# Ranking of Downstream Fish Passage Designs for a Hydroelectric Project under Spherical Fuzzy Bipolar Soft Framework

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## Abstract

**:**

## 1. Introduction

- The properties, operations, and results of the proposed model are provided and supported with illustrative examples.
- An algorithm for SFBSSs is provided to deal with MADM problems efficiently.
- A reality-based problem, i.e., the ranking of different downstream fish passage designs for a new hydroelectric project is modeled and solved using the initiated algorithm based on SFBSSs.

#### 1.1. Literature Review

#### 1.2. Organization

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

- 1.
- $\mathcal{F}\subseteq \mathcal{G}$ if $\forall \phantom{\rule{3.33333pt}{0ex}}\mathfrak{u}\in \mathcal{U}$, ${\mu}_{\mathcal{F}}\left(\mathfrak{u}\right)\le {\mu}_{\mathcal{G}}\left(\mathfrak{u}\right),{\tau}_{\mathcal{F}}\left(\mathfrak{u}\right)\le {\tau}_{\mathcal{G}}\left(\mathfrak{u}\right)$ and ${\nu}_{\mathcal{F}}\left(\mathfrak{u}\right)\ge {\nu}_{\mathcal{G}}\left(\mathfrak{u}\right).$
- 2.
- $\mathcal{F}=\mathcal{G}$⇔$\mathcal{F}\subseteq \mathcal{G}$ and $\mathcal{G}\subseteq \mathcal{F}.$
- 3.
- $\mathcal{F}\cup \mathcal{G}=\left\{(\mathfrak{u},max\{{\mu}_{\mathcal{F}}\left(\mathfrak{u}\right),{\mu}_{\mathcal{G}}\left(\mathfrak{u}\right)\},min\{{\tau}_{\mathcal{F}}\left(\mathfrak{u}\right),{\tau}_{\mathcal{G}}\left(\mathfrak{u}\right)\},min\{{\nu}_{\mathcal{F}}\left(\mathfrak{u}\right),{\nu}_{\mathcal{G}}\left(\mathfrak{u}\right)\})\right|\mathfrak{u}\in \mathcal{U}\}.$
- 4. .
- $\mathcal{F}\cap \mathcal{G}=\left\{(\mathfrak{u},min\{{\mu}_{\mathcal{F}}\left(\mathfrak{u}\right),{\mu}_{\mathcal{G}}\left(\mathfrak{u}\right)\},min\{{\tau}_{\mathcal{F}}\left(\mathfrak{u}\right),{\tau}_{\mathcal{G}}\left(\mathfrak{u}\right)\},max\{{\nu}_{\mathcal{F}}\left(\mathfrak{u}\right),{\nu}_{\mathcal{G}}\left(\mathfrak{u}\right)\})\right|\mathfrak{u}\in \mathcal{U}\}.$
- 5. .
- ${\left(\mathcal{F}\right)}^{c}=\left\{(\mathfrak{u},{\nu}_{\mathcal{F}}\left(\mathfrak{u}\right),{\tau}_{\mathcal{F}}\left(\mathfrak{u}\right),{\mu}_{\mathcal{F}}\left(\mathfrak{u}\right))\right|\mathfrak{u}\in \mathcal{U}\}.$

**Definition**

**3**

**Definition**

**4**

**Definition**

**5**

- 1.
- Mid-level Threshold Function ($mi{d}_{\mathsf{{\rm Y}}}$):The function $mi{d}_{\mathsf{{\rm Y}}}:\mathcal{A}\to {[0,1]}^{3}$ for the SFSS $\mathsf{{\rm Y}}=(\mathcal{F},\mathcal{A})$ is given by$$mi{d}_{\mathsf{{\rm Y}}}\left(\alpha \right)=\left({\mathtt{p}}_{mi{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right),{\mathtt{q}}_{mi{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right),{\mathtt{r}}_{mi{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right)\right)\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}\alpha \in \mathcal{A},$$$${\mathtt{p}}_{mi{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\frac{1}{\left|\mathcal{U}\right|}\sum _{\mathfrak{u}\in \mathcal{U}}{\mu}_{\mathcal{F}\left(\alpha \right)}\left(\mathfrak{u}\right);\phantom{\rule{3.33333pt}{0ex}}{\mathtt{q}}_{mi{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\frac{1}{\left|\mathcal{U}\right|}\sum _{\mathfrak{u}\in \mathcal{U}}{\tau}_{\mathcal{F}\left(\alpha \right)}\left(\mathfrak{u}\right),$$$${\mathtt{r}}_{mi{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\frac{1}{\left|\mathcal{U}\right|}\sum _{\mathfrak{u}\in \mathcal{U}}{\nu}_{\mathcal{F}\left(\alpha \right)}\left(\mathfrak{u}\right).$$
- 2.
- Top-bottom-bottom-level Threshold Function ($tb{b}_{\mathsf{{\rm Y}}}$):The function $tb{b}_{\mathsf{{\rm Y}}}:\mathcal{A}\to {[0,1]}^{3}$ for the SFSS $\mathsf{{\rm Y}}=(\mathcal{F},\mathcal{A})$ is given by$$tb{b}_{\mathsf{{\rm Y}}}\left(\alpha \right)=\left({\mathtt{p}}_{tb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right),{\mathtt{q}}_{tb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right),{\mathtt{r}}_{tb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right)\right)\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}\alpha \in \mathcal{A},$$$${\mathtt{p}}_{tb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\underset{\mathfrak{u}\in \mathcal{U}}{max}\phantom{\rule{3.33333pt}{0ex}}{\mu}_{\mathcal{F}\left(\alpha \right)}\left(\mathfrak{u}\right);\phantom{\rule{3.33333pt}{0ex}}{\mathtt{q}}_{tb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\underset{\mathfrak{u}\in \mathcal{U}}{min}\phantom{\rule{3.33333pt}{0ex}}{\tau}_{\mathcal{F}\left(\alpha \right)}\left(\mathfrak{u}\right);\phantom{\rule{3.33333pt}{0ex}}{\mathtt{r}}_{tb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\underset{\mathfrak{u}\in \mathcal{U}}{min}\phantom{\rule{3.33333pt}{0ex}}{\nu}_{\mathcal{F}\left(\alpha \right)}\left(\mathfrak{u}\right).$$
- 3.
- Bottom-bottom-bottom-level Threshold Function ($bb{b}_{\mathsf{{\rm Y}}}$):The function $bb{b}_{\mathsf{{\rm Y}}}:\mathcal{A}\to {[0,1]}^{3}$ for the SFSS $\mathsf{{\rm Y}}=(\mathcal{F},\mathcal{A})$ is given by$$bb{b}_{\mathsf{{\rm Y}}}\left(\alpha \right)=\left({\mathtt{p}}_{bb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right),{\mathtt{q}}_{bb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right),{\mathtt{r}}_{bb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right)\right)\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}\alpha \in \mathcal{A},$$$${\mathtt{p}}_{bb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\underset{\mathfrak{u}\in \mathcal{U}}{min}\phantom{\rule{3.33333pt}{0ex}}{\mu}_{\mathcal{F}\left(\alpha \right)}\left(\mathfrak{u}\right);\phantom{\rule{3.33333pt}{0ex}}{\mathtt{q}}_{bb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\underset{\mathfrak{u}\in \mathcal{U}}{min}\phantom{\rule{3.33333pt}{0ex}}{\tau}_{\mathcal{F}\left(\alpha \right)}\left(\mathfrak{u}\right);\phantom{\rule{3.33333pt}{0ex}}{\mathtt{r}}_{bb{b}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\underset{\mathfrak{u}\in \mathcal{U}}{min}\phantom{\rule{3.33333pt}{0ex}}{\nu}_{\mathcal{F}\left(\alpha \right)}\left(\mathfrak{u}\right).$$
- 4.
- Med-level Threshold Function ($me{d}_{\mathsf{{\rm Y}}}$):The function $me{d}_{\mathsf{{\rm Y}}}:\mathcal{A}\to {[0,1]}^{3}$ for the SFSS $\mathsf{{\rm Y}}=(\mathcal{F},\mathcal{A})$ is given by$$me{d}_{\mathsf{{\rm Y}}}\left(\alpha \right)=\left({\mathtt{p}}_{me{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right),{\mathtt{q}}_{me{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right),{\mathtt{r}}_{me{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right)\right)\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}\alpha \in \mathcal{A},$$$$\begin{array}{cc}\hfill {\mathtt{p}}_{me{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\phantom{\rule{1.em}{0ex}}& \left\{\begin{array}{cc}{\mu}_{\mathcal{F}\left(\alpha \right)}\left({\mathfrak{u}}_{\left(\frac{\left|\mathcal{U}\right|+1}{2}\right)}\right),\hfill & \phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}\left|\mathcal{U}\right|\phantom{\rule{3.33333pt}{0ex}}is\phantom{\rule{3.33333pt}{0ex}}odd,\hfill \\ {\displaystyle \frac{{\mu}_{\mathcal{F}\left(\alpha \right)}\left({\mathfrak{u}}_{\left(\frac{\left|\mathcal{U}\right|}{2}\right)}\right)+{\mu}_{\mathcal{F}\left(\alpha \right)}\left({\mathfrak{u}}_{\left(\frac{\left|\mathcal{U}\right|}{2}+1\right)}\right)}{2}},\hfill & \phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}\left|\mathcal{U}\right|\phantom{\rule{3.33333pt}{0ex}}is\phantom{\rule{3.33333pt}{0ex}}even.\hfill \end{array}\right.\hfill \\ \hfill {\mathtt{q}}_{me{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\phantom{\rule{1.em}{0ex}}& \left\{\begin{array}{cc}{\tau}_{\mathcal{F}\left(\alpha \right)}\left({\mathfrak{u}}_{\left(\frac{\left|\mathcal{U}\right|+1}{2}\right)}\right),\hfill & \phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}\left|\mathcal{U}\right|\phantom{\rule{3.33333pt}{0ex}}is\phantom{\rule{3.33333pt}{0ex}}odd,\hfill \\ {\displaystyle \frac{{\tau}_{\mathcal{F}\left(\alpha \right)}\left({\mathfrak{u}}_{\left(\frac{\left|\mathcal{U}\right|}{2}\right)}\right)+{\tau}_{\mathcal{F}\left(\alpha \right)}\left({\mathfrak{u}}_{\left(\frac{\left|\mathcal{U}\right|}{2}+1\right)}\right)}{2}},\hfill & \phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}\left|\mathcal{U}\right|\phantom{\rule{3.33333pt}{0ex}}is\phantom{\rule{3.33333pt}{0ex}}even.\hfill \end{array}\right.\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathtt{r}}_{me{d}_{\mathsf{{\rm Y}}}}\left(\alpha \right)=\phantom{\rule{1.em}{0ex}}& \left\{\begin{array}{cc}{\nu}_{\mathcal{F}\left(\alpha \right)}\left({\mathfrak{u}}_{\left(\frac{\left|\mathcal{U}\right|+1}{2}\right)}\right),\hfill & \phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}\left|\mathcal{U}\right|\phantom{\rule{3.33333pt}{0ex}}is\phantom{\rule{3.33333pt}{0ex}}odd,\hfill \\ {\displaystyle \frac{{\nu}_{\mathcal{F}\left(\alpha \right)}\left({\mathfrak{u}}_{\left(\frac{\left|\mathcal{U}\right|}{2}\right)}\right)+{\nu}_{\mathcal{F}\left(\alpha \right)}\left({\mathfrak{u}}_{\left(\frac{\left|\mathcal{U}\right|}{2}+1\right)}\right)}{2}},\hfill & \phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}\left|\mathcal{U}\right|\phantom{\rule{3.33333pt}{0ex}}is\phantom{\rule{3.33333pt}{0ex}}even.\hfill \end{array}\right.\hfill \end{array}$$

**Definition**

**6**

## 3. Spherical Fuzzy Bipolar Soft Sets

**Definition**

**7.**

**Example**

**1.**

**Definition**

**8.**

- 1.
- $\mathcal{A}\subseteq \mathcal{B}.$
- 2.
- $\forall \phantom{\rule{3.33333pt}{0ex}}\mathfrak{u}\in \mathcal{U}$ and $\forall \phantom{\rule{3.33333pt}{0ex}}\alpha \in \mathcal{A},\phantom{\rule{3.33333pt}{0ex}}{\mu}_{\zeta}\left(\mathfrak{u}\right)\le {\mu}_{\pi}\left(\mathfrak{u}\right),{\tau}_{\zeta}\left(\mathfrak{u}\right)\le {\tau}_{\pi}\left(\mathfrak{u}\right)$ and ${\nu}_{\zeta}\left(\mathfrak{u}\right)\ge {\nu}_{\pi}\left(\mathfrak{u}\right).$
- 3. .
- $\forall \phantom{\rule{3.33333pt}{0ex}}\mathfrak{u}\in \mathcal{U}$ and $\forall \phantom{\rule{3.33333pt}{0ex}}\neg \alpha \in \neg \mathcal{A},\phantom{\rule{3.33333pt}{0ex}}{\mu}_{\psi}\left(\mathfrak{u}\right)\le {\mu}_{\eta}\left(\mathfrak{u}\right),{\tau}_{\psi}\left(\mathfrak{u}\right)\le {\tau}_{\eta}\left(\mathfrak{u}\right)$ and ${\nu}_{\psi}\left(\mathfrak{u}\right)\ge {\nu}_{\eta}\left(\mathfrak{u}\right).$

**Example**

**2.**

**Definition**

**9.**

**Definition**

**10.**

**Example**

**3.**

**Proposition**

**1.**

- 1.
- ${\left({\mathsf{\Xi}}^{c}\right)}^{c}=\mathsf{\Xi}$

**Proof.**

- 1.
- From Definition 10, we have ${\mathsf{\Xi}}^{c}={(\zeta ,\eta ,\mathcal{A})}^{c}=({\zeta}^{c},{\eta}^{c},\mathcal{A})$ such that$$\begin{array}{cc}\hfill {\zeta}^{c}=& \left\{(\mathfrak{u},{\nu}_{\zeta}\left(\mathfrak{u}\right),{\tau}_{\zeta}\left(\mathfrak{u}\right),{\mu}_{\zeta}\left(\mathfrak{u}\right))\right|\phantom{\rule{3.33333pt}{0ex}}\mathfrak{u}\in \mathcal{U}\},\hfill \\ \hfill {\eta}^{c}=& \left\{(\mathfrak{u},{\nu}_{\eta}\left(\mathfrak{u}\right),{\tau}_{\eta}\left(\mathfrak{u}\right),{\mu}_{\eta}\left(\mathfrak{u}\right))\right|\phantom{\rule{3.33333pt}{0ex}}\mathfrak{u}\in \mathcal{U}\}.\hfill \end{array}$$$$\begin{array}{cc}\hfill {\left({\zeta}^{c}\right)}^{c}=& \left\{(\mathfrak{u},{\mu}_{\zeta}\left(\mathfrak{u}\right),{\tau}_{\zeta}\left(\mathfrak{u}\right),{\nu}_{\zeta}\left(\mathfrak{u}\right))\right|\phantom{\rule{3.33333pt}{0ex}}\mathfrak{u}\in \mathcal{U}\}=\zeta ,\hfill \\ \hfill {\left({\eta}^{c}\right)}^{c}=& \left\{(\mathfrak{u},{\mu}_{\eta}\left(\mathfrak{u}\right),{\tau}_{\eta}\left(\mathfrak{u}\right),{\nu}_{\eta}\left(\mathfrak{u}\right))\right|\phantom{\rule{3.33333pt}{0ex}}\mathfrak{u}\in \mathcal{U}\}=\eta .\hfill \end{array}$$

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13.**

**Definition**

**14.**

**Example**

**4.**

**Proposition**

**2.**

- 1.
- ${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}$
- 2.
- ${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}$

**Proof.**

- 1.
- Let $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=(\zeta ,\eta ,\mathcal{A})\tilde{\wedge}({\zeta}_{1},{\eta}_{1},\mathcal{B})=(\varkappa ,\varpi ,\mathcal{A}\times \mathcal{B})$. Then, for all $(\alpha ,\beta )\in \mathcal{A}\times \mathcal{B}$,$${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={(\varkappa ,\varpi ,\mathcal{A}\times \mathcal{B})}^{c}=({\varkappa}^{c},{\varpi}^{c},\mathcal{A}\times \mathcal{B}),$$and ${\varpi}^{c}={(\eta (\neg \alpha )\cup {\eta}_{1}(\neg \beta ))}^{c}={\eta}^{c}(\neg \alpha )\cap {\eta}_{1}^{c}(\neg \beta )$.Now consider for all $(\alpha ,\beta )\in \mathcal{A}\times \mathcal{B},$$${\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}={(\zeta ,\eta ,\mathcal{A})}^{c}\tilde{\vee}{({\zeta}_{1},{\eta}_{1},\mathcal{B})}^{c}=(\chi ,\psi ,\mathcal{A}\times \mathcal{B}),$$Clearly, we have $\chi ={\varkappa}^{c}$ and $\psi ={\varpi}^{c}$. Hence ${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}$.
- 2.
- Let $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=(\zeta ,\eta ,\mathcal{A})\tilde{\vee}({\zeta}_{1},{\eta}_{1},\mathcal{B})=(\varkappa ,\varpi ,\mathcal{A}\times \mathcal{B})$. Then for all $(\alpha ,\beta )\in \mathcal{A}\times \mathcal{B}$,$${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={(\varkappa ,\varpi ,\mathcal{A}\times \mathcal{B})}^{c}=({\varkappa}^{c},{\varpi}^{c},\mathcal{A}\times \mathcal{B}),$$and ${\varpi}^{c}={(\eta (\neg \alpha )\cap {\eta}_{1}(\neg \beta ))}^{c}={\eta}^{c}(\neg \alpha )\cup {\eta}_{1}^{c}(\neg \beta )$.Now consider for all $(\alpha ,\beta )\in \mathcal{A}\times \mathcal{B},$$${\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}={(\zeta ,\eta ,\mathcal{A})}^{c}\tilde{\wedge}{({\zeta}_{1},{\eta}_{1},\mathcal{B})}^{c}=(\chi ,\psi ,\mathcal{A}\times \mathcal{B}),$$Clearly, we have $\chi ={\varkappa}^{c}$ and $\psi ={\varpi}^{c}$. Hence ${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}$.

**Proposition**

**3.**

- 1.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}.$
- 2.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}.$
- 3.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right).$
- 4.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right).$

**Proof.**

- 1.
- By Definition 13, we have$$\begin{array}{cc}\hfill \mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)& =({\zeta}_{1}\left(\alpha \right),{\eta}_{1}(\neg \alpha ),\mathcal{A})\tilde{\wedge}\left({\zeta}_{2}\left(\beta \right)\cap {\zeta}_{3}\left(\gamma \right),{\eta}_{2}(\neg \beta )\cup {\eta}_{3}(\neg \gamma ),\mathcal{B}\times \mathcal{C}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left({\zeta}_{1}\left(\alpha \right)\cap ({\zeta}_{2}\left(\beta \right)\cap {\zeta}_{3}\left(\gamma \right)),{\eta}_{1}(\neg \alpha )\cup ({\eta}_{2}(\neg \beta )\cup {\eta}_{3}(\neg \gamma )),\mathcal{A}\times (\mathcal{B}\times \mathcal{C})\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left(({\zeta}_{1}\left(\alpha \right)\cap {\zeta}_{2}\left(\beta \right))\cap {\zeta}_{3}\left(\gamma \right),({\eta}_{1}(\neg \alpha )\cup {\eta}_{2}(\neg \beta ))\cup {\eta}_{3}(\neg \gamma ),(\mathcal{A}\times \mathcal{B})\times \mathcal{C}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left({\zeta}_{1}\left(\alpha \right)\cap {\zeta}_{2}\left(\beta \right),{\eta}_{1}(\neg \alpha )\cup {\eta}_{2}\left(\beta \right),\mathcal{A}\times \mathcal{B}\right)\tilde{\wedge}({\zeta}_{3}\left(\gamma \right),{\eta}_{3}(\neg \gamma ),\mathcal{C})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}.\hfill \end{array}$$
- 2.
- By Definition 14, we have$$\begin{array}{cc}\hfill \mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)& =({\zeta}_{1}\left(\alpha \right),{\eta}_{1}(\neg \alpha ),\mathcal{A})\tilde{\vee}\left({\zeta}_{2}\left(\beta \right)\cup {\zeta}_{3}\left(\gamma \right),{\eta}_{2}(\neg \beta )\cap {\eta}_{3}(\neg \gamma ),\mathcal{B}\times \mathcal{C}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left({\zeta}_{1}\left(\alpha \right)\cup ({\zeta}_{2}\left(\beta \right)\cup {\zeta}_{3}\left(\gamma \right)),{\eta}_{1}(\neg \alpha )\cap ({\eta}_{2}(\neg \beta )\cap {\eta}_{3}(\neg \gamma )),\mathcal{A}\times (\mathcal{B}\times \mathcal{C})\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left(({\zeta}_{1}\left(\alpha \right)\cup {\zeta}_{2}\left(\beta \right))\cup {\zeta}_{3}\left(\gamma \right),({\eta}_{1}(\neg \alpha )\cap {\eta}_{2}(\neg \beta ))\cap {\eta}_{3}(\neg \gamma ),(\mathcal{A}\times \mathcal{B})\times \mathcal{C}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left({\zeta}_{1}\left(\alpha \right)\cup {\zeta}_{2}\left(\beta \right),{\eta}_{1}(\neg \alpha )\cap {\eta}_{2}\left(\beta \right),\mathcal{A}\times \mathcal{B}\right)\tilde{\vee}({\zeta}_{3}\left(\gamma \right),{\eta}_{3}(\neg \gamma ),\mathcal{C})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}.\hfill \end{array}$$

**Definition**

**15.**

**Definition**

**16.**

**Example**

**5.**

**Proposition**

**4.**

- 1.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}$
- 2.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}$

**Proof.**

**Proposition**

**5.**

- 1.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}.$
- 2.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}.$
- 3.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right).$
- 4.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right).$

**Proof.**

- 1.
- By Definition 15, for all $\delta \in \mathcal{K}=\mathcal{A}\cap \mathcal{B}\cap \mathcal{C},$ we have$$\begin{array}{cc}\hfill \mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)& =({\zeta}_{1},{\eta}_{1},\mathcal{A}){\cap}_{R}\left({\zeta}_{2}\cap {\zeta}_{3},{\eta}_{2}\cup {\eta}_{3},\mathcal{B}\cap \mathcal{C}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left({\zeta}_{1}\cap ({\zeta}_{2}\cap {\zeta}_{3}),{\eta}_{1}\cup ({\eta}_{2}\cup {\eta}_{3}),\mathcal{K}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left(({\zeta}_{1}\cap {\zeta}_{2})\cap {\zeta}_{3},({\eta}_{1}\cup {\eta}_{2})\cup {\eta}_{3},\mathcal{K}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left({\zeta}_{1}\cap {\zeta}_{2},{\eta}_{1}\cup {\eta}_{2},\mathcal{A}\cap \mathcal{B}\right){\cap}_{R}({\zeta}_{3},{\eta}_{3},\mathcal{C})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}.\hfill \end{array}$$
- 2.
- By Definition 16, for all $\delta \in \mathcal{K}=\mathcal{A}\cap \mathcal{B}\cap \mathcal{C},$ we have$$\begin{array}{cc}\hfill \mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)& =({\zeta}_{1},{\eta}_{1},\mathcal{A}){\cup}_{R}\left({\zeta}_{2}\cup {\zeta}_{3},{\eta}_{2}\cap {\eta}_{3},\mathcal{B}\cup \mathcal{C}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left({\zeta}_{1}\cup ({\zeta}_{2}\cup {\zeta}_{3}),{\eta}_{1}\cap ({\eta}_{2}\cap {\eta}_{3}),\mathcal{K}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left(({\zeta}_{1}\cup {\zeta}_{2})\cup {\zeta}_{3},({\eta}_{1}\cap {\eta}_{2})\cap {\eta}_{3},\mathcal{K}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left({\zeta}_{1}\cup {\zeta}_{2},{\eta}_{1}\cap {\eta}_{2},\mathcal{A}\cup \mathcal{B}\right){\cup}_{R}({\zeta}_{3},{\eta}_{3},\mathcal{C})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}.\hfill \end{array}$$

**Proposition**

**6.**

- 1.
- ${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}$
- 2.
- ${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}$

**Proof.**

- 1.
- Let $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=(\zeta ,\eta ,\mathcal{A}){\cap}_{R}({\zeta}_{1},{\eta}_{1},\mathcal{B})=(\varkappa ,\varpi ,\mathcal{A}\times \mathcal{B})$. Then for all $\delta \in \mathcal{K}=\mathcal{A}\cap \mathcal{B}\ne \varnothing $,$${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={(\varkappa ,\varpi ,\mathcal{K})}^{c}=({\varkappa}^{c},{\varpi}^{c},\mathcal{K}),$$and ${\varpi}^{c}={(\eta (\neg \delta )\cup {\eta}_{1}(\neg \delta ))}^{c}={\eta}^{c}(\neg \delta )\cap {\eta}_{1}^{c}(\neg \delta )$.Now consider for all $\delta \in \mathcal{K},$$${\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}=(\chi ,\psi ,\mathcal{K}),$$Clearly, we have $\chi ={\varkappa}^{c}$ and $\psi ={\varpi}^{c}$. Hence ${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}$.
- 2.
- Let $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=(\zeta ,\eta ,\mathcal{A}){\cup}_{R}({\zeta}_{1},{\eta}_{1},\mathcal{B})=(\varkappa ,\varpi ,\mathcal{K})$. Then for all $\delta \in \mathcal{K}=\mathcal{A}\cap \mathcal{B}\ne \varnothing $,$${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={(\varkappa ,\varpi ,\mathcal{K})}^{c}=({\varkappa}^{c},{\varpi}^{c},\mathcal{K}),$$and ${\varpi}^{c}={(\eta (\neg \delta )\cap {\eta}_{1}(\neg \delta ))}^{c}={\eta}^{c}(\neg \delta )\cup {\eta}_{1}^{c}(\neg \delta )$.Now consider for all $\delta \in \mathcal{K},$$${\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}=(\chi ,\psi ,\mathcal{K}),$$Clearly, we have $\chi ={\varkappa}^{c}$ and $\psi ={\varpi}^{c}$. Hence ${\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)}^{c}={\mathsf{\Xi}}^{c}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Psi}}^{c}$.

**Definition**

**17.**

**Definition**

**18.**

**Example**

**6.**

**Proposition**

**7.**

- 1.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}$
- 2.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}$
- 3.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}.$
- 4.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}.$
- 5.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right).$
- 6.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right)=\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}\right)\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Delta}\right).$

**Proof.**

- 1.
- Let $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=(\xi ,\chi ,\mathcal{A}\cup \mathcal{B})$. By Definition 17, $\forall \phantom{\rule{3.33333pt}{0ex}}\delta \in \mathcal{A}\cup \mathcal{B}$ we have$$\begin{array}{cc}\hfill \xi \left(\delta \right)=\phantom{\rule{1.em}{0ex}}& \left\{\begin{array}{cc}{\zeta}_{1}\left(\delta \right)\hfill & if\phantom{\rule{4pt}{0ex}}\delta \in \mathcal{A}-\mathcal{B}\hfill \\ {\zeta}_{2}\left(\delta \right)\hfill & if\phantom{\rule{4pt}{0ex}}\delta \in \mathcal{B}-\mathcal{A}\hfill \\ {\zeta}_{1}\left(\delta \right)\cap {\zeta}_{2}\left(\delta \right)\hfill & if\phantom{\rule{4pt}{0ex}}\delta \in \mathcal{A}\cap \mathcal{B}\hfill \end{array}\right.\hfill \\ \hfill \chi (\neg \delta )=\phantom{\rule{1.em}{0ex}}& \left\{\begin{array}{cc}{\eta}_{1}(\neg \delta )\hfill & if\phantom{\rule{4pt}{0ex}}\neg \delta \in (\neg \mathcal{A})-(\neg \mathcal{B})\hfill \\ {\eta}_{2}(\neg \delta )\hfill & if\phantom{\rule{4pt}{0ex}}\neg \delta \in (\neg \mathcal{B})-(\neg \mathcal{A})\hfill \\ {\eta}_{1}(\neg \delta )\cup {\eta}_{2}(\neg \delta ))\hfill & if\phantom{\rule{4pt}{0ex}}\neg \delta \in (\neg \mathcal{A})\cap (\neg \mathcal{B})\hfill \end{array}\right.\hfill \end{array}$$

**Proposition**

**8.**

- 1.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}=\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}$
- 2.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}$
- 3.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Psi}=\mathsf{\Psi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}$
- 4.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}=\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{R}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}=\mathsf{\Xi}$
- 5.
- $\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}=\mathsf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{E}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Xi}=\mathsf{\Xi}$

**Proof.**

## 4. Application of SFBSSs in MADM Problem

**Definition**

**19.**

**Definition**

**20.**

Algorithm 1: Ranking alternatives under SFBSSs environment |

- (1)
**Input:**- (a)
- The universe $\mathcal{U}$ of alternatives,
- (b)
- The set $\mathcal{A}$ of parameters,
- (c)
- The not-set $\neg \mathcal{A}$ of parameters opposite to those in $\mathcal{A}$,
- (d)
- Insert the SFBSS $\mathsf{\Xi}=(\zeta ,\eta ,\mathcal{A})$ using the expert’s opinions.
- (2)
- Find the corresponding favor SFSS $\widehat{\mathsf{\Xi}}=(\zeta ,\mathcal{A})$ and non-favor SFSS $\stackrel{\u02da}{\mathsf{\Xi}}=(\eta ,\neg \mathcal{A})$ for $\mathsf{\Xi}$.
- (3)
- Calculate the favor and non-favor thresholds based on any one of the threshold functions $\lambda $ from the mid, top-bottom-bottom, bottom-bottom- bottom, or med-threshold functions.
- (4)
- Calculate the level favor SS table corresponding to the level favor threshold of $\widehat{\mathsf{\Xi}}$.
- (5)
- Calculate the level non-favor SS table corresponding to the level non-favor threshold of $\stackrel{\u02da}{\mathsf{\Xi}}$.
- (6)
- For ${a}_{ij}$ and ${b}_{ij}$ being the entries of level favor SS and level non-favor SS, respectively, calculate the focus level table with entries ${c}_{ij}={a}_{ij}-{b}_{ij}$.
- (7)
- Input the FL scores ${k}_{j}=\sum _{i}{c}_{ij}$ in the last row in focus level table.
- (8)
- Find s such that ${k}_{s}=max\left({k}_{j}\right)$.
Output:${\mathfrak{u}}_{s}$ corresponding to s found in step 8 is the best alternative. For ascending values of ${k}_{j}$s, corresponding ${\mathfrak{u}}_{j}$s in ascending orders give the required ranking of alternatives. |

**Example**

**7.**

- ${\alpha}_{1}:$ Safe fish passage ensuring that fishes are able to pass through the passage without injury.
- ${\alpha}_{2}:$ Economic ensuring that the design will be low-cost and will not make a negative economic impact on the project.
- ${\alpha}_{3}:$ Good fish guidance considering the effectiveness of measures such as angled bars, racks and walls in guiding the fishes towards passage.
- ${\alpha}_{4}:$ Effective complementing technology considering the aiding components such as bypass chutes for the procedure.
- ${\alpha}_{5}:$ Behavioral consistency, ensuring that the design is highly consistent with the species behavior including the swimming velocity, clustering and size.
- ${\alpha}_{6}:$ Good behavioral guidance considering the alternative behavioral guidance aids such as underwater lights, pulses, etc., to direct the fish through the passage.

- $\neg {\alpha}_{1}:$ Fish injury and mortality.
- $\neg {\alpha}_{2}:$ High cost.
- $\neg {\alpha}_{3}:$ Bad fish guidance.
- $\neg {\alpha}_{4}:$ No complementing aids.
- $\neg {\alpha}_{5}:$ Behavioral inconsistence.
- $\neg {\alpha}_{6}:$ Bad behavioral guidance.

## 5. Comparison and Discussion

#### 5.1. Advantages

#### 5.2. Comparison

- reducing the SFBSS to FBSS by dropping the neutral and negative membership degrees, and then finding the focus level set as the difference of fuzzy soft sets for the two sets of parameters;
- reducing the SFBSS to SFSS by ignoring the not-set of parameters (and corresponding opinions), and then finding the mid-level SS (in the place of focus set in Algorithm 1.

#### 5.3. Limitations

## 6. Conclusions and Future Orientations

- Spherical fuzzy bipolar soft expert sets;
- Spherical fuzzy cubic soft expert sets;
- Complex spherical fuzzy bipolar soft sets;
- Rough spherical fuzzy bipolar soft sets.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Decision Model | Negative Membership | Neutral Membership | Parameterization | Bipolarity |
---|---|---|---|---|

Fuzzy sets [1] | • | • | • | • |

IFSs [11] | ✓ | • | • | • |

PFSs [5,6] | ✓ | • | • | • |

SFSs [4] | ✓ | ✓ | • | • |

Soft sets [2] | • | • | ✓ | • |

BSSs [3] | • | • | ✓ | ✓ |

FBSSs [10] | • | • | ✓ | ✓ |

SFSSs [7] | ✓ | ✓ | ✓ | • |

Proposed SFBSSs | ✓ | ✓ | ✓ | ✓ |

$(\mathit{\zeta},\mathcal{A})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{1}$ | $(0.94,0.01,0.15)$ | $(0.50,0.25,0.40)$ | $(0.80,0.10,0.20)$ |

${\alpha}_{2}$ | $(0.70,0.20,0.35)$ | $(0.82,0.15,0.30)$ | $(0.60,0.30,0.40)$ |

${\alpha}_{3}$ | $(0.92,0.05,0.10)$ | $(0.90,0.10,0.10)$ | $(0.70,0.10,0.40)$ |

${\alpha}_{4}$ | $(0.80,0.10,0.25)$ | $(0.60,0.20,0.30)$ | $(0.80,0.10,0.20)$ |

${\alpha}_{5}$ | $(0.80,0.10,0.30)$ | $(0.90,0.05,0.15)$ | $(0.60,0.10,0.50)$ |

${\alpha}_{6}$ | $(0.70,0.10,0.20)$ | $(0.90,0.05,0.10)$ | $(0.85,0.10,0.20)$ |

${\alpha}_{7}$ | $(0.85,0.10,0.20)$ | $(0.75,0.15,0.35)$ | $(0.63,0.12,0.43)$ |

$(\mathit{\eta},\neg \mathcal{A})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

$\neg {\alpha}_{1}$ | $(0.15,0.01,0.80)$ | $(0.60,0.15,0.40)$ | $(0.25,0.10,0.50)$ |

$\neg {\alpha}_{2}$ | $(0.35,0.05,0.60)$ | $(0.24,0.10,0.55)$ | $(0.66,0.22,0.57)$ |

$\neg {\alpha}_{3}$ | $(0.14,0.15,0.90)$ | $(0.18,0.10,0.88)$ | $(0.39,0.14,0.67)$ |

$\neg {\alpha}_{4}$ | $(0.27,0.17,0.83)$ | $(0.50,0.18,0.79)$ | $(0.27,0.17,0.87)$ |

$\neg {\alpha}_{5}$ | $(0.35,0.10,0.50)$ | $(0.17,0.08,0.87)$ | $(0.45,0.17,0.58)$ |

$\neg {\alpha}_{6}$ | $(0.20,0.10,0.70)$ | $(0.10,0.07,0.80)$ | $(0.15,0.09,0.86)$ |

$\neg {\alpha}_{7}$ | $(0.24,0.10,0.73)$ | $(0.35,0.16,0.87)$ | $(0.36,0.12,0.34)$ |

$(\mathit{\zeta},\mathit{\eta},\mathcal{A})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{1}$ | $\left(\begin{array}{c}(0.94,0.01,0.15)\\ (0.15,0.01,0.80)\end{array}\right)$ | $\left(\begin{array}{c}(0.50,0.25,0.40)\\ (0.60,0.15,0.40)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.20)\\ (0.25,0.10,0.50)\end{array}\right)$ |

${\alpha}_{2}$ | $\left(\begin{array}{c}(0.70,0.20,0.35)\\ (0.35,0.05,0.60)\end{array}\right)$ | $\left(\begin{array}{c}(0.82,0.15,0.30)\\ (0.24,0.10,0.55)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.30,0.40)\\ (0.66,0.22,0.57)\end{array}\right)$ |

${\alpha}_{3}$ | $\left(\begin{array}{c}(0.92,0.05,0.10)\\ (0.14,0.15,0.90)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.10,0.10)\\ (0.18,0.10,0.88)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.39,0.14,0.67)\end{array}\right)$ |

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.27,0.17,0.83)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.20,0.30)\\ (0.50,0.18,0.79)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.20)\\ (0.27,0.17,0.87)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.35,0.10,0.50)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.15)\\ (0.17,0.08,0.87)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.45,0.17,0.58)\end{array}\right)$ |

${\alpha}_{6}$ | $\left(\begin{array}{c}(0.70,0.10,0.20)\\ (0.20,0.10,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.10)\\ (0.10,0.07,0.80)\end{array}\right)$ | $\left(\begin{array}{c}(0.85,0.10,0.20)\\ (0.15,0.09,0.86)\end{array}\right)$ |

${\alpha}_{7}$ | $\left(\begin{array}{c}(0.85,0.10,0.20)\\ (0.24,0.10,0.73)\end{array}\right)$ | $\left(\begin{array}{c}(0.75,0.15,0.35)\\ (0.35,0.16,0.87)\end{array}\right)$ | $\left(\begin{array}{c}(0.63,0.12,0.43)\\ (0.36,0.12,0.34)\end{array}\right)$ |

$(\mathit{\zeta},\mathit{\eta},\mathcal{A})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ⋯ | ${\mathfrak{u}}_{\mathit{n}}$ |
---|---|---|---|---|

${\alpha}_{1}$ | ${\left(\begin{array}{c}({\mu}_{\zeta},{\tau}_{\zeta},{\nu}_{\zeta})\\ ({\mu}_{\eta},{\tau}_{\eta},{\nu}_{\eta})\end{array}\right)}_{11}$ | ${\left(\begin{array}{c}({\mu}_{\zeta},{\tau}_{\zeta},{\nu}_{\zeta})\\ ({\mu}_{\eta},{\tau}_{\eta},{\nu}_{\eta})\end{array}\right)}_{12}$ | ⋯ | ${\left(\begin{array}{c}({\mu}_{\zeta},{\tau}_{\zeta},{\nu}_{\zeta})\\ ({\mu}_{\eta},{\tau}_{\eta},{\nu}_{\eta})\end{array}\right)}_{1n}$ |

${\alpha}_{2}$ | ${\left(\begin{array}{c}({\mu}_{\zeta},{\tau}_{\zeta},{\nu}_{\zeta})\\ ({\mu}_{\eta},{\tau}_{\eta},{\nu}_{\eta})\end{array}\right)}_{21}$ | ${\left(\begin{array}{c}({\mu}_{\zeta},{\tau}_{\zeta},{\nu}_{\zeta})\\ ({\mu}_{\eta},{\tau}_{\eta},{\nu}_{\eta})\end{array}\right)}_{22}$ | ⋯ | ${\left(\begin{array}{c}({\mu}_{\zeta},{\tau}_{\zeta},{\nu}_{\zeta})\\ ({\mu}_{\eta},{\tau}_{\eta},{\nu}_{\eta})\end{array}\right)}_{2n}$ |

⋮ | ⋮ | ⋮ | ⋱ | ⋮ |

${\alpha}_{m}$ | ${\left(\begin{array}{c}({\mu}_{\zeta},{\tau}_{\zeta},{\nu}_{\zeta})\\ ({\mu}_{\eta},{\tau}_{\eta},{\nu}_{\eta})\end{array}\right)}_{m1}$ | ${\left(\begin{array}{c}({\mu}_{\zeta},{\tau}_{\zeta},{\nu}_{\zeta})\\ ({\mu}_{\eta},{\tau}_{\eta},{\nu}_{\eta})\end{array}\right)}_{m2}$ | ⋯ | ${\left(\begin{array}{c}({\mu}_{\zeta},{\tau}_{\zeta},{\nu}_{\zeta})\\ ({\mu}_{\eta},{\tau}_{\eta},{\nu}_{\eta})\end{array}\right)}_{mn}$ |

$(\mathit{\zeta},\mathit{\eta},\mathcal{B})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{3}$ | $\left(\begin{array}{c}(0.92,0.05,0.10)\\ (0.14,0.15,0.90)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.10,0.10)\\ (0.18,0.10,0.88)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.39,0.14,0.67)\end{array}\right)$ |

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.27,0.17,0.83)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.20,0.30)\\ (0.50,0.18,0.79)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.20)\\ (0.27,0.17,0.87)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.35,0.10,0.50)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.15)\\ (0.17,0.08,0.87)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.45,0.17,0.58)\end{array}\right)$ |

${\alpha}_{6}$ | $\left(\begin{array}{c}(0.70,0.10,0.20)\\ (0.20,0.10,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.10)\\ (0.10,0.07,0.80)\end{array}\right)$ | $\left(\begin{array}{c}(0.85,0.10,0.20)\\ (0.15,0.09,0.86)\end{array}\right)$ |

$(\mathit{\pi},\mathit{\psi},\mathcal{C})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{3}$ | $\left(\begin{array}{c}(0.80,0.05,0.20)\\ (0.24,0.50,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.05,0.30)\\ (0.25,0.10,0.60)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.50)\\ (0.45,0.15,0.60)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.30,0.10,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.86,0.05,0.25)\\ (0.37,0.08,0.77)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.55,0.17,0.48)\end{array}\right)$ |

${\alpha}_{6}$ | $\left(\begin{array}{c}(0.67,0.10,0.30)\\ (0.35,0.10,0.65)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.05,0.20)\\ (0.30,0.07,0.60)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.25,0.10,0.75)\end{array}\right)$ |

$({\mathit{\zeta}}^{\mathit{c}},{\mathit{\eta}}^{\mathit{c}},\mathcal{A})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{1}$ | $\left(\begin{array}{c}(0.15,0.01,0.94)\\ (0.80,0.01,0.15)\end{array}\right)$ | $\left(\begin{array}{c}(0.40,0.25,0.50)\\ (0.40,0.15,0.60)\end{array}\right)$ | $\left(\begin{array}{c}(0.20,0.10,0.80)\\ (0.50,0.10,0.25)\end{array}\right)$ |

${\alpha}_{2}$ | $\left(\begin{array}{c}(0.35,0.20,0.70)\\ (0.60,0.05,0.35)\end{array}\right)$ | $\left(\begin{array}{c}(0.30,0.15,0.82)\\ (0.55,0.10,0.24)\end{array}\right)$ | $\left(\begin{array}{c}(0.40,0.30,0.60)\\ (0.57,0.22,0.66)\end{array}\right)$ |

${\alpha}_{3}$ | $\left(\begin{array}{c}(0.10,0.05,0.92)\\ (0.90,0.15,0.14)\end{array}\right)$ | $\left(\begin{array}{c}(0.10,0.10,0.90)\\ (0.88,0.10,0.18)\end{array}\right)$ | $\left(\begin{array}{c}(0.40,0.10,0.70)\\ (0.67,0.14,0.39)\end{array}\right)$ |

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.25,0.10,0.80)\\ (0.83,0.17,0.27)\end{array}\right)$ | $\left(\begin{array}{c}(0.30,0.20,0.60)\\ (0.79,0.18,0.50)\end{array}\right)$ | $\left(\begin{array}{c}(0.20,0.10,0.80)\\ (0.87,0.17,0.27)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.30,0.10,0.80)\\ (0.50,0.10,0.35)\end{array}\right)$ | $\left(\begin{array}{c}(0.15,0.05,0.90)\\ (0.87,0.08,0.17)\end{array}\right)$ | $\left(\begin{array}{c}(0.50,0.10,0.60)\\ (0.58,0.17,0.45)\end{array}\right)$ |

${\alpha}_{6}$ | $\left(\begin{array}{c}(0.20,0.10,0.70)\\ (0.70,0.10,0.20)\end{array}\right)$ | $\left(\begin{array}{c}(0.10,0.05,0.90)\\ (0.80,0.07,0.10)\end{array}\right)$ | $\left(\begin{array}{c}(0.20,0.10,0.85)\\ (0.86,0.09,0.15)\end{array}\right)$ |

${\alpha}_{7}$ | $\left(\begin{array}{c}(0.20,0.10,0.85)\\ (0.73,0.10,0.24)\end{array}\right)$ | $\left(\begin{array}{c}(0.35,0.15,0.75)\\ (0.87,0.16,0.35)\end{array}\right)$ | $\left(\begin{array}{c}(0.43,0.12,0.63)\\ (0.34,0.12,0.36)\end{array}\right)$ |

$(\mathit{\zeta},\mathit{\eta},\mathcal{B})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{2}$ | $\left(\begin{array}{c}(0.70,0.20,0.35)\\ (0.35,0.05,0.60)\end{array}\right)$ | $\left(\begin{array}{c}(0.82,0.15,0.30)\\ (0.24,0.10,0.55)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.30,0.40)\\ (0.66,0.22,0.57)\end{array}\right)$ |

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.27,0.17,0.83)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.20,0.30)\\ (0.50,0.18,0.79)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.20)\\ (0.27,0.17,0.87)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.35,0.10,0.50)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.15)\\ (0.17,0.08,0.87)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.45,0.17,0.58)\end{array}\right)$ |

$(\mathit{\pi},\mathit{\psi},\mathcal{C})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.70,0.10,0.35)\\ (0.38,0.10,0.63)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.20,0.30)\\ (0.60,0.38,0.21)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.50)\\ (0.32,0.17,0.80)\end{array}\right)$ |

${\alpha}_{7}$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.45,0.10,0.75)\end{array}\right)$ | $\left(\begin{array}{c}(0.89,0.05,0.10)\\ (0.27,0.08,0.81)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.45)\\ (0.50,0.20,0.50)\end{array}\right)$ |

$\mathbf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\wedge}\phantom{\rule{3.33333pt}{0ex}}\mathbf{\Psi}$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

$({\alpha}_{2},{\alpha}_{4})$ | $\left(\begin{array}{c}(0.70,0.10,0.35)\\ (0.38,0.05,0.60)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.15,0.30)\\ (0.60,0.10,0.21)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.66,0.17,0.57)\end{array}\right)$ |

$({\alpha}_{2},{\alpha}_{7})$ | $\left(\begin{array}{c}(0.70,0.10,0.35)\\ (0.45,0.05,0.60)\end{array}\right)$ | $\left(\begin{array}{c}(0.82,0.05,0.30)\\ (0.27,0.08,0.55)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.45)\\ (0.66,0.20,0.50)\end{array}\right)$ |

$({\alpha}_{4},{\alpha}_{4})$ | $\left(\begin{array}{c}(0.70,0.10,0.35)\\ (0.38,0.10,0.63)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.20,0.30)\\ (0.60,0.18,0.21)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.50)\\ (0.32,0.17,0.80)\end{array}\right)$ |

$({\alpha}_{4},{\alpha}_{7})$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.45,0.10,0.75)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.05,0.30)\\ (0.50,0.08,0.79)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.45)\\ (0.50,0.17,0.50)\end{array}\right)$ |

$({\alpha}_{5},{\alpha}_{4})$ | $\left(\begin{array}{c}(0.70,0.10,0.35)\\ (0.38,0.10,0.50)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.05,0.30)\\ (0.60,0.08,0.21)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.45,0.17,0.58)\end{array}\right)$ |

$({\alpha}_{5},{\alpha}_{7})$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.45,0.10,0.50)\end{array}\right)$ | $\left(\begin{array}{c}(0.89,0.05,0.15)\\ (0.27,0.08,0.81)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.50,0.17,0.50)\end{array}\right)$ |

$\mathbf{\Xi}\phantom{\rule{3.33333pt}{0ex}}\tilde{\vee}\phantom{\rule{3.33333pt}{0ex}}\mathbf{\Psi}$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

$({\alpha}_{2},{\alpha}_{4})$ | $\left(\begin{array}{c}(0.70,0.10,0.35)\\ (0.35,0.05,0.63)\end{array}\right)$ | $\left(\begin{array}{c}(0.82,0.15,0.30)\\ (0.24,0.10,0.55)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.32,0.17,0.80)\end{array}\right)$ |

$({\alpha}_{2},{\alpha}_{7})$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.35,0.05,0.75)\end{array}\right)$ | $\left(\begin{array}{c}(0.89,0.05,0.10)\\ (0.24,0.08,0.81)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.50,0.20,0.57)\end{array}\right)$ |

$({\alpha}_{4},{\alpha}_{4})$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.27,0.10,0.83)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.20,0.30)\\ (0.50,0.18,0.79)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.20)\\ (0.27,0.17,0.87)\end{array}\right)$ |

$({\alpha}_{4},{\alpha}_{7})$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.27,0.10,0.83)\end{array}\right)$ | $\left(\begin{array}{c}(0.89,0.05,0.10)\\ (0.27,0.08,0.81)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.20)\\ (0.27,0.17,0.87)\end{array}\right)$ |

$({\alpha}_{5},{\alpha}_{4})$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.35,0.10,0.63)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.15)\\ (0.17,0.08,0.87)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.50)\\ (0.32,0.17,0.80)\end{array}\right)$ |

$({\alpha}_{5},{\alpha}_{7})$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.35,0.10,0.75)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.10)\\ (0.17,0.08,0.87)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.45)\\ (0.45,0.17,0.58)\end{array}\right)$ |

$(\mathit{\zeta},\mathit{\eta},\mathcal{G})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{3}$ | $\left(\begin{array}{c}(0.92,0.05,0.10)\\ (0.14,0.15,0.90)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.10,0.10)\\ (0.18,0.10,0.88)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.39,0.14,0.67)\end{array}\right)$ |

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.27,0.17,0.83)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.20,0.30)\\ (0.50,0.18,0.79)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.20)\\ (0.27,0.17,0.87)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.35,0.10,0.50)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.15)\\ (0.17,0.08,0.87)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.45,0.17,0.58)\end{array}\right)$ |

${\alpha}_{6}$ | $\left(\begin{array}{c}(0.70,0.10,0.20)\\ (0.20,0.10,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.10)\\ (0.10,0.07,0.80)\end{array}\right)$ | $\left(\begin{array}{c}(0.85,0.10,0.20)\\ (0.15,0.09,0.86)\end{array}\right)$ |

$(\mathit{\pi},\mathit{\psi},\mathcal{H})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{2}$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.48,0.10,0.53)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.20,0.40)\\ (0.70,0.18,0.31)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.20)\\ (0.42,0.17,0.70)\end{array}\right)$ |

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.70,0.10,0.35)\\ (0.38,0.10,0.63)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.20,0.30)\\ (0.60,0.38,0.21)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.50)\\ (0.32,0.17,0.80)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.30,0.10,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.86,0.05,0.25)\\ (0.37,0.08,0.77)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.55,0.17,0.48)\end{array}\right)$ |

${\alpha}_{7}$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.45,0.10,0.75)\end{array}\right)$ | $\left(\begin{array}{c}(0.89,0.05,0.10)\\ (0.27,0.08,0.81)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.45)\\ (0.50,0.20,0.50)\end{array}\right)$ |

**Table 15.**Restricted SFBS Intersection between SFBSSs $\mathsf{\Xi}$ and $\mathsf{\Psi}$ in Example 5.

$\mathbf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{\mathit{R}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{\Psi}$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.70,0.10,0.35)\\ (0.38,0.10,0.63)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.20,0.30)\\ (0.60,0.18,0.21)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.50)\\ (0.32,0.17,0.80)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.35,0.10,0.50)\end{array}\right)$ | $\left(\begin{array}{c}(0.86,0.05,0.25)\\ (0.37,0.08,0.77)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.55,0.17,0.48)\end{array}\right)$ |

$\mathbf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{\mathit{R}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{\Psi}$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.27,0.10,0.83)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.20,0.30)\\ (0.50,0.18,0.79)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.20)\\ (0.27,0.17,0.87)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.30,0.10,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.15)\\ (0.17,0.08,0.87)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.45,0.17,0.58)\end{array}\right)$ |

**Table 17.**Extended SFBS Intersection between SFBSSs $\mathsf{\Xi}$ and $\mathsf{\Psi}$ in Example 5.

$\mathbf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cap}_{\mathit{E}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{\Psi}$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{2}$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.48,0.10,0.53)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.20,0.40)\\ (0.70,0.18,0.31)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.20)\\ (0.42,0.17,0.70)\end{array}\right)$ |

${\alpha}_{3}$ | $\left(\begin{array}{c}(0.92,0.05,0.10)\\ (0.14,0.15,0.90)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.10,0.10)\\ (0.18,0.10,0.88)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.39,0.14,0.67)\end{array}\right)$ |

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.70,0.10,0.35)\\ (0.38,0.10,0.63)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.20,0.30)\\ (0.60,0.18,0.21)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.50)\\ (0.32,0.17,0.80)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.35,0.10,0.50)\end{array}\right)$ | $\left(\begin{array}{c}(0.86,0.05,0.25)\\ (0.37,0.08,0.77)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.55,0.17,0.48)\end{array}\right)$ |

${\alpha}_{6}$ | $\left(\begin{array}{c}(0.70,0.10,0.20)\\ (0.20,0.10,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.10)\\ (0.10,0.07,0.80)\end{array}\right)$ | $\left(\begin{array}{c}(0.85,0.10,0.20)\\ (0.15,0.09,0.86)\end{array}\right)$ |

${\alpha}_{7}$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.45,0.10,0.75)\end{array}\right)$ | $\left(\begin{array}{c}(0.89,0.05,0.10)\\ (0.27,0.08,0.81)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.45)\\ (0.50,0.20,0.50)\end{array}\right)$ |

$\mathbf{\Xi}\phantom{\rule{3.33333pt}{0ex}}{\cup}_{\mathit{E}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{\Psi}$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ |
---|---|---|---|

${\alpha}_{2}$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.48,0.10,0.53)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.20,0.40)\\ (0.70,0.18,0.31)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.20)\\ (0.42,0.17,0.70)\end{array}\right)$ |

${\alpha}_{3}$ | $\left(\begin{array}{c}(0.92,0.05,0.10)\\ (0.14,0.15,0.90)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.10,0.10)\\ (0.18,0.10,0.88)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.40)\\ (0.39,0.14,0.67)\end{array}\right)$ |

${\alpha}_{4}$ | $\left(\begin{array}{c}(0.80,0.10,0.25)\\ (0.27,0.10,0.83)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.20,0.30)\\ (0.50,0.18,0.79)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.20)\\ (0.27,0.17,0.87)\end{array}\right)$ |

${\alpha}_{5}$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.30,0.10,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.15)\\ (0.17,0.08,0.87)\end{array}\right)$ | $\left(\begin{array}{c}(0.60,0.10,0.50)\\ (0.45,0.17,0.58)\end{array}\right)$ |

${\alpha}_{6}$ | $\left(\begin{array}{c}(0.70,0.10,0.20)\\ (0.20,0.10,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.05,0.10)\\ (0.10,0.07,0.80)\end{array}\right)$ | $\left(\begin{array}{c}(0.85,0.10,0.20)\\ (0.15,0.09,0.86)\end{array}\right)$ |

${\alpha}_{7}$ | $\left(\begin{array}{c}(0.80,0.10,0.30)\\ (0.45,0.10,0.75)\end{array}\right)$ | $\left(\begin{array}{c}(0.89,0.05,0.10)\\ (0.27,0.08,0.81)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.10,0.45)\\ (0.50,0.20,0.50)\end{array}\right)$ |

$(\mathit{\zeta},\mathit{\eta},\mathcal{A})$ | ${\mathfrak{u}}_{1}$ | ${\mathfrak{u}}_{2}$ | ${\mathfrak{u}}_{3}$ | ${\mathfrak{u}}_{4}$ |
---|---|---|---|---|

${\alpha}_{1}$ | $\left(\begin{array}{c}(0.93,0.01,0.25)\\ (0.20,0.02,0.86)\end{array}\right)$ | $\left(\begin{array}{c}(0.70,0.20,0.40)\\ (0.50,0.25,0.40)\end{array}\right)$ | $\left(\begin{array}{c}(0.83,0.10,0.40)\\ (0.25,0.15,0.70)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.25,0.30)\\ (0.30,0.01,0.90)\end{array}\right)$ |

${\alpha}_{2}$ | $\left(\begin{array}{c}(0.50,0.10,0.50)\\ (0.60,0.05,0.30)\end{array}\right)$ | $\left(\begin{array}{c}(0.82,0.05,0.20)\\ (0.34,0.10,0.85)\end{array}\right)$ | $\left(\begin{array}{c}(0.68,0.20,0.45)\\ (0.45,0.22,0.57)\end{array}\right)$ | $\left(\begin{array}{c}(0.82,0.15,0.30)\\ (0.34,0.10,0.55)\end{array}\right)$ |

${\alpha}_{3}$ | $\left(\begin{array}{c}(0.92,0.05,0.10)\\ (0.14,0.15,0.90)\end{array}\right)$ | $\left(\begin{array}{c}(0.80,0.10,0.50)\\ (0.40,0.10,0.68)\end{array}\right)$ | $\left(\begin{array}{c}(0.86,0.07,0.30)\\ (0.36,0.14,0.60)\end{array}\right)$ | $\left(\begin{array}{c}(0.90,0.10,0.10)\\ (0.16,0.10,0.80)\end{array}\right)$ |