 3.25.

Lemma 3.28

Suppose ξ₁ and ξ₂ are separated nPR-FSs in an nPR-FTS (A,ℑ). If ξ ⊂ ξ₁ ∪ ξ₂ and ξ is connected nPR-FS, then ξ ⊂ ξ₁ or ξ ⊂ ξ₂.

Proof

Since ξ ∩ ξ₁ ⊂ ξ₁ and ξ ∩ ξ₂ ⊂ ξ₂, then ξ ∩ ξ₁ and ξ ∩ ξ₂ are separated nPR-FSs and by Theorem 2.10 (1) we have ξ = ξ ∩ (ξ₁ ∪ ξ₂) = (ξ ∩ ξ₁) ∪ (ξ ∩ ξ₂). Since ξ is connected nPR-FS, hence either ξ∩ξ1=0A, so ξ ⊂ ξ₂ or ξ∩ξ2=0A, so ξ ⊂ ξ₁.

Theorem 3.29

Suppose ξ and ξ_i (i ∈ ℕ) are connected nPR-FSs in an nPR-FTS (A,ℑ) and that for each i, ξ_i and ξ are not separated nPR-FSs. Then, ξ ∪ ξ_i is connected nPR-FS.

Proof

Suppose that ξ ∪ ξ_i = ξ₁₁ ∪ ξ₁₂, where ξ₁₁ and ξ₁₂ are separated nPR-FSs such that ξ11≠0A and ξ12≠0A. By Lemma 3.28, either ξ ⊂ ξ₁₁ or ξ ⊂ ξ₁₂. Without loss of generality, assume ξ ⊂ ξ₁₁. By the same reasoning we conclude that for each i, either ξ_i ⊂ ξ₁₁ or ξ_i ⊂ ξ₁₂. If some ξ_i ⊂ ξ₁₂, then ξ and ξ_i would be separated nPR-FSs. Thus every ξ_i ⊂ ξ₁₁ and so ξ ∪ ξ_i ⊂ ξ₁₁. Therefore, ξ₁₂ ⊂ ξ₁₁ ∪ ξ₁₂ = ξ ∪ ξ_i ⊂ ξ₁₁ and hence ξ12⊂ξ11⊂cl(ξ11), so ξ12=0A which is contradiction, therefore ξ ∪ ξ_i is connected nPR-FS.

3.4 nPR-Fuzzy Continuous Maps

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Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

In this section we study the continuity of a function defined among two nPR-FTSs.

Let f:A→B be a function. Let ξ₁ and ξ₂ be nPR-fuzzy subsets of A and B, respectively. Then, the membership and non-membership functions of the image of ξ₁ denoted by f[ξ1] are, respectively, calculated by

ϖf[ξ1](b)={supa∈f-1(b)ϖξ1(a),if f-1(b)≠φ,0,otherwise,

and

ωf[ξ1](b)={infa∈f-1(b)ωξ1(a),if f-1(b)≠φ,1,otherwise.

The membership and non-membership functions of pre-image of ξ₂ denoted by f-1[ξ2], are, respectively, calculated by ϖf-1[ξ2](w)=ϖξ2(f(w)), and ϖf-1[ξ2](w)=ϖξ2(f(w)).

Remark 3.30

To show that f[ξ1] and f-1[ξ2] are nPR-fuzzy sets. Consider θξ1(a)=(ϖξ1(a))n+(ωξ1(a))1n and b∈B, then we obtain

(ϖf[ξ1](b))n+(ωf[ξ1](b))1n=(supa∈f-1(b)ϖξ1(a))n+(infa∈f-1(b)ωξ1(a))1n=supa∈f-1(b)(ϖξ1(a))n+infa∈f-1(b)(ωξ1(a))1n=supa∈f-1(b)(θξ1(a)-(ωξ1(a))1n)+infa∈f-1(b)(ωξ1(a))1n≤supa∈f-1(b)(1-(ωξ1(a))1n)+infa∈f-1(b)(ωξ1(a))1n=1,

whenever f-1(b) is non-empty. On the other hand if f-1(b)=φ, then we have (ϖf[ξ1](b))n+(ωf[ξ1](b))1n=1. Thus, f[ξ1] is an nPR-fuzzy set of B.

Now, let w∈A, then (ϖf-1[ξ2](w))n+(ωf-1[ξ2](w))1n=(ϖξ2(f(w)))n+(ωξ2(f(w)))1n≤1. Therefore, f-1[ξ2] is an nPR-fuzzy set of A.

Following theorems give some properties of image and pre-image.

Theorem 3.31

Let f:A→B be a function such that ξ₁ and ξ₂ are nPR-fuzzy subsets of A and B, respectively. Then,

f-1[ξ2c]=f-1[ξ2]c.
f[ξ1]c⊂f[ξ1c].
f[f-1[ξ2]]⊂ξ2.
ξ1⊂f-1[f[ξ1]].

Proof

For any w∈A and for any nPR-fuzzy subset ξ₂ of B we get from the definition of the complement that
ϖf-1[ξ2c](w)=ϖξ2c(f(w))=ωξ2(f(w))n2=ωf-1[ξ2](w)n2=ϖf-1[ξ2]c(w).
Similarly we have
ωf-1[ξ2c](w)=ωξ2c(f(w))=(ϖξ2(f(w)))n2=(ϖf-1[ξ2](w))n2=ωf-1[ξ2]c(w).
Thus, we have f-1[ξ2c]=f-1[ξ2]c.
For any b∈B such that f(b)=φ and for any nPR-fuzzy subset ξ₁ of A, we can write
(*)θf[ξ1](b)=(ϖf[ξ1](b))n+(ωf[ξ1](b))1n=supa∈f-1(b)(ϖξ1(a))n+infa∈f-1(b)(ωξ1(a))1n=supa∈f-1(b)(θξ1(a)-(ωξ1(a))1n)+infa∈f-1(b)(ωξ1(a))1n≤supa∈f-1(b)θξ1(a)-infa∈f-1(b)(ωξ1(a))1n+infa∈f-1(b)(ωξ1(a))1n=supa∈f-1(b)θξ1(a).
Now from (*), we have
ϖf[ξ1c](b)=supa∈f-1(b)ϖξ1c(a)=supa∈f-1(b)ωξ1(a)n2=supa∈f-1(b)θξ1(a)-(ϖξ1(a))nn≥supa∈f-1(b)θξ1(a)-supa∈f-1(b)(ϖξ1(a))nn≥θf[ξ1](b)-(ϖf[ξ1](b))nn=ωf[ξ1](b)n2=ϖf[ξ1]c(b).
Hence, we obtain ϖf[ξ1]c(b)≤ϖf[ξ1c](b).
Similarly we have
(**)θf[ξ1](b)=(ϖf[ξ1](b))n+(ωf[ξ1](b))1n=supa∈f-1(b)(ϖξ1(a))n+infa∈f-1(b)(ωξ1(a))1n=supa∈f-1(b)(ϖξ1(a))n+infa∈f-1(b)(θξ1(a)-(ϖξ1(a))n)≥supa∈f-1(b)(ϖξ1(a))n+infa∈f-1(b)θξ1(a)+infa∈f-1(b)(-(ϖξ1(a))n)=infa∈f-1(b)θξ1(a).
Now, (**) which implies that
ωf[ξ1c](b)=infa∈f-1(b)ωξ1c(a)=infa∈f-1(b)(ϖξ1(a))n2=infa∈f-1(b)(θξ1(a)-(ωξ1(a))1n)n≤(infa∈f-1(b)θξ1(a)-infa∈f-1(b)(ωξ1(a))1n)n≤(θf[ξ1](b)-(ωf[ξ1](b))1n)n=(ϖf[ξ1](b))n2=ωf[ξ1]c(b).
Hence, we obtain ωf[ξ1c](b)≤ωf[ξ1]c(b).
On the other hand, the proof is trivial for each b∈B such that f(b)=φ.
Therefore, we obtain f[ξ1]c⊂f[ξ1c].
For any b∈B such that f(b)=φ, we can write
ϖf[f-1[ξ2]](b)=supa∈f-1(b)ϖf-1[ξ2](a)=supa∈f-1(b)ϖξ2(f(a))≤ϖξ2(b).
Similarly, we have
ωf[f-1[ξ2]](b)=infa∈f-1(b)ωf-1[ξ2](a)=infa∈f-1(b)ωξ2(f(a))≥ωξ2(b).
On the other hand if f(b)=φ, then we have ϖf[f-1[ξ2]](b)=0≤ϖξ2(b), and ωf[f-1[ξ2]](b)=1≥ωξ2(b). Therefore, f[f-1[ξ2]]⊂ξ2.
For any w∈A, we have ϖf-1[f[ξ1]](w)=ϖf[ξ1](f(w))=supa∈f-1(f(w))ϖξ1(a)≥ϖξ1(w).
Similarly, we have ωf-1[f[ξ1]](w)≤ωξ1(w).
Therefore, ξ1⊂f-1[f[ξ1]].

Theorem 3.32

Let f:A→B be a function such that ξ₁_i and ξ₂_i are nPR-fuzzy subsets of A and B, respectively, for i = 1, 2. If

ξ₂₁ ⊂ ξ₂₂, then f-1[ξ21]⊂f-1[ξ22].
ξ₁₁ ⊂ ξ₁₂, then f[ξ11]⊂f[ξ12].

Proof

Assume that ξ₂₁ ⊂ ξ₂₂. Then, for any w∈A we have
ϖf-1[ξ21](w)=ϖξ21(f(w))≤ϖξ22(f(w))=ϖf-1[ξ22](w).
Thus, ϖf-1[ξ21](w)≤ϖf-1[ξ22](w). Similarly, we have
ωf-1[ξ22](w)=ωξ22(f(w))≤ωξ21(f(w))=ωf-1[B1ξ21](w).
Thus, ωf-1[ξ21](w)≥ωf-1[ξ22](w). Therefore, f-1[ξ21]⊂f-1[ξ22].
Assume that ξ₁₁ ⊂ ξ₁₂ and b∈B. If f(b)=φ, then the proof is trivial. Assume that f(b)≠φ. Then, we have
ϖf[ξ11](b)=supa∈f-1(b)ϖξ11(a)≤supa∈f-1(b)ϖξ12(a)=ϖf[ξ12](b),
and
ωf[ξ12](b)=infa∈f-1(b)ωξ12(a)≤infa∈f-1(b)ωξ11(a)=ωf[ξ11](b).
Therefore, f[ξ11]⊂f[ξ12].

The proof of the following result is easy and hence it is omitted.

Corollary 3.33

Let f:A→B be a function. Then, the following statements are true:

f[∪i∈Iξ1i]=∪i∈If[ξ1i] for any nPR-fuzzy subset ξ₁_i of A.
f-1[∪i∈Iξ2i]=∪i∈If-1[ξ2i] for any nPR-fuzzy subset ξ₂_i of B.
f[ξ11∩ξ12]⊂f[ξ11]∩f[ξ12] for any two nPR-fuzzy subsets ξ₁₁ and ξ₁₂ of A.
f-1[∩i∈Iξ2i]=∩i∈If-1[ξ2i] for any nPR-fuzzy subset ξ₂_i of B.

Definition 3.34

Let (A,ℑ1) and (B,ℑ2) be two nPR-FTSs and f:A→B be a function. Then, f is said to be nPR-fuzzy continuous if for any nPR-fuzzy subset ξ₁ of A and for any neighbourhood ξ₄ of f[ξ1] there exists a neighbourhood ξ₃ of ξ₁ such that f[ξ3]⊂ξ4.

Theorem 3.35

Let (A,ℑ1) and (B,ℑ2) be two nPR-FTSs and f:A→B be a function. Then, the following statements are equivalent:

f is nPR-fuzzy continuous.
For any nPR-fuzzy subset ξ₁ of A and for any neighbourhood ξ₄ of f[ξ1], there exists a neighbourhood ξ₃ of ξ₁ such that for any ξ₂ ⊂ ξ₃ we have f[ξ2]⊂ξ4.
For any nPR-fuzzy subset ξ₁ of A and for any neighbourhood ξ₄ of f[ξ1], there exists a neighbourhood ξ₃ of ξ₁ such that ξ3⊂f-1[ξ4].
For any nPR-fuzzy subset ξ₁ of A and for any neighbourhood ξ₄ of f[ξ1],f-1[ξ4] is a neighbourhood of ξ₁.

Proof

(1) ⇒ (2): Assume that f is nPR-fuzzy continuous. Let ξ₁ be an nPR-fuzzy subset of A and ξ₄ be a neighbourhood of f[ξ1]. Then by (1), there exists a neighbourhood ξ₃ of ξ₁ such that f[ξ3]⊂ξ4. Now, if ξ₂ ⊂ ξ₃, then we get f[ξ2]⊂f[ξ3]⊂ξ4. (2) ⇒ (3): Let ξ₁ be an nPR-fuzzy set of A and ξ₄ be a neighbourhood of f[ξ1]. From (2), there exists a neighbourhood ξ₃ of ξ₁ such that for any ξ₂ ⊂ ξ₃ we have f[ξ2]⊂ξ4. Then, we can write ξ2⊂f-1[f[ξ2]]⊂f-1[ξ4]. As ξ₂ is an arbitrary subset of ξ₃, we have ξ3⊂f-1[ξ4].

(3) ⇒ (4): Let ξ₁ be an nPR-fuzzy subset of A and ξ₄ be a neighbourhood of f[ξ1]. Then from (3), there exists a neighbourhood ξ₃ of ξ₁ such that ξ3⊂f-1[ξ4]. Since ξ₃ is a neighbourhood of ξ₁ there exists an open nPR-fuzzy subset ξ₅ of A such that ξ₁ ⊂ ξ₅ ⊂ ξ₃. On the other hand as ξ3⊂f-1[ξ4], one can get ξ1⊂ξ5⊂f-1[ξ4] which implies f-1[ξ4] is a neighbourhood of ξ₁.

(4) ⇒ (1): Let ξ₁ be an nPR-fuzzy subset of A and ξ₄ be a neighbourhood of f[ξ1]. From the hypothesis, we have f-1[ξ4] is a neighbourhood of ξ₁. Therefore, there exists an open nPR-fuzzy subset ξ₅ of A such that ξ1⊂ξ5⊂f-1[ξ4] which implies f[ξ5]⊂f[f-1[ξ4]]⊂ξ4. Moreover, as ξ₅ is open it is a neighbourhood of ξ₁. Hence, f is nPR-fuzzy continuous.

Theorem 3.36

Let (A,I1) and (B,I2) be two nPR-FTSs. A function f:A→B is nPR-fuzzy continuous if and only if for each open nPR-fuzzy subset ξ₂ of B we have f-1[ξ2] is an open nPR-fuzzy subset of A.

Proof

Assume that f is nPR-fuzzy continuous. Let ξ₂ be an open nPR-fuzzy subset of B and ξ1⊂f-1[ξ2]. Then, we get f[ξ1]⊂ξ2. Since ξ₂ is open nPR-fuzzy, then by Theorem 3.7, there exists a neighbourhood ξ₄ of f[ξ1] such that ξ₄ ⊂ ξ₂. Thus, nPR-fuzzy continuity of f and (4) of Theorem 3.35 imply that f-1[ξ4] is a neighbourhood of ξ₁. On the other hand from (1) of Theorem 3.32 we have f-1[ξ4]⊂f-1[ξ2]. Therefore, f-1[ξ2] is a neighbourhood of ξ₁ as well. As ξ₁ is an arbitrary subset of f-1[ξ2], then by Theorem 3.7, the nPR-fuzzy subset f-1[ξ2] is open.

Conversely, let ξ₁ be an nPR-fuzzy subset of A and ξ₄ be a neighbourhood of f[ξ1]. Then, there exists an open nPR-fuzzy subset ξ₆ of B such that f[ξ1]⊂ξ6⊂ξ4. Now, from the hypothesis f-1[ξ6] is open. On the other hand, we can write ξ1⊂f-1[f[ξ1]]⊂f-1[ξ6]⊂f-1[ξ4]. Hence, f-1[ξ4] is a neighbourhood of ξ₁. Then by Theorem 3.35 (4), f is nPR-fuzzy continuous.

Proposition 3.37

Let (B,I*) be an nPR-FTS and f:A→B be a function such that I1 and I2 are two nPR-fuzzy topologies on A. If I2⊂I1 and f:(A,I2)→(B,I*) is nPR-fuzzy continuous, then f:(A,I1)→(B,I*) is nPR-fuzzy continuous.

Proof

From Theorem 3.36 and the fact I2⊂I1.

We build the following two examples such that the first one provides an 6PR-fuzzy continuous map, whereas the second one presents a fuzzy map that is not 6PR-fuzzy continuous.

Example 3.38

Consider A={w1,w2} with the 6PR-FT I2={1A,0A,ξ1} and B={b1,b2} with the 6PR-FT I2={1B,0B,ξ2}, where ξ1={〈w1,0.61,0.71〉,〈w2,0.73,0.31〉} and ξ₂ = {〈b₁, 0.73, 0.31〉, 〈b₂, 0.61, 0.71〉}.

Let f:A→B defined as follows:

f(w)={b2,if w=w1,b1,if w=m2.

Since 1B,0B and ξ₂ are open 6PR-fuzzy subsets of B, then f-1[1B]={〈w1,1,0〉,〈w2,1,0〉},f-1[0B]={〈w1,0,1〉,〈w2,0,1〉} and f-1[ξ2]={〈w1,0.61,0.71〉,〈w2,0.73,0.31〉} are open 6PR-fuzzy subsets of A. Thus, f is 6PR-fuzzy continuous.

Example 3.39

Consider A={w1,w2} with the 6PR-FT I1={1A,0A} and B={b1,b2} with the 6PR-FT I2={1B,0B,ξ2}, where ξ₂ = {〈b₁, 0.67, 0.54〉, 〈b₂, 0.49, 0.89〉}.

Let f:A→B defined as follows:

f(w)={b1,if w=w1,b2,if w=m2.

Since ξ₂ is open 6PR-fuzzy subset of B, but f-1[ξ2]={〈w1,0.67,0.54〉,〈w2,0.49,0.89〉} is not open 6PR-fuzzy subset of A. Thus, f is not 6PR-fuzzy continuous.

Corollary 3.40

A function f:A→B is nPR-fuzzy continuous if and only if for each closed nPR-fuzzy subset ξ₂ of B we have f-1[ξ2] is a closed nPR-fuzzy subset of A.

Proof

Assume that ξ₂ is a closed nPR-fuzzy subset of B, then (ξ2)c is an open nPR-fuzzy set in B. Since f is nPR-fuzzy continuous, then from Theorem 3.36, we get f-1[(ξ2)c] is an open nPR-fuzzy subset of A. On the other hand, from Theorem 3.31 (1) we know that f-1[(ξ2)c]=(f-1[ξ2]c). Then (f-1[ξ2])c is an open nPR-fuzzy set. Thus, f-1[ξ2] is a closed nPR-fuzzy subset of A.

Conversely, let ξ be an open nPR-fuzzy subset of B, then ξ^c is a closed nPR-fuzzy set in B. By hypothesis, f-1[ξc] is a closed nPR-fuzzy subset of A and hence f-1[ξc]=(f-1[ξ])c is closed nPR-fuzzy. Therefore, f-1[ξ] is an open nPR-fuzzy subset of A and by Theorem 3.36 we get that f is nPR-fuzzy continuous.

Corollary 3.41

The following are equivalent to each other:

f:(A,I1)→(B,I2) is nPR-fuzzy continuous.
cl(f-1[ξ2])⊂f-1[cl(ξ2)] for each nPR-fuzzy set ξ₂ in B.
f-1[int(ξ2)]⊂int(f-1[ξ2]) for each nPR-fuzzy set ξ₂ in B.

Proof

They can be easily proved using Remark 3.9, Theorems 3.11, 3.13, 3.31, 3.32, 3.36 and Corollary 3.40.

Now, we prove that an nPR-FT can be established on A, whenever an nPR-FT on B and a function f:A→B are given. This topology is the coarsest topology on A.

Theorem 3.42

Let (B,I) be an nPR-FTS and f:A→B be a function. Then, there exists a coarsest nPR-FT I1 over A such that f is nPR-fuzzy continuous.

Proof

Let us define a class of nPR-fuzzy subsets I1 of A by I1={f-1[ξ4]:ξ4∈I}.

We prove that I1 is the coarsest nPR-FT over A such that f is continuous.

We can write for any w∈A that
ϖf-1[0B](w)=ϖ0B(f(w))=0=ϖ0A(w).
Similarly, we immediately have ωf-1[0B](w)=ω0A(w) for any w∈A which implies f-1[0B]=0A. Now, as 0B∈I we have 0A=f-1[0B]∈I1. In similar manner, it is easy to see that 1A=f-1[1B]∈I1.
Assume that ξ1,ξ2∈I1. Then, for i = 1, 2 there exists ξ2i∈I such that f-1[ξ2i]=ξi which implies ϖf-1[ξ2i]=ϖξi and ωf-1[ξ2i]=ωξi. Thus, we obtain for any w∈A that
ϖξ1∩ξ2(w)=min{ϖξ1(w),ϖξ2(w)}=min{ϖf-1[ξ21](w),ϖf-1[ξ22](w)}=min{ϖξ21(f(w)),ϖξ22(f(w))}=ϖξ21∩ξ22(f(w))=ϖf-1[ξ21∩ξ22](w).
Similarly, it is not difficult to see that ωξ1∩ξ2=ωf-1[ξ21∩ξ22]. Hence, we get ξ1∩ξ2∈I1.
Assume that {ξ_i}_i_∈_I is an arbitrary sub-family of I1. Then for any i ∈ I, there exists ξ2i∈I1 such that f-1[ξ2i]=ξi which implies ϖf-1[ξ2i]=ϖξi and ϖf-1[ξ2i]=ωξi. Therefore, one can get for any w∈A that
ϖ∪i∈Iξi(w)=supi∈Iϖξi(w)=supi∈Iϖf-1[ξ2i](w)=supi∈Iϖξ2i(f(w))=ϖ∪i∈Iξ2i(f(w))=ϖf-1[∪i∈Iξ2i](w).
On the other hand, it is easy to see that ω∪i∈Iξi=ωf-1[∪i∈Iξ2i]. Thus, we have ∪i∈Iξi∈I1.

Hence, I1 is an nPR-FT on A, and from Theorem 3.36, f is nPR-fuzzy continuous. Now, we prove that I1 is the coarsest nPR-FT over A such that f is nPR-fuzzy continuous. Let I2⊂I1 be an nPR-FT over A such that f is nPR-fuzzy continuous. If ξ2∈I1 then there exists ξ4∈I such that f-1[ξ4]=ξ2. Since f is nPR-fuzzy continuous with respect to I2 we have ξ2=f-1[ξ4]∈I2. Hence, we have I2=I1.

The following corollary says that we can construct an nPR-FT on a given set B whenever there exists an nPR-FTS A and a function f:A→B. This topology is the finest topology that makes the function f is nPR-fuzzy continuous.

Corollary 3.43

Let (A,I*) be an nPR-FTS and f:A→B be a function. Then, there exists a finest nPR-FT I** over B such that f is nPR-fuzzy continuous.

Proof

Can be proved similar to Theorem 3.42.

3.5 nPR-Fuzzy Separation Axioms

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Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
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Definition 3.44

Let A≠φ and w∈A be a fixed element. Suppose ε₁ ∈ (0, 1] and ε₂ ∈ [0, 1) are two fixed real numbers such that ɛ1n+ɛ21n≤1. Then, an nPR-fuzzy point χ(ɛ1,ɛ2)w={〈w,ϖχ(w),ωχ(w)〉} is defined to be an nPR-fuzzy set of A as follows:

χ(ɛ1,ɛ2)w(b)={(ɛ1,ɛ2),if b=w,(0,1),otherwise,

for b∈A. In this case, w is called the support of χ(ɛ1,ɛ2)w. An nPR-fuzzy point χ(ɛ1,ɛ2)w is said to belong to an nPR-fuzzy set ξ={〈w,ϖξ(w),ωξ(w)〉} of A denoted by χ(ɛ1,ɛ2)w∈ξ if ɛ1≤ϖξ(w) and ɛ2≤ϖξ(w).

Two nPR-fuzzy points are said to be distinct if their supports are distinct.

Remark 3.45

Let ξ1={〈w,ϖξ1(w),ωξ1(w)〉} and ɛ2={〈w,ϖξ2(w),ωξ2(w)〉} be two nPR-fuzzy sets of A. Then, ξ₁ ⊂ ξ₂ if and only if χ(ɛ1,ɛ2)w∈ξ1 implies χ(ɛ1,ɛ2)w∈ξ2 for any nPR-fuzzy point χ(ɛ1,ɛ2)w in A.

Definition 3.46

Let ε₁, ε₃ ∈ (0, 1], ε₂, ε₄ ∈ [0, 1) and w,b∈A. An nPR-FTS (A,I) is said to be:

T0 if for each pair of distinct nPR-fuzzy points χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b in A, there exist two open nPR-fuzzy sets ξ₆ and ξ₅ such that
ξ6={〈w,1,0〉,〈b,0,1〉},
or
ξ5={〈w,0,1〉,〈b,1,0〉}.
T1 if for each pair of distinct nPR-fuzzy points χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b in A, there exist two open nPR-fuzzy sets ξ₆ and ξ₅ such that
ξ6={〈w,1,0〉,〈b,0,1〉},
and
ξ5={〈w,0,1〉,〈b,1,0〉}.

Example 3.47

Consider A={w1,w2} with the nPR-FT I={1A,0A,ξ1,ξ2}, where ξ1={〈w1,1,0〉,〈w2,0,1〉} and ξ2={〈w1,0,1〉,〈w2,1,0〉}. Then, (A,I) is To and T1.

Corollary 3.48

Let (A,I) be an nPR-FTS. If (A,I) is T1, then (A,I) is To.

Proof

The proof is straightforward from the Definition 3.46.

Here is an example which shows that the converse of above corollary is not true in general.

Example 3.49

Consider A={w1,w2} with the nPR-FT I={1A,0A,ξ}, where ξ={〈w1,1,0〉,〈w2,0,1〉}. Then, (A,I) is T0 but not T1 because there exists no open nPR-fuzzy set ξ₅ such that ξ5={〈w1,0,1〉,〈w2,1,0〉}.

Theorem 3.50

Let (A,I) be an nPR-FTS, ε₁, ε₃ ∈ (0, 1] and ε₂, ε₄ ∈ [0, 1). If (A,I) is To, then for each pair of distinct nPR-fuzzy points χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b of A,cl(χ(ɛ1,ɛ2)w)≠cl(χ(ɛ3,ɛ4)b).

Proof

Let (A,I) be To and χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b be any two distinct nPR-fuzzy points of A. Then, there exist two open nPR-fuzzy sets ξ₆ and ξ₅ such that

ξ6={〈w,1,0〉,〈b,0,1〉},

ξ5={〈w,0,1〉,〈b,1,0〉}.

Let ξ6={〈w,1,0〉,〈b,0,1〉} be exists. Then, ξ6c={〈w,0,1〉,〈b,1,0〉} is a closed nPR-fuzzy set which does not contain χ(ɛ1,ɛ2)w but contains χ(ɛ3,ɛ4)b. Since cl(χ(ɛ3,ɛ4)b) is the smallest closed nPR-fuzzy set containing χ(ɛ3,ɛ4)b, then cl(χ(ɛ3,ɛ4)b)⊂ξ6c and therefore χ(ɛ1,ɛ2)w∉cl(χ(ɛ3,ɛ4)b). Consequently cl(χ(ɛ1,ɛ2)x)≠cl(χ(ɛ3,ɛ4)b).

Theorem 3.51

Let (A,I) be an nPR-FTS. Then, (A,I) is T1 if χ(1,0)w is closed nPR-fuzzy set for every w∈A.

Proof

Suppose χ(1,0)w is a closed nPR-fuzzy set for every w∈A. Let χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b be any two distinct nPR-fuzzy points of A, then w≠b implies χ(1,0)wc and χ(1,0)bc are two open nPR-fuzzy sets such that

χ(1,0)bc={〈w,1,0〉,〈b,0,1〉},

and

χ(1,0)wc={〈w,0,1〉,〈b,1,0〉}.

Thus, (A,I) is T1.

4. Conclusions

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Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
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References
Biographies

In this article, we have proposed a new generalized fuzzy set called “nPR-fuzzy sets” and compared their relationship with other kinds of generalizations of fuzzy sets. Moreover, some operators on nPR-fuzzy sets have been studied and their relationship have been reveled. The concepts of topology, neighborhood, connectedness, and continuous mapping via nPR-fuzzy sets have been investigated. Finally, we have presented the notion of nPR-fuzzy points and introduced separation axioms via nPR-fuzzy topological space.

In future works, more applications of nPR-fuzzy sets may be investigated, and also nPR-fuzzy soft sets may be introduced. In addition, we will try to present some types of weighted aggregated operators over nPR-FSs and study a MCDM methods depending on these operators.

Funding:

Go to

Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

This research has received no external funding.

Conflicts of interest:

Go to

Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

The authors declare that they have no competing interests.

Data Availability statement:

Go to

Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

No data were used to support this study.

Figures

Go to

Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

Fig. 1.

Grades spaces of some kinds of nPR-fuzzy sets.

Fig. 2.

Comparison of grades space of IFSs, SR-FSs, CR-FSs, 4-ROFSs and 4PR-FSs

References

Go to

Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

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Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions
Funding:
Conflicts of interest:
Data Availability statement:
Figures
References
Biographies

TareqM. Al-shami received the M.S. and Ph.D. degrees in pure mathematics from the Department of Mathematics, Faculty of Science, Mansoura University, Egypt. He is currently an Assistant Professor with the department of Mathematics, Sana’a University, Sana’a, Yemen. He has published more than 136 research articles in international peer-reviewed SCIE and ESCI journals such as Information Sciences, Knowledge-Based Systems, Applied and Computational Mathematics, Computational and Applied Mathematics, Artificial Intelligence Review, AIMS mathematics, Soft Computing journal among others. His research interests include pure mathematics, topology and its extensions, ordered topology, soft set theory, rough set theory and fuzzy set theory with applications in decision-making, medical diagnosis, information measures, and information aggregation. Dr. Tareq received Obada-Prize for postgraduate students in Feb. 2019. Also, he obtained the first rank at Sana’a University -Yemen according to Scopus data of the region of Yemen for the period from 2018 to 2121.

Hariwan Z. Ibrahim is currently working as an assistant professor in the Department of Mathematics at the University of Zakho, Kurdistan Region-Iraq. He received his Ph.D. degree from the same university. His area of interest includes Fuzzy Topology, Topological Algebra, Ditopology, Ideal Topology and Soft Topology. He has published more than 80 research papers in international journals.

Abdelwaheb Mhemdi is a doctor in pure Mathematics. He was awarded a PhD. from the Faculty of Sciences, Tunis Elmanar University. He was an assistant professor from 2011 to 2016 in Tunis Elmanar University and Gafsa University. Since 2016, he has been teaching at Prince Sattam bin Abdulaziz University in Saudi Arabia. Now, he is an associate professor, and he is the head of the department of Mathematics in the College of Sciences and Humanities in Al Aflaj. Mhemdi produced many original articles in general Topology, especially in Separation Axioms, Soft Topological spaces, and Category of Topological Spaces.

Radwan Abu-Gdairi is an assistant professor of Mathematics at Department of Mathematics, Faculty of Science, Zarqa University, Jordan. His research interests are in the areas of pure and applied mathematics including Topology, Fuzzy topology, Rough set theory and it’s applications. He has published research articles in reputed international journals of mathematical sciences. He is referee of some mathematical journals.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(4): 350-365

Published online December 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.4.350

n Power Root Fuzzy Sets and Its Topology

Tareq M. Al-shami¹, Hariwan Z. Ibrahim², Abdelwaheb Mhemdi³, and Radwan Abu-Gdairi⁴

Correspondence to:Abdelwaheb Mhemdi (mhemdiabd@gmail.com)

Received: October 15, 2022; Revised: November 23, 2022; Accepted: December 12, 2022

Abstract

Keywords: nPR-fuzzy sets, Operations, nPR-fuzzy topology, Separated nPR-FSs, Connected nPR-FS, nPR-fuzzy continuous maps, 𝕿,₀, 𝕿,₁

1. Introduction

2. nPR-Fuzzy Sets

In this section, we study the notion of n^th power root fuzzy sets, and establish main properties. For computations, we use only three decimal places in whole paper.

Definition 2.1

Let ℕ be the set of natural numbers and $A$ be a universal set. Then, the n^th power root fuzzy set (briefly, nPR-FS) ξ, which is a set of ordered pairs over $A$ , is defined as following:

ξ = {〈 w, ϖ_{ξ} (w), ω_{ξ} (w) 〉 : w \in A},

where $ϖ_{ξ} : A \to [0, 1]$ is the degree of membership and $ω_{ξ} : A \to [0, 1]$ is the degree of non-membership functions such that

0 \leq {(ϖ_{ξ} (w))}^{n} + \sqrt[n]{ω_{ξ} (w)} \leq 1,

for each $w \in A$ and n ∈ ℕ \ {1}.

Then, there is a degree of indeterminacy of $w \in A$ to ξ defined by

π_{ξ} (w) = 1 - [{(ϖ_{ξ} (w))}^{n} + \sqrt[n]{ω_{ξ} (w)}] .

It is clear that ${(ϖ_{ξ} (w))}^{n} + \sqrt[n]{ω_{ξ} (w)} + π_{ξ} (w) = 1$ . Otherwise, $π_{ξ} (w) = 0$ whenever ${(ϖ_{ξ} (w))}^{n} + \sqrt[n]{ω_{ξ} (w)} = 1$ .

For the sake of simplicity, we shall mention the symbol ξ = (ω̄_ξ, ω_ξ) for the nPR-FS $ξ = {〈 w, ϖ_{ξ} (w), ω_{ξ} (w) 〉 : w \in A}$ . We noticed that 2PR-fuzzy is called SR-FS in [?], and 3PR-fuzzy is called CR-FS in [34]. If $0 \leq ϖ_{ξ} (w) + ω_{ξ} (w) \leq 1$ (resp. $0 \leq ϖ_{ξ} {(w)}^{n} + ω_{ξ} {(w)}^{n} \leq 1$ ), then ξ = (ω̄_ξ, ω_ξ) is called intuitionistic fuzzy set (IFS) [17](resp. n-rung orthopair fuzzy set (n-ROFS) [29]).

The spaces of some kinds of nPR-fuzzy membership grades are displayed in Figure 1.

Remark 2.2

From Figure 2, we get that

the space of 4-rung orthopair fuzzy membership grades is larger than the space of 4PR-fuzzy membership grades.
ξ = (ω̄_ξ ≈ 0.724, ω_ξ ≈ 0.276) is a point of intersection between 4PR-fuzzy and intuitionistic fuzzy sets.
for ω̄_ξ ∈ (0, 0.724) and ω_ξ ∈ (0.276, 1) the space of 4PR-fuzzy membership grades starts to be larger than the space of intuitionistic membership grades.
for ω̄_ξ ∈ (0.724, 1) and ω_ξ ∈ (0, 0.276) the space of 4PR-fuzzy membership grades starts to be smaller than the space of intuitionistic membership grades.
$ξ = (ϖ_{ξ} = \sqrt{\frac{\sqrt{5} - 1}{2}}, ω_{ξ} = \frac{7 - 3 \sqrt{5}}{2})$ is a point of intersection between 4PR-fuzzy and SR-fuzzy.
for $ϖ_{ξ} \in (0, \sqrt{\frac{\sqrt{5} - 1}{2}})$ and $ω_{ξ} \in (\frac{7 - 3 \sqrt{5}}{2}, 1)$ the space of 4PR-fuzzy membership grades starts to be larger than the space of SR-fuzzy membership grades.
for $ϖ_{ξ} \in (\sqrt{\frac{\sqrt{5} - 1}{2}}, 1)$ and $ω_{ξ} \in (0, \frac{7 - 3 \sqrt{5}}{2})$ the space of 4PR-fuzzy membership grades starts to be smaller than the space of SR-fuzzy membership grades.
ξ = (ω̄_ξ ≈ 0.819, ω_ξ ≈ 0.091) is a point of intersection between 4PR-fuzzy and CR-fuzzy sets.
for ω̄_ξ ∈ (0, 0.819) and ω_ξ ∈ (0.091, 1) the space of 4PR-fuzzy membership grades starts to be larger than the space of CR-fuzzy membership grades.
for ω̄_ξ ∈ (0.819, 1) and ω_ξ ∈ (0, 0.091) the space of 4PR-fuzzy membership grades starts to be smaller than the space of CR-fuzzy membership grades.

Remark 2.3

It is clear that for any nPR-FS ξ = (ω̄_ξ, ω_ξ) we have $0 \leq ϖ_{ξ}^{n} + ω_{ξ}^{n} \leq ϖ_{ξ}^{n} + \sqrt[n]{ω_{ξ}} \leq 1$ , then ξ is an n-rung orthopair fuzzy set. Therefore, every nPR-fuzzy set is an n-rung orthopair fuzzy set.

Definition 2.4

Let ξ = (ω̄_ξ, ω_ξ), ξ₁ = (ω̄_ξ_₁, ω_ξ_₁) and ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) be three nPR-fuzzy sets (nPR-FSs), then

$ξ_{1} \cap ξ_{2} = (\min {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, \max {ω_{ξ_{1}}, ω_{ξ_{2}}})$ .
$ξ_{1} \cup ξ_{2} = (\max {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, \min {ω_{ξ_{1}}, ω_{ξ_{2}}})$ .
$ξ^{c} = (\sqrt[n^{2}]{ω_{ξ}}, {(ϖ_{ξ})}^{n^{2}})$ .

Example 2.5

Suppose that ξ₁ = (0.6, 0.5) and ξ₂ = (0.5, 0.6) are both 4PR-FSs for $A = {w}$ . Then,

$ξ_{1} \cap ξ_{2} = (\min {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, \max {ω_{ξ_{1}}, ω_{ξ_{2}}}) = (\min {0.6, 0.5}, \max {0.5, 0.6}) = (0.5, 0.6)$ .
$ξ_{1} \cup ξ_{2} = (\max {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, \min {ω_{ξ_{1}}, ω_{ξ_{2}}}) = (\max {0.6, 0.5}, \min {0.5, 0.6}) = (0.6, 0.5)$ .
$ξ_{1}^{c} \approx (0.958, 0.000)$ .

Theorem 2.6

If ξ = (ω̄_ξ, ω_ξ) is a nPR-FS, then $ξ^{c}$ is also a nPR-FS and ${(ξ^{c})}^{c} = ξ$ .

Proof

Since $0 \leq ϖ_{ξ}^{n} \leq 1, 0 \leq \sqrt[n]{ω_{ξ}} \leq 1$ and $0 \leq {(ϖ_{ξ})}^{n} + \sqrt[n]{ω_{ξ}} \leq 1$ , then

0 \leq {(\sqrt[n^{2}]{ω_{ξ}})}^{n} + \sqrt[n]{{(ϖ_{ξ})}^{n^{2}}} = {(ϖ_{ξ})}^{n} + \sqrt[n]{ω_{ξ}} \leq 1,

and hence

0 \leq {(\sqrt[n^{2}]{ω_{ξ}})}^{n} + \sqrt[n]{{(ϖ_{ξ})}^{n^{2}}} \leq 1.

Thus, $ξ^{c}$ is a nPR-FS and it is obvious that ${(ξ^{c})}^{c} = {(\sqrt[n^{2}]{ω_{ξ}}, {(ϖ_{ξ})}^{n^{2}})}^{c} = (ϖ_{ξ}, ω_{ξ})$ .

Theorem 2.7

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁) and ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) be two nPR-FSs, then

ξ₁ ∩ ξ₂ = ξ₂ ∩ ξ₁.
ξ₁ ∪ ξ₂ = ξ₂ ∪ ξ₁.

Proof

From Definition 2.4, we can obtain:

$ξ_{1} \cap ξ_{2} = (\min {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, \max {ω_{ξ_{1}}, ω_{ξ_{2}}}) = (\min {ϖ_{ξ_{2}}, ϖ_{ξ_{1}}}, \max {ω_{ξ_{2}}, ω_{ξ_{1}}}) = ξ_{2} \cap ξ_{1}$ .
The proof is similar to (1).

Theorem 2.8

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁) and ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) be two nPR-FSs, then

(ξ₁ ∩ ξ₂) ∪ ξ₂ = ξ₂.
(ξ₁ ∪ ξ₂) ∩ ξ₂ = ξ₂.

Proof

From Definition 2.4, we can obtain:

$\begin{array}{l} (ξ_{1} \cap ξ_{2}) \cup ξ_{2} = (\min {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, \max {ω_{ξ_{1}}, ω_{ξ_{2}}}) \cup (ϖ_{ξ_{2}}, ω_{ξ_{2}}) \\ = (\max {\min {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, ϖ_{ξ_{2}}}, \min {\max {ω_{ξ_{1}}, ω_{ξ_{2}}}, ω_{ξ_{2}}}) \\ = (ϖ_{ξ_{2}}, ω_{ξ_{2}}) = ξ_{2} . \end{array}$
$\begin{array}{l} (ξ_{1} \cup ξ_{2}) \cap ξ_{2} = (\max {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, \min {ω_{ξ_{1}}, ω_{ξ_{2}}}) \cap (ϖ_{ξ_{2}}, ω_{ξ_{2}}) \\ = (\min {\max {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, ϖ_{ξ_{2}}}, \max {\min {ω_{ξ_{1}}, ω_{ξ_{2}}}, ω_{ξ_{2}}}) \\ = (ϖ_{ξ_{2}}, ω_{ξ_{2}}) = ξ_{2} . \end{array}$

Theorem 2.9

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁), ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) and ξ₃ = (ω̄_ξ_₃, ω_ξ_₃) be three nPR-FSs, then

ξ₁ ∩ (ξ₂ ∩ ξ₃) = (ξ₁ ∩ ξ₂) ∩ ξ₃.
ξ₁ ∪ (ξ₂ ∪ ξ₃) = (ξ₁ ∪ ξ₂) ∪ ξ₃.

Proof

For the three nPR-FSs ξ₁, ξ₂ and ξ₃, according to Definition 2.4, we can obtain

1. $\begin{array}{l} ξ_{1} \cap (ξ_{2} \cap ξ_{3}) = (ϖ_{ξ_{1}}, ω_{ξ_{1}}) \cap (\min {ϖ_{ξ_{2}}, ϖ_{ξ_{3}}}, \max {ω_{ξ_{2}}, ω_{ξ_{3}}}) \\ = (\min {ϖ_{ξ_{1}}, \min {ϖ_{ξ_{2}}, ϖ_{ξ_{3}}}}, \max {ω_{ξ_{1}}, \max {ω_{ξ_{2}}, ω_{ξ_{3}}}}) \\ = (\min {\min {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, ϖ_{ξ_{3}}}, \max {\max {ω_{ξ_{1}}, ω_{ξ_{2}}}, ω_{ξ_{3}}}) \\ = (\min {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, \max {ω_{ξ_{1}}, ω_{ξ_{2}}}) \cap (ϖ_{ξ_{3}}, ω_{ξ_{3}}) \\ = (ξ_{1} \cap ξ_{2}) \cap ξ_{3} . \end{array}$
2. The proof is similar to (1).

Theorem 2.10

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁), ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) and ξ₃ = (ω̄_ξ_₃, ω_ξ_₃) be three nPR-FSs, then

(ξ₁ ∪ ξ₂) ∩ ξ₃ = (ξ₁ ∩ ξ₃) ∪ (ξ₂ ∩ ξ₃).
(ξ₁ ∩ ξ₂) ∪ ξ₃ = (ξ₁ ∪ ξ₃) ∩ (ξ₂ ∪ ξ₃).

Proof

It can be proved following similar arguments given to the proof of its counterpart in [7].

Theorem 2.11

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁) and ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) be two nPR-FSs, then

${(ξ_{1} \cap ξ_{2})}^{c} = ξ_{1}^{c} \cup ξ_{2}^{c}$ .
${(ξ_{1} \cup ξ_{2})}^{c} = ξ_{1}^{c} \cap ξ_{2}^{c}$ .

Proof

For the two nPR-FSs ξ₁ and ξ₂, according to Definition 2.4, we can obtain:

1. $\begin{array}{l} {(ξ_{1} \cap ξ_{2})}^{c} = {(\min {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}}, \max {ω_{ξ_{1}}, ω_{ξ_{2}}})}^{c} \\ = (\max {\sqrt[n^{2}]{ω_{ξ_{1}}}, \sqrt[n^{2}]{ω_{ξ_{2}}}}, \min {{(ϖ_{ξ_{1}})}^{n^{2}}, {(ϖ_{ξ_{2}})}^{n^{2}}}) \\ = (\sqrt[n^{2}]{ω_{ξ_{1}}}, {(ϖ_{ξ_{1}})}^{n^{2}}) \cup (\sqrt[n^{2}]{ω_{ξ_{2}}}, {(ϖ_{ξ_{2}})}^{n^{2}}) \\ = ξ_{1}^{c} \cup ξ_{2}^{c} . \end{array}$
2. The proof is similar to (1).

Definition 2.12

Let ξ₁ = (ω̄_ξ_₁, ω_ξ_₁) and ξ₂ = (ω̄_ξ_₂, ω_ξ_₂) be two nPR-FSs, then

ξ₁ = ξ₂ if and only if ω̄_ξ_₁ = ω̄_ξ_₂ and ω_ξ_₁ = ω_ξ_₂.
ξ₁ ≥ ξ₂ if and only if ω̄_ξ_₁ ≥ ω̄_ξ_₂ and ω_ξ_₁ ≤ ω_ξ_₂.
ξ₂ ⊂ ξ₁ or ξ₁ ⊃ ξ₂ if ξ₁ ≥ ξ₂.

Example 2.13

Let $A = {w_{1}, w_{2}}$ , then

ξ₁ = ξ₂ for two 4PR-FSs $ξ_{1} = {〈 w_{1}, 0.62, 0.51 〉, 〈 w_{2}, 0.45, 0.76 〉}$ and $ξ_{2} = {〈 w_{1}, 0.62, 0.51 〉, 〈 w_{2}, 0.45, 0.76 〉}$ .
ξ₂ ≤ ξ₁ and ξ₂ ⊂ ξ₁ for two 4PR-FSs $ξ_{1} = {〈 w_{1}, 0.62, 0.51 〉, 〈 w_{2}, 0.45, 0.76 〉}$ and $ξ_{2} = {〈 w_{1}, 0.61, 0.52 〉, 〈 w_{2}, 0.44, 0.77 〉}$ .

3. Some Topological Concepts via nPR-Fuzzy Topology

3.1 nPR-Fuzzy Topology

Definition 3.1

Let $ℑ$ be a family of nPR-fuzzy subsets of a non-empty set $A$ . If

$1_{A}, 0_{A} \in ℑ$ where $1_{A} = (1, 0)$ and $0_{A} = (0, 1)$ ,
$ξ_{1} \cap ξ_{2} \in ℑ$ , for any ξ₁, $ξ_{2} \in ℑ$ ,
$\cup_{i \in I} ξ_{i} \in ℑ$ , for any ${ξ_{i}}_{i \in N} \subset ℑ$ ,

then $ℑ$ is called an nPR-fuzzy topology (briefly, nPR-FT) on $A$ and ( $A, ℑ$ ) is an nPR-fuzzy topological space (briefly, nPR-FTS). Each member of $ℑ$ is called an open nPR-FS. We call ξ a closed nPR-FS if its complement is an open nPR-FS.

Now, we give an example of 7PR-FTS.

Example 3.2

Let $A = {w_{1}, w_{2}}$ and ξ_i be 7PR-fuzzy subsets of $A$ such that ω̄_ξ_{_i} and ω_ξ_{_i} are corresponding membership and non-membership functions of ξ_i for each i = 1, 2, 3, 4, 5. If

\begin{array}{l} ϖ_{ξ_{1}} (w_{1}) = 0.52, ω_{ξ_{1}} (w_{1}) = 0.72, \\ ϖ_{ξ_{1}} (w_{2}) = 0.61, ω_{ξ_{1}} (w_{2}) = 0.72, \\ ϖ_{ξ_{2}} (w_{1}) = 0.55, ω_{ξ_{2}} (w_{1}) = 0.68, \\ ϖ_{ξ_{2}} (w_{2}) = 0.62, ω_{ξ_{2}} (w_{2}) = 0.73, \\ ϖ_{ξ_{3}} (w_{1}) = 0.51, ω_{ξ_{3}} (w_{1}) = 0.73, \\ ϖ_{ξ_{3}} (w_{2}) = 0.59, ω_{ξ_{3}} (w_{2}) = 0.74, \\ ϖ_{ξ_{4}} (w_{1}) = 0.55, ω_{ξ_{4}} (w_{1}) = 0.68, \\ ϖ_{ξ_{4}} (w_{2}) = 0.62, ω_{ξ_{4}} (w_{2}) = 0.72, \\ ϖ_{ξ_{5}} (w_{1}) = 0.52, ω_{ξ_{5}} (w_{1}) = 0.72, \\ ϖ_{ξ_{5}} (w_{2}) = 0.61, ω_{ξ_{5}} (w_{2}) = 0.73, \end{array}

then the family of 7PR-fuzzy sets $ℑ = {1_{A}, 0_{A}, ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}, ξ_{5}}$ is 7PR-FT on $A$ .

Remark 3.3

Example 3.4

Let $A = {w_{1}, w_{2}}$ and ξ_i be 2-rung orthopair fuzzy subsets of $A$ such that ω̄_ξ_{_i} and ω_ξ_{_i} are corresponding membership and non-membership functions of ξ_i for each i = 1, 2. If

\begin{array}{l} ϖ_{ξ_{1}} (w_{1}) = 0.96, ω_{ξ_{1}} (w_{1}) = 0.25, \\ ϖ_{ξ_{1}} (w_{2}) = 0.92, ω_{ξ_{1}} (w_{2}) = 0.35, \\ ϖ_{ξ_{2}} (w_{1}) = 0.95, ω_{ξ_{2}} (w_{1}) = 0.25, \\ ϖ_{ξ_{2}} (w_{2}) = 0.91, ω_{ξ_{2}} (w_{2}) = 0.35, \end{array}

then $ℑ = {1_{A}, 0_{A}, ξ_{1}, ξ_{2}}$ is 2-rung orthopair fuzzy topology but $ℑ$ is not 2PR-FT.

Remark 3.5

We call $ℑ = {1_{A}, 0_{A}}$ the indiscrete nPR-FT on $A$ .
If $ℑ$ contains all nPR-fuzzy subsets, then we call $ℑ$ the discrete nPR-FT on $A$ .

Definition 3.6

Let ξ₁ and ξ₂ be two nPR-fuzzy subsets in an nPR-FTS ( $A, ℑ$ ). Then, ξ₂ is called a neighbourhood of ξ₁ if there exists an open nPR-fuzzy subset ξ₃ such that ξ₁ ⊂ ξ₃ ⊂ ξ₂.

Theorem 3.7

An nPR-fuzzy set ξ₁ is open in an nPR-FTS ( $A, ℑ$ ) if and only if it contains a neighbourhood of its each subset.

Proof

3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators

Definition 3.8

Let ( $A, ℑ$ ) be an nPR-FTS and ξ = (ω̄_ξ, ω_ξ) be an nPR-FS in $A$ . Then, the nPR-fuzzy interior and nPR-fuzzy closure of ξ are, respectively, defined by

$i n t (ξ) = \cup {ξ_{2} : ξ_{2} is open nPR-FS in A and ξ_{2} \subset ξ}$ .
$c l (ξ) = \cap {ξ_{1} : ξ_{1} is closed nPR-FS in A and ξ \subset ξ_{1}}$ .

Remark 3.9

Let ( $A, ℑ$ ) be an nPR-FTS and ξ be any nPR-FS in $A$ . Then,

$i n t (ξ)$ is an open nPR-FS.
$c l (ξ)$ is a closed nPR-FS.
$i n t (1_{A}) = c l (1_{A}) = 1_{A}$ and $i n t (0_{A}) = c l (0_{A}) = 0_{A}$ .

Example 3.10

Consider the 7PR-FTS ( $A, ℑ$ ) in Example 3.2. If $ξ = {〈 w_{1}, 0.53, 0.73 〉, 〈 w_{2}, 0.63, 0.74 〉}$ , then $i n t (ξ) = ξ_{3}$ and $c l (ξ) = 1_{A}$ .

Theorem 3.11

Let ( $A, ℑ$ ) be an nPR-FTS and ξ₁, ξ₂ be two nPR-FSs in $A$ . Then,

$i n t (ξ_{1}) \subset ξ_{1} \subset c l (ξ_{1})$ .
$i n t (ξ_{1}) \subset i n t (ξ_{2})$ and $c l (ξ_{1}) \subset c l (ξ_{2})$ if ξ₁ ⊂ ξ₂.
ξ₁ is an open nPR-FS if and only if $ξ_{1} = i n t (ξ_{1})$ .
ξ₁ is a closed nPR-FS if and only if $ξ_{1} = c l (ξ_{1})$ .

Proof

(1) and (2) are obvious.

(3) and (4) follow from Definition 3.8.

Theorem 3.12

Let ( $A, ℑ$ ) be an nPR-FTS and ξ₁, ξ₂ be two nPR-FSs in $A$ . Then,

$i n t (ξ_{1}) \cup i n t (ξ_{2}) \subset i n t (ξ_{1} \cup ξ_{2})$ .
$c l (ξ_{1} \cap ξ_{2}) \subset c l (ξ_{1}) \cap c l (ξ_{2})$ .
$i n t (ξ_{1} \cap ξ_{2}) = i n t (ξ_{1}) \cap i n t (ξ_{2})$ .
$c l (ξ_{1}) \cup c l (ξ_{2}) = c l (ξ_{1} \cup ξ_{2})$ .

Proof

Since ξ₁ ⊂ ξ₁ ∪ ξ₂ and ξ₂ ⊂ ξ₁ ∪ ξ₂, then by Theorem 3.11 (2), we have $i n t (ξ_{1}) \subset i n t (ξ_{1} \cup ξ_{2})$ and $i n t (ξ_{2}) \subset i n t (ξ_{1} \cup ξ_{2})$ . Therefore, $i n t (ξ_{1}) \cup i n t (ξ_{2}) \subset i n t (ξ_{1} \cup ξ_{2}) \cup i n t (ξ_{1} \cup ξ_{2}) = i n t (ξ_{1} \cup ξ_{2})$ .
Since ξ₁∩ξ₂ ⊂ ξ₁ and ξ₁∩ξ₂ ⊂ ξ₂, then by Theorem 3.11 (2), we have $c l (ξ_{1} \cap ξ_{2}) \subset c l (ξ_{1})$ and $c l (ξ_{1} \cap ξ_{2}) \subset c l (ξ_{2})$ . Therefore, $c l (ξ_{1} \cap ξ_{2}) = c l (ξ_{1} \cap ξ_{2}) \cap c l (ξ_{1} \cap ξ_{2}) \subset c l (ξ_{1}) \cap c l (ξ_{2})$ .
Since $i n t (ξ_{1} \cap ξ_{2}) \subset i n t (ξ_{1})$ and $i n t (ξ_{1} \cap ξ_{2}) \subset i n t (ξ_{2})$ , we obtain $i n t (ξ_{1} \cap ξ_{2}) \subset i n t (ξ_{1}) \cap i n t (ξ_{2})$ . On the other hand, from the facts $i n t (ξ_{1}) \subset ξ_{1}$ and $i n t (ξ_{2}) \subset ξ_{2}$ we have $i n t (ξ_{1}) \cap i n t (ξ_{2}) \subset ξ_{1} \cap ξ_{2}$ and $i n t (ξ_{1}) \cap i n t (ξ_{2}) \in ℑ$ ; we see that $i n t (ξ_{1}) \cap i n t (ξ_{2}) \subset i n t (ξ_{1} \cap ξ_{2})$ , and hence $i n t (ξ_{1} \cap ξ_{2}) = i n t (ξ_{1}) \cap i n t (ξ_{2})$ .
Can be proved similar to (3).

Theorem 3.13

Let ( $A, ℑ$ ) be an nPR-FTS and ξ be nPR-FS in $A$ . Then,

$c l (ξ^{c}) = i n t {(ξ)}^{c}$ .
$c l {(ξ^{c})}^{c} = i n t (ξ)$ .
$i n t {(ξ^{c})}^{c} = c l (ξ)$ .
$i n t (ξ^{c}) = c l {(ξ)}^{c}$ .

Proof

Let $ξ = {〈 w, ϖ_{ξ} (w), ω_{ξ} (w) 〉 : w \in A}$ and suppose that the family of open nPR-fuzzy sets contained in ξ are indexed by the family { $〈 w, ϖ_{ξ_{i}} (w), ω_{ξ_{i}} (w) 〉 : i \in J$ }. Then, $i n t (ξ) = {〈 w, ⋁ ϖ_{ξ_{i}} (w), ⋀ ω_{ξ_{i}} (w) 〉}$ and hence

$i n t {(ξ)}^{c} = {〈 w, \sqrt[n^{2}]{⋀ ω_{ξ_{i}} (w)}, {(⋁ ϖ_{ξ_{i}} (w))}^{n^{2}} 〉}$ . Since $ξ^{c} = {〈 w, \sqrt[n^{2}]{ω_{ξ} (w)}, {(ϖ_{ξ} (w))}^{n^{2}} 〉}$ and ω̄_ξ_{_i} ≤ ω̄_ξ, ω_ξ_{_i} ≥ ω_ξ for each i ∈ J, so we obtain that $〈 w, \sqrt[n^{2}]{ω_{ξ_{i}} (w)}, {(ϖ_{ξ_{i}} (w))}^{n^{2}} 〉$ is the family of closed nPR-fuzzy sets containing $ξ^{c}$ , that is,
$c l (ξ^{c}) = {〈 w, \sqrt[n^{2}]{⋀ ω_{ξ_{i}} (w)}, {(⋁ ϖ_{ξ_{i}} (w))}^{n^{2}} 〉} .$
Thus, $c l (ξ^{c}) = i n t {(ξ)}^{c}$ .
${〈 w, \sqrt[n^{2}]{ω_{ξ_{i}} (w)}, {(ϖ_{ξ_{i}} (w))}^{n^{2}} 〉}$ is the family of closed nPR-fuzzy sets containing $ξ^{c}$ , so,
$c l (ξ^{c}) = {〈 m, ⋀ \sqrt[n^{2}]{ω_{ξ_{i}} (m)}, {(⋁ ϖ_{ξ_{i}} (w))}^{n^{2}} 〉}$
and
$\begin{array}{l} c l {(ξ^{c})}^{c} = {〈 m, \sqrt[n^{2}]{(⋁ ϖ_{ξ_{i}} (m))^{n^{2}}}, {(\sqrt[n^{2}]{⋀ ω_{ξ_{i}} (m)})}^{n^{2}} 〉} \\ = {〈 m, ⋁ ϖ_{ξ_{i}} (m), ⋀ ω_{ξ_{i}} (m) 〉} = int (ξ) . \end{array}$
from (1), we can get $c l (ξ) = i n t {(ξ^{c})}^{c}$ .
from (2), we can get $c l {(ξ)}^{c} = i n t (ξ^{c})$ .

3.3 nPR-Fuzzy Connectedness

Definition 3.14

Two nPR-FSs ξ₁ and ξ₂ of an nPR-FTS ( $A, ℑ$ ) are called separated nPR-FSs if $c l (ξ_{1}) ⋀ ξ_{2} = 0_{A}$ and $ξ_{1} ⋁ c l (ξ_{2}) = 0_{A}$ .

If ξ₁ and ξ₂ are separated nPR-FSs, then $ξ_{1} ⋀ ξ_{2} = 0_{A}$ . The converse may not be true in general, as it is shown in the following example.

Example 3.15

Consider the 7PR-FTS ( $A, ℑ$ ) in Example 3.2. If $ξ_{1} = {〈 w_{1}, 0, 1 〉, 〈 w_{2}, 0.64, 0.71 〉}$ and $ξ_{2} = {〈 w_{1}, 0.52, 0.87 〉, 〈 w_{2}, 0, 1 〉}$ , then $ξ_{1} \cap ξ_{2} = 0_{A}$ but ξ₁ and ξ₂ are not separated nPR-FSs.

Corollary 3.16

If ξ₁ and ξ₂ are two nPR-FSs in an nPR-FTS ( $A, ℑ$ ) such that $ξ_{1} \cap ξ_{2} = 0_{A}$ , then $ξ_{1} \subset ξ_{2}^{c}$ .

Proof

From $ξ_{1} \cap ξ_{2} = 0_{A}$ , we get $\min {ϖ_{ξ_{1}}, ϖ_{ξ_{2}}} = 0$ and $\max {ω_{ξ_{1}}, ω_{ξ_{2}}} = 1$ . If the inequality $ϖ_{ξ_{1}} \leq \sqrt[n^{2}]{ω_{ξ_{2}}}$ were not true, then there would exist $w_{o} \in A$ such that $ϖ_{ξ_{1}} (w_{o}) > \sqrt[n^{2}]{ω_{ξ_{2}} (w_{o})}$ , then, since $ϖ_{ξ_{1}} (w_{o}) > 0$ and hence, $ω_{ξ_{1}} (w_{o}) < 1$ , we obtain $ω_{ξ_{2}} (w_{o}) = 1$ which is an obvious contradiction. Similarly, if the inequality ω_ξ_₁ ≥ (ω̄_ξ_₂)^n² were not true, then there would exist $w_{o o} \in A$ such that $ω_{ξ_{1}} (w_{o o}) < {(ϖ_{ξ_{2}} (w_{o o}))}^{n^{2}}$ , then, since $ϖ_{ξ_{2}} (w_{o o}) > 0$ and hence, $ω_{ξ_{2}} (w_{o o}) < 1$ , we obtain $ω_{ξ_{1}} (w_{o o}) = 1$ which is an obvious contradiction. So, we get ω_ξ_₁ ≥ (ω̄_ξ_₂ )^n², and thus $ξ_{1} \subset ξ_{2}^{c}$ .

Theorem 3.17

If $ξ_{1} \neq 0_{A}$ and $ξ_{2} \neq 0_{A}$ are any two nPR-FSs in an nPR-FTS ( $A, ℑ$ ), then the following statements are true:

If ξ₁ and ξ₂ are separated nPR-FSs, ξ₁₁ ⊂ ξ₁ and ξ₂₁ ⊂ ξ₂, then ξ₁₁ and ξ₂₁ are also separated nPR-FSs.
If $ξ_{1} \cap ξ_{2} = 0_{A}$ such that ξ₁ and ξ₂ are both closed nPR-FSs, then ξ₁ and ξ₂ are separated nPR-FSs.

Proof

Since ξ₁₁ ⊂ ξ₁, then $c l (ξ_{11}) \subset c l (ξ_{1})$ . Then, $ξ_{2} \cap c l (ξ_{1}) = 0_{A}$ implies $ξ_{21} \cap c l (ξ_{1}) = 0_{A}$ and so $ξ_{21} \cap c l (ξ_{11}) = 0_{A}$ . Similarly $ξ_{11} \cap c l (ξ_{21}) = 0_{A}$ . Hence, ξ₁₁ and ξ₂₁ are separated nPR-FSs.
Since $ξ_{1} = c l (ξ_{1}), ξ_{2} = c l (ξ_{2})$ and $ξ_{1} \cap ξ_{2} = 0_{A}$ , then $c l (ξ_{1}) \cap ξ_{2} = 0_{A}$ and $c l (ξ_{2}) \cap ξ_{1} = 0_{A}$ . Hence, ξ₁ and ξ₂ are separated nPR-FSs.

Remark 3.18

If $ξ_{1} \cap ξ_{2} = 0_{A}$ such that ξ₁ and ξ₂ are both open nPR-FSs in an nPR-FTS ( $A, ℑ$ ), then ξ₁ and ξ₂ are not separated nPR-FSs in general.

Example 3.19

Let $A = {w_{1}, w_{2}}$ and ξ_i be 4PR-fuzzy subsets of $A$ such that ω̄_ξ_{_i} and ω_ξ_{_i} are corresponding membership and non-membership functions of ξ_i for each i = 1, 2, 3. If

\begin{array}{l} ϖ_{ξ_{1}} (w_{1}) = 0, ω_{ξ_{1}} (w_{1}) = 1, \\ ϖ_{ξ_{1}} (w_{2}) = 0.25, ω_{ξ_{1}} (w_{2}) = 0.86, \\ ϖ_{ξ_{2}} (w_{1}) = 0.26, ω_{ξ_{2}} (w_{1}) = 0.85, \\ ϖ_{ξ_{2}} (w_{2}) = 0, ω_{ξ_{2}} (w_{2}) = 1, \\ ϖ_{ξ_{3}} (w_{1}) = 0.26, ω_{ξ_{3}} (w_{1}) = 0.85, \\ ϖ_{ξ_{3}} (w_{2}) = 0.25, ω_{ξ_{3}} (w_{2}) = 0.86, \end{array}

then $ℑ = {1_{A}, 0_{A}, ξ_{1}, ξ_{2}, ξ_{3}}$ is 4PR-FT and $ξ_{1} \cap ξ_{2} = 0_{A}$ , but ξ₁ and ξ₂ are not separated 4PR-FSs.

Corollary 3.20

If ξ₁ and ξ₂ are separated nPR-FSs in an nPR-FTS ( $A, ℑ$ ), then there exist two open nPR-FSs ξ₃ and ξ₄ in ( $A, ℑ$ ) such that ξ₁ ⊂ ξ₃ and ξ₂ ⊂ ξ₄.

Proof

Let ξ₁ and ξ₂ be separated nPR-FSs. Set $ξ_{4} = c l {(ξ_{1})}^{c}$ and $ξ_{3} = c l {(ξ_{2})}^{c}$ , then ξ₃ and ξ₄ are open nPR-FSs such that ξ₁ ⊂ ξ₃ and ξ₂ ⊂ ξ₄.

Theorem 3.21

Let the following statements in an nPR-FTS ( $A, ℑ$ ),

$0_{A}$ and $1_{A}$ are the only open nPR-FSs and closed nPR-FSs in ( $A, ℑ$ ).
there are no two open nPR-FSs ξ₁ and ξ₂ such that $ξ_{1} \neq 0_{A}, ξ_{2} \neq 0_{A}, ξ_{1} \cap ξ_{2} = 0_{A}$ and $1_{A} = ξ_{1} \cup ξ_{2}$ .
there are no two closed nPR-FSs ξ₁ and ξ₂ such that $ξ_{1} \neq 0_{A}, ξ_{2} \neq 0_{A}, ξ_{1} \cap ξ_{2} = 0_{A}$ and $1_{A} = ξ_{1} \cup ξ_{2}$ .
there are no two separated nPR-FSs ξ₁ and ξ₂ such that $ξ_{1} \neq 0_{A}, ξ_{2} \neq 0_{A}$ and $1_{A} \neq ξ_{1} \cup ξ_{2}$ .

Then, (1) ⇒ (2) ⇔ (3) ⇔ (4) hold.

Proof

(1) ⇒ (2): Suppose (2) is false and that $1_{A} = ξ_{1} \cup ξ_{2}$ , where ξ₁ and ξ₂ are two open nPR-FSs such that $ξ_{1} \neq 0_{A}, ξ_{2} \neq 0_{A}$ and $ξ_{1} \cap ξ_{2} = 0_{A}$ . By Theorems 2.6, 2.11 (2) and Corollary 3.16, we have $ξ_{1}^{c} = ξ_{2}$ and hence ξ₂ is open nPR-FS and closed nPR-FS in ( $A, ℑ$ ) such that $1_{A} \neq ξ_{2} \neq 0_{A}$ , which shows that (1) is false.

(2) ⇔ (3): This is clear.

(3) ⇒ (4): If (4) is false, then $1_{A} = ξ_{1} \cup ξ_{2}$ , where ξ₁ and ξ₂ are two separated nPR-FSs such that $ξ_{1} \neq 0_{A}$ and $ξ_{2} \neq 0_{A}$ . By Theorems 2.11 (2), we have $0_{A} = 1_{A}^{c} = {(ξ_{1} \cup ξ_{2})}^{c} = ξ_{1}^{c} \cap ξ_{2}^{c}$ . By Corollary 3.16 and Theorem 2.6, we get $ξ_{1}^{c} \subset {(ξ_{2}^{c})}^{c} = ξ_{2}$ . Since $c l (ξ_{2}) \cap ξ_{1} = 0_{A}$ , we conclude that $c l (ξ_{2}) \subset ξ_{2}$ so ξ₂ is closed nPR-FS. Similarly, ξ₁ must be closed nPR-FS. Therefore, (3) is false.

(4) ⇒ (3): If (3) is false, then $1_{A} = ξ_{1} \cup ξ_{2}$ , where ξ₁ and ξ₂ are two closed nPR-FSs such that $ξ_{1} \neq 0_{A}, ξ_{2} \neq 0_{A}$ and $ξ_{1} \cap ξ_{2} = 0_{A}$ . Then, ξ₁ and ξ₂ are separated nPR-FSs by Theorem 3.11 (4). Therefore, (4) is false.

Definition 3.22

Let ( $A, ℑ$ ) be an nPR-FTS and ξ be an nPR-FS in $A$ , then ξ is called disconnected nPR-FS if there are two separated nPR-FSs ξ₁ and ξ₂ in $A$ such that $ξ_{1} \neq 0_{A}, ξ_{2} \neq 0_{A}$ and ξ = ξ₁ ∪ ξ₂. If ξ is not disconnected nPR-FS, then ξ is called connected nPR-FS.

Example 3.23

Let $A = {w_{1}, w_{2}}$ and ξ_i be nPR-fuzzy subsets of $A$ such that ω̄_ξ_{_i} and ω_ξ_{_i} are corresponding membership and non-membership functions of ξ_i for each i = 1, 2. If

\begin{array}{l} ϖ_{ξ_{1}} (w_{1}) = 0, ω_{ξ_{1}} (w_{1}) = 1, ϖ_{ξ_{1}} (w_{2}) = 1, ω_{ξ_{1}} (w_{2}) = 0, \\ ϖ_{ξ_{2}} (w_{1}) = 1, ω_{ξ_{2}} (w_{1}) = 0, ϖ_{ξ_{2}} (w_{2}) = 0, ω_{ξ_{2}} (w_{2}) = 1, \end{array}

then $ℑ = {1_{A}, 0_{A}, ξ_{1}, ξ_{2}}$ is nPR-FT and hence $1_{A}$ is disconnected nPR-FS because there are two nPR-FSs ξ₁ and ξ₂ such that $ξ_{1} \neq 0_{A}, ξ_{2} \neq 0_{A}, c l (ξ_{1}) \cap ξ_{2} = 0_{A}, ξ_{1} \cap c l (ξ_{2}) = 0_{A}$ and $1_{A} = ξ_{1} \cup ξ_{2}$ .

Remark 3.24

An nPR-FS $1_{A}$

is connected in every indiscrete nPR-FTS ( $A, ℑ$ ).

Remark 3.25

An nPR-FS $1_{A}$ is connected if any (therefore all) of the conditions (2) – (4) in Theorem 3.21 hold.

Remark 3.26

According to the Definition 3.22 and Remark 3.25, $1_{A}$ is disconnected nPR-FS if we can write $1_{A} = ξ_{1} \cup ξ_{2}$ , where the following (equivalent) statements are true:

ξ₁ and ξ₂ are open nPR-FSs such that $ξ_{1} \neq 0_{A}, ξ_{2} \neq 0_{A}$ and $ξ_{1} \cap ξ_{2} = 0_{A}$ .
ξ₁ and ξ₂ are closed nPR-FSs such that $ξ_{1} \neq 0_{A}, ξ_{2} \neq 0_{A}$ and $ξ_{1} \cap ξ_{2} = 0_{A}$ .
ξ₁ and ξ₂ are separated nPR-FSs such that $ξ_{1} \neq 0_{A}$ and $ξ_{2} \neq 0_{A}$ .

Corollary 3.27

Let ( $A, ℑ$ ) be an nPR-FTS. Then, $1_{A}$ is connected nPR-FS if $0_{A}$ and $1_{A}$ are the only open nPR-FSs and closed nPR-FSs in ( $A, ℑ$ ).

Proof

Follows from Theorem 3.21 and Remark 3.25.

Lemma 3.28

Suppose ξ₁ and ξ₂ are separated nPR-FSs in an nPR-FTS ( $A, ℑ$ ). If ξ ⊂ ξ₁ ∪ ξ₂ and ξ is connected nPR-FS, then ξ ⊂ ξ₁ or ξ ⊂ ξ₂.

Proof

Since ξ ∩ ξ₁ ⊂ ξ₁ and ξ ∩ ξ₂ ⊂ ξ₂, then ξ ∩ ξ₁ and ξ ∩ ξ₂ are separated nPR-FSs and by Theorem 2.10 (1) we have ξ = ξ ∩ (ξ₁ ∪ ξ₂) = (ξ ∩ ξ₁) ∪ (ξ ∩ ξ₂). Since ξ is connected nPR-FS, hence either $ξ \cap ξ_{1} = 0_{A}$ , so ξ ⊂ ξ₂ or $ξ \cap ξ_{2} = 0_{A}$ , so ξ ⊂ ξ₁.

Theorem 3.29

Suppose ξ and ξ_i (i ∈ ℕ) are connected nPR-FSs in an nPR-FTS ( $A, ℑ$ ) and that for each i, ξ_i and ξ are not separated nPR-FSs. Then, ξ ∪ ξ_i is connected nPR-FS.

Proof

Suppose that ξ ∪ ξ_i = ξ₁₁ ∪ ξ₁₂, where ξ₁₁ and ξ₁₂ are separated nPR-FSs such that $ξ_{11} \neq 0_{A}$ and $ξ_{12} \neq 0_{A}$ . By Lemma 3.28, either ξ ⊂ ξ₁₁ or ξ ⊂ ξ₁₂. Without loss of generality, assume ξ ⊂ ξ₁₁. By the same reasoning we conclude that for each i, either ξ_i ⊂ ξ₁₁ or ξ_i ⊂ ξ₁₂. If some ξ_i ⊂ ξ₁₂, then ξ and ξ_i would be separated nPR-FSs. Thus every ξ_i ⊂ ξ₁₁ and so ξ ∪ ξ_i ⊂ ξ₁₁. Therefore, ξ₁₂ ⊂ ξ₁₁ ∪ ξ₁₂ = ξ ∪ ξ_i ⊂ ξ₁₁ and hence $ξ_{12} \subset ξ_{11} \subset c l (ξ_{11})$ , so $ξ_{12} = 0_{A}$ which is contradiction, therefore ξ ∪ ξ_i is connected nPR-FS.

3.4 nPR-Fuzzy Continuous Maps

In this section we study the continuity of a function defined among two nPR-FTSs.

Let $f : A \to B$ be a function. Let ξ₁ and ξ₂ be nPR-fuzzy subsets of $A$ and $B$ , respectively. Then, the membership and non-membership functions of the image of ξ₁ denoted by $f [ξ_{1}]$ are, respectively, calculated by

ϖ_{f [ξ_{1}]} (b) = {\begin{array}{l} s u p_{a \in f^{- 1} (b)} ϖ_{ξ_{1}} (a), & if f^{- 1} (b) \neq φ, \\ 0, & otherwise, \end{array}

and

ω_{f [ξ_{1}]} (b) = {\begin{array}{l} i n f_{a \in f^{- 1} (b)} ω_{ξ_{1}} (a), & if f^{- 1} (b) \neq φ, \\ 1, & otherwise. \end{array}

The membership and non-membership functions of pre-image of ξ₂ denoted by $f^{- 1} [ξ_{2}]$ , are, respectively, calculated by $ϖ_{f^{- 1} [ξ_{2}]} (w) = ϖ_{ξ_{2}} (f (w))$ , and $ϖ_{f^{- 1} [ξ_{2}]} (w) = ϖ_{ξ_{2}} (f (w))$ .

Remark 3.30

To show that $f [ξ_{1}]$ and $f^{- 1} [ξ_{2}]$ are nPR-fuzzy sets. Consider $θ_{ξ_{1}} (a) = {(ϖ_{ξ_{1}} (a))}^{n} + {(ω_{ξ_{1}} (a))}^{\frac{1}{n}}$ and $b \in B$ , then we obtain

\begin{array}{l} {(ϖ_{f [ξ_{1}]} (b))}^{n} + {(ω_{f [ξ_{1}]} (b))}^{\frac{1}{n}} = {(s u p_{a \in f^{- 1} (b)} ϖ_{ξ_{1}} (a))}^{n} + {(i n f_{a \in f^{- 1} (b)} ω_{ξ_{1}} (a))}^{\frac{1}{n}} \\ = s u p_{a \in f^{- 1} (b)} {(ϖ_{ξ_{1}} (a))}^{n} + i n f_{a \in f^{- 1} (b)} {(ω_{ξ_{1}} (a))}^{\frac{1}{n}} \\ = s u p_{a \in f^{- 1} (b)} (θ_{ξ_{1}} (a) - {(ω_{ξ_{1}} (a))}^{\frac{1}{n}}) + i n f_{a \in f^{- 1} (b)} {(ω_{ξ_{1}} (a))}^{\frac{1}{n}} \\ \leq s u p_{a \in f^{- 1} (b)} (1 - {(ω_{ξ_{1}} (a))}^{\frac{1}{n}}) + i n f_{a \in f^{- 1} (b)} {(ω_{ξ_{1}} (a))}^{\frac{1}{n}} = 1, \end{array}

whenever $f^{- 1} (b)$ is non-empty. On the other hand if $f^{- 1} (b) = φ$ , then we have ${(ϖ_{f [ξ_{1}]} (b))}^{n} + {(ω_{f [ξ_{1}]} (b))}^{\frac{1}{n}} = 1$ . Thus, $f [ξ_{1}]$ is an nPR-fuzzy set of $B$ .

Now, let $w \in A$ , then ${(ϖ_{f^{- 1} [ξ_{2}]} (w))}^{n} + {(ω_{f^{- 1} [ξ_{2}]} (w))}^{\frac{1}{n}} = {(ϖ_{ξ_{2}} (f (w)))}^{n} + {(ω_{ξ_{2}} (f (w)))}^{\frac{1}{n}} \leq 1$ . Therefore, $f^{- 1} [ξ_{2}]$ is an nPR-fuzzy set of $A$ .

Following theorems give some properties of image and pre-image.

Theorem 3.31

Let $f : A \to B$ be a function such that ξ₁ and ξ₂ are nPR-fuzzy subsets of $A$ and $B$ , respectively. Then,

$f^{- 1} [ξ_{2}^{c}] = f^{- 1} {[ξ_{2}]}^{c}$ .
$f {[ξ_{1}]}^{c} \subset f [ξ_{1}^{c}]$ .
$f [f^{- 1} [ξ_{2}]] \subset ξ_{2}$ .
$ξ_{1} \subset f^{- 1} [f [ξ_{1}]]$ .

Proof

For any $w \in A$ and for any nPR-fuzzy subset ξ₂ of $B$ we get from the definition of the complement that
$\begin{array}{l} ϖ_{f^{- 1} [ξ_{2}^{c}]} (w) = ϖ_{ξ_{2}^{c}} (f (w)) = \sqrt[n^{2}]{ω_{ξ_{2}} (f (w))} \\ = \sqrt[n^{2}]{ω_{f^{- 1} [ξ_{2}]} (w)} = ϖ_{f^{- 1} {[ξ_{2}]}^{c}} (w) . \end{array}$
Similarly we have
$\begin{array}{l} ω_{f^{- 1} [ξ_{2}^{c}]} (w) = ω_{ξ_{2}^{c}} (f (w)) = {(ϖ_{ξ_{2}} (f (w)))}^{n^{2}} \\ = {(ϖ_{f^{- 1} [ξ_{2}]} (w))}^{n^{2}} = ω_{f^{- 1} {[ξ_{2}]}^{c}} (w) . \end{array}$
Thus, we have $f^{- 1} [ξ_{2}^{c}] = f^{- 1} {[ξ_{2}]}^{c}$ .
For any $b \in B$ such that $f (b) = φ$ and for any nPR-fuzzy subset ξ₁ of $A$ , we can write
(*) $\begin{array}{l} θ_{f [ξ_{1}]} (b) = {(ϖ_{f [ξ_{1}]} (b))}^{n} + {(ω_{f [ξ_{1}]} (b))}^{\frac{1}{n}} \\ = s u p_{a \in f^{- 1} (b)} {(ϖ_{ξ_{1}} (a))}^{n} + i n f_{a \in f^{- 1} (b)} {(ω_{ξ_{1}} (a))}^{\frac{1}{n}} \\ = s u p_{a \in f^{- 1} (b)} (θ_{ξ_{1}} (a) - {(ω_{ξ_{1}} (a))}^{\frac{1}{n}}) + i n f_{a \in f^{- 1} (b)} {(ω_{ξ_{1}} (a))}^{\frac{1}{n}} \\ \leq s u p_{a \in f^{- 1} (b)} θ_{ξ_{1}} (a) - i n f_{a \in f^{- 1} (b)} {(ω_{ξ_{1}} (a))}^{\frac{1}{n}} + i n f_{a \in f^{- 1} (b)} {(ω_{ξ_{1}} (a))}^{\frac{1}{n}} \\ = s u p_{a \in f^{- 1} (b)} θ_{ξ_{1}} (a) . \end{array}$
Now from (*), we have
$\begin{array}{l} ϖ_{f [ξ_{1}^{c}]} (b) = s u p_{a \in f^{- 1} (b)} ϖ_{ξ_{1}^{c}} (a) \\ = s u p_{a \in f^{- 1} (b)} \sqrt[n^{2}]{ω_{ξ_{1}} (a)} \\ = s u p_{a \in f^{- 1} (b)} \sqrt[n]{θ_{ξ_{1}} (a) - {(ϖ_{ξ_{1}} (a))}^{n}} \\ \geq \sqrt[n]{s u p_{a \in f^{- 1} (b)} θ_{ξ_{1}} (a) - s u p_{a \in f^{- 1} (b)} {(ϖ_{ξ_{1}} (a))}^{n}} \\ \geq \sqrt[n]{θ_{f [ξ_{1}]} (b) - {(ϖ_{f [ξ_{1}]} (b))}^{n}} \\ = \sqrt[n^{2}]{ω_{f [ξ_{1}]} (b)} \\ = ϖ_{f {[ξ_{1}]}^{c}} (b) . \end{array}$
Hence, we obtain $ϖ_{f {[ξ_{1}]}^{c}} (b) \leq ϖ_{f [ξ_{1}^{c}]} (b)$ .
Similarly we have
(**) $\begin{array}{l} θ_{f [ξ_{1}]} (b) = {(ϖ_{f [ξ_{1}]} (b))}^{n} + {(ω_{f [ξ_{1}]} (b))}^{\frac{1}{n}} \\ = s u p_{a \in f^{- 1} (b)} {(ϖ_{ξ_{1}} (a))}^{n} + i n f_{a \in f^{- 1} (b)} {(ω_{ξ_{1}} (a))}^{\frac{1}{n}} \\ = s u p_{a \in f^{- 1} (b)} {(ϖ_{ξ_{1}} (a))}^{n} + i n f_{a \in f^{- 1} (b)} (θ_{ξ_{1}} (a) - {(ϖ_{ξ_{1}} (a))}^{n}) \\ \geq s u p_{a \in f^{- 1} (b)} {(ϖ_{ξ_{1}} (a))}^{n} + i n f_{a \in f^{- 1} (b)} θ_{ξ_{1}} (a) + i n f_{a \in f^{- 1} (b)} (- {(ϖ_{ξ_{1}} (a))}^{n}) \\ = i n f_{a \in f^{- 1} (b)} θ_{ξ_{1}} (a) . \end{array}$
Now, (**) which implies that
$\begin{array}{l} ω_{f [ξ_{1}^{c}]} (b) = i n f_{a \in f^{- 1} (b)} ω_{ξ_{1}^{c}} (a) \\ = i n f_{a \in f^{- 1} (b)} {(ϖ_{ξ_{1}} (a))}^{n^{2}} \\ = i n f_{a \in f^{- 1} (b)} {(θ_{ξ_{1}} (a) - {(ω_{ξ_{1}} (a))}^{\frac{1}{n}})}^{n} \\ \leq {(i n f_{a \in f^{- 1} (b)} θ_{ξ_{1}} (a) - i n f_{a \in f^{- 1} (b)} {(ω_{ξ_{1}} (a))}^{\frac{1}{n}})}^{n} \\ \leq {(θ_{f [ξ_{1}]} (b) - {(ω_{f [ξ_{1}]} (b))}^{\frac{1}{n}})}^{n} \\ = {(ϖ_{f [ξ_{1}]} (b))}^{n^{2}} \\ = ω_{f {[ξ_{1}]}^{c}} (b) . \end{array}$
Hence, we obtain $ω_{f [ξ_{1}^{c}]} (b) \leq ω_{f {[ξ_{1}]}^{c}} (b)$ .
On the other hand, the proof is trivial for each $b \in B$ such that $f (b) = φ$ .
Therefore, we obtain $f {[ξ_{1}]}^{c} \subset f [ξ_{1}^{c}]$ .
For any $b \in B$ such that $f (b) = φ$ , we can write
$\begin{array}{l} ϖ_{f [f^{- 1} [ξ_{2}]]} (b) = s u p_{a \in f^{- 1} (b)} ϖ_{f^{- 1} [ξ_{2}]} (a) \\ = s u p_{a \in f^{- 1} (b)} ϖ_{ξ_{2}} (f (a)) \leq ϖ_{ξ_{2}} (b) . \end{array}$
Similarly, we have
$\begin{array}{l} ω_{f [f^{- 1} [ξ_{2}]]} (b) = i n f_{a \in f^{- 1} (b)} ω_{f^{- 1} [ξ_{2}]} (a) \\ = i n f_{a \in f^{- 1} (b)} ω_{ξ_{2}} (f (a)) \geq ω_{ξ_{2}} (b) . \end{array}$
On the other hand if $f (b) = φ$ , then we have $ϖ_{f [f^{- 1} [ξ_{2}]]} (b) = 0 \leq ϖ_{ξ_{2}} (b)$ , and $ω_{f [f^{- 1} [ξ_{2}]]} (b) = 1 \geq ω_{ξ_{2}} (b)$ . Therefore, $f [f^{- 1} [ξ_{2}]] \subset ξ_{2}$ .
For any $w \in A$ , we have $ϖ_{f^{- 1} [f [ξ_{1}]]} (w) = ϖ_{f [ξ_{1}]} (f (w)) = s u p_{a \in f^{- 1} (f (w))} ϖ_{ξ_{1}} (a) \geq ϖ_{ξ_{1}} (w)$ .
Similarly, we have $ω_{f^{- 1} [f [ξ_{1}]]} (w) \leq ω_{ξ_{1}} (w)$ .
Therefore, $ξ_{1} \subset f^{- 1} [f [ξ_{1}]]$ .

Theorem 3.32

Let $f : A \to B$ be a function such that ξ₁_i and ξ₂_i are nPR-fuzzy subsets of $A$ and $B$ , respectively, for i = 1, 2. If

ξ₂₁ ⊂ ξ₂₂, then $f^{- 1} [ξ_{21}] \subset f^{- 1} [ξ_{22}]$ .
ξ₁₁ ⊂ ξ₁₂, then $f [ξ_{11}] \subset f [ξ_{12}]$ .

Proof

Assume that ξ₂₁ ⊂ ξ₂₂. Then, for any $w \in A$ we have
$\begin{array}{l} ϖ_{f^{- 1} [ξ_{21}]} (w) = ϖ_{ξ_{21}} (f (w)) \leq ϖ_{ξ_{22}} (f (w)) \\ = ϖ_{f^{- 1} [ξ_{22}]} (w) . \end{array}$
Thus, $ϖ_{f^{- 1} [ξ_{21}]} (w) \leq ϖ_{f^{- 1} [ξ_{22}]} (w)$ . Similarly, we have
$\begin{array}{l} ω_{f^{- 1} [ξ_{22}]} (w) = ω_{ξ_{22}} (f (w)) \leq ω_{ξ_{21}} (f (w)) \\ = ω_{f^{- 1} [B_{1} ξ_{21}]} (w) . \end{array}$
Thus, $ω_{f^{- 1} [ξ_{21}]} (w) \geq ω_{f^{- 1} [ξ_{22}]} (w)$ . Therefore, $f^{- 1} [ξ_{21}] \subset f^{- 1} [ξ_{22}]$ .
Assume that ξ₁₁ ⊂ ξ₁₂ and $b \in B$ . If $f (b) = φ$ , then the proof is trivial. Assume that $f (b) \neq φ$ . Then, we have
$\begin{array}{l} ϖ_{f [ξ_{11}]} (b) = s u p_{a \in f^{- 1} (b)} ϖ_{ξ_{11}} (a) \\ \leq s u p_{a \in f^{- 1} (b)} ϖ_{ξ_{12}} (a) \\ = ϖ_{f [ξ_{12}]} (b), \end{array}$
and
$\begin{array}{l} ω_{f [ξ_{12}]} (b) = i n f_{a \in f^{- 1} (b)} ω_{ξ_{12}} (a) \\ \leq i n f_{a \in f^{- 1} (b)} ω_{ξ_{11}} (a) \\ = ω_{f [ξ_{11}]} (b) . \end{array}$
Therefore, $f [ξ_{11}] \subset f [ξ_{12}]$ .

The proof of the following result is easy and hence it is omitted.

Corollary 3.33

Let $f : A \to B$ be a function. Then, the following statements are true:

$f [\cup_{i \in I} ξ_{1 i}] = \cup_{i \in I} f [ξ_{1 i}]$ for any nPR-fuzzy subset ξ₁_i of $A$ .
$f^{- 1} [\cup_{i \in I} ξ_{2 i}] = \cup_{i \in I} f^{- 1} [ξ_{2 i}]$ for any nPR-fuzzy subset ξ₂_i of $B$ .
$f [ξ_{11} \cap ξ_{12}] \subset f [ξ_{11}] \cap f [ξ_{12}]$ for any two nPR-fuzzy subsets ξ₁₁ and ξ₁₂ of $A$ .
$f^{- 1} [\cap_{i \in I} ξ_{2 i}] = \cap_{i \in I} f^{- 1} [ξ_{2 i}]$ for any nPR-fuzzy subset ξ₂_i of $B$ .

Definition 3.34

Let ( $A, ℑ_{1}$ ) and ( $B, ℑ_{2}$ ) be two nPR-FTSs and $f : A \to B$ be a function. Then, $f$ is said to be nPR-fuzzy continuous if for any nPR-fuzzy subset ξ₁ of $A$ and for any neighbourhood ξ₄ of $f [ξ_{1}]$ there exists a neighbourhood ξ₃ of ξ₁ such that $f [ξ_{3}] \subset ξ_{4}$ .

Theorem 3.35

Let ( $A, ℑ_{1}$ ) and ( $B, ℑ_{2}$ ) be two nPR-FTSs and $f : A \to B$ be a function. Then, the following statements are equivalent:

$f$ is nPR-fuzzy continuous.
For any nPR-fuzzy subset ξ₁ of $A$ and for any neighbourhood ξ₄ of $f [ξ_{1}]$ , there exists a neighbourhood ξ₃ of ξ₁ such that for any ξ₂ ⊂ ξ₃ we have $f [ξ_{2}] \subset ξ_{4}$ .
For any nPR-fuzzy subset ξ₁ of $A$ and for any neighbourhood ξ₄ of $f [ξ_{1}]$ , there exists a neighbourhood ξ₃ of ξ₁ such that $ξ_{3} \subset f^{- 1} [ξ_{4}]$ .
For any nPR-fuzzy subset ξ₁ of $A$ and for any neighbourhood ξ₄ of $f [ξ_{1}], f^{- 1} [ξ_{4}]$ is a neighbourhood of ξ₁.

Proof

(1) ⇒ (2): Assume that $f$ is nPR-fuzzy continuous. Let ξ₁ be an nPR-fuzzy subset of $A$ and ξ₄ be a neighbourhood of $f [ξ_{1}]$ . Then by (1), there exists a neighbourhood ξ₃ of ξ₁ such that $f [ξ_{3}] \subset ξ_{4}$ . Now, if ξ₂ ⊂ ξ₃, then we get $f [ξ_{2}] \subset f [ξ_{3}] \subset ξ_{4}$ . (2) ⇒ (3): Let ξ₁ be an nPR-fuzzy set of $A$ and ξ₄ be a neighbourhood of $f [ξ_{1}]$ . From (2), there exists a neighbourhood ξ₃ of ξ₁ such that for any ξ₂ ⊂ ξ₃ we have $f [ξ_{2}] \subset ξ_{4}$ . Then, we can write $ξ_{2} \subset f^{- 1} [f [ξ_{2}]] \subset f^{- 1} [ξ_{4}]$ . As ξ₂ is an arbitrary subset of ξ₃, we have $ξ_{3} \subset f^{- 1} [ξ_{4}]$ .

(3) ⇒ (4): Let ξ₁ be an nPR-fuzzy subset of $A$ and ξ₄ be a neighbourhood of $f [ξ_{1}]$ . Then from (3), there exists a neighbourhood ξ₃ of ξ₁ such that $ξ_{3} \subset f^{- 1} [ξ_{4}]$ . Since ξ₃ is a neighbourhood of ξ₁ there exists an open nPR-fuzzy subset ξ₅ of $A$ such that ξ₁ ⊂ ξ₅ ⊂ ξ₃. On the other hand as $ξ_{3} \subset f^{- 1} [ξ_{4}]$ , one can get $ξ_{1} \subset ξ_{5} \subset f^{- 1} [ξ_{4}]$ which implies $f^{- 1} [ξ_{4}]$ is a neighbourhood of ξ₁.

(4) ⇒ (1): Let ξ₁ be an nPR-fuzzy subset of $A$ and ξ₄ be a neighbourhood of $f [ξ_{1}]$ . From the hypothesis, we have $f^{- 1} [ξ_{4}]$ is a neighbourhood of ξ₁. Therefore, there exists an open nPR-fuzzy subset ξ₅ of $A$ such that $ξ_{1} \subset ξ_{5} \subset f^{- 1} [ξ_{4}]$ which implies $f [ξ_{5}] \subset f [f^{- 1} [ξ_{4}]] \subset ξ_{4}$ . Moreover, as ξ₅ is open it is a neighbourhood of ξ₁. Hence, $f$ is nPR-fuzzy continuous.

Theorem 3.36

Let ( $A, I_{1}$ ) and ( $B, I_{2}$ ) be two nPR-FTSs. A function $f : A \to B$ is nPR-fuzzy continuous if and only if for each open nPR-fuzzy subset ξ₂ of $B$ we have $f^{- 1} [ξ_{2}]$ is an open nPR-fuzzy subset of $A$ .

Proof

Assume that $f$ is nPR-fuzzy continuous. Let ξ₂ be an open nPR-fuzzy subset of $B$ and $ξ_{1} \subset f^{- 1} [ξ_{2}]$ . Then, we get $f [ξ_{1}] \subset ξ_{2}$ . Since ξ₂ is open nPR-fuzzy, then by Theorem 3.7, there exists a neighbourhood ξ₄ of $f [ξ_{1}]$ such that ξ₄ ⊂ ξ₂. Thus, nPR-fuzzy continuity of $f$ and (4) of Theorem 3.35 imply that $f^{- 1} [ξ_{4}]$ is a neighbourhood of ξ₁. On the other hand from (1) of Theorem 3.32 we have $f^{- 1} [ξ_{4}] \subset f^{- 1} [ξ_{2}]$ . Therefore, $f^{- 1} [ξ_{2}]$ is a neighbourhood of ξ₁ as well. As ξ₁ is an arbitrary subset of $f^{- 1} [ξ_{2}]$ , then by Theorem 3.7, the nPR-fuzzy subset $f^{- 1} [ξ_{2}]$ is open.

Conversely, let ξ₁ be an nPR-fuzzy subset of $A$ and ξ₄ be a neighbourhood of $f [ξ_{1}]$ . Then, there exists an open nPR-fuzzy subset ξ₆ of $B$ such that $f [ξ_{1}] \subset ξ_{6} \subset ξ_{4}$ . Now, from the hypothesis $f^{- 1} [ξ_{6}]$ is open. On the other hand, we can write $ξ_{1} \subset f^{- 1} [f [ξ_{1}]] \subset f^{- 1} [ξ_{6}] \subset f^{- 1} [ξ_{4}]$ . Hence, $f^{- 1} [ξ_{4}]$ is a neighbourhood of ξ₁. Then by Theorem 3.35 (4), $f$ is nPR-fuzzy continuous.

Proposition 3.37

Let ( $B, I^{*}$ ) be an nPR-FTS and $f : A \to B$ be a function such that $I_{1}$ and $I_{2}$ are two nPR-fuzzy topologies on $A$ . If $I_{2} \subset I_{1}$ and $f : (A, I_{2}) \to (B, I^{*})$ is nPR-fuzzy continuous, then $f : (A, I_{1}) \to (B, I^{*})$ is nPR-fuzzy continuous.

Proof

From Theorem 3.36 and the fact $I_{2} \subset I_{1}$ .

We build the following two examples such that the first one provides an 6PR-fuzzy continuous map, whereas the second one presents a fuzzy map that is not 6PR-fuzzy continuous.

Example 3.38

Consider $A = {w_{1}, w_{2}}$ with the 6PR-FT $I_{2} = {1_{A}, 0_{A}, ξ_{1}}$ and $B = {b_{1}, b_{2}}$ with the 6PR-FT $I_{2} = {1_{B}, 0_{B}, ξ_{2}}$ , where $ξ_{1} = {〈 w_{1}, 0.61, 0.71 〉, 〈 w_{2}, 0.73, 0.31 〉}$ and ξ₂ = {〈b₁, 0.73, 0.31〉, 〈b₂, 0.61, 0.71〉}.

Let $f : A \to B$ defined as follows:

f (w) = {\begin{array}{l} b_{2}, & if w = w_{1}, \\ b_{1}, & if w = m_{2} . \end{array}

Since $1_{B}, 0_{B}$ and ξ₂ are open 6PR-fuzzy subsets of $B$ , then $f^{- 1} [1_{B}] = {〈 w_{1}, 1, 0 〉, 〈 w_{2}, 1, 0 〉}, f^{- 1} [0_{B}] = {〈 w_{1}, 0, 1 〉, 〈 w_{2}, 0, 1 〉}$ and $f^{- 1} [ξ_{2}] = {〈 w_{1}, 0.61, 0.71 〉, 〈 w_{2}, 0.73, 0.31 〉}$ are open 6PR-fuzzy subsets of $A$ . Thus, $f$ is 6PR-fuzzy continuous.

Example 3.39

Consider $A = {w_{1}, w_{2}}$ with the 6PR-FT $I_{1} = {1_{A}, 0_{A}}$ and $B = {b_{1}, b_{2}}$ with the 6PR-FT $I_{2} = {1_{B}, 0_{B}, ξ_{2}}$ , where ξ₂ = {〈b₁, 0.67, 0.54〉, 〈b₂, 0.49, 0.89〉}.

Let $f : A \to B$ defined as follows:

f (w) = {\begin{array}{l} b_{1}, & if w = w_{1}, \\ b_{2}, & if w = m_{2} . \end{array}

Since ξ₂ is open 6PR-fuzzy subset of $B$ , but $f^{- 1} [ξ_{2}] = {〈 w_{1}, 0.67, 0.54 〉, 〈 w_{2}, 0.49, 0.89 〉}$ is not open 6PR-fuzzy subset of $A$ . Thus, $f$ is not 6PR-fuzzy continuous.

Corollary 3.40

A function $f : A \to B$ is nPR-fuzzy continuous if and only if for each closed nPR-fuzzy subset ξ₂ of $B$ we have $f^{- 1} [ξ_{2}]$ is a closed nPR-fuzzy subset of $A$ .

Proof

Assume that ξ₂ is a closed nPR-fuzzy subset of $B$ , then ${(ξ_{2})}^{c}$ is an open nPR-fuzzy set in $B$ . Since $f$ is nPR-fuzzy continuous, then from Theorem 3.36, we get $f^{- 1} [{(ξ_{2})}^{c}]$ is an open nPR-fuzzy subset of $A$ . On the other hand, from Theorem 3.31 (1) we know that $f^{- 1} [{(ξ_{2})}^{c}] = (f^{- 1} {[ξ_{2}]}^{c})$ . Then ${(f^{- 1} [ξ_{2}])}^{c}$ is an open nPR-fuzzy set. Thus, $f^{- 1} [ξ_{2}]$ is a closed nPR-fuzzy subset of $A$ .

Conversely, let ξ be an open nPR-fuzzy subset of $B$ , then ξ^c is a closed nPR-fuzzy set in $B$ . By hypothesis, $f^{- 1} [ξ^{c}]$ is a closed nPR-fuzzy subset of $A$ and hence $f^{- 1} [ξ^{c}] = {(f^{- 1} [ξ])}^{c}$ is closed nPR-fuzzy. Therefore, $f^{- 1} [ξ]$ is an open nPR-fuzzy subset of $A$ and by Theorem 3.36 we get that $f$ is nPR-fuzzy continuous.

Corollary 3.41

The following are equivalent to each other:

$f : (A, I_{1}) \to (B, I_{2})$ is nPR-fuzzy continuous.
$c l (f^{- 1} [ξ_{2}]) \subset f^{- 1} [c l (ξ_{2})]$ for each nPR-fuzzy set ξ₂ in $B$ .
$f^{- 1} [i n t (ξ_{2})] \subset i n t (f^{- 1} [ξ_{2}])$ for each nPR-fuzzy set ξ₂ in $B$ .

Proof

They can be easily proved using Remark 3.9, Theorems 3.11, 3.13, 3.31, 3.32, 3.36 and Corollary 3.40.

Now, we prove that an nPR-FT can be established on $A$ , whenever an nPR-FT on $B$ and a function $f : A \to B$ are given. This topology is the coarsest topology on $A$ .

Theorem 3.42

Let ( $B, I$ ) be an nPR-FTS and $f : A \to B$ be a function. Then, there exists a coarsest nPR-FT $I_{1}$ over $A$ such that $f$ is nPR-fuzzy continuous.

Proof

Let us define a class of nPR-fuzzy subsets $I_{1}$ of $A$ by $I_{1} = {f^{- 1} [ξ_{4}] : ξ_{4} \in I}$ .

We prove that $I_{1}$ is the coarsest nPR-FT over $A$ such that $f$ is continuous.

We can write for any $w \in A$ that
$ϖ_{f^{- 1} [0_{B}]} (w) = ϖ_{0_{B}} (f (w)) = 0 = ϖ_{0_{A}} (w) .$
Similarly, we immediately have $ω_{f^{- 1} [0_{B}]} (w) = ω_{0_{A}} (w)$ for any $w \in A$ which implies $f^{- 1} [0_{B}] = 0_{A}$ . Now, as $0_{B} \in I$ we have $0_{A} = f^{- 1} [0_{B}] \in I_{1}$ . In similar manner, it is easy to see that $1_{A} = f^{- 1} [1_{B}] \in I_{1}$ .
Assume that $ξ_{1}, ξ_{2} \in I_{1}$ . Then, for i = 1, 2 there exists $ξ_{2 i} \in I$ such that $f^{- 1} [ξ_{2 i}] = ξ_{i}$ which implies $ϖ_{f^{- 1} [ξ_{2 i}]} = ϖ_{ξ_{i}}$ and $ω_{f^{- 1} [ξ_{2 i}]} = ω_{ξ_{i}}$ . Thus, we obtain for any $w \in A$ that
$\begin{array}{l} ϖ_{ξ_{1} \cap ξ_{2}} (w) = \min {ϖ_{ξ_{1}} (w), ϖ_{ξ_{2}} (w)} \\ = \min {ϖ_{f^{- 1} [ξ_{21}]} (w), ϖ_{f^{- 1} [ξ_{22}]} (w)} \\ = \min {ϖ_{ξ_{21}} (f (w)), ϖ_{ξ_{22}} (f (w))} \\ = ϖ_{ξ_{21} \cap ξ_{22}} (f (w)) \\ = ϖ_{f^{- 1} [ξ_{21} \cap ξ_{22}]} (w) . \end{array}$
Similarly, it is not difficult to see that $ω_{ξ_{1} \cap ξ_{2}} = ω_{f^{- 1} [ξ_{21} \cap ξ_{22}]}$ . Hence, we get $ξ_{1} \cap ξ_{2} \in I_{1}$ .
Assume that {ξ_i}_i_∈_I is an arbitrary sub-family of $I_{1}$ . Then for any i ∈ I, there exists $ξ_{2 i} \in I_{1}$ such that $f^{- 1} [ξ_{2 i}] = ξ_{i}$ which implies $ϖ_{f^{- 1} [ξ_{2 i}]} = ϖ_{ξ_{i}}$ and $ϖ_{f^{- 1} [ξ_{2 i}]} = ω_{ξ_{i}}$ . Therefore, one can get for any $w \in A$ that
$\begin{array}{l} ϖ_{\cup_{i \in I} ξ_{i}} (w) = s u p_{i \in I^{ϖ} ξ_{i}} (w) \\ = s u p_{i \in I^{ϖ} f^{- 1} [ξ_{2 i}]} (w) \\ = s u p_{i \in I^{ϖ} ξ_{2 i}} (f (w)) \\ = ϖ_{\cup_{i \in I} ξ_{2 i}} (f (w)) \\ = ϖ_{f^{- 1} [\cup_{i \in I} ξ_{2 i}]} (w) . \end{array}$
On the other hand, it is easy to see that $ω_{\cup_{i \in I} ξ_{i}} = ω_{f^{- 1} [\cup_{i \in I} ξ_{2 i}]}$ . Thus, we have $\cup_{i \in I} ξ_{i} \in I_{1}$ .

Hence, $I_{1}$ is an nPR-FT on $A$ , and from Theorem 3.36, $f$ is nPR-fuzzy continuous. Now, we prove that $I_{1}$ is the coarsest nPR-FT over $A$ such that $f$ is nPR-fuzzy continuous. Let $I_{2} \subset I_{1}$ be an nPR-FT over $A$ such that $f$ is nPR-fuzzy continuous. If $ξ_{2} \in I_{1}$ then there exists $ξ_{4} \in I$ such that $f^{- 1} [ξ_{4}] = ξ_{2}$ . Since $f$ is nPR-fuzzy continuous with respect to $I_{2}$ we have $ξ_{2} = f^{- 1} [ξ_{4}] \in I_{2}$ . Hence, we have $I_{2} = I_{1}$ .

The following corollary says that we can construct an nPR-FT on a given set $B$ whenever there exists an nPR-FTS $A$ and a function $f : A \to B$ . This topology is the finest topology that makes the function $f$ is nPR-fuzzy continuous.

Corollary 3.43

Let ( $A, I^{*}$ ) be an nPR-FTS and $f : A \to B$ be a function. Then, there exists a finest nPR-FT $I^{* *}$ over $B$ such that $f$ is nPR-fuzzy continuous.

Proof

Can be proved similar to Theorem 3.42.

3.5 nPR-Fuzzy Separation Axioms

Definition 3.44

Let $A \neq φ$ and $w \in A$ be a fixed element. Suppose ε₁ ∈ (0, 1] and ε₂ ∈ [0, 1) are two fixed real numbers such that $ɛ_{1}^{n} + ɛ_{2}^{\frac{1}{n}} \leq 1$ . Then, an nPR-fuzzy point $χ_{(ɛ_{1}, ɛ_{2})}^{w} = {〈 w, ϖ_{χ} (w), ω_{χ} (w) 〉}$ is defined to be an nPR-fuzzy set of $A$ as follows:

χ_{(ɛ_{1}, ɛ_{2})}^{w} (b) = {\begin{array}{l} (ɛ_{1}, ɛ_{2}), & if b = w, \\ (0, 1), & otherwise, \end{array}

for $b \in A$ . In this case, $w$ is called the support of $χ_{(ɛ_{1}, ɛ_{2})}^{w}$ . An nPR-fuzzy point $χ_{(ɛ_{1}, ɛ_{2})}^{w}$ is said to belong to an nPR-fuzzy set $ξ = {〈 w, ϖ_{ξ} (w), ω_{ξ} (w) 〉}$ of $A$ denoted by $χ_{(ɛ_{1}, ɛ_{2})}^{w} \in ξ$ if $ɛ_{1} \leq ϖ_{ξ} (w)$ and $ɛ_{2} \leq ϖ_{ξ} (w)$ .

Two nPR-fuzzy points are said to be distinct if their supports are distinct.

Remark 3.45

Let $ξ_{1} = {〈 w, ϖ_{ξ_{1}} (w), ω_{ξ_{1}} (w) 〉}$ and $ɛ_{2} = {〈 w, ϖ_{ξ_{2}} (w), ω_{ξ_{2}} (w) 〉}$ be two nPR-fuzzy sets of $A$ . Then, ξ₁ ⊂ ξ₂ if and only if $χ_{(ɛ_{1}, ɛ_{2})}^{w} \in ξ_{1}$ implies $χ_{(ɛ_{1}, ɛ_{2})}^{w} \in ξ_{2}$ for any nPR-fuzzy point $χ_{(ɛ_{1}, ɛ_{2})}^{w}$ in $A$ .

Definition 3.46

Let ε₁, ε₃ ∈ (0, 1], ε₂, ε₄ ∈ [0, 1) and $w, b \in A$ . An nPR-FTS ( $A, I$ ) is said to be:

$T_{0}$ if for each pair of distinct nPR-fuzzy points $χ_{(ɛ_{1}, ɛ_{2})}^{w}, χ_{(ɛ_{3}, ɛ_{4})}^{b}$ in $A$ , there exist two open nPR-fuzzy sets ξ₆ and ξ₅ such that
$ξ_{6} = {〈 w, 1, 0 〉, 〈 b, 0, 1 〉},$
or
$ξ_{5} = {〈 w, 0, 1 〉, 〈 b, 1, 0 〉} .$
$T_{1}$ if for each pair of distinct nPR-fuzzy points $χ_{(ɛ_{1}, ɛ_{2})}^{w}, χ_{(ɛ_{3}, ɛ_{4})}^{b}$ in $A$ , there exist two open nPR-fuzzy sets ξ₆ and ξ₅ such that
$ξ_{6} = {〈 w, 1, 0 〉, 〈 b, 0, 1 〉},$
and
$ξ_{5} = {〈 w, 0, 1 〉, 〈 b, 1, 0 〉} .$

Example 3.47

Consider $A = {w_{1}, w_{2}}$ with the nPR-FT $I = {1_{A}, 0_{A}, ξ_{1}, ξ_{2}}$ , where $ξ_{1} = {〈 w_{1}, 1, 0 〉, 〈 w_{2}, 0, 1 〉}$ and $ξ_{2} = {〈 w_{1}, 0, 1 〉, 〈 w_{2}, 1, 0 〉}$ . Then, ( $A, I$ ) is $T_{o}$ and $T_{1}$ .

Corollary 3.48

Let ( $A, I$ ) be an nPR-FTS. If ( $A, I$ ) is $T_{1}$ , then ( $A, I$ ) is $T_{o}$ .

Proof

The proof is straightforward from the Definition 3.46.

Here is an example which shows that the converse of above corollary is not true in general.

Example 3.49

Consider $A = {w_{1}, w_{2}}$ with the nPR-FT $I = {1_{A}, 0_{A}, ξ}$ , where $ξ = {〈 w_{1}, 1, 0 〉, 〈 w_{2}, 0, 1 〉}$ . Then, ( $A, I$ ) is $T_{0}$ but not $T_{1}$ because there exists no open nPR-fuzzy set ξ₅ such that $ξ_{5} = {〈 w_{1}, 0, 1 〉, 〈 w_{2}, 1, 0 〉}$ .

Theorem 3.50

Let ( $A, I$ ) be an nPR-FTS, ε₁, ε₃ ∈ (0, 1] and ε₂, ε₄ ∈ [0, 1). If ( $A, I$ ) is $T_{o}$ , then for each pair of distinct nPR-fuzzy points $χ_{(ɛ_{1}, ɛ_{2})}^{w}, χ_{(ɛ_{3}, ɛ_{4})}^{b}$ of $A, c l (χ_{(ɛ_{1}, ɛ_{2})}^{w}) \neq c l (χ_{(ɛ_{3}, ɛ_{4})}^{b})$ .

Proof

Let ( $A, I$ ) be $T_{o}$ and $χ_{(ɛ_{1}, ɛ_{2})}^{w}, χ_{(ɛ_{3}, ɛ_{4})}^{b}$ be any two distinct nPR-fuzzy points of $A$ . Then, there exist two open nPR-fuzzy sets ξ₆ and ξ₅ such that

ξ_{6} = {〈 w, 1, 0 〉, 〈 b, 0, 1 〉},

ξ_{5} = {〈 w, 0, 1 〉, 〈 b, 1, 0 〉} .

Let $ξ_{6} = {〈 w, 1, 0 〉, 〈 b, 0, 1 〉}$ be exists. Then, $ξ_{6}^{c} = {〈 w, 0, 1 〉, 〈 b, 1, 0 〉}$ is a closed nPR-fuzzy set which does not contain $χ_{(ɛ_{1}, ɛ_{2})}^{w}$ but contains $χ_{(ɛ_{3}, ɛ_{4})}^{b}$ . Since $c l (χ_{(ɛ_{3}, ɛ_{4})}^{b})$ is the smallest closed nPR-fuzzy set containing $χ_{(ɛ_{3}, ɛ_{4})}^{b}$ , then $c l (χ_{(ɛ_{3}, ɛ_{4})}^{b}) \subset ξ_{6}^{c}$ and therefore $χ_{(ɛ_{1}, ɛ_{2})}^{w} \notin c l (χ_{(ɛ_{3}, ɛ_{4})}^{b})$ . Consequently $c l (χ_{(ɛ_{1}, ɛ_{2})}^{x}) \neq c l (χ_{(ɛ_{3}, ɛ_{4})}^{b})$ .

Theorem 3.51

Let ( $A, I$ ) be an nPR-FTS. Then, ( $A, I$ ) is $T_{1}$ if $χ_{(1, 0)}^{w}$ is closed nPR-fuzzy set for every $w \in A$ .

Proof

Suppose $χ_{(1, 0)}^{w}$ is a closed nPR-fuzzy set for every $w \in A$ . Let $χ_{(ɛ_{1}, ɛ_{2})}^{w}, χ_{(ɛ_{3}, ɛ_{4})}^{b}$ be any two distinct nPR-fuzzy points of $A$ , then $w \neq b$ implies ${χ_{(1, 0)}^{w}}^{c}$ and ${χ_{(1, 0)}^{b}}^{c}$ are two open nPR-fuzzy sets such that

{χ_{(1, 0)}^{b}}^{c} = {〈 w, 1, 0 〉, 〈 b, 0, 1 〉},

and

{χ_{(1, 0)}^{w}}^{c} = {〈 w, 0, 1 〉, 〈 b, 1, 0 〉} .

Thus, ( $A, I$ ) is $T_{1}$ .

4. Conclusions

Abstract
1. Introduction
2. nPR-Fuzzy Sets
3. Some Topological Concepts via nPR-Fuzzy Topology
3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators
3.3 nPR-Fuzzy Connectedness
3.4 nPR-Fuzzy Continuous Maps
3.5 nPR-Fuzzy Separation Axioms
4. Conclusions

Fig 1.

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Figure 1.

Grades spaces of some kinds of nPR-fuzzy sets.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 350-365https://doi.org/10.5391/IJFIS.2022.22.4.350

Fig 2.

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Figure 2.

Comparison of grades space of IFSs, SR-FSs, CR-FSs, 4-ROFSs and 4PR-FSs

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 350-365https://doi.org/10.5391/IJFIS.2022.22.4.350

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SCOPUS 11

International Journalof Fuzzy Logic andIntelligent Systems

Original Article

n Power Root Fuzzy Sets and Its Topology

Abstract

1. Introduction

2. nPR-Fuzzy Sets

Definition 2.1

Remark 2.2

Remark 2.3

Definition 2.4

Example 2.5

Theorem 2.6

Theorem 2.7

Theorem 2.8

Theorem 2.9

Theorem 2.10

Theorem 2.11

Definition 2.12

Example 2.13

3. Some Topological Concepts via nPR-Fuzzy Topology

3.1 nPR-Fuzzy Topology

3.2 nPR-Fuzzy Interior and nPR-Fuzzy Closure Operators

Definition 3.8

Remark 3.9

Example 3.10

Theorem 3.11

Theorem 3.12

Theorem 3.13

3.3 nPR-Fuzzy Connectedness

Definition 3.14

Example 3.15

Corollary 3.16

Theorem 3.17

Remark 3.18

Example 3.19

Corollary 3.20

Theorem 3.21

Definition 3.22

Example 3.23

Remark 3.24

Remark 3.25

Remark 3.26

Corollary 3.27

Lemma 3.28

Theorem 3.29

3.4 nPR-Fuzzy Continuous Maps

Remark 3.30

Theorem 3.31

Proof

Theorem 3.32

Corollary 3.33

Definition 3.34

Theorem 3.35

Theorem 3.36

Proposition 3.37

Example 3.38

Example 3.39

Corollary 3.40

Corollary 3.41

Theorem 3.42

Corollary 3.43

3.5 nPR-Fuzzy Separation Axioms

Definition 3.44

Remark 3.45

Definition 3.46

Example 3.47

Corollary 3.48

Example 3.49

Theorem 3.50

Theorem 3.51

4. Conclusions

Funding:

Conflicts of interest:

Data Availability statement:

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Article

Original Article

n Power Root Fuzzy Sets and Its Topology

International Journal
of Fuzzy Logic and
Intelligent Systems