# The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fuzzy Discrimination Is Consistent with Distance Measure

**Definition**

**1**

**.**Let X be a universe course, and fuzzy set $\tilde{A}$ defined on X be given as:

- (1)
- if $M\subseteq N$, then ${\mu}_{M}(x)\le {\mu}_{N}(x)$;
- (2)
- if $M\subseteq N,N\subseteq M$, then $M=N$;
- (3)
- if $M\subseteq N,N\subseteq T$, then $M\subseteq T$.

**Definition**

**2.**

**Definition**

**3.**

- (1)
- $0\le D(A,B)\le 1$;
- (2)
- $D(A,B)=0$, if and only if $A=B$;
- (3)
- $D(A,B)=D(B,A)$;
- (4)
- if $A\subseteq B\subseteq C$, then $D(A,C)\ge D(A,B),D(A,C)\ge D(B,C)$.

**Definition**

**4.**

- (1)
- $FD(A,B)\ge 0$;
- (2)
- $FD(A,B)=0$, if and only if $A=B$;
- (3)
- $FD(A,B)=FD(B,A)$;
- (4)
- if $A\subseteq B\subseteq C$, then $FD(A,C)\ge FD(A,B),FD(A,C)\ge FD(B,C)$.

- (1)
- $E(M,N)\ge 0$
- (2)
- $E(M,N)=0$, if and only if $M=N$;
- (3)
- $E(M,N)=E(N,M)$.

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Example**

**1.**

**Theorem**

**3.**

## 3. Fuzzy Cross-Entropy Is Consistent with Distance Measure

**Definition**

**5**

**.**Let $M,N\in FS(X)$, where we can define a fuzzy cross-entropy as:

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Example**

**2.**

**Theorem**

**6.**

## 4. Neutrosophic Cross-Entropy Is a Distance Measure

**Definition**

**6**

**.**Let X be a universe course, where a neutrosophic set A in X is comprised of the truth-membership function ${T}_{A}(x)$, indeterminacy-membership function ${I}_{A}(x)$, and falsity-membership function ${F}_{A}(x)$, in which ${T}_{A}(x),{I}_{A}(x),{F}_{A}(x):X\to ]{0}^{-},{1}^{+}[$.

**Definition**

**7**

**.**Let X be a space of points, where a single-value netrosophic set A in X is comprised of the truth-membership function ${T}_{A}(x)$, indeterminacy-membership function ${I}_{A}(x)$, and falsity-membership function ${F}_{A}(x)$. For each point x in X, ${T}_{A}(x),{I}_{A}(x),{F}_{A}(x)\in [0,1]$. Therefore, a SVNS A can be denoted by:

- (1)
- $M\subseteq N$ if, and only if ${T}_{M}(x)\le {T}_{N}(x),{I}_{M}(x)\ge {I}_{N}(x),{F}_{M}(x)\ge {F}_{N}(x)$ for every x in X [5];
- (2)
- (3)
- If $M\subseteq N,N\subseteq T$, then $M\subseteq T$.

**Definition**

**8**

**.**The single-value neutrosophic cross-entropy about M and N where $M,N\in SVNS(X)$ can be defined as follows:

**Theorem**

**7.**

**Example**

**3.**

**Theorem**

**8.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Wang, Y.; Yang, H.; Qin, K.
The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets. *Symmetry* **2019**, *11*, 386.
https://doi.org/10.3390/sym11030386

**AMA Style**

Wang Y, Yang H, Qin K.
The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets. *Symmetry*. 2019; 11(3):386.
https://doi.org/10.3390/sym11030386

**Chicago/Turabian Style**

Wang, Yameng, Han Yang, and Keyun Qin.
2019. "The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets" *Symmetry* 11, no. 3: 386.
https://doi.org/10.3390/sym11030386