# Extended Fuzzy Sets and Their Applications

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## Abstract

**:**

## 1. Introduction

- The introduction of the concept of E-FS, as an overarching concept of FS to include the concepts of IFS, PFS and p-ROFS;
- The study of the fundamental principles of E-FSs in comparison to FSs, IFSs, PFSs and p-ROFSs;
- The definition of some algebraic and set operations on E-FSs, including the average and geometric operations on E-FSs;
- The presentation of a score function for E-FSs.

## 2. Fundamental Principles

- A is a classical set (CS) when ${\mu}_{A}$ and ${\nu}_{A}$ range is $\{0,1\}$ and verify the property$${\mu}_{A}\left(x\right)+{\nu}_{A}\left(x\right)=1,\phantom{\rule{1.em}{0ex}}\forall x\in X.$$In this case, the membership function is known as a characteristic function and usually denoted by ${\delta}_{A}$; the non-membership function is uniquely defined by (1).
- A is a fuzzy set (FS) when ${\mu}_{A}$ and ${\nu}_{A}$ range is $[0,1]$ and verify the property$${\mu}_{A}\left(x\right)+{\nu}_{A}\left(x\right)=1,\phantom{\rule{1.em}{0ex}}\forall x\in X.$$Consequently, as with CSs, the non-membership function ${\nu}_{A}$ is uniquely defined from the membership function.
- A is an intuitionistic fuzzy set (IFS) when ${\mu}_{A}$ and ${\nu}_{A}$ range is $[0,1]$ and verify the property$${\mu}_{A}\left(x\right)+{\nu}_{A}\left(x\right)\le 1,\phantom{\rule{1.em}{0ex}}\forall x\in X.$$In this case, given a membership function, multiple non-membership functions verifying (3) may exist. The concept of hesitancy is therefore present in an IFS, which is modelled in this framework via the hesitancy function ${\pi}_{A}=1-\left({\mu}_{A}\left(x\right)+{\nu}_{A}\left(x\right)\right)$. For IFSs, membership and non-membership functions are specifically denoted as ${\mu}_{{A}_{IFS}}$ and ${\nu}_{{A}_{IFS}}$, respectively.
- A is a Pythagorean fuzzy set (PFS) when ${\mu}_{A}$ and ${\nu}_{A}$ range is $[0,1]$ and verify the property$${\mu}_{A}^{2}\left(x\right)+{\nu}_{A}^{2}\left(x\right)\le 1,\phantom{\rule{1.em}{0ex}}\forall x\in X.$$As with IFSs, a membership function may have associated more than one non-membership functions as per (4). For PFSs, membership and non-membership functions are specifically denoted as ${\mu}_{{A}_{PFS}}$ and ${\nu}_{{A}_{PFS}}$, respectively.
- A is a p-rung orthopair fuzzy set (p-ROFS) ($p\ge 1$) when ${\mu}_{A}$ and ${\nu}_{A}$ range is $[0,1]$ and verify the property$${\mu}_{A}^{p}\left(x\right)+{\nu}_{A}^{p}\left(x\right)\le 1,\phantom{\rule{1.em}{0ex}}\forall x\in X.$$

**Definition**

**1.**

- $\odot (x,0)=x$ (boundary condition);
- $\forall x,y,z\in [0,1]$, if $y\le z$, then $\odot (x,y)\le \odot (x,z)$ (monotonicity);
- $\forall x,y\in [0,1]$, $\odot (x,y)=\odot (y,x)$ (commutativity);
- $\forall x,y,z\in [0,1]$, $\odot (x,\odot (y,z\left)\right)=\odot (\odot (x,y),z)$ (associativity).

**Proposition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

## 3. Set and Algebraic Operations on E-FNs

**Definition**

**2.**

**Definition**

**3.**

## 4. Average and Geometric Operators of E-FNs

**Definition**

**4.**

**Theorem**

**1.**

**Theorem 2.**

**Theorem 3.**

**Theorem 4.**

**Definition**

**5.**

## 5. Score Function of E-FNs

**Definition**

**6.**

- $S{c}_{{\odot}_{1}}\left({A}_{E-FN}\right)=1$ if ${A}_{E-FN}=(1,0)$;
- $S{c}_{{\odot}_{1}}\left({A}_{E-FN}\right)=0$ if ${A}_{E-FN}=(0,1)$.

**Theorem**

**5.**

**Theorem**

**6.**

**Lemma**

**1.**

## 6. Decision Making with E-FSs

#### 6.1. A Critical Analysis of all Existing p-ROFS Score Functions

- Yager’s [4] score function:$$S{c}_{Y}\left({A}_{p-ROFS}\right)={\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right);$$
- Wei et al.’s [13] score function:$$S{c}_{W}\left({A}_{p-ROFS}\right)={\displaystyle \frac{1}{2}}\left(1+{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)\right);$$
- Peng et al.’s [14] score function:$$\begin{array}{cc}\hfill S{c}_{PDG}\left({A}_{p-ROFS}\right)={\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)& +\left({\displaystyle \frac{{e}^{{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}}{{e}^{{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}+1}}-{\displaystyle \frac{1}{2}}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times (1-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right));\hfill \end{array}$$
- Mi et al.’s [26] score function:$$S{c}_{MLL}\left({A}_{p-ROFS}\right)={\displaystyle \frac{2+{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}{\left(2-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)+{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)\right)\times \left(2-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)\right)}};$$
- Farhadinia and Liao’s [22] score function:$$S{c}_{FL}\left({A}_{p-ROFS}\right)={\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)+\lambda \left(1-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)\right),\phantom{\rule{1.em}{0ex}}0\le \lambda \le 1.$$
- Peng and Huang’s [27] score function:$$S{c}_{PH}\left({A}_{p-ROFS}\right)={\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)+ln\left(2-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)\right);$$
- Peng and Dai’s [14] score function:$$S{c}_{PD}\left({A}_{p-ROFS}\right)={\displaystyle \frac{{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)-2{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)-1}{3}}+{\displaystyle \frac{\lambda}{3}}({\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)+{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)+2);$$

#### 6.2. E-FS-Based EDAS Technique for MCGDM

**Step****1.**- Construct the individual experts’ evaluation matrices ${A}^{t}={\left[{A}_{{ijt}_{E-FN}}\right]}_{m\times n}$$t=1,2,\dots ,l$.
**Step****2.**- Apply a weighted aggregation operator, for instance operator $E-IFWA$ (16), to compute the E-FN group matrix ${A}_{E-FN}={\left[{A}_{{ij}_{E-FN}}\right]}_{m\times n}$.
**Step****3.**- Compute the following average value for each alternative$${\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{i}=\left(\sqrt[p]{1-{\prod}_{j=1}^{n}{\left(1-{\mu}_{{A}_{E-FN}}^{p}\right)}^{\frac{1}{n}}},\prod _{j=1}^{n}{\left({\nu}_{{A}_{E-FN}}\right)}^{\frac{1}{n}}\right),\phantom{\rule{1.em}{0ex}}i=1,2,\dots ,m.$$
**Step****4.**- Apply $S{c}_{{\odot}_{1}}$ (23) to derive the positive distance (PD) and negative distance (ND) from ${A}_{E-FN}$:$$PD{\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{ij}={\displaystyle \frac{max\left\{0,Sc\left({\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{ij}\right)-\mathbf{Sc}\left({\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{i}\right)\right\}}{\mathbf{Sc}\left({\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{i}\right)}},$$$$ND{\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{ij}={\displaystyle \frac{max\left\{0,\mathbf{Sc}\left({\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{i}\right)-Sc\left({\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{ij}\right)\right\}}{\mathbf{Sc}\left({\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{i}\right)}},\phantom{\rule{1.em}{0ex}}$$
**Step****5.**- Compute the positive weighted distance ${P}_{i}$$(i=1,2,\dots ,m)$ and the negative weighted distance ${N}_{i}$$(i=1,2,\dots ,m)$:$${P}_{i}=\sum _{j=1}^{n}{\omega}_{{C}_{j}}PD{\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{ij},$$$${N}_{i}=\sum _{j=1}^{n}{\omega}_{{C}_{j}}ND{\left({\mu}_{{A}_{E-FN}},{\nu}_{{A}_{E-FN}}\right)}_{ij},$$
**Step****6.**- Normalise ${P}_{i}$$(i=1,2,\dots ,m)$ and ${N}_{i}$$(i=1,2,\dots ,m)$$${\overline{P}}_{i}=\frac{{P}_{i}}{max\{{P}_{1},{P}_{2},\dots ,{P}_{m}\}};$$$${\overline{N}}_{i}=\frac{{N}_{i}}{max\{{N}_{1},{N}_{2},\dots ,{N}_{m}\}}.$$
**Step****7.**- Compute the integrative appraisal scores$$IS{c}_{i}=\frac{1}{2}\left({\overline{P}}_{i}+1-{\overline{N}}_{i}\right),\phantom{\rule{1.em}{0ex}}i=1,2,\dots ,m.$$
**Step****8.**- Produce the ranking of alternatives, with best alternative being the one with maximum $IS{c}_{i}$ value.

#### 6.3. An EDAS-Based Case Study

## 7. Conclusions

- Development of E-FS algebraic and set operations;
- Presentation of E-FS average and geometric aggregating operations;
- Introduction of an E-FS score function.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

A | Classical set (CS) |

${A}_{FS}$ | Fuzzy set (FS) |

${A}_{IFS}$ | Intuitionistic fuzzy set (IFS) |

${A}_{PFS}$ | Pythagorean fuzzy set (PFS) |

${A}_{p-ROFS}$ | p-rung orthopair fuzzy set (p-ROFS) |

${A}_{E-FS}$ | Extended fuzzy set (E-FS) |

${A}_{E-FN}$ | Extended fuzzy number (E-FN) |

## Appendix A. Proofs of Main Results

#### Appendix A.1. Proof of Proposition 1

**Proof.**

**Case****1****(**$\odot :={\odot}_{1}$**):**$$\phantom{\rule{-34.14322pt}{0ex}}\begin{array}{cc}\hfill {\mu}_{{A}_{E-FS}}\left(x\right){\odot}_{1}{\nu}_{{A}_{E-FS}}\left(x\right)& :={\mu}_{{A}_{E-FS}}\left(x\right)+{\nu}_{{A}_{E-FS}}\left(x\right)-{\mu}_{{A}_{E-FS}}\left(x\right){\nu}_{{A}_{E-FS}}\left(x\right)\hfill \\ & =1-(1-{\mu}_{{A}_{E-FS}}\left(x\right))\times (1-{\nu}_{{A}_{E-FS}}\left(x\right))\hfill \\ & =1-\left(1-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)\right)\times \left(1-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)\right)\le 1.\hfill \end{array}$$**Case****2****(**$\odot :={\odot}_{2}$**):**$$\phantom{\rule{-28.45274pt}{0ex}}\begin{array}{c}{\mu}_{{A}_{E-FS}}\left(x\right){\odot}_{2}{\nu}_{{A}_{E-FS}}\left(x\right):={\displaystyle \frac{{\mu}_{{A}_{E-FS}}\left(x\right)+{\nu}_{{A}_{E-FS}}\left(x\right)}{1+{\mu}_{{A}_{E-FS}}\left(x\right){\nu}_{{A}_{E-FS}}\left(x\right)}}\hfill \\ ={\displaystyle \frac{{\mu}_{{A}_{E-FS}}\left(x\right)+{\nu}_{{A}_{E-FS}}\left(x\right)-{\mu}_{{A}_{E-FS}}\left(x\right){\nu}_{{A}_{E-FS}}\left(x\right)+{\mu}_{{A}_{E-FS}}\left(x\right){\nu}_{{A}_{E-FS}}\left(x\right)}{1+{\mu}_{{A}_{E-FS}}\left(x\right){\nu}_{{A}_{E-FS}}\left(x\right)}}\hfill \\ ={\displaystyle \frac{{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)+{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right){\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)+{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right){\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}{1+{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right){\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}}.\hfill \end{array}$$$${\mu}_{{A}_{E-FS}}\left(x\right){\odot}_{2}{\nu}_{{A}_{E-FS}}\left(x\right)\le {\displaystyle \frac{1+{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right){\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}{1+{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right){\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}}=1.$$**Case****3****(**$\odot :={\odot}_{3}$**):**$$\phantom{\rule{-34.14322pt}{0ex}}\begin{array}{c}{\mu}_{{A}_{E-FS}}\left(x\right){\odot}_{3}{\nu}_{{A}_{E-FS}}\left(x\right):=\hfill \\ {\displaystyle \frac{{\mu}_{{A}_{E-FS}}\left(x\right)+{\nu}_{{A}_{E-FS}}\left(x\right)-{\mu}_{{A}_{E-FS}}\left(x\right){\nu}_{{A}_{E-FS}}\left(x\right)-(1-\u03f5){\mu}_{{A}_{E-FS}}\left(x\right){\nu}_{{A}_{E-FS}}\left(x\right)}{1-(1-\u03f5){\mu}_{{A}_{E-FS}}\left(x\right){\nu}_{{A}_{E-FS}}\left(x\right)}}\hfill \\ ={\displaystyle \frac{{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)+{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right){\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)-(1-\u03f5){\mu}_{{A}_{p-ROFS}}^{p}\left(x\right){\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}{1-(1-\u03f5){\mu}_{{A}_{p-ROFS}}^{p}\left(x\right){\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}}.\hfill \end{array}$$Again, applying equation (A1), we deduce that$${\mu}_{{A}_{E-FS}}\left(x\right){\odot}_{3}{\nu}_{{A}_{E-FS}}\left(x\right)\le {\displaystyle \frac{1-(1-\u03f5){\mu}_{{A}_{p-ROFS}}^{p}\left(x\right){\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}{1-(1-\u03f5){\mu}_{{A}_{p-ROFS}}^{p}\left(x\right){\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}}=1$$**Case****4****(**$\odot :={\odot}_{4}$**):**$$\phantom{\rule{-28.45274pt}{0ex}}\begin{array}{cc}\hfill {\mu}_{{A}_{E-FS}}\left(x\right){\odot}_{4}{\nu}_{{A}_{E-FS}}\left(x\right)& :=1-lo{g}_{\u03f5}\left(1+{\displaystyle \frac{\left({\u03f5}^{1-{\mu}_{{A}_{E-FS}}\left(x\right)}-1\right)\left({\u03f5}^{1-{\nu}_{{A}_{E-FS}}\left(x\right)}-1\right)}{\u03f5-1}}\right)\hfill \\ & =1-lo{g}_{\u03f5}\left(1+{\displaystyle \frac{\left({\u03f5}^{1-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)}-1\right)\left({\u03f5}^{1-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}-1\right)}{\u03f5-1}}\right).\hfill \end{array}$$The fact that $\u03f5>1$ gives rise to $\u03f5-1>0$ together with ${\u03f5}^{1-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)}-1\ge 0$ and ${\u03f5}^{1-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}-1\ge 0$ for any ${\mu}_{{A}_{p-ROFS}}^{p}\left(x\right),{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)\le 1$. By taking these outcomes into account, we get $lo{g}_{\u03f5}\left(1+{\displaystyle \frac{\left({\u03f5}^{1-{\mu}_{{A}_{p-ROFS}}^{p}\left(x\right)}-1\right)\left({\u03f5}^{1-{\nu}_{{A}_{p-ROFS}}^{p}\left(x\right)}-1\right)}{\u03f5-1}}\right)\ge lo{g}_{\u03f5}1=0$, and$${\mu}_{{A}_{E-FS}}\left(x\right){\odot}_{4}{\nu}_{{A}_{E-FS}}\left(x\right)\le 1,$$

#### Appendix A.2. Proof of Definition 2 Operations Are Well Defined

**Proof.**

**Proof of****(8).**- It is obvious for any E-FN ${A}_{E-FN}$, we have $0\le {\mu}_{{A}_{E-FS}}{\odot}_{1}{\nu}_{{A}_{E-FS}}\le 1$. Therefore, from the fact that ${\mu}_{{A}_{E-FN}^{c}}{\odot}_{1}{\nu}_{{A}_{E-FN}^{c}}={\nu}_{{A}_{E-FN}}{\odot}_{1}{\mu}_{{A}_{E-FN}}$, we conclude that $0\le {\mu}_{{A}_{E-FS}^{c}}{\odot}_{1}{\nu}_{{A}_{E-FS}^{c}}\le 1$, that is, ${A}_{E-FN}^{c}=({\mu}_{{A}_{E-FN}^{c}},{\nu}_{{A}_{E-FN}^{c}})$ is an E-FN.
**Proof of****(9).**- Since ${A}_{E-FN}$ and ${B}_{E-FN}$ are two E-FNs, that is,$$\begin{array}{cc}\hfill 0\le {\mu}_{{A}_{E-FS}}{\odot}_{1}{\nu}_{{A}_{E-FS}}:=& {\mu}_{{A}_{E-FS}}+{\nu}_{{A}_{E-FS}}-{\mu}_{{A}_{E-FS}}{\nu}_{{A}_{E-FS}}\hfill \\ \hfill =& 1-[1-{\mu}_{{A}_{E-FS}}]\times [1-{\nu}_{{A}_{E-FS}}]\le 1\hfill \end{array}$$$$\begin{array}{cc}\hfill 0\le {\mu}_{{B}_{E-FS}}{\odot}_{1}{\nu}_{{B}_{E-FS}}:=& {\mu}_{{B}_{E-FS}}+{\nu}_{{B}_{E-FS}}-{\mu}_{{B}_{E-FS}}{\nu}_{{B}_{E-FS}}\hfill \\ \hfill =& 1-[1-{\mu}_{{B}_{E-FS}}]\times [1-{\nu}_{{B}_{E-FS}}]\le 1\hfill \end{array}$$$$\phantom{\rule{-5.69046pt}{0ex}}\begin{array}{c}{\mu}_{{A}_{E-FN}\cap {B}_{E-FN}}{\odot}_{1}{\nu}_{{A}_{E-FN}\cup {B}_{E-FN}}=min\{{\mu}_{{A}_{E-FN}},{\mu}_{{B}_{E-FN}}\}{\odot}_{1}max\{{\nu}_{{A}_{E-FN}},{\nu}_{{B}_{E-FN}}\}\hfill \\ \phantom{\rule{56.9055pt}{0ex}}=1-\left[1-min\{{\mu}_{{A}_{E-FN}},{\mu}_{{B}_{E-FN}}\}\right]\times \left[1-max\{{\mu}_{{A}_{E-FN}},{\mu}_{{B}_{E-FN}}\}\right]\hfill \\ \phantom{\rule{56.9055pt}{0ex}}=1-max\left\{1-{\mu}_{{A}_{E-FN}},1-{\mu}_{{B}_{E-FN}}\right\}\times min\left\{1-{\mu}_{{A}_{E-FN}},1-{\mu}_{{B}_{E-FN}}\right\}.\hfill \end{array}$$By taking the non-negativity property of all the terms $1-{\mu}_{{A}_{E-FN}}$, $1-{\mu}_{{B}_{E-FN}}$, $1-{\mu}_{{A}_{E-FN}}$ and $1-{\mu}_{{B}_{E-FN}}$ into consideration, we are able to get that$$\begin{array}{c}0\le {\mu}_{{A}_{E-FN}\cap {B}_{E-FN}}{\odot}_{1}{\nu}_{{A}_{E-FN}\cup {B}_{E-FN}}\hfill \\ \phantom{\rule{42.67912pt}{0ex}}=1-max\left\{1-{\mu}_{{A}_{E-FN}},1-{\mu}_{{B}_{E-FN}}\right\}\times min\left\{1-{\mu}_{{A}_{E-FN}},1-{\mu}_{{B}_{E-FN}}\right\}\le 1.\hfill \end{array}$$
**Proof of****(10).**- The proof is much like that of (9).
**Proof of****(11).**- Follows from definition ${A}_{E-FN}\oplus {B}_{E-FN}$, we conclude that$$\phantom{\rule{-28.45274pt}{0ex}}\begin{array}{c}{\mu}_{{A}_{E-FN}\oplus {B}_{E-FN}}{\odot}_{1}{\nu}_{{A}_{E-FN}\oplus {B}_{E-FN}}=\left[1-(1-{\mu}_{{A}_{E-FN}})(1-{\mu}_{{B}_{E-FN}})\right]{\odot}_{1}\left[{\nu}_{{A}_{E-FN}}{\nu}_{{B}_{E-FN}}\right]\hfill \\ \phantom{\rule{72.26999pt}{0ex}}=1-\left[1-\left[1-\left(1-{\mu}_{{A}_{E-FN}}\right)\left(1-{\mu}_{{B}_{E-FN}}\right)\right]\right]\times \left[1-\left[{\nu}_{{A}_{E-FN}}{\nu}_{{B}_{E-FN}}\right]\right]\hfill \\ \phantom{\rule{72.26999pt}{0ex}}=1-\left(1-{\mu}_{{A}_{E-FN}}\right)\left(1-{\mu}_{{B}_{E-FN}}\right)\times \left(1-{\nu}_{{A}_{E-FN}}{\nu}_{{B}_{E-FN}}\right).\hfill \end{array}$$Again, the non-negativity property of all the terms $1-{\mu}_{{A}_{E-FN}}$, $1-{\mu}_{{B}_{E-FN}}$ and $1-{\nu}_{{A}_{E-FN}}{\nu}_{{B}_{E-FN}}$ gives rise to$$\begin{array}{c}0\le {\mu}_{{A}_{E-FN}\oplus {B}_{E-FN}}{\odot}_{1}{\nu}_{{A}_{E-FN}\oplus {B}_{E-FN}}\hfill \\ \phantom{\rule{72.26999pt}{0ex}}=1-(1-{\mu}_{{A}_{E-FN}})(1-{\mu}_{{B}_{E-FN}})\times [1-{\nu}_{{A}_{E-FN}}{\nu}_{{B}_{E-FN}}]\le 1.\hfill \end{array}$$
**Proof of****(12).**- The proof is much like that of (11).
**Proof of****(13).**- The proof is immediate from the fact that$$\begin{array}{cc}\hfill 0\le {\mu}_{\lambda {A}_{E-FN}}{\odot}_{1}{\nu}_{\lambda {A}_{E-FN}}& =1-\left(1-\left[1-{\left(1-{\mu}_{{A}_{E-FN}}\right)}^{\lambda}\right]\right)\times \left[1-{\left({\nu}_{{A}_{E-FN}}\right)}^{\lambda}\right]\hfill \\ & =1-{\left(1-{\mu}_{{A}_{E-FN}}\right)}^{\lambda}\times \left(1-{\left({\nu}_{{A}_{E-FN}}\right)}^{\lambda}\right)\le 1.\hfill \end{array}$$
**Proof of****(14).**- The proof is much like that of (13).

#### Appendix A.3. Proof of Theorem 1

**Proof.**

#### Appendix A.4. Proof of Theorem 2

**Proof.**

#### Appendix A.5. Proof of Theorem 3

**Proof.**

#### Appendix A.6. Proof of Theorem 4

**Proof.**

#### Appendix A.7. Proof of Theorem 5

**Proof.**

#### Appendix A.8. Proof of Theorem 6

**Proof.**

#### Appendix A.9. Proof of Lemma 1

**Proof.**

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**Table 1.**The ranking of $p-ROFS$$A=(0.3,0.3)$ and $B=(0.2,0.2)$ with existing score functions with $p=1$.

Score Function | A–Score | B–Score | Ranking |
---|---|---|---|

$S{c}_{Y}$ | 0 | 0 | $A=B$ |

$S{c}_{W}$ | $0.5$ | $0.5$ | $A=B$ |

$S{c}_{PDG}$ | 0 | 0 | $A=B$ |

$S{c}_{MLL}$ | $0.7143$ | $0.625$ | $A>B$ |

$S{c}_{FL}\phantom{\rule{3.33333pt}{0ex}}(\lambda =1)$ | $0.7$ | $0.8$ | $A<B$ |

$S{c}_{PH}$ | $-0.3365$ | $-0.47$ | $A>B$ |

$S{c}_{PD}$ | 0 | 0 | $A=B$ |

**Table 2.**The ranking of $p-ROFS$$A=(0.3,0.3)$ and $B=(0.2,0.2)$ with existing score functions with $p=2$.

Score Function | A–Score | B–Score | Ranking |
---|---|---|---|

$S{c}_{Y}$ | $A=B$ | ||

$S{c}_{W}$ | $0.5$ | $0.5$ | $A=B$ |

$S{c}_{PDG}$ | 0 | 0 | $A=B$ |

$S{c}_{MLL}$ | $0.5495$ | $0.5208$ | $A>B$ |

$S{c}_{FL}\phantom{\rule{3.33333pt}{0ex}}(\lambda =1)$ | $0.91$ | $0.96$ | $A<B$ |

$S{c}_{PH}$ | $-0.5988$ | $-0.6523$ | $A>B$ |

$S{c}_{PD}$ | 0 | 0 | $A=B$ |

**Table 3.**The ranking of $p-ROFS$$A=(0.3,0.3)$ and $B=(0.2,0.2)$ with existing score functions with $p=3$.

Score Function | A–Score | B–Score | Ranking |
---|---|---|---|

$S{c}_{Y}$ | 0 | 0 | $A=B$ |

$S{c}_{W}$ | $0.5$ | $0.5$ | $A=B$ |

$S{c}_{PDG}$ | 0 | 0 | $A=B$ |

$S{c}_{MLL}$ | $0.5139$ | $0.504$ | $A>B$ |

$S{c}_{FL}\phantom{\rule{3.33333pt}{0ex}}(\lambda =1)$ | $0.973$ | $0.992$ | $A<B$ |

$S{c}_{PH}$ | $-0.6658$ | $-0.6851$ | $A>B$ |

$S{c}_{PD}$ | 0 | 0 | $A=B$ |

**Table 4.**The ranking of $p-ROFS$$A=(0.3,0.3)$ and $B=(0.2,0.2)$ as extended fuzzy sets (E-FSs) with new score function.

${\mathbf{Sc}}_{\mathbf{FC}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathit{\lambda}\right)$ | A–Score | B–Score | Ranking |
---|---|---|---|

$S{c}_{FC}\phantom{\rule{3.33333pt}{0ex}}(\lambda =1)$ | $0.21$ | $0.16$ | $A>B$ |

$S{c}_{FC}\phantom{\rule{3.33333pt}{0ex}}(\lambda =1/2)$ | $0.455$ | $0.48$ | $A<B$ |

$S{c}_{FC}\phantom{\rule{3.33333pt}{0ex}}(\lambda =0)$ | $0.7$ | $0.8$ | $A<B$ |

${\mathbf{Sc}}_{\mathbf{FC}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathit{\lambda}\right)$ | A–Score | B–Score | Ranking |
---|---|---|---|

$S{c}_{FC}\phantom{\rule{3.33333pt}{0ex}}(\lambda =1)$ | $0.18$ | $0.16$ | $A>B$ |

$S{c}_{FC}\phantom{\rule{3.33333pt}{0ex}}(\lambda =1/2)$ | $0.19$ | $0.48$ | $A<B$ |

$S{c}_{FC}\phantom{\rule{3.33333pt}{0ex}}(\lambda =0)$ | $0.2$ | $0.8$ | $A<B$ |

Score Function | ${\left({\mathit{\mu}}_{\mathit{A}},{\mathit{\nu}}_{\mathit{A}}\right)}_{1}$ | ${\left({\mathit{\mu}}_{\mathit{A}},{\mathit{\nu}}_{\mathit{A}}\right)}_{2}$ | ${\left({\mathit{\mu}}_{\mathit{A}},{\mathit{\nu}}_{\mathit{A}}\right)}_{3}$ | ${\left({\mathit{\mu}}_{\mathit{A}},{\mathit{\nu}}_{\mathit{A}}\right)}_{4}$ | ${\left({\mathit{\mu}}_{\mathit{A}},{\mathit{\nu}}_{\mathit{A}}\right)}_{5}$ | ${\left({\mathit{\mu}}_{\mathit{A}},{\mathit{\nu}}_{\mathit{A}}\right)}_{6}$ |
---|---|---|---|---|---|---|

${\mathbf{Sc}}_{Y}$ | 0.1228 | 0.1284 | 0.1469 | 0.1147 | 0.0668 | 0.1004 |

${\mathbf{Sc}}_{W}$ | 0.5614 | 0.5642 | 0.5735 | 0.5574 | 0.5334 | 0.5502 |

${\mathbf{Sc}}_{MLL}$ | 0.6242 | 0.6342 | 0.6441 | 0.6182 | 0.5761 | 0.6017 |

${\mathbf{Sc}}_{PDG}$ | 0.1477 | 0.1539 | 0.1762 | 0.1381 | 0.0811 | 0.1214 |

${\mathbf{Sc}}_{FL}$ | 0.9672 | 0.9608 | 0.9729 | 0.9647 | 0.9613 | 0.9691 |

${\mathbf{Sc}}_{PH}$ | −0.4715 | −0.4555 | −0.4402 | −0.4811 | −0.5515 | −0.5081 |

${\mathbf{Sc}}_{PD}$ | 0.0614 | 0.0642 | 0.0735 | 0.0574 | 0.0334 | 0.0502 |

${\mathbf{Sc}}_{FC}\phantom{\rule{3.33333pt}{0ex}}(\lambda =1)$ | 0.3657 | 0.3641 | 0.3906 | 0.3570 | 0.3127 | 0.3488 |

Score-Based EDAS Method | ${\mathbf{ISc}}_{1}$ | ${\mathbf{ISc}}_{2}$ | ${\mathbf{ISc}}_{3}$ | ${\mathbf{ISc}}_{4}$ | ${\mathbf{ISc}}_{5}$ | Ranking Order |
---|---|---|---|---|---|---|

$S{c}_{Y}$ (Equation (25)) | 0.392 | 0.95 | 0.3776 | 0.1382 | 0.0868 | ${R}_{2}>{R}_{1}>{R}_{3}>{R}_{4}>{R}_{5}$ |

$S{c}_{W}$ (Equation (26)) | 0.3489 | 0.9387 | 0.3317 | 0.2326 | 0.013 | ${R}_{2}>{R}_{1}>{R}_{3}>{R}_{4}>{R}_{5}$ |

$S{c}_{MLL}$ (Equation (28)) | 0.3067 | 0.9874 | 0.2584 | 0.2657 | 0.0096 | ${R}_{2}>{R}_{1}>{R}_{4}>{R}_{3}>{R}_{5}$ |

$S{c}_{PDG}$ (Equation (27)) | 0.3944 | 0.9488 | 0.3829 | 0.1419 | 0.0902 | ${R}_{2}>{R}_{1}>{R}_{3}>{R}_{4}>{R}_{5}$ |

$S{c}_{FL}$ (Equation (29)) | 0.4911 | 0.0733 | 0.876 | 0.3131 | 0.2953 | ${R}_{3}>{R}_{1}>{R}_{4}>{R}_{5}>{R}_{2}$ |

$S{c}_{PH}$ (Equation (30)) | −6.6229 | 23.0165 | −9.6394 | −6.9174 | −20.1849 | ${R}_{2}>{R}_{1}>{R}_{4}>{R}_{3}>{R}_{5}$ |

$S{c}_{PD}$ (Equation (31)) | 0.392 | 0.95 | 0.3776 | 0.1382 | 0.0868 | ${R}_{2}>{R}_{1}>{R}_{3}>{R}_{4}>{R}_{5}$ |

$S{c}_{FC}\phantom{\rule{3.33333pt}{0ex}}(\lambda =1)$ | 0.3408 | 0.9132 | 0.4978 | 0.3201 | 0.0233 | ${R}_{2}>{R}_{3}>{R}_{1}>{R}_{4}>{R}_{5}$ |

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**MDPI and ACS Style**

Farhadinia, B.; Chiclana, F.
Extended Fuzzy Sets and Their Applications. *Mathematics* **2021**, *9*, 770.
https://doi.org/10.3390/math9070770

**AMA Style**

Farhadinia B, Chiclana F.
Extended Fuzzy Sets and Their Applications. *Mathematics*. 2021; 9(7):770.
https://doi.org/10.3390/math9070770

**Chicago/Turabian Style**

Farhadinia, Bahram, and Francisco Chiclana.
2021. "Extended Fuzzy Sets and Their Applications" *Mathematics* 9, no. 7: 770.
https://doi.org/10.3390/math9070770