# Dual Hesitant Fuzzy Probability

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Some Basic Concepts about DHFS

**Definition**

**1.**

**Example**

**1.**

- ${h}_{D}\left({x}_{1}\right)=\{\{0.7,0.5,0.3\},\{0.2,0.1\}\}$,
- ${h}_{D}\left({x}_{2}\right)=\{\{0.4,0.3\},\{0.2,0.4\}\}$,
- ${h}_{D}\left({x}_{3}\right)=\{\{0.6,0.3\},\{0.3,0.1,0.2\}\}$,

- (1)
- $\oplus -union$: ${d}_{1}\oplus {d}_{2}=$ ${\cup}_{{\gamma}_{{d}_{1}}\in {h}_{{d}_{1}},{\gamma}_{{d}_{2}}\in {h}_{{d}_{2}}}$ $\{\{{\gamma}_{{d}_{1}}+{\gamma}_{{d}_{2}}-{\gamma}_{{d}_{1}}{\gamma}_{{d}_{2}}\},\left\{{\eta}_{{d}_{1}}{\eta}_{{d}_{2}}\right\}\}$;
- (2)
- $\otimes -intersection$: ${d}_{1}\otimes {d}_{2}=$ ${\cup}_{{\gamma}_{{d}_{1}}\in {h}_{{d}_{1}},{\gamma}_{{d}_{2}}\in {h}_{{d}_{2}}}$ $\{\left\{{\gamma}_{{d}_{1}}{\gamma}_{{d}_{2}}\right\},\{{\eta}_{{d}_{1}}+{\eta}_{{d}_{2}}-{\eta}_{{d}_{1}}{\eta}_{{d}_{2}}\}\}$;
- (3)
- $nd={\cup}_{{\gamma}_{d}\in {h}_{d},{\eta}_{d}\in {g}_{d}}\{1-{(1-{\gamma}_{d})}^{n},{\left({\eta}_{d}\right)}^{n}\}$;
- (4)
- ${d}^{n}={\cup}_{{\gamma}_{d}\in {h}_{d},{\eta}_{d}\in {g}_{d}}\{{\left({\gamma}_{d}\right)}^{n},1-{(1-{\eta}_{d})}^{n}\}$, where n is a positive integral and all the results are also DHFEs.

**Definition**

**2.**

- (i)
- if $s\left({d}_{1}\right)>s\left({d}_{2}\right)$, then ${d}_{1}$ is superior to ${d}_{2}$, denoted by ${d}_{1}\succ {d}_{2}$;
- (ii)
- if $s\left({d}_{1}\right)=s\left({d}_{2}\right)$, then
- (1)
- if $p\left({d}_{1}\right)=p\left({d}_{2}\right)$, then ${d}_{1}$ is equivalent to ${d}_{2}$, denoted by ${d}_{1}\sim {d}_{2}$;
- (2)
- if $p\left({d}_{1}\right)>p\left({d}_{2}\right)$, then ${d}_{2}$ is superior to ${d}_{1}$, denoted by ${d}_{2}\succ {d}_{1}$.

## 3. Dual Hesitant Fuzzy Numbers

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**1.**

- (1)
- D is a quasi-convex DHFS;
- (2)
- any $(\alpha ,\beta )-cuts$ of DHFS D are convex crisp sets, for any $\alpha ,\beta \in (0,1]$;
- (3)
- ${h}_{D}$ is the membership function of a convex hesitant fuzzy set with respect to $\alpha -cuts$ [16].

**Proof.**

**Definition**

**6.**

**Definition**

**7.**

**Remark**

**2.**

**Theorem**

**2.**

**Lemma**

**2.**

**Lemma**

**3.**

**Theorem**

**3.**

- (i)
- If $A\u2aafB$ then $\frac{A}{C}\u2aaf\frac{B}{C}$;
- (ii)
- If $B\subseteq C$ then $\frac{A}{B}\subseteq \frac{A}{C}$;
- (iii)
- $B\subseteq \frac{A\xb7B}{A}$.

**Proof.**

- (i)
- If $A\u2aafB$, we have $r\left({A}_{(\alpha ,\beta )}\right)<r\left({B}_{(\alpha ,\beta )}\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}r\left({A}_{(\alpha ,\beta )}\right)=r\left({B}_{(\alpha ,\beta )}\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}l\left({B}_{(\alpha ,\beta )}\right)<l\left({A}_{(\alpha ,\beta )}\right)$, thus $\frac{r\left({A}_{(\alpha ,\beta )}\right)}{l\left({C}_{(\alpha ,\beta )}\right)}<\frac{r\left({B}_{(\alpha ,\beta )}\right)}{l\left({C}_{(\alpha ,\beta )}\right)}$ or $\frac{r\left({A}_{(\alpha ,\beta )}\right)}{l\left({C}_{(\alpha ,\beta )}\right)}=\frac{r\left({B}_{(\alpha ,\beta )}\right)}{l\left({C}_{(\alpha ,\beta )}\right)}$ and $\frac{l\left({A}_{(\alpha ,\beta )}\right)}{r\left({C}_{(\alpha ,\beta )}\right)}\le \frac{l\left({B}_{(\alpha ,\beta )}\right)}{r\left({C}_{(\alpha ,\beta )}\right)}$. Then $\frac{A}{C}\u2aaf\frac{B}{C}$;
- (ii)
- If $B\subseteq C$, we have $[l\left({B}_{(\alpha ,\beta )}\right),r\left({B}_{(\alpha ,\beta )}\right)]$ $\subseteq [l\left({C}_{(\alpha ,\beta )}\right),r\left({C}_{(\alpha ,\beta )}\right)]$, thus $[\frac{l\left({A}_{(\alpha ,\beta )}\right)}{r\left({B}_{(\alpha ,\beta )}\right)},\frac{r\left({A}_{(\alpha ,\beta )}\right)}{l\left({B}_{(\alpha ,\beta )}\right)}]$ $\subseteq [\frac{l\left({A}_{(\alpha ,\beta )}\right)}{r\left({C}_{(\alpha ,\beta )}\right)},\frac{r\left({A}_{(\alpha ,\beta )}\right)}{l\left({C}_{(\alpha ,\beta )}\right)}]$, namely, $\frac{A}{B}\subseteq \frac{A}{C}$;
- (iii)
- As $[l\left({B}_{(\alpha ,\beta )}\right),r\left({B}_{(\alpha ,\beta )}\right)]$ $\subseteq [\frac{l\left({A}_{(\alpha ,\beta )}\right)l\left({B}_{(\alpha ,\beta )}\right)}{r\left({A}_{(\alpha ,\beta )}\right)},\frac{r\left({A}_{(\alpha ,\beta )}\right)r\left({B}_{(\alpha ,\beta )}\right)}{l\left({A}_{(\alpha ,\beta )}\right)}]$, namely, $B\subseteq \frac{A\xb7B}{A}$.

## 4. Dual Hesitant Fuzzy Probability

**Theorem**

**4.**

**Proof.**

**Remark**

**3.**

**Theorem**

**5.**

- (i)
- If $A\cap B=\u2300$ then $\overline{P}(A\cup B)\subseteq \overline{P}\left(A\right)+\overline{P}\left(B\right);$
- (ii)
- If $A\subseteq B$ then $\overline{P}\left(A\right)\u2aaf\overline{P}\left(B\right);$
- (iii)
- $\overline{P}(\u2300)=\overline{0}\u2aaf\overline{P}\left(A\right)\u2aaf\overline{P}\left(X\right)=\overline{1};$
- (iv)
- $\overline{1}\u2aaf\overline{P}\left(A\right)+\overline{P}\left({A}^{c}\right);$
- (v)
- If $A\cap B\ne \u2300$ then $\overline{P}(A\cup B)\subseteq \overline{P}\left(A\right)+\overline{P}\left(B\right)-\overline{P}(A\cap B).$

**Proof.**

- (i)
- Notice that $A\cap B=\u2300$ if and only if ${I}_{A}\cap {I}_{B}=\u2300$. In order to prove (i), we only need to prove that:$$\overline{P}{(A\cup B)}_{\alpha}\subseteq \overline{P}{\left(A\right)}_{\alpha}+\overline{P}{\left(B\right)}_{\alpha}.$$Namely, for each $(\alpha ,\beta )\in L,$$\{{\sum}_{i\in {I}_{A}}{d}_{i}+$ ${\sum}_{j\in {I}_{B}}{d}_{j}|({d}_{1},{d}_{2},...,{d}_{n})\in {S}_{\alpha}^{\beta}$ $and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}$ lies in$\{{\sum}_{i\in {I}_{A}}{d}_{i}|({d}_{1},{d}_{2},...,{d}_{n})\in {S}_{\alpha}^{\beta}$ $and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}$ + $\{{\sum}_{j\in {I}_{B}}{d}_{j}|({d}_{1},{d}_{2},...,{d}_{n})$ $\in {S}_{\alpha}^{\beta}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}$, which is obvious;
- (ii)
- If $A\subseteq B$, we have $\{{\sum}_{i\in {I}_{A}}{d}_{i}|({d}_{1},{d}_{2},...,{d}_{n})\in {S}_{\alpha}^{\beta}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}\subseteq \{{\sum}_{i\in {I}_{B}}{d}_{j}|({d}_{1},{d}_{2},...,{d}_{n})\in {S}_{\alpha}^{\beta}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}$, thus $\overline{P}\left(A\right)\u2aaf\overline{P}\left(B\right);$
- (iii)
- If $\varnothing \subseteq A\subseteq X$, $\varnothing \subseteq \{{\sum}_{i\in {I}_{A}}{d}_{i}|({d}_{1},{d}_{2},...,{d}_{n})\in {S}_{\alpha}^{\beta}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}\subseteq \{{\sum}_{i\in {I}_{X}}{d}_{j}|({d}_{1},{d}_{2},...,{d}_{n})\in {S}_{\alpha}^{\beta}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}$, thus $\overline{P}(\u2300)=\overline{0}\u2aaf\overline{P}\left(A\right)\u2aaf\overline{P}\left(X\right)=\overline{1}.$
- (iv)
- As (i), we donate $B={A}^{c}$, then $\overline{P}\left(A\right)+\overline{P}\left({A}^{c}\right)$ $\u2ab0\overline{P}(A\cup {A}^{c}))=\overline{P}\left(X\right)=\overline{1}$;
- (v)
- If $A\cap B\ne \u2300$, for each $(\alpha ,\beta )\in L$, $\{{\sum}_{u\in {I}_{A}\cup {I}_{B}}{d}_{u}|({d}_{1},{d}_{2},...,{d}_{n})\in {S}_{\alpha}^{\beta}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}$ lies in $\{{\sum}_{i\in {I}_{A}}{d}_{i}|({d}_{1},{d}_{2},...,{d}_{n})\in {S}_{\alpha}^{\beta}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}$ + $\{{\sum}_{j\in {I}_{B}}{d}_{j}|({d}_{1},{d}_{2},...,{d}_{n})\in {S}_{\alpha}^{\beta}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}$ − $\{{\sum}_{k\in {I}_{A}\cap {I}_{B}}{d}_{k}|({d}_{1},{d}_{2},...,{d}_{n})\in {S}_{\alpha}^{\beta}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sum}_{l=1}^{n}{d}_{l}=1\}$, which proves that $\overline{P}(A\cup B)\subseteq \overline{P}\left(A\right)+\overline{P}\left(B\right)-\overline{P}(A\cap B).$☐

## 5. Dual Hesitant Fuzzy Conditional Probability

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

- 1.
- If ${A}_{1}\cap {A}_{2}=\u2300$ then $\overline{P}({A}_{1}\cup {A}_{2}|B)\subseteq \overline{P}\left({A}_{1}\right|B)+\overline{P}\left({A}_{2}\right|B)$;
- 2.
- $\overline{0}\u2aaf\overline{P}\left(A\right|B)\u2aaf\overline{1}$;
- 3.
- $\overline{P}\left(A\right|A)=\overline{1}$, $\overline{P}\left(A\right|{A}^{c})=\overline{0}$;
- 4.
- If $B\subseteq A$ then $\overline{P}\left(A\right|B)=\overline{1}$;
- 5.
- If $A\cap B=\u2300$ then $\overline{P}\left(A\right|B)=\overline{0}$.

**Proof.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

## 6. Application of Color Blindness

**Example**

**2.**

- ${p}_{1}=\{\{p(M\cap C\},\{p(M\cap {C}^{\prime})\}\}$ $=\{<\{0.04,0.06,0.02\}$ $,\{0.4,0.35,0.45\}>\}$;
- ${p}_{2}=\{\{p(F\cap C\},\{p(F\cap {C}^{\prime})\}\}$ $=\{<\{0.08,0.06,0.07\},$ $\{0.3,0.4,0.2\}>\}$.

- ${p}_{1}=\{s(p(M\cap C),s\left(p(M\cap {C}^{\prime})\right)\}$ $=\{<0.04,0.4>\}$;
- ${p}_{2}=\{s(p(F\cap C),s(p(F\cap {C}^{\prime})\}$ $=\{<0.07,0.3>\}$.

## 7. Conclusions and Further Study

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Uncertainty in Membership Degree h | Uncertainty in Nonmembership Degree g | |
---|---|---|

${p}_{1}$ | (0.02/0.04/0.06) | (0.3/0.4/0.5) |

${p}_{2}$ | (0.04/0.07/0.1) | (0.2/0.3/0.4) |

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**MDPI and ACS Style**

Chen, J.; Huang, X.
Dual Hesitant Fuzzy Probability. *Symmetry* **2017**, *9*, 52.
https://doi.org/10.3390/sym9040052

**AMA Style**

Chen J, Huang X.
Dual Hesitant Fuzzy Probability. *Symmetry*. 2017; 9(4):52.
https://doi.org/10.3390/sym9040052

**Chicago/Turabian Style**

Chen, Jianjian, and Xianjiu Huang.
2017. "Dual Hesitant Fuzzy Probability" *Symmetry* 9, no. 4: 52.
https://doi.org/10.3390/sym9040052