# Representations of a Comparison Measure between Two Fuzzy Sets

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## Abstract

**:**

## 1. Introduction

## 2. Fuzzy Sets and Comparison Measures

**Definition**

**1.**

**Definition**

**2.**

- ${\mathsf{\mu}}_{A{\displaystyle \cap}B}\left(x\right)=min\left\{{\mathsf{\mu}}_{A}\left(x\right),{\mathsf{\mu}}_{B}\left(x\right)\right\}$.
- ${\mathsf{\mu}}_{A\cup B}\left(x\right)=max\left\{{\mathsf{\mu}}_{A}\left(x\right),{\mathsf{\mu}}_{B}\left(x\right)\right\}$.
- ${\mathsf{\mu}}_{A\backslash B}\left(x\right)=min\left\{{\mathsf{\mu}}_{A}\left(x\right),1-{\mathsf{\mu}}_{B}\left(x\right)\right\}$.

**Definition**

**3.**

- G1:$0\le m\left(A,B\right)\le 1$, $\forall A,B\in \mathcal{F}\left(U\right)$.
- G1*:$0\le m\left(A,B\right)\le 1$, $\forall A,B\in \mathcal{F}\left(U\right)$and there exists two FSs$C,D\in \mathcal{F}\left(U\right)$such that$m\left(C,D\right)=1$.
- G2:$m\left(A,B\right)=m\left(B,A\right)$, $\forall A,B\in \mathcal{F}\left(U\right)$.
- G3:Let$\rho :U\to U$be a permutation for finite U. Define${A}^{\rho}\in \mathcal{F}\left(U\right)$with membership function${\mathsf{\mu}}_{{A}^{\rho}}\left(x\right)={\mathsf{\mu}}_{A}\left(\rho \left(x\right)\right)$for$A\in \mathcal{F}\left(U\right)$. Then$m\left(A,B\right)=m\left({A}^{\rho},{B}^{\rho}\right)$.
- G3*:For finite set U, there exists a function$h:\left[0,1\right]\times \left[0,1\right]\to \mathcal{R}$such that$m\left(A,B\right)={\displaystyle \sum}_{x\in U}h\left({\mathsf{\mu}}_{A}\left(x\right),{\mathsf{\mu}}_{B}\left(x\right)\right)$,$\forall A,B\in \mathcal{F}\left(U\right)$.
- G4:There exists a function$f:\mathcal{F}{(\mathrm{U})}^{3}\to \mathcal{R}$such that$m\left(A,B\right)=f\left(A{\displaystyle \cap}B,A\backslash B,B\backslash A\right)$,$\forall A,B\in \mathcal{F}\left(U\right)$.
- G4*:There exists a function$F:{\mathcal{R}}^{3}\to \mathcal{R}$and a fuzzy measure$M:\mathcal{F}\left(U\right)\to \mathcal{R}$such that$m\left(A,B\right)=F\left(M\left(A{\displaystyle \cap}B\right),M\left(A\backslash B\right),M\left(B\backslash A\right)\right)$,$\forall A,B\in \mathcal{F}\left(U\right)$.
- G5:If$A{\displaystyle \cap}B=\varnothing $,${A}^{\prime}{\displaystyle \cap}{B}^{\prime}=\varnothing $,$m\left(A,\varnothing \right)\le m\left({A}^{\prime},\varnothing \right)$and$m\left(B,\varnothing \right)\le m\left({B}^{\prime},\varnothing \right)$, then$m\left(A,B\right)\le m\left({A}^{\prime},{B}^{\prime}\right)$,$\forall A,B,{A}^{\prime},{B}^{\prime}\in \mathcal{F}\left(U\right)$.

## 3. Representations of a Comparison Measure for Two Disjoint Union Fuzzy Sets

## 4. Representations of a Comparison Measure for Two Disjoint Fuzzy Sets

## 5. Representations of a Comparison Measure for Two General Fuzzy Sets

## 6. Conclusions and Future Research

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Chen, J.-H.; Tang, H.-C.
Representations of a Comparison Measure between Two Fuzzy Sets. *Symmetry* **2020**, *12*, 2008.
https://doi.org/10.3390/sym12122008

**AMA Style**

Chen J-H, Tang H-C.
Representations of a Comparison Measure between Two Fuzzy Sets. *Symmetry*. 2020; 12(12):2008.
https://doi.org/10.3390/sym12122008

**Chicago/Turabian Style**

Chen, Juin-Han, and Hui-Chin Tang.
2020. "Representations of a Comparison Measure between Two Fuzzy Sets" *Symmetry* 12, no. 12: 2008.
https://doi.org/10.3390/sym12122008