# Axiomatic of Fuzzy Complex Numbers

## Abstract

**:**

## 1. Introduction

## 2. History of Complex Numbers

## 3. Essentials of Fuzzy Sets and Fuzzy Numbers

**A**, on the Universe of Discourse

**U**, with images on the lattice L, may be defined [8,9,10] as an application

**A: U → L**

**m**

_{A}: U → L_{A}(x))}

_{x}

_{∈U}

**U**}

_{A }(x), may be called the membership function value for the element x into the fuzzy set A.

**R**.

_{A }(x)) or (x | A (x))

_{A}(x)”.

^{*}, with

_{A}(x

^{*}) = 1

**R**, will be convex, if for any real numbers, x, y, and any t, the following is true:

_{A }[tx + (1 − t) y] ≥ min {m

_{A}(x), m

_{A }(y)}

**N,**on the real line (

**R**) that satisfies the above mentioned conditions of normality and convexity. I.e., a fuzzy number is merely a particular case of a fuzzy set,

**R**→ [0, 1]

^{α}, the α-cut of N, will be a closed interval, and it is true for all the values of αcomprised between 0 and 1:

- - the Fuzzy Trapezoidal Numbers,
- and
- - the Fuzzy Triangular Number, as the limiting case;
- - bell-shaped membership function (Gaussian);
- - L-R fuzzy numbers, etc.

**FGN**(FuzzyGeneralized Number, by acronym), F, is any fuzzy subset of the real line whose membership function, m

_{F}, verifies:

- (1) m
_{F}(x) = 0, if −∞ < x ≤ a; - (2) m
_{F }(x) = L(x) strictly increases on [a, b]; - (3) m
_{F}(x) = w; if b ≤ x ≤ c; - (4) m
_{F}(x) = R(x) strictly decreases on [c, d]; - (5) m
_{F}(x) = 0, if d ≤ x < +∞.

_{LR}

**Fuzzy Generalized Trapezoidal Numbers**may be denoted by

**Fuzzy Generalized Triangular Numbers**, by

- - It will be easiest for calculation.
- - It is physically accurate.
- - It is very useful in applications, by using complex algebra.

**Fuzzy Complex Set**(

**FCS**, by acronym), also denoted by A, based on its membership function,

_{A }(x) = r

_{A}(x) exp [i ɸ

_{A }(x)]

_{A}and ɸ

_{A}are both real-valued functions, with an important restriction on r

_{A}such that

_{A }(x) ≤ 1

_{A}(x), as composed by two factors,

_{A}(x)

_{A}(x).

_{A}(x) = 0; for instance in the case where either r

_{A}(x), or r

_{A }(x) and ɸ

_{A}(x) = 0; therefore, null phase and amplitude.

_{A∪B }(x) = [r

_{A}(x) ▸ r

_{B}(x)] exp {i ɸ

_{A∪B}(x)}

_{A∩B }(x) = [r

_{A}(x) ◄ r

_{B}(x)] exp {i ɸ

_{A∩B}(x)}

_{A∪B}(x) and ɸ

_{A∩B}(x)

_{u}= R

_{j}= {a ∈

**C**: |a| = 1} x {b ∈

**C**: |b| = 1} → {d ∈

**C**: |d| = 1}

**C**represent the field of complex numbers.

**Axioms defining the u application**

- (1) u (a, 0) = a
- (2) if |b| ≤ |d|, then |u(a, b)| ≤ |u(a, d)|
- (3) u (a, b) = u (b, a)
- (4) u (a, u (b, d)) = u (u (a, b), d)

- (1) boundary conditions;
- (2) monotonicity;
- (3) commutativity, and
- (4) associativity, respectively.

- (5) u is a continuous function
- (6) |u (a, a)| > |a|
- (7) If |a| ≤ |c|, and |b| ≤ |d |, then |u (a, b)| ≤ u (c, d)

- (5) continuity;
- (6) supidempotency (with “p”); and
- (7) strict monotonicity, respectively.

**Axioms defining the v application**

- (1) v (a, 0) = a
- (2) if |b| ≤ |d| then |v (a, b)| ≤ | v (a, d)|
- (3) v (a, b) = v (b, a)
- (4) v (a, v (b, d)) = v (v (a, b), d)

- (1) boundary conditions;
- (2) monotonicity;
- (3) commutativity;
- (4) associativity, respectively.

- (5) v is a continuous function;
- (6) |v (a, a)| < |a|;
- (7) If |a| ≤ |c|, and |b| ≤ |d|, then |v (a, b)| ≤ v (c, d).

- (5) continuity;
- (6) subidempotency (with “b”);
- (7) strict monotonicity, respectively.

## 4. Lattice of Fuzzy Numbers

**Interval Operations**. So, we have

**(**+) [−3, 5] = [0 + (−3), 1 + 5] = [−3, 6]

**Properties of Interval Operations**

^{α}, being 0 < α ≤ 1

_{α∈(}

_{0, 1]}[A (⊥) B]

^{α}

_{ z = x (⊥) y }[min {A(x), B(y)}]

_{(x, y)}{min [A(x), B (y)]} = MEET (A, B)

_{(x, y)}{min [A(x), B (y)]} = JOIN (A, B)

_{F}, MIN, MAX >

_{F}family, or collection, of all fuzzy sets.

## 5. Conclusions

## Acknowledgements

## References

- Ramot, D.; Milo, R.; Friedman, M.; Kandel, A. Complex fuzzy sets. IEEE Trans. Fuzzy Syst.
**2002**, 10, 171–186. [Google Scholar] [CrossRef] - Nóvak, V.; Perfilieva, I.; Mockor, J. Mathematical Principles of Fuzzy Logic; Kluwer Academic Publishers: Boston, MA, USA, 1999; Chapters 1–2. [Google Scholar]
- Wu, Y.; Zhang, B.; Lu, J. Fuzzy logic and neuro-fuzzy systems: A systematic introduction. Int. J. AI Expert Syst.
**2011**, 2, 47–80. [Google Scholar] - Buckley, J.J. Fuzzy complex numbers. Fuzzy Sets Syst.
**1989**, 33, 333–345. [Google Scholar] [CrossRef] - Van der Waerden, B.L. A History of Algebra; Springer-Verlag: New York, NY, USA, 1985. [Google Scholar]
- Buckley, J.J.; Qu, Y. Fuzzy complex analysis I: Diﬀerentiation. Fuzzy Sets Syst.
**1991**, 41, 269–284. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy Sets, Fuzzy Logic and Fuzzy Systems: Selected Papers. In Advances in Fuzzy Systems—Applications and Theory; Zadeh, L.A., George, J.K., Bo, Y., Eds.; World Scientific: Hackensack, NJ, USA, 1996; Volume 6, pp. 324–376. [Google Scholar]
- Yuan, B.; Klir, G.J. Fuzzy Sets and Fuzzy Logic: Theory and Applications; Prentice Hall PTR: Upper Saddle River, NJ, USA, 1995. [Google Scholar]
- Wang, Z.; Klir, G.J. Fuzzy Measure Theory; Plenum Press: New York, NY, USA, 1992. [Google Scholar]
- Wang, Z.; Klir, G.J. Generalized Measure Theory. In IFSR International Series on Systems; Springer-Verlag: New York, NY, USA, 2009. [Google Scholar]
- Hsieh, C.H.; Chen, S.H. Similarity of Generalized Fuzzy Numbers with Graded Mean Integration Representation. In Proceedings 8th International Fuzzy Systems Association World Congress, Taipei, Taiwan, 10 January 2009; 2, pp. 551–555.
- Garrido, A. Classifying fuzzy numbers. Adv. Model. Optim. (AMO)
**2011**, 13, 89–96. [Google Scholar] - Garrido, A. Searching the arcane origins of fuzzy logic. Brain
**2011**, 2, 51–57. [Google Scholar] - Garrido, A. Fuzzy boolean algebras and Lukasiewicz logic. Acta Univ. Apulensis (AUA)
**2010**, 22, 101–112. [Google Scholar] - Nguyen, H.T. Fuzzy Mathematics and Statistical Applications; Wu-Nan Book Company: Taipei, Taiwan, 2000. [Google Scholar]
- Fu, X.; Shen, Q. Fuzzy Complex Numbers and their Application for Classifiers Performance Evaluation. Pattern Recognit.
**2011**, 44, 1403–1417. [Google Scholar] [CrossRef] - Wu, C.; Qiu, J. Some remarks for fuzzy complex analysis. Fuzzy Sets Syst.
**1999**, 106, 231–238. [Google Scholar] [CrossRef] - Tamir, D.E.; Kandel, A. Axiomatic theory of complex fuzzy logic and complex fuzzy classes. Int. J. Comput. Commun. Control
**2011**, 6, 562–576. [Google Scholar] - Tamir, D.E.; Lin, J.; Kandel, A. A new interpretation of complex membership grade. Int. J. Intell. Syst.
**2011**, 26, 283–312. [Google Scholar] - Qiu, J.; Wu, C.; Li, F. On the restudy of fuzzy complex analysis. Fuzzy Sets Syst.
**2001**, 120, 517–521. [Google Scholar] [CrossRef] - Dick, S. Toward complex fuzzy logic. IEEE Trans. Fuzzy Syst.
**2005**, 13, 405–414. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Garrido, A.
Axiomatic of Fuzzy Complex Numbers. *Axioms* **2012**, *1*, 21-32.
https://doi.org/10.3390/axioms1010021

**AMA Style**

Garrido A.
Axiomatic of Fuzzy Complex Numbers. *Axioms*. 2012; 1(1):21-32.
https://doi.org/10.3390/axioms1010021

**Chicago/Turabian Style**

Garrido, Angel.
2012. "Axiomatic of Fuzzy Complex Numbers" *Axioms* 1, no. 1: 21-32.
https://doi.org/10.3390/axioms1010021