Equivalence of Common Metrics on Trapezoidal Fuzzy Numbers

Abstract

From both theoretical and applied perspectives, the trapezoidal fuzzy numbers are widely relevant fuzzy sets. In this paper, we show that the four kinds of common metrics—the supremum metric, the $L_p$-type $d_p$ metrics, the sendograph metric, and the endograph metric—are equivalent on the trapezoidal fuzzy numbers. In fact, we obtain a stronger result: the convergence induced by these four kinds of metrics on the trapezoidal fuzzy numbers is equivalent to the convergence of the corresponding representation quadruples of the trapezoidal fuzzy numbers in ${\mathbb{R}}^4$. The latter convergence is very easy to verify. Our results give a fundamental understanding of these four kinds of common metrics on the trapezoidal fuzzy numbers and provide a quick judgment condition for the convergence induced by them.

1. Introduction

The trapezoidal fuzzy numbers and its special case triangular fuzzy numbers have been widely used and discussed in theory and applications [1,2,3,4,5,6,7,8,9,10]. The supremum distance, the $L_p$-type $d_p$ distances, the sendograph distance and the endograph distance on fuzzy sets have been attracted much attention [3,11,12,13,14,15,16,17,18,19]. These four kinds of distances are metrics on the set of trapezoidal fuzzy numbers.

It is natural and important to consider the relationship of these four kinds of metrics on the set of trapezoidal fuzzy numbers. From the already available results, what is known is that, among them, the supremum metric is the strongest and the endograph metric is the weakest on the trapezoidal fuzzy numbers.

In this paper, we show that each one of the convergences induced by these four kinds of common metrics on the trapezoidal fuzzy numbers is equivalent to the convergence of the corresponding representation quadruples of the trapezoidal fuzzy numbers in ${\mathbb{R}}^4$. So, they are equivalent metrics on the trapezoidal fuzzy numbers.

In the theory of metric spaces, convergences induced by the metrics play a key role. Our results give a fundamental understanding of the four kinds of common metrics on the trapezoidal fuzzy numbers. As the convergence in ${\mathbb{R}}^4$ is very easy to verify, we provide a quick judgment condition for the convergence induced by them.

The remainder part of this paper is organized as follows: Section 2 reviews some basic concepts and fundamental conclusions of fuzzy sets and the extended metrics on them. In Section 3, we recall some basic concepts and properties related to the triangular fuzzy numbers and the trapezoidal fuzzy numbers. Section 4 recalls and gives some properties of the trapezoidal fuzzy numbers and triangular fuzzy numbers. Section 5 presents the main results of this paper. Finally, in Section 6, we give our conclusions.

2. Fuzzy Sets and Extended Metrics on Them

In this section, we review some basic concepts and fundamental conclusions of fuzzy sets and the extended metrics on them. For fuzzy theory and applications, we refer the readers to [1,2,3,8,11,16,17,20,21,22,23,24,25].

Let $\mathbb{N}$ be the set of all positive integers and let ${\mathbb{R}}^m$ be the m-dimensional Euclidean space. ${\mathbb{R}}^1$ is also written as $\mathbb{R}$. We use ${\mathbb{R}}^{+}$ to denote the set $\{r\in \mathbb{R}:r\ge 0\}$.

Let Y be a nonempty set. The symbol $P\left(Y\right)$ denotes the power set of Y, i.e., the set of all subsets of Y. The symbol $F\left(Y\right)$ denotes the set of all fuzzy sets in Y. We see a fuzzy set in Y as a function from Y to $[0,1]$. Given $u\in F\left(Y\right)$ and $\alpha \in (0,1]$, the $\alpha$-cut ${\left[u\right]}_{\alpha }$ of u is defined by ${\left[u\right]}_{\alpha }:=\{x\in Y:u\left(x\right)\ge \alpha \}$.

Let Y be a topological space. The symbol $C\left(Y\right)$ denotes the set of all nonempty closed subsets of Y. $K\left(Y\right)$ denotes the set of all nonempty compact subsets of Y. For $u\in F\left(Y\right)$, the 0-cut ${\left[u\right]}_0$ of u is defined by ${\left[u\right]}_0:=\overline{\{x\in Y:u(x)>0\}}$, where $\overline{S}$ denotes the topological closure of S in Y. ${\left[u\right]}_0$ is called the support of u, and is also denoted by supp u.

Let Y be a nonempty set. For $u\in F\left(Y\right)$, define

$$\begin{array}{c}\mathrm{end}u:=\left\{\right(x,t)\in Y\times [0,1]:u(x)\ge t\},\\ \mathrm{send}u:=\mathrm{end}u\cap ({\left[u\right]}_0\times [0,1]),\end{array}$$

where $\mathrm{send}u$ is well-defined if and only if Y is a topological space. $\mathrm{end}u$ is called the endograph of u, and $\mathrm{send}u$ is called the sendograph of u. Clearly $\mathrm{end}u=\mathrm{send}u\cup (X\times \{0\left\}\right)$.

In this paper, we suppose that X is a nonempty set, and it is equipped with a metric d. For convenience, X is also used to denote the metric space $(X,d)$.

The symbol $\mathit{H}$ is used to denote the Hausdorff extended metric on $C\left(X\right)$ induced by d, i.e.,

$$\mathit{H}(\mathit{U},\mathit{V})=max\{H^{*}(U,V),H^{*}(V,U)\}$$

for any $U,V\in C\left(X\right)$, where

$$H^{*}(U,V)=\underset{u\in U}{sup}d(u,V)=\underset{u\in U}{sup}\underset{v\in V}{inf}d(u,v).$$

For convenience, we often refer to the Hausdorff extended metric as the Hausdorff metric. See also Remark 2.5 of [19].

Let $[a,b]$ and $[c,d]$ be two intervals. Then

$$H\left(\right[a,b],[c,d\left]\right)=max\left\{\right|a-c|,|b-d\left|\right\}.$$

(1)

The metric $\overline{d}$ on $X\times [0,1]$ induced by d is defined as follows: for any $(x,\alpha ),(y,\beta )\in X\times [0,1]$, $\overline{d}((x,\alpha ),(y,\beta ))=d(x,y)+|\alpha -\beta |.$ When there is no risk of confusion, H is also used to denote the Hausdorff extended metric on $C(X\times [0,1\left]\right)$ induced by $\overline{d}$.

The symbol $F_{USC}\left(X\right)$ denotes the set of all upper semi-continuous fuzzy sets in X; that is

$$F_{USC}\left(X\right):=\{u\in F\left(X\right):{\left[u\right]}_{\alpha }\in C\left(X\right)\cup \{Ø\}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\alpha \in [0,1]\}.$$

Define $F_{USC}^1\left(X\right):=\{u\in F_{USC}\left(X\right):{\left[u\right]}_1\ne Ø\}.$ Clearly $F_{USC}^1\left(X\right)⫋F_{USC}\left(X\right).$

The supremum distance $d_{\infty }$, the sendograph distance $H_{\mathrm{send}}$, and the endograph distance $H_{\mathrm{end}}$ on $F_{USC}^1\left(X\right)$ are defined as follows, respectively (for each $u,v\in F_{USC}^1\left(X\right)$):

$$\begin{array}{c}{d}_{\infty }(u,v)=\underset{\alpha \in [0,1]}{sup}H({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha }),\\ H_{\mathrm{send}}(u,v)=H(\mathrm{send}u,\mathrm{send}v),\\ H_{\mathrm{end}}(u,v)=H(\mathrm{end}u,\mathrm{end}v),\end{array}$$

where H in the definition of $d_{\infty }$ denotes the Hausdorff extended metric on $C\left(X\right)$ induced by d, and H in the definitions of $H_{\mathrm{send}}$ and $H_{\mathrm{end}}$ denote the Hausdorff extended metric on $C(X\times [0,1\left]\right)$ induced by $\overline{d}$.

The sendograph distance $H_{\mathrm{send}}$ was introduced by Kloeden [26]. Both $d_{\infty }$ and $H_{\mathrm{send}}$ on $F_{USC}^1\left(X\right)$ are extended metrics, but each one of them is not necessarily a metric. $H_{\mathrm{end}}$ on $F_{USC}^1\left(X\right)$ is a metric. See also Remark 2.7 of [19].

The $L_p$-type $d_p$ distance, $1\le p<+\infty$, of each $u,v\in F_{USC}^1\left(X\right)$ is defined by

$$d_p(u,v)={\left({\int }_0^{1}H{({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha })}^{p}d\alpha \right)}^{1/p},$$

where $d_p(u,v)$ is well-defined if and only if $H({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha })$ is a measurable function of $\alpha$ on $[0,1]$.

We assume that, in the sequel, “p” appearing in mathematical expressions, such as $d_p$, etc., is an arbitrary number satisfying $1\le p<+\infty$.

For some metric spaces Y, $d_p$ distances could be not well-defined on $F_{USC}^1\left(Y\right)$ (see Example 3.25 of [19]). So, the following $d_p^{*}$ extended metrics on $F_{USC}^1\left(X\right)$ are introduced in [27] (for each $u,v\in F_{USC}^1\left(X\right)$):

$$\begin{array}{c}{d}_p^{*}(u,v):=inf\{{\left({\int }_0^{1}f{\left(\alpha \right)}^{p}d\alpha \right)}^{1/p}:f\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}[0,1]\mathrm{t}\mathrm{o}{\mathbb{R}}^{+}\cup \{+\infty \},\\ \mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{g}f\left(\alpha \right)\ge H({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha })\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\alpha \in [0,1]\}.\end{array}$$

Theorem 1

( [18,19,27]). Let $u,v\in F_{USC}^1\left(X\right)$.

(i) 

$d_{\infty }(u,v)\ge H_{\mathrm{send}}(u,v)\ge H_{\mathrm{end}}(u,v)$.

(ii) 

$d_{\infty }(u,v)\ge d_p^{*}(u,v)$.

(iii) 

$d_p^{*}(u,v)\ge {\left({\displaystyle \frac{H_{\mathrm{end}}{(u,v)}^{p+1}}{p+1}}\right)}^{1/p}$.

(iv) 

If $d_p(u,v)$ is well-defined, then $d_p^{*}(u,v)=d_p(u,v)$.

(v) 

If $X={\mathbb{R}}^m$, then $d_p(u,v)$ is well-defined; so, $d_p^{*}(u,v)=d_p(u,v)$.

(vi) 

If $X={\mathbb{R}}^m$, then $d_p^{*}$ in (ii) and (iii) can be replaced by $d_p$.

Proof. 

Clearly (i) holds. (i) is (1) of [18]. (i) may have been a known conclusion before the appearance of [18]. (ii) is (13) of [18]. (iii) is Theorem 4.1(i) of [19]. (iv) is given in Remark 3.2 of [27]. (iv) is obvious. A routine proof of (iv) is given below.

Put $S:=\{f:f\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}[0,1]\mathrm{t}\mathrm{o}{\mathbb{R}}^{+}\cup \{+\infty \},\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{g}f\left(\alpha \right)\ge H({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha })\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\alpha \in [0,1]\}$. Suppose that $d_p(u,v)$ is well-defined; that is, $H({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha })$ is a measurable function of $\alpha$ on $[0,1]$. Then, the function $H({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha })$ of $\alpha$ on $[0,1]$ belongs to S. (The converse is also true.) Hence, we have (a) ${inf}_{f\in S}{\left({\int }_0^{1}f{\left(\alpha \right)}^{p}d\alpha \right)}^{1/p}\le {\left({\int }_0^{1}H{({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha })}^{p}d\alpha \right)}^{1/p}$. On the other hand, for each $f\in S$, ${\left({\int }_0^{1}f{\left(\alpha \right)}^{p}d\alpha \right)}^{1/p}\ge {\left({\int }_0^1{\left(H({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha }\right)}^{p}d\alpha \right)}^{1/p}$, as $f\left(\alpha \right)\ge H({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha })$ for all $\alpha \in [0,1]$. Thus, we obtain (b) ${inf}_{f\in S}{\left({\int }_0^{1}f{\left(\alpha \right)}^{p}d\alpha \right)}^{1/p}\ge {\left({\int }_0^{1}H{({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha })}^{p}d\alpha \right)}^{1/p}$. So, $d_p^{*}(u,v)={inf}_{f\in S}{\left({\int }_0^{1}f{\left(\alpha \right)}^{p}d\alpha \right)}^{1/p}=\left(\mathrm{b}\mathrm{y}\right(\mathrm{a}\left)\mathrm{a}\mathrm{n}\mathrm{d}\right(\mathrm{b}\left)\right){\left({\int }_0^{1}H{({\left[u\right]}_{\alpha },{\left[v\right]}_{\alpha })}^{p}d\alpha \right)}^{1/p}=d_p(u,v)$.

Theorem 3.8 of [19] says that if $X={\mathbb{R}}^m$, then $d_p(u,v)$ is well-defined. Obviously, by this and (iv), we obtain that if $X={\mathbb{R}}^m$, then $d_p^{*}(u,v)=d_p(u,v)$. So, (v) holds. (vi) follows immediately from (v). □

Let Y be a nonempty set, Z a subset of Y, and ${\rho }_1$ and ${\rho }_2$ two extended metrics on Y. We say that ${\rho }_1$ is stronger than ${\rho }_2$ on Z, denoted by ${\rho }_1⪰{\rho }_2\left(Z\right)$, if for each sequence $\left\{y_n\right\}$ in Z and each $y\in Z$, ${lim}_{n\to \infty }{\rho }_1(y_n,y)=0$ implies that ${lim}_{n\to \infty }{\rho }_2(y_n,y)=0$.

${\rho }_1$ is stronger than ${\rho }_2$ on Z is also known as ${\rho }_2$ is weaker than ${\rho }_1$ on Z and written as ${\rho }_2⪯{\rho }_1\left(Z\right)$. ${\rho }_1$ is said to be equivalent to ${\rho }_2$ on Z, denoted by ${\rho }_1\approx {\rho }_2\left(Z\right)$, if ${\rho }_1⪰{\rho }_2\left(Z\right)$ and ${\rho }_2⪰{\rho }_1\left(Z\right)$.

The expression ${\rho }_1⪰S⪰{\rho }_2\left(Z\right)$, where S is a set of extended metrics on Y, means that for each $\rho \in S$, ${\rho }_1⪰\rho ⪰{\rho }_2\left(Z\right)$.

Theorem 2

([18]). $d_{\infty }⪰\{H_{\mathrm{send}},d_p^{*}\}⪰H_{\mathrm{end}}\left(F_{USC}^1\left(X\right)\right)$.

Proof. 

By Theorem 1(i), $d_{\infty }⪰H_{\mathrm{send}}⪰H_{\mathrm{end}}\left(F_{USC}^1\left(X\right)\right)$. By Theorem 1(ii), $d_{\infty }⪰d_p^{*}\left(F_{USC}^1\left(X\right)\right)$. By Theorem 1(iii), $d_p^{*}⪰H_{\mathrm{end}}\left(F_{USC}^1\left(X\right)\right)$. Theorem 6.2 of [18] also says that $d_p^{*}⪰H_{\mathrm{end}}\left(F_{USC}^1\left(X\right)\right)$. So, this theorem is indeed given in [18]. □

By Theorems 1(v) and 2, we make the following conclusion:

Corollary 1

([18,19,27]). $d_{\infty }⪰\{H_{\mathrm{send}},d_p\}⪰H_{\mathrm{end}}\left(F_{USC}^1\left({\mathbb{R}}^m\right)\right)$.

$d_{\infty }$, $d_p$, $H_{\mathrm{send}}$, and $H_{\mathrm{end}}$ are metrics of $F_{USCB}^1\left(X\right)$.

The corresponding author of this paper independently provided Section 2.

3. Triangular Fuzzy Numbers and Trapezoidal Fuzzy Numbers

In this section, we review some basic concepts and properties related to the triangular fuzzy numbers and the trapezoidal fuzzy numbers. The readers may also refer to [8,14,28,29,30] for several types of fuzzy sets.

Usually, the symbols $(a,b,c,d)$ with $a,b,c,d$ in $\mathbb{R}$ represent the elements in ${\mathbb{R}}^4$, and the symbols $(a,b,c)$ with $a,b,c$ in $\mathbb{R}$ represent the elements in ${\mathbb{R}}^3$. In this paper, for each $a,b,c,d$ in $\mathbb{R}$, we use $[a,b,c,d]$ instead of $(a,b,c,d)$ to represent the corresponding element in ${\mathbb{R}}^4$ and use $[a,b,c]$ instead of $(a,b,c)$ to represent the corresponding element in ${\mathbb{R}}^3$.

We use T to denote the set $\{[a,b,c,d]\in {\mathbb{R}}^4:a\le b\le c\le d\}$ and $T_0$ to denote the set $\{[a,b,c,d]\in {\mathbb{R}}^4:a<b\le c<d\}$. Clearly, $T_0⫋T$.

We use G to denote the set $\{[a,b,c]\in {\mathbb{R}}^3:a\le b\le c\}$ and $G_0$ to denote the set $\{[a,b,c]\in {\mathbb{R}}^3:a<b<c\}$. Clearly, $G_0⫋G$.

Definition 1.

We use Tag to denote the set of all regular triangular fuzzy numbers. $Tag:=\{(a,b,c):[a,b,c]in{G}_0\}$, where, for any $[a,b,c]$ in $G_0$, the regular triangular fuzzy number $(a,b,c)$ is defined to be the fuzzy set u in $F\left(\mathbb{R}\right)$, given by

$$u\left(x\right)=\left\{\begin{array}{cc}\hfill {\displaystyle \frac{x-a}{b-a}},& ifa\le x\le b,\hfill \\ \hfill {\displaystyle \frac{c-x}{c-b}},& ifb\le x\le c,\hfill \\ \hfill 0,& ifx\in \mathbb{R}\setminus [a,c].\hfill \end{array}\right.$$

Definition 2.

We use Tap to denote the set of all regular trapezoidal fuzzy numbers. $Tap:=\{(a,b,c,d):[a,b,c,d]in{T}_0\}$, where, for any $[a,b,c,d]$ in $T_0$, the regular trapezoidal fuzzy number $(a,b,c,d)$ is defined to be the fuzzy set u in $F\left(\mathbb{R}\right)$, given by

$$u\left(x\right)=\left\{\begin{array}{cc}\hfill {\displaystyle \frac{x-a}{b-a}},& ifa\le x\le b,\hfill \\ \hfill 1,& ifb\le x\le c,\hfill \\ \hfill {\displaystyle \frac{d-x}{d-c}},& ifc\le x\le d,\hfill \\ \hfill 0,& ifx\in \mathbb{R}\setminus [a,d].\hfill \end{array}\right.$$

Remark 1.

(i) $u\in$ Tag means that there is an $[a,b,c]\in G_0$ satisfying $u=(a,b,c)$. (ii) $u\in$ Tap means that there is an $[a,b,c,d]\in T_0$ satisfying $u=(a,b,c,d)$. (iii) Each regular triangular fuzzy number $(a,b,c)$ is the regular trapezoidal fuzzy number $(a,b,b,c)$. So, Tag ⊆ Tap.

We say that two fuzzy sets are equal if they have the same membership function.

Definition 3.

We use Trag to denote the set of all triangular fuzzy numbers. $Trag:=\left\{\right(a,b,c):[a,b,c]inG\}$, where, for any $[a,b,c]$ in G, the triangular fuzzy number $(a,b,c)$ is defined to be the fuzzy set u in $F\left(\mathbb{R}\right)$ in the following way:

$$\begin{array}{c}uistheregulartriangularfuzzynumber(a,b,c)whena<b<c;\\ u\left(x\right)=\left\{\begin{array}{cc}\hfill {\displaystyle \frac{c-x}{c-b}},& ifb\le x\le c,\hfill \\ \hfill 0,& ifx\in \mathbb{R}\setminus [b,c],\hfill \end{array}\right.whena=b<c;\\ u\left(x\right)=\left\{\begin{array}{cc}\hfill {\displaystyle \frac{x-a}{b-a}},& ifa\le x\le b,\hfill \\ \hfill 0,& ifx\in \mathbb{R}\setminus [a,b],\hfill \end{array}\right.whena<b=c;\\ u\left(x\right)=\left\{\begin{array}{cc}\hfill 1,& ifx=b,\hfill \\ \hfill 0,& ifx\in \mathbb{R}\setminus \left\{b\right\},\hfill \end{array}\right.whena=b=c.\end{array}$$

Clearly, each $(a,b,c)$ in Tag is the $(a,b,c)$ in Trag. This means that the concept of triangular fuzzy numbers is a kind of generalization of the concept of regular triangular fuzzy numbers. Hence, Tag ⊆ Trag.

Definition 4.

We use Trap to denote the set of all trapezoidal fuzzy numbers. $Trap:=\left\{\right(a,b,c,d):[a,b,c,d]inT\}$, where, for any $[a,b,c,d]$ in T, the trapezoidal fuzzy number $(a,b,c,d)$ is defined to be the fuzzy set u in $F\left(\mathbb{R}\right)$ in the following way:

$$\begin{array}{c}uistheregulartrapezoidalfuzzynumber(a,b,c,d)whena<b\le c<d;\\ u\left(x\right)=\left\{\begin{array}{cc}\hfill 1,& ifb\le x\le c,\hfill \\ \hfill {\displaystyle \frac{d-x}{d-c}},& ifc\le x\le d,\hfill \\ \hfill 0,& ifx\in \mathbb{R}\setminus [b,d],\hfill \end{array}\right.whena=b\le c<d;\\ u\left(x\right)=\left\{\begin{array}{cc}\hfill {\displaystyle \frac{x-a}{b-a}},& ifa\le x\le b,\hfill \\ \hfill 1,& ifb\le x\le c,\hfill \\ \hfill 0,& ifx\in \mathbb{R}\setminus [a,c],\hfill \end{array}\right.whena<b\le c=d;\\ u\left(x\right)=\left\{\begin{array}{cc}\hfill 1,& ifb\le x\le c,\hfill \\ \hfill 0,& ifx\in \mathbb{R}\setminus [b,c],\hfill \end{array}\right.whena=b\le c=d.\end{array}$$

Clearly, each $(a,b,c,d)$ in Tap is the $(a,b,c,d)$ in Trap. This means that the concept of trapezoidal fuzzy numbers is a kind of generalization of the concept of regular trapezoidal fuzzy numbers. Hence, Tap ⊆ Trap.

Remark 2.

(i) $u\in$ Trag means that there is an $[a,b,c]\in G$ satisfying $u=(a,b,c)$. (ii) $u\in$ Trap means that there is an $[a,b,c,d]\in T$ satisfying $u=(a,b,c,d)$.

Remark 3.

Each triangular fuzzy number $(a,b,c)$ is the trapezoidal fuzzy number $(a,b,b,c)$. So, Trag ⊆ Trap.

The readers may also refer to the corresponding contents in [31] for details.

4. Some Properties of Trapezoidal Fuzzy Numbers and Triangular Fuzzy Numbers

In this section, we recall and give some properties of the trapezoidal fuzzy numbers and triangular fuzzy numbers. These properties are useful to obtain and understand the main results of this paper.

The following Propositions 1 and 2 should be known. See [9] and related works. Clearly, Proposition 2 is a corollary of Proposition 1.

Proposition 1.

Let $u\in F\left(\mathbb{R}\right)$ and $(a,b,c,d)\in$ Trap. Then, $u=(a,b,c,d)$ if and only if

$$foreach\xi \in [0,1],{\left[u\right]}_{\xi }=[\xi (b-a)+a,c+(1-\xi )(d-c)].$$

(2)

Proposition 2.

Let $u\in F\left(\mathbb{R}\right)$ and $(a,b,c)\in$ Trag. Then, $u=(a,b,c)$ if and only if $foreach\xi \in [0,1],{\left[u\right]}_{\xi }=[\xi (b-a)+a,b+(1-\xi )(c-b)]$.

For any $[a,b,c,d]$ and $[a_1,b_1,c_1,d_1]$ in ${\mathbb{R}}^4$, $[a,b,c,d]=[a_1,b_1,c_1,d_1]$ means that $a=a_1$, $b=b_1$, $c=c_1$ and $d=d_1$. For any $(a,b,c,d)$ and $(a_1,b_1,c_1,d_1)$ in Trap, $(a,b,c,d)=(a_1,b_1,c_1,d_1)$ means that $(a,b,c,d)$ and $(a_1,b_1,c_1,d_1)$ are the same fuzzy set.

For any $[a,b,c]$ and $[a_1,b_1,c_1]$ in ${\mathbb{R}}^3$, $[a,b,c]=[a_1,b_1,c_1]$ means that $a=a_1$, $b=b_1$ and $c=c_1$. For any $(a,b,c)$ and $(a_1,b_1,c_1)$ in Trag, $(a,b,c)=(a_1,b_1,c_1)$ means that $(a,b,c)$ and $(a_1,b_1,c_1)$ are the same fuzzy set.

The following Theorem 3(ii) states the representation uniqueness of the trapezoidal fuzzy numbers.

Theorem 3.

(i) Let $u=(a,b,c,d)$ be in Trap. Then, ${\left[u\right]}_0=[a,d]$ and ${\left[u\right]}_1=[b,c]$. (ii) Let $(a,b,c,d)$ and $(a_1,b_1,c_1,d_1)$ be in Trap. Then, $(a,b,c,d)=(a_1,b_1,c_1,d_1)$ if and only if $[a,b,c,d]=[a_1,b_1,c_1,d_1]$. (iii) Let $u=(a,b,c,d)$ be in Tap. Then, ${\left[u\right]}_0=[a,d]$ and ${\left[u\right]}_1=[b,c]$. (iv) Let $(a,b,c,d)$ and $(a_1,b_1,c_1,d_1)$ be in Tap. Then, $(a,b,c,d)=(a_1,b_1,c_1,d_1)$ if and only if $[a,b,c,d]=[a_1,b_1,c_1,d_1]$.

Proof. 

By Definition 4 and easy calculations, we obtain (i). (One way to perform these calculations is by watching the graphs of the membership functions of $(a,b,c,d)$ in the four cases $a<b\le c<d$, $a=b\le c<d$, $a<b\le c=d$, and $a=b\le c=d$.) (i) follows immediately from Proposition 1.

Now, we show (ii). If $[a,b,c,d]=[a_1,b_1,c_1,d_1]$, i.e., $a=a_1,b=b_1,c=c_1$ and $d=d_1$, then, by Definition 4, $(a,b,c,d)=(a_1,b_1,c_1,d_1)$.

Suppose that $(a,b,c,d)=(a_1,b_1,c_1,d_1)$. Then, ${\left[(a,b,c,d)\right]}_1={\left[(a_1,b_1,c_1,d_1)\right]}_1$ and ${\left[(a,b,c,d)\right]}_0={\left[(a_1,b_1,c_1,d_1)\right]}_0$. By (i), this means that $[b,c]=[b_1,c_1]$ and $[a,d]=[a_1,d_1]$. This is equivalent to $a=a_1$, $b=b_1$, $c=c_1$ and $d=d_1$; that is, $[a,b,c,d]=[a_1,b_1,c_1,d_1]$. So, (ii) is proven.

As Tap is a subset of Trap, (iii) follows immediately from (i), and (iv) follows immediately from (ii). (iii) is easy and should be known. □

Proposition 3(ii) gives the representation uniqueness of the triangular fuzzy numbers.

Proposition 3.

(i) Let $u=(a,b,c)$ be in Trag. Then, ${\left[u\right]}_0=[a,c]$ and ${\left[u\right]}_1=\left\{b\right\}$. (ii) Let $(a,b,c)$ and $(a_1,b_1,c_1)$ be in Trag. Then, $(a,b,c)=(a_1,b_1,c_1)$ if and only if $[a,b,c]=[a_1,b_1,c_1]$. (iii) Let $u=(a,b,c)$ be in Tag. Then, ${\left[u\right]}_0=[a,c]$ and ${\left[u\right]}_1=\left\{b\right\}$. (iv) Let $(a,b,c)$ and $(a_1,b_1,c_1)$ be in Tag. Then, $(a,b,c)=(a_1,b_1,c_1)$ if and only if $[a,b,c]=[a_1,b_1,c_1]$.

Proof. 

By Definition 3 and easy calculations, we obtain (i). (One way to perform these calculations is by watching the graphs of the membership functions of $(a,b,c)$ in the four cases $a<b<c$, $a=b<c$, $a<b=c$, and $a=b=c$.) (i) follows immediately from Proposition 2.

Now, we show (ii). If $[a,b,c]=[a_1,b_1,c_1]$, i.e., $a=a_1$, $b=b_1$ and $c=c_1$, then, by Definition 3, $(a,b,c)=(a_1,b_1,c_1)$.

Suppose that $(a,b,c)=(a_1,b_1,c_1)$. Then, ${\left[(a,b,c)\right]}_1={\left[(a_1,b_1,c_1)\right]}_1$ and ${\left[(a,b,c)\right]}_0={\left[(a_1,b_1,c_1)\right]}_0$. By (i), this means that $\left\{b\right\}=\left\{b_1\right\}$ and $[a,c]=[a_1,c_1]$. This is equivalent to $a=a_1$, $b=b_1$ and $c=c_1$; that is, $[a,b,c]=[a_1,b_1,c_1]$. So, (ii) is proven.

As Tag is a subset of Trag, (iii) follows immediately from (i), and (iv) follows immediately from (ii). (iii) is easy and should be known. □

The above proofs of Theorem 3 and Proposition 3 are similar. Clearly, for k = i, ii, iii, and iv, Proposition 3(k) is a corollary of Theorem 3(k) (see also Remark 4.3 in [31] for details).

We know that Tag ⊆ Trag, Tap ⊆ Trap, Tag ⊆ Tap, and Trag ⊆ Trap (see Section 3). Based on Theorem 3(ii) and Proposition 3(ii), it is easy to see that Tag ⫋ Trag, Tap ⫋ Trap, Tag ⫋ Tap, and Trag ⫋ Trap (see also Remarks 4.7 and 4.8 in [31] for details).

Define

$$\begin{array}{c}{F}_{USCB}\left(X\right):=\{u\in F_{USC}\left(X\right):{\left[u\right]}_0\in K\left(X\right)\cup \{Ø\}\},\\ F_{USCB}^1\left(X\right):=\{u\in F_{USCB}\left(X\right):{\left[u\right]}_1\ne Ø\}.\end{array}$$

Clearly, $F_{USCB}^1\left(X\right)⫋F_{USCB}\left(X\right)$, $F_{USCB}^1\left(X\right)\subseteq F_{USC}^1\left(X\right)$ and $F_{USCB}\left(X\right)\subseteq F_{USC}\left(X\right)$.

For $u\in F\left(\mathbb{R}\right)$, we call u a 1-dimensional compact fuzzy number if u has the following properties:

(i)

${\left[u\right]}_1\ne Ø$; and

(ii)

for each $\alpha \in [0,1]$, ${\left[u\right]}_{\alpha }$ is a compact interval of $\mathbb{R}$.

The set of all one-dimensional compact fuzzy numbers is denoted by E. For $u\in E$ and $\alpha \in [0,1]$, ${\left[u\right]}_{\alpha }$ is denoted by $[u^{-}\left(\alpha \right),u^{+}\left(\alpha \right)]$.

Let $u\in$ Trap. Denote $u=(a,b,c,d)$. By Proposition 1, ${\left[u\right]}_1=[b,c]\ne Ø$ and for each $\xi \in [0,1]$, ${\left[u\right]}_{\xi }=[\xi (b-a)+a,c+(1-\xi )(d-c)]$ is a compact interval of $\mathbb{R}$. Also $u\in F\left(\mathbb{R}\right)$. Thus, $u\in E$. So, Trap $\subseteq E$. Clearly, Trap $⫋E$ (see also the corresponding contents in [32]), and $E⫋F_{USCB}^1\left(\mathbb{R}\right)⫋F_{USC}^1\left(\mathbb{R}\right)$. So, Trap $⫋F_{USC}^1\left(\mathbb{R}\right)$, and then, by Corollary 1, we find that

Corollary 2

([18,19,27]). $d_{\infty }⪰\{H_{\mathrm{send}},d_p\}⪰H_{\mathrm{end}}\left(Trap\right)$.

As Trap $⫋F_{USC}^1\left(\mathbb{R}\right)$ is a quite obvious fact and this fact should be known, we think it is reasonable to cite Corollary 1 when we use the fact given in Corollary 2.

5. Main Results

In this section, we prove that each one of the convergences induced by the four kinds of common metrics $d_{\infty }$, $d_p$, $H_{\mathrm{send}}$, and $H_{\mathrm{end}}$ on the trapezoidal fuzzy numbers is equivalent to the convergence of the corresponding representation quadruples of the trapezoidal fuzzy numbers in ${\mathbb{R}}^4$. So, we obtain that these four kinds of common metrics are equivalent on the trapezoidal fuzzy numbers.

The following Proposition 4(iii) gives a calculation formula of the supremum metric $d_{\infty }$ between two trapezoidal fuzzy numbers, which is expressed in terms of these two trapezoidal fuzzy numbers’ representation quadruples in ${\mathbb{R}}^4$. This conclusion is useful in this paper.

Proposition 4.

Let $u=(a_1,b_1,c_1,d_1)$ and $v=(a,b,c,d)$ be two trapezoidal fuzzy numbers.

(i) 

${sup}_{\xi \in [0,1]}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|=max\{|a_1-a|,|b_1-b\left|\right\}$.

(ii) 

${sup}_{\xi \in [0,1]}|u^{+}\left(\xi \right)-v^{+}\left(\xi \right)|=max\{|c_1-c|,|d_1-d\left|\right\}$.

(iii) 

$d_{\infty }(u,v)=max\left\{\right|a_1-a|,|b_1-b|,|c_1-c|,|d_1-d\left|\right\}$.

Proof. 

Set $A:=max\left\{\right|a_1-a|,|b_1-b\left|\right\}$ and $B:=max\left\{\right|c_1-c|,|d_1-d\left|\right\}$. Notice that, for each $\xi \in [0,1]$

$$\begin{array}{cc}\hfill & |u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|\hfill \\ \hfill & =|\xi (b_1-a_1)+a_1-(\xi (b-a)+a)|\left(by\right(),see(I\left)below\right)\hfill \\ \hfill & =|\xi (b_1-b)+(1-\xi )(a_1-a)|\hfill \\ \hfill & \le |\xi A+(1-\xi \left)A\right|=\left|A\right|=A.\hfill \end{array}$$

Thus, ${sup}_{\xi \in [0,1]}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|\le A$. On the other hand, since, by Theorem 3(i), $|u^{-}\left(0\right)-v^{-}\left(0\right)|=|a_1-a|$ and $|u^{-}\left(1\right)-v^{-}\left(1\right)|=|b_1-b|$, we have that ${sup}_{\xi \in [0,1]}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|\ge A$. So, ${sup}_{\xi \in [0,1]}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|=A$. Hence, (i) is proven.

The proof of (ii) is similar to that of (i). Notice that for each $\xi \in [0,1]$,

$$\begin{array}{cc}\hfill & |u^{+}\left(\xi \right)-v^{+}\left(\xi \right)|\hfill \\ \hfill & =|c_1+(1-\xi )(d_1-c_1)-(c+(1-\xi )(d-c))|\left(by\right(\left)\right)\hfill \\ \hfill & =|\xi (c_1-c)+(1-\xi )(d_1-d)|\hfill \\ \hfill & \le |\xi B+(1-\xi \left)B\right|=\left|B\right|=B.\hfill \end{array}$$

Thus, ${sup}_{\xi \in [0,1]}|u^{+}\left(\xi \right)-v^{+}\left(\xi \right)|\le B$. On the other hand, since, by Theorem 3(i), $|u^{+}\left(0\right)-v^{+}\left(0\right)|=|d_1-d|$ and $|u^{+}\left(1\right)-v^{+}\left(1\right)|=|c_1-c|$, we have that ${sup}_{\xi \in [0,1]}|u^{+}\left(\xi \right)-v^{+}\left(\xi \right)|\ge B$. So, ${sup}_{\xi \in [0,1]}|u^{+}\left(\xi \right)-v^{+}\left(\xi \right)|=B$. Hence, (ii) is proven.

Note that

$$\begin{array}{cc}\hfill & d_{\infty }(u,v)=\underset{\xi \in [0,1]}{sup}H({\left[u\right]}_{\xi },{\left[v\right]}_{\xi })=\left(by\right(\left)\right)\underset{\xi \in [0,1]}{sup}\left(\right|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|\vee |u^{+}\left(\xi \right)-v^{+}\left(\xi \right)\left|\right)\hfill \\ \hfill & =\underset{\xi \in [0,1]}{sup}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|\vee \underset{\xi \in [0,1]}{sup}|u^{+}\left(\xi \right)-v^{+}\left(\xi \right)|\hfill \\ \hfill & =\left(\mathrm{b}\mathrm{y}\right(\mathrm{i}\left)\mathrm{a}\mathrm{n}\mathrm{d}\right(\mathrm{i}\mathrm{i}\left)\right)A\vee B=max\left\{\right|a_1-a|,|b_1-b|,|c_1-c|,|d_1-d\left|\right\}.\hfill \end{array}$$

So, (iii) is proven.

(I) We think (2) can also be used without citing since it is easy to see. □

Remark 4.

In the proof of Lemma 5.1 of [33] (${\mathbb{R}}^4$ should be replaced by A in the proof), the conclusion is proven that for all $u=(a,b,c,d),v=(a_1,b_1,c_1,d_1)$ in Trap, $d_{\infty }(u,v)\le max\left\{\right|a-a_1|,|b-b_1|,|c-c_1|,|d-d_1\left|\right\}$. Clearly, Proposition 4(iii) improves this conclusion.

Remark 5.

Let $u=(a,b,c,d)=(a_1,b_1,c_1,d_1)\in$ Trap. Then, $0=d_{\infty }(u,u)=\left(byProposition\right(iii\left)\right)max\left\{\right|a_1-a|,|b_1-b|,|c_1-c|,|d_1-d\left|\right\}$. Thus, $a_1=a$, $b_1=b$, $c_1=c$ and $d_1=d$. Hence, “⇒” of Theorem 3(ii) holds. “⇐” of Theorem 3(ii) holds obviously. So, Proposition 4(iii) implies Theorem 3(ii). As Proposition 3(ii) is a corollary of Theorem 3(ii), Proposition 4(iii) also implies Proposition 3(ii).

By Remark 3, Proposition 4(iii) implies (a) for each $u=(a,b,c)$ and $v=(a_1,b_1,c_1)$ in Trag, $d_{\infty }(u,v)=max\left\{\right|a_1-a|,|b_1-b|,|c_1-c\left|\right\}$. Clearly, (a) implies Proposition 3(ii).

Remark 6.

The conclusions in this remark are easy to see. The symbols in this remark are consistent with those in the proof of Proposition 4.

(i) 

${sup}_{\xi \in [0,1]}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|={max}_{\xi \in [0,1]}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|$.

(ii) 

${sup}_{\xi \in [0,1]}|u^{+}\left(\xi \right)-v^{+}\left(\xi \right)|={max}_{\xi \in [0,1]}|u^{+}\left(\xi \right)-v^{+}\left(\xi \right)|$.

(iii-1) 

(a) If $d_{\infty }(u,v)=|a_1-a|$, then $d_{\infty }(u,v)=H({\left[u\right]}_0,{\left[v\right]}_0)={max}_{\xi \in [0,1]}H({\left[u\right]}_{\xi },{\left[v\right]}_{\xi })$

(b) If $d_{\infty }(u,v)=|a_1-a|$, then $d_{\infty }(u,v)=|u^{-}\left(0\right)-v^{-}\left(0\right)|={max}_{\xi \in [0,1]}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|$.

(iii-2) 

If $d_{\infty }(u,v)=|b_1-b|$, then $d_{\infty }(u,v)=H({\left[u\right]}_1,{\left[v\right]}_1)={max}_{\xi \in [0,1]}H({\left[u\right]}_{\xi },{\left[v\right]}_{\xi })$ and $d_{\infty }(u,v)=|u^{-}\left(1\right)-v^{-}\left(1\right)|={max}_{\xi \in [0,1]}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|$.

(iii-3) 

If $d_{\infty }(u,v)=|c_1-c|$, then $d_{\infty }(u,v)=H({\left[u\right]}_1,{\left[v\right]}_1)={max}_{\xi \in [0,1]}H({\left[u\right]}_{\xi },{\left[v\right]}_{\xi })$ and $d_{\infty }(u,v)=|u^{+}\left(1\right)-v^{+}\left(1\right)|={max}_{\xi \in [0,1]}|u^{+}\left(\xi \right)-v^{+}\left(\xi \right)|$.

(iii-4) 

If $d_{\infty }(u,v)=|d_1-d|$, then $d_{\infty }(u,v)=H({\left[u\right]}_0,{\left[v\right]}_0)={max}_{\xi \in [0,1]}H({\left[u\right]}_{\xi },{\left[v\right]}_{\xi })$ and $d_{\infty }(u,v)=|u^{+}\left(0\right)-v^{+}\left(0\right)|={max}_{\xi \in [0,1]}|u^{+}\left(\xi \right)-v^{+}\left(\xi \right)|$.

(iv) 

$d_{\infty }(u,v)={max}_{\xi \in [0,1]}H({\left[u\right]}_{\xi },{\left[v\right]}_{\xi })$.

Note that $A=|a_1-a|=|u^{-}\left(0\right)-v^{-}\left(0\right)|$ or $A=|b_1-b|=|u^{-}\left(1\right)-v^{-}\left(1\right)|$. Combining this and Proposition 4(i) yields that the supremum in (i) is attainable; that is, this supremum can be replaced by maximum. Hence, (i) holds.

Note that $B=|d_1-d|=|u^{+}\left(0\right)-v^{+}\left(0\right)\left)\right|$ or $B=|c_1-c|=|u^{+}\left(1\right)-v^{+}\left(1\right)\left)\right|$. Combining this and Proposition 4(ii) yields that the supremum in (ii) is attainable. Hence, (ii) holds.

Note that $d_{\infty }(u,v)={sup}_{\xi \in [0,1]}H({\left[u\right]}_{\xi },{\left[v\right]}_{\xi })\ge H({\left[u\right]}_0,{\left[v\right]}_0)\ge |u^{-}\left(0\right)-v^{-}\left(0\right)|=|a_1-a|$. So, if $d_{\infty }(u,v)=|a_1-a|$, then $d_{\infty }(u,v)={sup}_{\xi \in [0,1]}H({\left[u\right]}_{\xi },{\left[v\right]}_{\xi })=H({\left[u\right]}_0,{\left[v\right]}_0)$, and hence, this supremum is attainable. Thus, (iii-1)(a) holds.

Note that $d_{\infty }(u,v)={sup}_{\xi \in [0,1]}\left(\right|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|\vee |u^{+}\left(\xi \right)-v^{+}\left(\xi \right)\left|\right)\ge {sup}_{\xi \in [0,1]}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|$$\ge |u^{-}\left(0\right)-v^{-}\left(0\right)|=|a_1-a|$. So, if $d_{\infty }(u,v)=|a_1-a|$, then $d_{\infty }(u,v)={sup}_{\xi \in [0,1]}|u^{-}\left(\xi \right)-v^{-}\left(\xi \right)|=|u^{-}\left(0\right)-v^{-}\left(0\right)|$, and hence, this supremum is attainable. Thus, (iii-1)(b) holds. So, (iii-1) is proven. The proof of any of (iii-2), (iii-3), and (iii-4) is similar to that of (iii-1).

By Proposition 4(iii), $d_{\infty }(u,v)$ is equal to some of $|a_1-a|$, $|b_1-b|$, $|c_1-c|$ and $|d_1-d|$. So, by (iii), (iv) is true.

Corollary 3.

Let $\{u_n=(a_n,b_n,c_n,d_n):n\in \mathbb{N}\}$ be a sequence of trapezoidal fuzzy numbers and $u=(a,b,c,d)$ a trapezoidal fuzzy number. Then, the following three statements are equivalent: (i) ${lim}_{n\to \infty }{d}_{\infty }(u_n,u)=0$; (ii) ${lim}_{n\to \infty }max\left\{\right|a_n-a|,|b_n-b|,|c_n-c|,|d_n-d\left|\right\}=0$; (iii) ${lim}_{n\to \infty }{a}_n=a$, ${lim}_{n\to \infty }{b}_n=b$, ${lim}_{n\to \infty }{c}_n=c$ and ${lim}_{n\to \infty }{d}_n=d$.

Proof. 

By Proposition 4(iii), (i)⇔(ii). Clearly (ii)⇔(iii). So the statements (i), (ii) and (iii) are equivalent. □

The following Theorem 4 is useful in this paper:

Theorem 4.

Let $\{u_n=(a_n,b_n,c_n,d_n):n\in \mathbb{N}\}$ be a sequence of trapezoidal fuzzy numbers and $u=(a,b,c,d)$ a trapezoidal fuzzy number. Then, the following statements are equivalent:

(i) 

${lim}_{n\to \infty }{a}_n=a$, ${lim}_{n\to \infty }{b}_n=b$, ${lim}_{n\to \infty }{c}_n=c$, and ${lim}_{n\to \infty }{d}_n=d$.

(ii) 

${lim}_{n\to \infty }{d}_{\infty }(u_n,u)=0$.

(iii) 

There exist ${\xi }_1$ and ${\xi }_2$ in $[0,1]$, with ${\xi }_1\ne {\xi }_2$ satisfying

$$\begin{array}{c}\underset{n\to \infty }{lim}H({\left[u_n\right]}_{{\xi }_1},{\left[u\right]}_{{\xi }_1})=0,\end{array}$$

(3)

$$\begin{array}{c}\underset{n\to \infty }{lim}H({\left[u_n\right]}_{{\xi }_2},{\left[u\right]}_{{\xi }_2})=0.\end{array}$$

(4)

(iv) 

There exist ${\xi }_1$ and ${\xi }_2$ in $[0,1]$, with ${\xi }_1\ne {\xi }_2$ satisfying

$$\begin{array}{cc}\hfill & (iv-1)\underset{n\to \infty }{lim}({\xi }_1(b_n-a_n)+a_n)={\xi }_1(b-a)+a,\hfill \\ \hfill & (iv-2)\underset{n\to \infty }{lim}({\xi }_2(b_n-a_n)+a_n)={\xi }_2(b-a)+a,\hfill \\ \hfill & (iv-3)\underset{n\to \infty }{lim}(c_n+(1-{\xi }_1)(d_n-c_n))=c+(1-{\xi }_1)(d-c),and\hfill \\ \hfill & (iv-4)\underset{n\to \infty }{lim}(c_n+(1-{\xi }_2)(d_n-c_n))=c+(1-{\xi }_2)(d-c).\hfill \end{array}$$

Proof. 

By Corollary 3, (i)⇔(ii). Clearly, (ii)⇒(iii), as for each $v,w\in$ Trap, $d_{\infty }(v,w)={sup}_{\alpha \in [0,1]}H({\left[v\right]}_{\alpha },{\left[w\right]}_{\alpha })$.

By (1) and (2), for each ${\xi }_1\in [0,1]$, (3) means that both (iv-1) and (iv-3) hold. By (1) and (2), for each ${\xi }_2\in [0,1]$, () means that both (iv-2) and (iv-4) hold. So, (iii)⇔(iv).

Now, we show that (iv)⇒(i). (Obviously, (i)⇒(iv).) Assume that (iv) is true. Computing (iv-1) − (iv-2), we obtain (a) ${lim}_{n\to \infty }({\xi }_1-{\xi }_2)(b_n-a_n)=({\xi }_1-{\xi }_2)(b-a)$. As ${\xi }_1-{\xi }_2\ne 0$, (a) is equivalent to (iv-5) ${lim}_{n\to \infty }(b_n-a_n)=(b-a)$. Computing (iv-1) $-{\xi }_1·$(iv-5), we have (iv-6) ${lim}_{n\to \infty }{a}_n=a$. Computing (iv-5) + (iv-6), we obtain ${lim}_{n\to \infty }{b}_n=b$. Similarly, from (iv-3) and (iv-4), we can deduce that ${lim}_{n\to \infty }{c}_n=c$ and ${lim}_{n\to \infty }{d}_n=d$ (see also (I) below). So, (i) is true. Hence, (iv)⇒(i) is proven.

Thus, (i), (ii), (iii), and (iv) are equivalent. This completes the proof.

(I) Computing (iv-3) − (iv-4), we obtain (b) ${lim}_{n\to \infty }({\xi }_2-{\xi }_1)(d_n-c_n)=({\xi }_2-{\xi }_1)(d-c)$. As ${\xi }_2-{\xi }_1\ne 0$, (b) is equivalent to (iv-7) ${lim}_{n\to \infty }(d_n-c_n)=(d-c)$. Computing (iv-3) $-(1-{\xi }_1)·$(iv-7), we have (iv-8) ${lim}_{n\to \infty }{c}_n=c$. Computing (iv-7) + (iv-8), we obtain ${lim}_{n\to \infty }{d}_n=d$. □

Let S be a subset of $\mathbb{R}$ and $P\left(x\right)$ a statement about real numbers x. If there exists a set $S_1$ of measure zero such that $P\left(x\right)$ holds for all $x\in S\setminus S_1$, then we say that $P\left(x\right)$ holds almost everywhere on $x\in S$. For simplicity, “almost everywhere” is also written as “a.e.”.

The result of the following Theorem 5 was first given in [34]. As $E⫋{\tilde{S}}_{nc}^1$, E is also written as $E^1$. See Page 57 of [35] for the definition of ${\tilde{S}}_{nc}^1$ and the relation of E and ${\tilde{S}}_{nc}^1$). The result of Theorem 5 is part of the result of Theorem 9.4 in [35]. Theorem 5 is useful in this paper.

Theorem 5

([34,35]). Suppose that u, $u_n$, $n=1,2,\dots$ are fuzzy sets in E. Then, the following statements are equivalent: (i) ${lim}_{n\to \infty }{H}_{\mathrm{end}}(u_n,u)=0$. (ii) ${lim}_{n\to \infty }H({\left[u_n\right]}_{\alpha },{\left[u\right]}_{\alpha })=0$ holds a.e. on $\alpha \in (0,1)$. (iii) $u={lim}_{n\to \infty }^{\left(\Gamma \right)}{u}_n.$

Putting together Corollary 2 and Theorems 4 and 5, we arrive at the following conclusion:

Theorem 6.

Let $\{u_n=(a_n,b_n,c_n,d_n):n\in \mathbb{N}\}$ be a sequence of trapezoidal fuzzy numbers and $u=(a,b,c,d)$ a trapezoidal fuzzy number. Then, the following statements are equivalent:

(i) 

${lim}_{n\to \infty }{d}_{\infty }(u_n,u)=0$.

(ii) 

${lim}_{n\to \infty }{d}_p(u_n,u)=0$.

(iii) 

${lim}_{n\to \infty }{H}_{\mathrm{send}}(u_n,u)=0$.

(iv) 

${lim}_{n\to \infty }{H}_{\mathrm{end}}(u_n,u)=0$.

(v) 

${lim}_{n\to \infty }{a}_n=a$, ${lim}_{n\to \infty }{b}_n=b$, ${lim}_{n\to \infty }{c}_n=c$, and ${lim}_{n\to \infty }{d}_n=d$.

Proof. 

By Theorem 4, (v)⇔(i). So, to show the desired result, it suffices to show that (i)⇔(ii)⇔(iii)⇔(iv). By Corollary 2, to show (i)⇔(ii)⇔(iii)⇔(iv), we only need to show that (iv)⇒(i).

Suppose that (iv) holds. As Trap $\subset E$, by Theorem 5, (iv) means that ${lim}_{n\to \infty }H$$({\left[u_n\right]}_{\alpha },{\left[u\right]}_{\alpha })=0$ holds a.e. on $\alpha \in (0,1)$. Then, there exists two distinct ${\xi }_1$ and ${\xi }_2$ in $(0,1)$ such that ${lim}_{n\to \infty }H({\left[u_n\right]}_{{\xi }_1},{\left[u\right]}_{{\xi }_1})=0$ and ${lim}_{n\to \infty }H({\left[u_n\right]}_{{\xi }_2},{\left[u\right]}_{{\xi }_2})=0$. Hence, by Theorem 4, (i) holds. Thus, (iv)⇒(i). This completes the proof. □

Theorem 7.

$d_{\infty }\approx d_p\approx H_{\mathrm{send}}\approx H_{\mathrm{end}}\left(\mathrm{Trap}\right)$.

Proof. 

The desired result follows immediately from Theorem 6. □

Corollary 4.

Let $\{u_n=(a_n,b_n,c_n,d_n):n\in \mathbb{N}\}$ be a sequence of trapezoidal fuzzy numbers and $u=(a,b,d)$ a triangular fuzzy number. Then, the following statements are equivalent:

(i) 

${lim}_{n\to \infty }{d}_{\infty }(u_n,u)=0$.

(ii) 

${lim}_{n\to \infty }{d}_p(u_n,u)=0$.

(iii) 

${lim}_{n\to \infty }{H}_{\mathrm{send}}(u_n,u)=0$.

(iv) 

${lim}_{n\to \infty }{H}_{\mathrm{end}}(u_n,u)=0$.

(v) 

${lim}_{n\to \infty }{a}_n=a$, ${lim}_{n\to \infty }{b}_n=b$, ${lim}_{n\to \infty }{c}_n=b$, and ${lim}_{n\to \infty }{d}_n=d$.

Proof. 

Note that, by Remark 3, u is the trapezoidal fuzzy number $(a,b,b,d)$. Thus, the desired result follows immediately from Theorem 6. □

Corollary 5.

Let $\{u_n=(a_n,b_n,c_n):n\in \mathbb{N}\}$ be a sequence of triangular fuzzy numbers and $u=(a,b,c)$ a triangular fuzzy number. Then, the following statements are equivalent:

(i) 

${lim}_{n\to \infty }{d}_{\infty }(u_n,u)=0$.

(ii) 

${lim}_{n\to \infty }{d}_p(u_n,u)=0$.

(iii) 

${lim}_{n\to \infty }{H}_{\mathrm{send}}(u_n,u)=0$.

(iv) 

${lim}_{n\to \infty }{H}_{\mathrm{end}}(u_n,u)=0$.

(v) 

${lim}_{n\to \infty }{a}_n=a$, ${lim}_{n\to \infty }{b}_n=b$, and ${lim}_{n\to \infty }{c}_n=c$.

Proof. 

Note that, by Remark 3, u is the trapezoidal fuzzy number $(a,b,b,c)$, and $\left\{u_n\right\}$ is the sequence of trapezoidal fuzzy numbers $\{(a_n,b_n,b_n,c_n):n\in \mathbb{N}\}$. Thus, the desired result follows immediately from Theorem 6.

Clearly, the desired result also follows from Corollary 4. □

6. Conclusions

In this paper, we find that each one of the convergences induced by $d_{\infty }$, $d_p$, $H_{\mathrm{send}}$, and $H_{\mathrm{end}}$ on the trapezoidal fuzzy numbers is equivalent to the convergence of the corresponding representation quadruples of the trapezoidal fuzzy numbers in ${\mathbb{R}}^4$. Therefore, these four kinds of metrics are equivalent on the trapezoidal fuzzy numbers.

As $d_{\infty }$, $d_p$, $H_{\mathrm{send}}$, and $H_{\mathrm{end}}$ are commonly used metrics on the trapezoidal fuzzy numbers, the results of this paper have potential effects on the analysis and applications of the triangular fuzzy numbers and the trapezoidal fuzzy numbers.

In the future, we will consider the applications of the results in this paper to fuzzy neural networks or fuzzy control systems.