# New Arithmetic Operations of Non-Normal Fuzzy Sets Using Compatibility

## Abstract

**:**

## 1. Introduction

## 2. Arithmetic Operations of Fuzzy Sets

**Proposition 1.**

**Definition 1.**

**Example 1.**

**Example 2.**

**Proposition 2.**

**Proof.**

## 3. Compatibility

**Definition 2.**

- The function ${\mathfrak{D}}_{n}:{[0,1]}^{n}\to [0,1]$ is said to be compatible with the arithmetic operations of α-level sets when the following equality is satisfied:$${\left(\right)}_{{\tilde{F}}^{\left(1\right)}}\alpha $$
- The function ${\mathfrak{D}}_{n}:{[0,1]}^{n}\to [0,1]$ is said to be strongly compatible with the arithmetic operations of α-level sets when the following equality is satisfied:$${\left(\right)}_{{\tilde{F}}^{\left(1\right)}}\alpha $$

**Lemma 1**

**Theorem 1.**

- (i)
- For any $\alpha \in {I}^{*}\cap {I}_{\tilde{F}}$ with $\alpha >0$, we assumed that the function ${\mathfrak{D}}_{n}$ satisfies the following condition:$${\alpha}_{i}\ge \alpha foralli=1,\cdots ,nimply{\mathfrak{D}}_{n}({\alpha}_{1},\cdots ,{\alpha}_{n})\ge \alpha .$$Then, the following inclusion:$${\tilde{F}}_{\alpha}^{\left(1\right)}{\circ}_{1}\cdots {\circ}_{n-1}{\tilde{F}}_{\alpha}^{\left(n\right)}\subseteq {\left(\right)}_{{\tilde{F}}^{\left(1\right)}}\alpha $$
- (ii)
- Suppose that the membership functions of ${\tilde{F}}^{\left(i\right)}$ are upper semi-continuous for all $i=1,\cdots ,n$. We also assumed that the function ${\mathfrak{D}}_{n}$ satisfies the following conditions:
- given any $\alpha \in {I}^{*}\cap {I}_{\tilde{F}}$ with $\alpha >0$,$${\mathfrak{D}}_{n}({\alpha}_{1},\cdots ,{\alpha}_{n})\ge \alpha ifandonlyif{\alpha}_{i}\ge \alpha foralli=1,\cdots ,n.$$
- Given any $\alpha \notin {I}^{*}$ with $\alpha \in (0,1]$,$${\alpha}_{i}<\alpha forsomei\in \{1,\cdots ,n\}imply{\mathfrak{D}}_{n}({\alpha}_{1},\cdots ,{\alpha}_{n})\alpha $$

Then, the following equality:$${\left(\right)}_{{\tilde{F}}^{\left(1\right)}}\alpha $$$${\left(\right)}_{{\tilde{F}}^{\left(1\right)}}0$$

**Proof.**

- Suppose that $\alpha \le 0$. Then, we have$$\left(\right)open="\{"\; close="\}">\left(\right)open="("\; close=")">{x}_{1},\cdots ,{x}_{n}\ge \alpha $$
- Suppose that $\alpha >1$. Then, we have$$\left(\right)open="\{"\; close="\}">\left(\right)open="("\; close=")">{x}_{1},\cdots ,{x}_{n}\ge \alpha $$
- Suppose that $\alpha \in {I}^{*}\cap {I}_{\tilde{F}}$ with $\alpha >0$, i.e., ${\tilde{F}}_{\alpha}^{\left(i\right)}\ne \xd8$ for all $i=1,\cdots ,n$. Then, we have$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \left(\right)open="\{"\; close="\}">\left(\right)open="("\; close=")">{x}_{1},\cdots ,{x}_{n}:f\left(\right)open="("\; close=")">{x}_{1},\cdots ,{x}_{n}\hfill & \ge \alpha \\ =\left(\right)open="\{"\; close="\}">\left(\right)open="("\; close=")">{x}_{1},\cdots ,{x}_{n}\end{array}\ge \alpha $$
- Suppose that $\alpha \notin {I}^{*}$ with $\alpha \in (0,1]$. Then, we have ${\tilde{F}}_{\alpha}^{\left(i\right)}=\xd8$ for some i, i.e., $\alpha \notin {I}_{{\tilde{F}}^{\left(i\right)}}$. By referring to (14), it follows that ${\tilde{F}}^{\left(i\right)}\left(x\right)<\alpha $ for all $x\in \mathbb{R}$. Therefore, using the assumption (22), we obtain$$f\left(\right)open="("\; close=")">{x}_{1},\cdots ,{x}_{n}\alpha \mathrm{for}\mathrm{all}\left(\right)open="("\; close=")">{x}_{1},\cdots ,{x}_{n}$$This shows$$\left(\right)open="\{"\; close="\}">\left(\right)open="("\; close=")">{x}_{1},\cdots ,{x}_{n}\ge \alpha $$
- Suppose that $\alpha \notin {I}_{\tilde{F}}$ with $\alpha \in (0,1]$. Then, we have$$\begin{array}{cc}\hfill \xd8& =\left(\right)open="\{"\; close="\}">\left(\right)open="("\; close=")">{x}_{1},\cdots ,{x}_{n}:{\mathfrak{D}}_{n}\left(\right)open="("\; close=")">{\tilde{F}}^{\left(1\right)}\left({x}_{1}\right),\cdots ,{\tilde{F}}^{\left(n\right)}\left({x}_{n}\right)\hfill & \ge \alpha \end{array}$$

**Theorem 2.**

- Given any $\alpha \in {I}^{*}\cap {I}_{\tilde{F}}$ with $\alpha >0$,$${\mathfrak{D}}_{n}({\alpha}_{1},\cdots ,{\alpha}_{n})\ge \alpha ifandonlyif{\alpha}_{i}\ge \alpha foralli=1,\cdots ,n.$$
- Given any $\alpha \notin {I}^{*}$ with $\alpha \in (0,1]$,$${\alpha}_{i}<\alpha forsomei\in \{1,\cdots ,n\}imply{\mathfrak{D}}_{n}({\alpha}_{1},\cdots ,{\alpha}_{n})\alpha $$

- (i)
- Suppose that the membership functions of ${\tilde{F}}^{\left(i\right)}$ are upper semi-continuous for all $i=1,\cdots ,n$. Then, the function ${\mathfrak{D}}_{n}$ is compatible with arithmetic operations of α-level sets. In other words, given any $\alpha \in {I}^{*}\cap {I}_{\tilde{F}}$ with $\alpha >0$, we have$${\left(\right)}_{{\tilde{F}}^{\left(1\right)}}\alpha $$In particular, if ${\tilde{F}}^{\left(1\right)},\cdots ,{\tilde{F}}^{\left(n\right)}$ are normal, the equality (32) holds true for all $\alpha \in (0,1]$.
- (ii)
- Suppose that the membership functions of ${\tilde{F}}^{\left(i\right)}$ are upper semi-continuous and that the supports ${\tilde{F}}_{0+}^{\left(i\right)}$ are bounded for all $i=1,\cdots ,n$. Then, the function ${\mathfrak{D}}_{n}$ is strongly compatible with the arithmetic operations of α-level sets. In other words, the equality (32) holds true for all $\alpha \in {I}^{*}\cap {I}_{\tilde{F}}$. In particular, if ${\tilde{F}}^{\left(1\right)},\cdots ,{\tilde{F}}^{\left(n\right)}$ are normal, the equality (32) holds true for all $\alpha \in [0,1]$.

**Proof.**

**Corollary 1.**

- (i)
- Suppose that the membership functions of ${\tilde{F}}^{\left(i\right)}$ are upper semi-continuous for all $i=1,\cdots ,n$. Then, given any $\alpha \in {I}^{*}\cap {I}_{\tilde{F}}$ with $\alpha >0$, we have$${\left(\right)}_{{\tilde{F}}^{\left(1\right)}}\alpha $$In particular, if ${\tilde{F}}^{\left(1\right)},\cdots ,{\tilde{F}}^{\left(n\right)}$ are normal, the equality (33) holds true for all $\alpha \in (0,1]$.
- (ii)
- Suppose that the membership functions of ${\tilde{F}}^{\left(i\right)}$ are upper semi-continuous and that the supports ${\tilde{F}}_{0+}^{\left(i\right)}$ are bounded for all $i=1,\cdots ,n$. Then, the equality (33) holds true for all $\alpha \in {I}^{*}\cap {I}_{\tilde{F}}$. In particular, if ${\tilde{F}}^{\left(1\right)},\cdots ,{\tilde{F}}^{\left(n\right)}$ are normal, the equality (33) holds true for all $\alpha \in [0,1]$.

**Proof.**

**Definition 3.**

- The supremum $sup{\mathcal{R}}_{\tilde{a}}$ is obtained, i.e., $sup{\mathcal{R}}_{\tilde{a}}=max{\mathcal{R}}_{\tilde{a}}$.
- The membership function of $\tilde{a}$ is upper semi-continuous and quasi-concave on $\mathbb{R}$.
- The 0-level set ${\tilde{a}}_{0}$ is a closed and bounded subset of $\mathbb{R}$.

**Proposition 3.**

**Proof.**

## 4. Conclusions

## Funding

## Conflicts of Interest

## References

- Dubois, D.; Prade, H. A Review of Fuzzy Set Aggregation Connectives. Inf. Sci.
**1985**, 36, 85–121. [Google Scholar] [CrossRef] - Weber, S. A General Concept of Fuzzy Connectives, Negations and Implications Based on t-Norms and t-Conorms. Fuzzy Sets Syst.
**1983**, 11, 115–134. [Google Scholar] [CrossRef] - Dubois, D.; Prade, H. Possibility Theory; Springer: New Yourk, NY, USA, 1988. [Google Scholar]
- Klir, G.J.; Yuan, B. Fuzzy Sets and Fuzzy Logic: Theory and Applications; Prentice-Hall: New York, NY, USA, 1995. [Google Scholar]
- Gebhardt, A. On Types of Fuzzy Numbers and Extension Principles. Fuzzy Sets Syst.
**1995**, 75, 311–318. [Google Scholar] [CrossRef] - Fullér, R.; Keresztfalvi, T. On Generalization of Nguyen’s Theorem. Fuzzy Sets Syst.
**1990**, 41, 371–374. [Google Scholar] [CrossRef] - Mesiar, R. Triangular-Norm-Based Addition of Fuzzy Intervals. Fuzzy Sets Syst.
**1997**, 91, 231–237. [Google Scholar] [CrossRef] - Ralescu, D.A. A generalization of the representation theorem. Fuzzy Sets Syst.
**1992**, 51, 309–311. [Google Scholar] [CrossRef] - Yager, R.R. A Characterization of the Extension Principle. Fuzzy Sets Syst.
**1986**, 18, 205–217. [Google Scholar] [CrossRef] - Wu, H.-C. Generalized Extension Principle for Non-Normal Fuzzy Sets. Fuzzy Optim. Decis. Mak.
**2019**, 18, 399–432. [Google Scholar] [CrossRef] - Coroianua, L.; Fuller, R. Nguyen Type Theorem For Extension Principle Based on a Joint Possibility Distribution. Int. J. Approx. Reason.
**2018**, 95, 22–35. [Google Scholar] [CrossRef] - Coroianua, L.; Fuller, R. Necessary and Sufficient Conditions for The Equality of Interactive and Non-Interactive Extensions of Continuous Functions. Fuzzy Sets Syst.
**2018**, 331, 116–130. [Google Scholar] [CrossRef] - Holčapek, M.; Štěpnixcxka, M. MI-Algebras: A New Framework for Arithmetics of (Extensional) Fuzzy Numbers. Fuzzy Sets Syst.
**2014**, 257, 102–131. [Google Scholar] [CrossRef] - Holčapek, M.; Škorupová, N.; xSxtěpnixcxka, M. Fuzzy Interpolation with Extensional Fuzzy Numbers. Symmetry
**2021**, 13, 170. [Google Scholar] [CrossRef] - Esmi, E.; Sánchez, D.E.; Wasques, V.F.; de Barros, L.C. Solutions of Higher Order Linear Fuzzy Differential Equations with Interactive Fuzzy Values. Fuzzy Sets Syst.
**2021**, 419, 122–140. [Google Scholar] [CrossRef] - Pedro, F.S.; de Barros, L.C.; Esmi, E. Population Growth Model via Interactive Fuzzy Differential Equation. Inf. Sci.
**2019**, 481, 160–173. [Google Scholar] [CrossRef] - Wu, H.-C. Decomposition and Construction of Fuzzy Sets and Their Applications to the Arithmetic Operations on Fuzzy Quantities. Fuzzy Sets Syst.
**2013**, 233, 1–25. [Google Scholar] [CrossRef] - Wu, H.-C. Compatibility between Fuzzy Set Operations and Level Set Operations: Applications to Fuzzy Difference. Fuzzy Sets Syst.
**2018**, 353, 1–43. [Google Scholar] [CrossRef] - Bede, B.; Stefanini, L. Generalized Differentiability of Fuzzy-Valued Functions. Fuzzy Sets Syst.
**2013**, 230, 119–141. [Google Scholar] [CrossRef] - Gomes, L.T.; Barros, L.C. A Note on the Generalized Difference and the Generalized Differentiability. Fuzzy Sets Syst.
**2015**, 280, 142–145. [Google Scholar] [CrossRef] - Zulqarnian, R.M.; Xin, X.L.; Jun, Y.B. Fuzzy axiom of choice, fuzzy Zorn’s lemma and fuzzy Hausdorff maximal principle. Soft Comput.
**2021**, 25, 11421–11428. [Google Scholar] [CrossRef] - Black, F.; Scholes, M. The Pricing of Options and Corporate Liabilities. J. Political Econ.
**1973**, 81, 637–659. [Google Scholar] [CrossRef] - Musiela, M.; Rutkowski, M. Martingale Methods in Financial Modelling; Springer: New York, NY, USA, 1997. [Google Scholar]
- Royden, H.L. Real Analysis, 2nd ed.; Macmillan: London, UK, 1968. [Google Scholar]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wu, H.-C.
New Arithmetic Operations of Non-Normal Fuzzy Sets Using Compatibility. *Axioms* **2023**, *12*, 277.
https://doi.org/10.3390/axioms12030277

**AMA Style**

Wu H-C.
New Arithmetic Operations of Non-Normal Fuzzy Sets Using Compatibility. *Axioms*. 2023; 12(3):277.
https://doi.org/10.3390/axioms12030277

**Chicago/Turabian Style**

Wu, Hsien-Chung.
2023. "New Arithmetic Operations of Non-Normal Fuzzy Sets Using Compatibility" *Axioms* 12, no. 3: 277.
https://doi.org/10.3390/axioms12030277