# Asymmetric Equivalences in Fuzzy Logic

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

- (i)
- Minimum t-norm and its residuum, Gödel implication$$a{\ast}_{G}b=a\wedge b,\phantom{\rule{4pt}{0ex}}a{\to}_{G}b=\left\{\begin{array}{cc}1,& \mathrm{if}\phantom{\rule{4pt}{0ex}}a\le b,\\ b,& \mathrm{if}\phantom{\rule{4pt}{0ex}}a>b.\end{array}\right.$$
- (ii)
- Łukasiewicz t-norm and its residuum, Łukasiewicz implication$$a{\ast}_{L}b=0\vee (a+b-1),\phantom{\rule{4pt}{0ex}}a{\to}_{L}b=1\wedge (1-a+b);$$
- (iii)
- Product t-norm and its residuum, product implication$$a{\ast}_{P}b=ab;\phantom{\rule{4pt}{0ex}}a{\to}_{P}b=\left\{\begin{array}{cc}1,& \mathrm{if}\phantom{\rule{4pt}{0ex}}a\le b,\\ b/a,& \mathrm{if}\phantom{\rule{4pt}{0ex}}a>b.\end{array}\right.$$

**Definition**

**3.**

**Lemma**

**1.**

- (i)
- $a\ast b\le c\iff a\le b\to c$;
- (ii)
- $a\ast (a\to b))\le a\wedge b$; furthermore, if ∗ is continuous, then $a\ast (a\to b))=a\wedge b$;
- (iii)
- $a\le b\iff a\to b=1$;
- (iv)
- $1\to a=a$;
- (v)
- $(a\to b)\ast (b\to c)\le a\to c$.

**Lemma**

**2.**

- (i)
- if ${a}_{1}+{b}_{1}\le {c}_{1},$ and ${a}_{2}+{b}_{2}\le {c}_{2}$, then ${a}_{1}\wedge {a}_{2}+{b}_{1}\wedge {b}_{2}\le {c}_{1}\wedge {c}_{2}.$
- (ii)
- if ${a}_{1}+{b}_{1}\ge {c}_{1},$ and ${a}_{2}+{b}_{2}\ge {c}_{2}$, then ${a}_{1}\vee {a}_{2}+{b}_{1}\vee {b}_{2}\ge {c}_{1}\vee {c}_{2}.$

**Proof.**

- (i)
- Since ${a}_{1}\wedge {a}_{2}+{b}_{1}\wedge {b}_{2}\le {a}_{1}+{b}_{1}\le {c}_{1}$ and ${a}_{1}\wedge {a}_{2}+{b}_{1}\wedge {b}_{2}\le {a}_{2}+{b}_{2}\le {c}_{2},$ then ${a}_{1}\wedge {a}_{2}+{b}_{1}\wedge {b}_{2}\le {c}_{1}\wedge {c}_{2}.$
- (ii)
- Since ${a}_{1}\vee {a}_{2}+{b}_{1}\vee {b}_{2}\ge {a}_{1}+{b}_{1}\ge {c}_{1}$ and ${a}_{1}\vee {a}_{2}+{b}_{1}\vee {b}_{2}\ge {a}_{2}+{b}_{2}\ge {c}_{2},$ then ${a}_{1}\vee {a}_{2}+{b}_{1}\vee {b}_{2}\ge {c}_{1}\vee {c}_{2}.$

## 3. Asymmetric Equivalence Induced by R-Implication

**Theorem**

**1.**

- (i)
- $a{\to}_{i}b=a{\leftrightarrow}_{ij}(a\wedge b)=(a\wedge b){\leftrightarrow}_{ji}a$;
- (ii)
- $a{\to}_{j}b=(a\wedge b){\leftrightarrow}_{ij}a=a{\leftrightarrow}_{ji}(a\wedge b)$;
- (iii)
- ${\neg}_{i}a=a{\leftrightarrow}_{ij}0$;
- (iv)
- ${\neg}_{j}a=0{\leftrightarrow}_{ij}a$.

**Proof.**

- (i)
- Since t-norm ${\ast}_{i}$ is continuous, from Lemma 1(ii), $a{\ast}_{i}\left(a{\to}_{i}b\right))=a\wedge b$, then using Lemma 1(i), we obtain $a{\to}_{i}b=a{\to}_{i}a\wedge b.$ In addition, from Lemma 1(iii), $a\wedge b{\to}_{j}a=1.$ Hence,$$a{\leftrightarrow}_{ij}(a\wedge b)=(a{\to}_{i}a\wedge b)\wedge (a\wedge b{\to}_{j}a)=\left(a{\to}_{i}b\right)\wedge 1=a{\to}_{i}b.$$
- (ii)
- can be proven similarly. Properties (iii) and (iv) are special cases of (i) and (ii), respectively, by setting $b=0$.

**Theorem**

**2.**

- (i)
- Reflexivity: $a{\leftrightarrow}_{ij}b$ if and only if $a=b$;
- (ii)
- Left monotonicity: $(a\wedge b\wedge c){\leftrightarrow}_{ij}a\ge (a\wedge b){\leftrightarrow}_{ij}a$;
- (iii)
- Right monotonicity: $a{\leftrightarrow}_{ij}(a\wedge b\wedge c)\le a{\leftrightarrow}_{ij}(a\wedge b)$.

**Proof.**

- (i)
- follows immediately from Lemma 1 (iii) and (iv).
- (ii)
- and (iii) follow immediately from the monotonicity of fuzzy implications.

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 4. Quasi-Metrics Induced by Asymmetric Equivalences

**Definition**

**4.**

- (i)
- $d(a,a)=0$;
- (ii)
- $d(a,b)=d(b,a)=0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}a=b$;
- (iii)
- $d(a,b)+d(b,c)\ge d(a,c)$.

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

- (i)
- Since $a\to a=1$, $\to \in \{{\to}_{i},{\to}_{j}\}$, then $d(a,a)=1-a{\to}_{i}a\wedge a{\to}_{j}a=0$.
- (ii)
- If $d(a,b)=1-a{\leftrightarrow}_{ij}b=1-a{\to}_{i}b\wedge b{\to}_{j}a=0$, then we have $a{\to}_{i}b=1$, $b{\to}_{j}a=1$, from Lemma 1(iii), $a\le b,\text{}b\le a$, thus $a=b$. Similarly, we obtain $d(b,a)=0\Rightarrow a=b$.
- (iii)
- Now, we prove that d satisfies the triangle inequality.

**Corollary**

**1.**

**Proof.**

## 5. Symmetrization of Asymmetric Equivalences

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

- Formula (3) can be proven from ${\to}_{j}\prec {\to}_{i}$ and the commutative law of ∧.
- Formula (4) can be proven from ${\to}_{j}\prec {\to}_{i}$ and distributivity of ∨ over ∧. ☐

**Corollary**

**2.**

**Proof.**

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**3D plots of two asymmetric equivalences, ${\leftrightarrow}_{LP}$ and ${\leftrightarrow}_{PL}$.

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Hu, B.; Bi, L.; Li, S.; Dai, S.
Asymmetric Equivalences in Fuzzy Logic. *Symmetry* **2017**, *9*, 224.
https://doi.org/10.3390/sym9100224

**AMA Style**

Hu B, Bi L, Li S, Dai S.
Asymmetric Equivalences in Fuzzy Logic. *Symmetry*. 2017; 9(10):224.
https://doi.org/10.3390/sym9100224

**Chicago/Turabian Style**

Hu, Bo, Lvqing Bi, Sizhao Li, and Songsong Dai.
2017. "Asymmetric Equivalences in Fuzzy Logic" *Symmetry* 9, no. 10: 224.
https://doi.org/10.3390/sym9100224