# Special Discrete Fuzzy Numbers on Countable Sets and Their Applications

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## Abstract

**:**

## 1. Introduction

- The endpoints function representation theorem of special discrete fuzzy numbers on countable sets is proven.
- Two metrics of special discrete fuzzy numbers on countable sets are defined and compared.
- The definitions and properties of t-norm operator and t-conorm operator of special discrete fuzzy numbers on countable sets are proposed and proven. In addition, these two operators are used in the practical application of image fusion and subjective evaluation.

## 2. Preliminaries

**Definition 1 ([2]).**

- (1)
- $u({x}_{i})=1$ for any natural number i with $s\le i\le t$;
- (2)
- $u({x}_{i})\le u({x}_{j})$ for any natural numbers i, j with $1\le i\le j\le s$;
- (3)
- $u({x}_{i})\ge u({x}_{j})$ for any natural numbers i, j with $t\le i\le j\le n$.

**Remark 1.**

**Theorem 1 ([7]).**

- (1)
- ${\left[u\right]}^{r}$ is a nonempty finite subset of $\mathbb{R}$ for any $r\in [0,1]$;
- (2)
- ${\left[u\right]}^{{r}_{2}}\subset {\left[u\right]}^{{r}_{1}}$ for any ${r}_{1}$,${r}_{2}\in [0,1]$ with ${r}_{1}\le {r}_{2}$;
- (3)
- For any ${r}_{1}$,${r}_{2}\in [0,1]$ with $0\le {r}_{1}\le {r}_{2}\le 1,$ if $x\in {\left[u\right]}^{{r}_{1}}\setminus {\left[u\right]}^{{r}_{2}},$ we have $x<y$ for all $y\in {\left[u\right]}^{{r}_{2}},$ or $x>y$ for all $y\in {\left[u\right]}^{{r}_{2}};$
- (4)
- For any ${r}_{0}\in (0,1]$, there exists a real number ${r}_{0}^{\prime}$ with $0<{r}_{0}^{\prime}<{r}_{0}$ such that ${\left[u\right]}^{{r}_{0}^{\prime}}={\left[u\right]}^{{r}_{0}}$ (i.e., ${\left[u\right]}^{r}={\left[u\right]}^{{r}_{0}}$ for any $r\in [{r}_{0}^{\prime},{r}_{0}]$).

- (i)
- ${A}_{r}$ is nonempty and finite for any $r\in [0,1]$;
- (ii)
- ${A}_{{r}_{2}}\subset {A}_{{r}_{1}}$ for any ${r}_{1}$,${r}_{2}\in [0,1]$ with ${r}_{1}\le {r}_{2}$;
- (iii)
- For any ${r}_{1}$,${r}_{2}\in [0,1]$ with $0\le {r}_{1}\le {r}_{2}\le 1,$ if $x\in {A}_{{r}_{1}}\setminus {A}_{{r}_{2}},$ then $x<y$ for all $y\in {A}_{{r}_{2}},$ or $x>y$ for all $y\in {A}_{{r}_{2}};$
- (iv)
- For any ${r}_{0}\in (0,1]$, there exists a real number ${r}_{0}^{\prime}$ with $0<{r}_{0}^{\prime}<{r}_{0}$ such that ${A}_{{r}_{0}^{\prime}}={A}_{{r}_{0}}$ (i.e., ${A}_{r}={A}_{{r}_{0}}$ for any $r\in [{r}_{0}^{\prime},{r}_{0}]$),

**Definition 2.**

## 3. Special Discrete Fuzzy Numbers on Countable Sets

**Definition 3 ([26]).**

- (1)
- ${\left[u\right]}^{0}\subset C$ and ${\left[u\right]}^{0}$ is finite;
- (2)
- There exists ${x}_{0}\in C$ such that $u({x}_{0})=1$;
- (3)
- For any ${x}_{s},{x}_{t}\in C$ with ${x}_{s}\le {x}_{t}\le {x}_{0}$, $u({x}_{s})\le u({x}_{t})$ is tenable;
- (4)
- For any ${x}_{s},{x}_{t}\in C$ with ${x}_{0}\le {x}_{s}\le {x}_{t}$, $u({x}_{s})\ge u({x}_{t})$ is tenable.

**Theorem 2 ([26]).**

- (1)
- For any $r\in [0,1]$, there exist ${x}_{r},{y}_{r}\in C$ with ${x}_{r}\le {y}_{r}$, such that ${\left[u\right]}^{r}={[{x}_{r},{y}_{r}]}_{C}$, and ${[{x}_{0},{y}_{0}]}_{C}$ is finite;
- (2)
- For any ${r}_{1},{r}_{2}\in [0,1]$ with $0\le {r}_{1}\le {r}_{2}\le 1$, ${\left[u\right]}^{{r}_{2}}\subset {\left[u\right]}^{{r}_{1}}$ is tenable;
- (3)
- For any ${r}_{0}\in (0,1]$, there exists a real number ${r}_{0}^{\prime}$ with $0<{r}_{0}^{\prime}<{r}_{0}$, such that ${\left[u\right]}^{{r}_{0}^{\prime}}={\left[u\right]}^{{r}_{0}}$, i.e., for any $r\in [{r}_{0}^{\prime},{r}_{0}]$, ${\left[u\right]}^{r}={\left[u\right]}^{{r}_{0}}$ is tenable.

- (i)
- There exist ${x}_{r},{y}_{r}\in C$ with ${x}_{r}\le {y}_{r}$, such that ${A}_{r}={[{x}_{r},{y}_{r}]}_{C}$ and ${[{x}_{0},{y}_{0}]}_{C}$ is finite;
- (ii)
- For any ${r}_{1}$,${r}_{2}\in [0,1]$ with $0\le {r}_{1}\le {r}_{2}\le 1$, ${A}_{{r}_{2}}\subset {A}_{{r}_{1}}$ is tenable;
- (iii)
- For any ${r}_{0}\in (0,1]$, there exists a real number ${r}_{0}^{\prime}$ with $0<{r}_{0}^{\prime}<{r}_{0}$, such that ${A}_{{r}_{0}^{\prime}}={A}_{{r}_{0}}$, i.e., for any $r\in [{r}_{0}^{\prime},{r}_{0}]$, ${A}_{r}={A}_{{r}_{0}}$ is tenable.

**Theorem 3.**

- (1)
- $\underline{u}(r)$ is monotone nondecreasing left continuous;
- (2)
- $\overline{u}(r)$ is monotone nonincreasing left continuous;
- (3)
- $\underline{u}(r)\le \overline{u}(r)$ for all $r\in [0,1]$;
- (4)
- $\underline{u}(r)$ and $\overline{u}(r)$ are right continuous at $r=0$.

- (i)
- $X(r)$ is monotone nondecreasing left continuous;
- (ii)
- $Y(r)$ is monotone nonincreasing left continuous;
- (iii)
- $X(r)\le Y(r)$ for all $r\in [0,1]$;
- (iv)
- $X(r)$ and $Y(r)$ are right continuous at $r=0$.

**Proof.**

**Theorem 4 ([26]).**

- (1)
- ${[u+v]}^{r}={\left[u\right]}^{r}+{\left[v\right]}^{r}$;
- (2)
- ${\left[ku\right]}^{r}=k{\left[u\right]}^{r}$;
- (3)
- ${\left[uv\right]}^{r}={\left[u\right]}^{r}{\left[v\right]}^{r}$.

**Theorem 5 ([26]).**

- (1)
- $ku\in {\mathcal{F}}_{\mathcal{DC}}$ if C satisfies $kx\in C$ for any $x\in C$;
- (2)
- $u+v\in {\mathcal{F}}_{\mathcal{DC}}$ if C preserves the closeness of the operations of addition and difference.

## 4. Metrics of Special Discrete Fuzzy Numbers on Countable Sets

**Definition 4.**

**Definition 5.**

**Theorem 6.**

- (1)
- $\dot{D}(u,v)=\dot{D}(v,u),\widehat{D}(u,v)=\widehat{D}(v,u)$;
- (2)
- $\dot{D}(u,v)\ge 0,\widehat{D}(u,v)\ge 0$;
- (3)
- $\dot{D}(u,v)=0\iff u=v,\widehat{D}(u,v)=0\iff u=v$;
- (4)
- $\dot{D}(u,v)\le \dot{D}(u,w)+\dot{D}(w,v)$, $\widehat{D}(u,v)\le \widehat{D}(u,w)+\widehat{D}(w,v)$;
- (5)
- $\dot{D}(u+w,v+w)=\dot{D}(u,v),\widehat{D}(u+w,v+w)=\widehat{D}(u,v)$;
- (6)
- $\dot{D}(ku,kv)=\left|k\right|\dot{D}(u,v),\widehat{D}(ku,kv)=\left|k\right|\widehat{D}(u,v)$.

**Proof.**

**Theorem 7.**

**Proof.**

## 5. The Triangular Norm and Triangular Conorm Operations of Special Discrete Fuzzy Numbers on Countable Sets

**Definition 6 ([28]).**

- (1)
- Commutativity: $T(x,y)=T(y,x)$;
- (2)
- Associativity: $T(T(x,y),z)=T(x,T(y,z))$;
- (3)
- Monotonicity: $T(x,y)\le T({x}^{\prime},{y}^{\prime})$ when $x\le {x}^{\prime},y\le {y}^{\prime}$;
- (4)
- Boundary condition: $T(x,m)=x$.

**Definition 7 ([28]).**

- (1)
- Commutativity: $S(x,y)=S(y,x)$;
- (2)
- Associativity: $S(S(x,y),z)=S(x,S(y,z))$;
- (3)
- Monotonicity: $S(x,y)\le S({x}^{\prime},{y}^{\prime})$ when $x\le {x}^{\prime},y\le {y}^{\prime}$;
- (4)
- Boundary condition: $T(x,e)=x$.

**Definition 8.**

**Proposition 1.**

**Proof.**

- (2)
- For any ${r}_{1},{r}_{2}\in [0,1]$ and $0\le {r}_{1}\le {r}_{2}\le 1$, ${\left[u\right]}^{{r}_{2}}\subset {\left[u\right]}^{{r}_{1}}$ and ${\left[v\right]}^{{r}_{2}}\subset {\left[v\right]}^{{r}_{1}}$ are tenable; therefore, ${x}_{u}^{{r}_{1}}\le {x}_{u}^{{r}_{2}},{y}_{u}^{{r}_{2}}\le {y}_{u}^{{r}_{1}},{x}_{v}^{{r}_{1}}\le {x}_{v}^{{r}_{2}},{y}_{v}^{{r}_{2}}\le {y}_{v}^{{r}_{1}}$, because T satisfies monotonicity,$$\begin{array}{ccc}\hfill T({x}_{u}^{{r}_{1}},{x}_{v}^{{r}_{1}})& \le & T({x}_{u}^{{r}_{2}},{x}_{v}^{{r}_{2}}),\hfill \\ \hfill T({y}_{u}^{{r}_{2}},{y}_{v}^{{r}_{2}})& \le & T({y}_{u}^{{r}_{1}},{y}_{v}^{{r}_{1}}),\hfill \\ \hfill T({x}_{u}^{{r}_{2}},{x}_{v}^{{r}_{2}})& \le & T({y}_{u}^{{r}_{2}},{y}_{v}^{{r}_{2}}).\hfill \end{array}$$These three inequalities are combined:$$T({x}_{u}^{{r}_{1}},{x}_{v}^{{r}_{1}})\le T({x}_{u}^{{r}_{2}},{x}_{v}^{{r}_{2}})\le T({y}_{u}^{{r}_{2}},{y}_{v}^{{r}_{2}})\le T({y}_{u}^{{r}_{1}},{y}_{v}^{{r}_{1}}).$$Therefore,$$T({\left[u\right]}^{{r}_{2}},{\left[v\right]}^{{r}_{2}})\subset T({\left[u\right]}^{{r}_{1}},{\left[v\right]}^{{r}_{1}}).$$
- (3)
- Because $u,v\in {\mathcal{F}}_{\mathcal{DC}}$, then for any ${r}_{0}\in [0,1]$, there exist ${r}_{1}^{\prime},{r}_{2}^{\prime}\in R$ that satisfy $0<{r}_{1}^{\prime}<{r}_{0}$ and $0<{r}_{2}^{\prime}<{r}_{0}$ such that ${\left[u\right]}^{{r}_{1}^{\prime}}={\left[u\right]}^{{r}_{0}}$ and ${\left[v\right]}^{{r}_{2}^{\prime}}={\left[v\right]}^{{r}_{0}}$ are tenable, i.e., ${\left[u\right]}^{{\alpha}_{1}}={\left[u\right]}^{{r}_{0}}$ is tenable for any ${\alpha}_{1}\in [{r}_{1}^{\prime},{r}_{0}]$, and ${\left[u\right]}^{{\alpha}_{2}}={\left[u\right]}^{{r}_{0}}$ is tenable for any ${\alpha}_{2}\in [{r}_{2}^{\prime},{r}_{0}]$. Therefore, if $\alpha ={\alpha}_{1}\vee {\alpha}_{2}$ then$$T({\left[u\right]}^{\alpha},{\left[v\right]}^{\alpha})=T({\left[u\right]}^{{r}_{0}},{\left[v\right]}^{{r}_{0}}).$$

**Theorem 8.**

**Proof.**

**Definition 9.**

**Proposition 2.**

**Theorem 9.**

**Remark 2.**

**Remark 3.**

**Example 1.**

- (1)
- When $r=0.2$, ${\left[u\right]}^{0.2}=\{0,2,3,5\}$, and ${\left[v\right]}^{0.2}=\{3,4,5\}$, then ${T}_{L}({\left[u\right]}^{0.2},{\left[v\right]}^{0.2})=\{0,1,2,3,4,5\}$,
- (2)
- When $r=0.5$, ${\left[u\right]}^{0.5}=\{2,3,5\}$, and ${\left[v\right]}^{0.5}=\{3,4,5\}$, then ${T}_{L}({\left[u\right]}^{0.5},{\left[v\right]}^{0.5})=\{0,1,2,3,4,5\}$,
- (3)
- When $r=0.6$, ${\left[u\right]}^{0.6}=\{3,5\}$, and ${\left[v\right]}^{0.6}=\{3,4,5\}$, then ${T}_{L}({\left[u\right]}^{0.6},{\left[v\right]}^{0.6})=\{1,2,3,4,5\}$,
- (4)
- When $r=0.8$, ${\left[u\right]}^{0.8}=\{3,5\}$, and ${\left[v\right]}^{0.8}=\{3,4\}$, then ${T}_{L}({\left[u\right]}^{0.8},{\left[v\right]}^{0.8})=\{1,2,3,4\}$,
- (5)
- When $r=1$, ${\left[u\right]}^{1}=\left\{3\right\}$, and ${\left[v\right]}^{1}=\left\{4\right\}$, then ${T}_{L}({\left[u\right]}^{1},{\left[v\right]}^{1})=\left\{2\right\}$.

**Example 2.**

**Example 3.**

**Example 4.**

**Example 5.**

**Proposition 3.**

- (1)
- Commutativity:$$T(x,y)=T(y,x),$$$$S(x,y)=S(y,x).$$
- (2)
- Associativity:$$T(T(x,y),z)=T(x,T(y,z)),$$$$S(S(x,y),z)=S(x,S(y,z)).$$

**Proof.**

**Theorem 10.**

- (1)
- Commutativity:$$\mathbb{T}(u,v)=\mathbb{T}(v,u),$$$$\mathbb{S}(u,v)=\mathbb{S}(v,u).$$
- (2)
- Associativity:$$\mathbb{T}(\mathbb{T}(u,v),w)=\mathbb{T}(u,\mathbb{T}(v,w)),$$$$\mathbb{S}(\mathbb{S}(u,v),w)=\mathbb{S}(u,\mathbb{S}(v,w)).$$

**Proof.**

- (1)
- In order to prove $\mathbb{T}(u,v)=\mathbb{T}(v,u)$, we need to prove that for any $r\in [0,1]$, both sides of the equation have the same r-level set.$$\begin{array}{ccc}\hfill {\left[\mathbb{T}(u,v)\right]}^{r}& =& T({\left[u\right]}^{r},{\left[v\right]}^{r})\hfill \\ \hfill & =& \left\{T(x,y)\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}x\in {\left[u\right]}^{r},y\in {\left[v\right]}^{r}\}\hfill \\ \hfill & =& \left\{T(y,x)\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}y\in {\left[v\right]}^{r},x\in {\left[u\right]}^{r}\}\hfill \\ \hfill & =& T({\left[v\right]}^{r},{\left[u\right]}^{r})\hfill \\ \hfill & =& {\left[\mathbb{T}(v,u)\right]}^{r}.\hfill \end{array}$$
- (2)
- In order to prove $\mathbb{T}(\mathbb{T}(u,v),w)=\mathbb{T}(u,\mathbb{T}(v,w))$, we need to prove that for any $r\in [0,1]$, both sides of the equation have the same r-level set.$$\begin{array}{ccc}\hfill {\left[\mathbb{T}(\mathbb{T}(u,v),w)\right]}^{r}& =& T({\left[\mathbb{T}(u,v)\right]}^{r},{\left[w\right]}^{r})\hfill \\ \hfill & =& T(T({\left[u\right]}^{r},{\left[v\right]}^{r}),{\left[w\right]}^{r})\hfill \\ \hfill & =& \left\{T(T(x,y),z)\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}x\in {\left[u\right]}^{r},y\in {\left[v\right]}^{r},z\in {\left[w\right]}^{r}\}\hfill \\ \hfill & =& \left\{T(x,T(y,z))\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}x\in {\left[u\right]}^{r},y\in {\left[v\right]}^{r},z\in {\left[w\right]}^{r}\}\hfill \\ \hfill & =& T({\left[u\right]}^{r},T({\left[v\right]}^{r},{\left[w\right]}^{r}))\hfill \\ \hfill & =& {\left[\mathbb{T}(u,\mathbb{T}(v,w))\right]}^{r}.\hfill \end{array}$$

## 6. Application to Image Fusion

#### 6.1. Interpretation of Gray Image as Special Discrete Fuzzy Numbers on Countable Sets

- (1)
- Let a gray image with 256 grayscale levels, i.e., $\{0,1,\dots ,255\}$, be I, and the size of I is $M\times N$. $I(x,y)$ represents a gray-scale value of $(x,y)$ in I, where $x\in \{1,2,\dots ,M\}$, $y\in \{1,2,\dots ,N\}$.
- (2)
- We take a point $({x}_{0},{y}_{0}),{x}_{0}\in \{2,3,\dots ,M-1\},{y}_{0}\in \{2,3,\dots ,N-1\}$ in I as the center and use the neighboring pixels around $({x}_{0},{y}_{0})$ to form a rectangle, we call this rectangle W. The size of W is ${n}_{W}\times {n}_{W}$. When ${n}_{W}=3$, the points of W are represented as $({x}_{0}+i,{y}_{0}+j),i,j=\{-1,0,1\}$ and the corresponding pixel value can be expressed as $I({x}_{0}+i,{y}_{0}+j),i,j=\{-1,0,1\}$.
- (3)
- In order to represent the gray-scale pixel value, the mean value $\overline{W}$ and standard deviation $\overline{S}$ of W are calculated.$$\overline{W}=\frac{{\sum}_{i=-1}^{1}{\sum}_{j=-1}^{1}I({x}_{0}+i,{y}_{0}+j)}{3\times 3},$$$$\overline{S}=\sqrt{\frac{{\sum}_{i=-1}^{1}{\sum}_{j=-1}^{1}{(I({x}_{0}+i,{y}_{0}+j)-\overline{W})}^{2}}{3\times 3-1}}.$$
- (4)
- We construct Gaussian discrete fuzzy numbers for $I({x}_{0},{y}_{0})$.$u:R\to [0,1]$ is defined by:$$u(I(x,y))=\left(\right)open="\{"\; close>\begin{array}{cc}exp(-\frac{{(I(x,y)-\overline{W})}^{2}}{2{\overline{S}}^{2}}),\hfill & if(x,y)\in W\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}$$Then, u is the special discrete fuzzy numbers on countable sets with ${\left[u\right]}^{0}=\{I({x}_{0}+i,{y}_{0}+j):i,j=\{-1,0,1\}\}$. In this case, the countable set is $C=\{0,1,\dots ,255\}$.

**Example 6.**

#### 6.2. Gray Image Fusion by Means of the Triangular Norm and Triangular Conorm Operations of Special Discrete Fuzzy Numbers on Countable Sets

**Definition 10.**

- (1)
- Let the point $(x,y)$ of f be the center and interpret it as special discrete fuzzy numbers on countable sets; this discrete fuzzy number is denoted as $u(f(x,y))$. Similarly, let the point $({x}^{\prime},{y}^{\prime})$ of g be the center and interpret it as special discrete fuzzy numbers on countable sets; this discrete fuzzy number is denoted as $v(g({x}^{\prime},{y}^{\prime}))$.
- (2)
- By using the triangular norm $\mathbb{T}$ or triangular conorm $\mathbb{S}$ defined in Section 5, two discrete fuzzy numbers $u(f(x,y))$ and $v(g({x}^{\prime},{y}^{\prime}))$ at corresponding positions are operated, and a new discrete fuzzy number $\mathbb{T}(u,v)$ or $\mathbb{S}(u,v)$ is obtained.
- (3)
- The mass center of the new discrete fuzzy number is calculated according to Equation (7) as the pixel gray value of the fused image.
- (4)
- Change the points $(x,y)$ and $({x}^{\prime},{y}^{\prime})$ to the same position and skip to step (1) until the points $(x,y)$ and $({x}^{\prime},{y}^{\prime})$ traverse the image f and g, respectively.

## 7. Application to Aggregation of Subjective Evaluation

**Example 7.**

**Example 8.**

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The steps of using the special discrete fuzzy numbers on countable sets to represent pixel value of gray images.

**Figure 2.**Using the special discrete fuzzy numbers on countable sets to represent pixel value of gray images.

**Figure 5.**The results of the fusion of thermal image and visible light image. (

**a**) ${\mathbb{T}}_{Min}$. (

**b**) ${\mathbb{S}}_{Max}$. (

**c**) ${\mathbb{T}}_{L}$. (

**d**) ${\mathbb{S}}_{L}$.

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## Share and Cite

**MDPI and ACS Style**

Qin, N.; Gong, Z.
Special Discrete Fuzzy Numbers on Countable Sets and Their Applications. *Symmetry* **2024**, *16*, 264.
https://doi.org/10.3390/sym16030264

**AMA Style**

Qin N, Gong Z.
Special Discrete Fuzzy Numbers on Countable Sets and Their Applications. *Symmetry*. 2024; 16(3):264.
https://doi.org/10.3390/sym16030264

**Chicago/Turabian Style**

Qin, Na, and Zengtai Gong.
2024. "Special Discrete Fuzzy Numbers on Countable Sets and Their Applications" *Symmetry* 16, no. 3: 264.
https://doi.org/10.3390/sym16030264