Operations on complex multi-fuzzy sets

Abstract

In this paper, we introduce the concept of complex multi-fuzzy sets (CM ^k FSs) as a generalization of the concept of multi-fuzzy sets by adding the phase term to the definition of multi-fuzzy sets. In other words, we extend the range of multi-membership function from the interval [0,1] to unit circle in the complex plane. The novelty of CM ^k FSs lies in the ability of complex multi-membership functions to achieve more range of values while handling uncertainty of data that is periodic in nature. The basic operations on CM ^k FSs, namely complement, union, intersection, product and Cartesian product are studied along with accompanying examples. Properties of these operations are derived. Finally, we introduce the intuitive definition of the distance measure between two complex multi-fuzzy sets which are used to define δ-equalities of complex multi-fuzzy sets.

Keywords

Complex multi-fuzzy set multi-fuzzy sets fuzzy set distance measure

1 Introduction

The concept of fuzzy set (FS) was used for the first time by Zadeh [1] to handle uncertainty in many fields of everyday life. Fuzzy set theory generalized the range values of classical set theory from the integer 0 and 1 to the interval [0, 1]. In addition, this concept has proven to be very useful in many different fields [2 –4].

The traditional fuzzy sets is characterized by the membership value or the grade of membership value. Atanassov [5] introduced intuitionistic fuzzy sets as a generalization of fuzzy set with truth-membership and falsity-membership functions. However, fuzzy sets and intuitionistic fuzzy sets can only process incomplete information but not the indeterminate and inconsistent information which exist commonly in belief systems. Smarandache [6] introduced neutrosophic set with truth-membership function (T), indeterminacy-membership function (I) and falsity-membership function (F) to deal with incomplete, indeterminate and inconsistent information. Sebastian and Ramakrishnan [7, 8] generalized fuzzy sets to multi-fuzzy sets in terms of multi dimensional membership functions and multi-level fuzziness. In the same line, Smarandache [9] extended the classical neutrosophic logic to n-valued refined neutrosophic logic, in which neutrosophic components T, I, F are refined (split) into T₁, T₂, . . . , T_p, I₁, I₂, . . . , I_q and F₁, F₂, . . . , F_r, respectively, where all p, q and r ≥ 1 are integers and p + q + r = n. These split neutrosophic components also constructed as n-valued refined neutrosophic set.

Ramot et al. [10] presented a new innovative concept called complex fuzzy sets (CFSs). The complex fuzzy sets is characterized by a membership function whose range is not limited to [0, 1] but extended to the unit circle in the complex plane, where the degree of membership function μ is traded by a complex-valued function of the form $r_{\tilde{s}} (x) e^{i ω_{\tilde{s}} (x)}$ $(i = \sqrt{- 1})$ , where $r_{\tilde{s}} (x)$ and $ω_{\tilde{s}} (x)$ are both real-valued functions and $r_{\tilde{s}} (x) e^{i ω_{\tilde{s}} (x)}$ the range are in a complex unit circle. They also added an additional term called the phase term to solve the enigma in translating some complex-valued functions on physical terms to human language and vice versa. Ramot et al. [10, 11] discussed several important operations such as complement, union, and intersection and discussed fuzzy relations for such complex fuzzy sets. The complex fuzzy sets are used to represent the information with uncertainty and periodicity simultaneously.

Alkouri and Salleh [12] introduced the concept of complex intuitionistic fuzzy set (CIFS) which is generalized from the innovative concept of a complex fuzzy set. The complex fuzzy sets and complex intuitionistic fuzzy sets cannot handle imprecise, indeterminate, inconsistent, and incomplete information of periodic nature. Ali and Smarandache [13] presented the concept of complex neutrosophic set. The complex neutrosophic set can handle the redundant nature of uncertainty, incompleteness, indeterminacy and inconsistency among others.

The concept of “proximity measure” was used for the first time by Pappis [14], with an attempt to show that precise membership values should normally be of no practical significance. Hong and Hwang [15] then proposed an important generalization. Moreover, Cai [16, 17] introduced and discussed δ-equalities of fuzzy sets and proposed that if two fuzzy sets are equal to an extent of δ, then they are said to be δ-equal. Zhang et al. [18] introduced the concept of δ-equalities of complex fuzzy set and applied it to signal processing.

We will extend the studies on multi-fuzzy sets [7] and complex fuzzy sets [10] and a novel notion called complex multi-fuzzy sets (CM ^k FSs) by adding a second dimension (phase term) to multi-membership function of multi-fuzzy sets. CM ^k FSs is constructed to hold the advantages of multi-fuzzy sets while keeping the features of complex numbers in complex fuzzy sets as follows: On the one hand, multi-fuzzy sets can completely and comprehensively characterize many real world problems that involve multi-agent, multi-attribute, multi-object, multi-index and uncertainty utilizing multi- membership functions. On the other hand, complex fuzzy sets has the complexity feature which handles information that has a periodic nature. All of these features together will be contained in our proposed complex multi-fuzzy sets.

To facilitate our discussion, we first review some background on fuzzy sets, multi-fuzzy sets and complex fuzzy sets in Section 2. In Section 3, we discuss comparisons, advantages and drawbacks of the various hybrid structures of fuzzy sets, and complex numbers which served as the motivation for this paper. In Section 4, we introduce the definition of complex multi-fuzzy sets which are a generalization of multi-fuzzy sets. In Section 5, we introduce some basic theoretic operations on complex multi-fuzzy sets, which are complement, union and intersection, derive their properties and give some examples. In Section 6, we give the structure of distance measure on complex multi-fuzzy sets by extending the structure of distance measure of complex fuzzy sets. Finally, conclusions are pointed out in Section 7.

2 Preliminaries

In the current section we will briefly recall the notions of fuzzy sets, multi-fuzzy sets and complex fuzzy sets which are relevant to this paper.

First we shall recall the basic definitions of fuzzy sets. The theory of fuzzy sets, first developed by Zadeh in 1965 [1] is as follows.

Definition 2.1. (see [1]) A fuzzy set $\tilde{S}$ in a universe of discourse X is characterised by a membership function $μ_{\tilde{S}} (x)$ that takes values in the interval [0, 1] for all x ∈ X.

Sebastian and Ramakrishnan [7] introduced the following definition of multi-fuzzy sets.

Definition 2.2. (see [7]) Let k be a positive integer and X be a non-empty set. A multi-fuzzy set $\tilde{S}$ in X is a set of ordered sequences $\tilde{S} = {〈 x, μ_{1} (x), ..., μ_{k} (x) 〉 : x \in X},$ where μ_i : X ⟶ L_i = [0, 1], i = 1, 2, . . . , k.

The function $μ_{\tilde{S}} (x) = (μ_{1} (x), . . ., μ_{k} (x))$ is called the multi-membership function of multi-fuzzy sets $\tilde{S}$ , k is called a dimension of $\tilde{S}$ . The set of all multi-fuzzy sets of dimension k in X is denoted by M ^k FS(x).

Definition 2.3. (see [7]) Let k be a positive integer and let A and B in M ^k FS(X), that is $\tilde{A} = {〈 x, μ_{1} (x), ..., μ_{k} (x) 〉 : x \in X}$ and $\tilde{B} = {〈 x, ν_{1} (x),$ . . , ν_k (x) 〉 : x ∈ X}, then we have the following relations and operations:

A ⊂ B if and only if μ_j (x) ≤ ν_j (x), for all x ∈ X and j = 1, 2, . . . , k ;

A = B if and only if μ_j (x) = ν_j (x), for all x ∈ X and j = 1, 2, . . . , k ;

A∪ B = {〈x, max(μ₁ (x) , ν₁ (x)) , . . . , max(μ_k (x) , ν_k (x)) 〉 : x ∈ X} ;

A ∩ B = {〈x, min(μ₁ (x) , ν₁ (x)) , . . . , min(μ_k (x) , ν_k (x)) 〉 : x ∈ X} .

Definition 2.4. (see [7]) Let $\tilde{A} = (μ_{1}, . . ., μ_{k})$ be a multi-fuzzy set of dimension k and $μ_{j}^{'}$ be the fuzzy complement of the ordinary fuzzy set $μ_{j}^{'}$ for j = 1, 2, . . . , k . The multi-fuzzy complement of the multi-fuzzy set $\tilde{A}$ is a multi-fuzzy set $(μ_{1}^{'}, . . ., μ_{k}^{'})$ and it is denoted by C (A) or A′ or A^c . That is, $\begin{array}{l} C (\tilde{A}) = {〈 x, c (μ_{1} (x)), ..., c (μ_{k} (x)) 〉 : x \in X} \\ = {〈 x, 1 - μ_{1} (x), ..., 1 - μ_{k} (x) 〉 : x \in X}, \end{array}$ where c is the fuzzy complement operation.

In the following, we give some basic definitions and set theoretic operations of complex fuzzy sets.

Definition 2.5. (see [10]) A complex fuzzy set (CFS) $\tilde{S}$ , defined on a universe of discourse X, is characterized by a membership function $μ_{\tilde{s}} (x)$ that assigns to any element x ∈ X a complex-valued grade of membership in $\tilde{S}$ . By definition, the values of $μ_{\tilde{s}} (x)$ may receive all lie within the unit circle in the complex plane and are thus of the form $μ_{\tilde{s}} (x) = r_{\tilde{s}} (x) . e^{i {Arg}_{\tilde{s}} (x)}$ , where $i = \sqrt{- 1}$ , $r_{\tilde{s}} (x)$ is a real-valued function such that $r_{\tilde{s}} (x) \in [0, 1]$ and $e^{{iArg}_{\tilde{s}} (x)}$ is a periodic function whose periodic law and principal period are, respectively, 2π and $0 < \arg_{\tilde{s}} (x) \leq 2 π$ , i.e., ${Arg}_{\tilde{s}} (x) = \arg_{\tilde{s}} (x) + 2 k π, k = 0, \pm 1, \pm 2, . . .,$ where $\arg_{\tilde{s}} (x)$ is the principal argument. The principal argument $\arg_{\tilde{s}} (x)$ will be used in the following text. The CFS $\tilde{S}$ may be represented as the set of ordered pairs $\begin{matrix} \tilde{S} & = & {(x, μ_{\tilde{s}} (x)) : x \in X} \\ = & {(x, r_{\tilde{s}} (x) . e^{i {arg}_{\tilde{s}} (x)}) : x \in X} . \end{matrix}$

Definition 2.6. (see [10]) Let $\tilde{A}$ and $\tilde{B}$ be two complex fuzzy sets on X, and let $μ_{\tilde{A}} (x) = r_{\tilde{A}} (x) . e^{i {arg}_{\tilde{A}} (x)}$ and $μ_{\tilde{B}} (x) = r_{\tilde{B}} (x) . e^{i {arg}_{\tilde{B}} (x)}$ be their membership functions, respectively.

The complex fuzzy complement of $\tilde{A}$ , denoted by ${\tilde{A}}^{c},$ is specified by a function $\begin{matrix} {\tilde{A}}^{c} & = & {(x, r_{{\tilde{A}}^{c}} (x) {. e}^{i {arg}_{{\tilde{A}}^{c}} (x)}) : x \in X} \\ = & {(x, (1 - r_{\tilde{A}} (x)) {. e}^{i ({2 π - arg}_{\tilde{A}} (x))}) : x \in X}, \end{matrix}$

The complex fuzzy union of $\tilde{A}$ and $\tilde{B}$ , denoted by $\tilde{A} \cup \tilde{B},$ is specified by a function $\begin{matrix} μ_{\tilde{A} \cup \tilde{B}} (x) & = & r_{\tilde{A} \cup \tilde{B}} (x) . e^{i {arg}_{\tilde{A} \cup \tilde{B}} (x)} \\ = & max (r_{\tilde{A}} (x), r_{\tilde{B}} (x)) . e^{i max ({arg}_{\tilde{A}} (x), {arg}_{\tilde{B}} (x))}, \end{matrix}$

The complex fuzzy intersection of $\tilde{A}$ and $\tilde{B}$ , denoted by $\tilde{A} \cap \tilde{B},$ is specified by a function $\begin{matrix} μ_{\tilde{A} \cap \tilde{B}} (x) & = & r_{\tilde{A} \cap \tilde{B}} (x) . e^{i {arg}_{\tilde{A} \cap \tilde{B}} (x)} \\ = & min (r_{\tilde{A}} (x), r_{\tilde{B}} (x)) . e^{i min ({arg}_{\tilde{A}} (x), {arg}_{\tilde{B}} (x))}, \end{matrix}$

We say that $\tilde{A}$ is greater than $\tilde{B}$ , denoted by $\tilde{A} \supseteq \tilde{B}$ or $\tilde{B} \subseteq \tilde{A}$ , if for any x ∈ X, $r_{\tilde{A}} (x) \leq r_{\tilde{B}} (x) and \arg_{\tilde{A}} (x) \leq \arg_{\tilde{B}} (x) .$

Definition 2.7. (see [18]) Let $\tilde{A}$ and $\tilde{B}$ be two complex fuzzy sets on X, and $μ_{\tilde{A}} (x) = r_{\tilde{A}} (x) . e^{i \arg_{\tilde{A}} (x)}$ and $μ_{B} (x) = r_{\tilde{B}} (x) . e^{i \arg_{\tilde{B}} (x)}$ their membership functions, respectively. The complex fuzzy product of $\tilde{A}$ and $\tilde{B}$ , denoted $\tilde{A} \circ \tilde{B},$ is specified by afunction $\begin{matrix} μ_{\tilde{A} \circ \tilde{B}} & = & r_{\tilde{A} \circ \tilde{B}} (x) {. e}^{i \arg_{\tilde{A} \circ \tilde{B}} (x)} \\ = & (r_{\tilde{A}} (x) \cdot r_{\tilde{B}} (x)) {. e}^{i 2 π (\frac{\arg_{\tilde{A}} (x)}{2 π} \cdot \frac{\arg_{\tilde{B}} (x)}{2 π})} . \end{matrix}$

Definition 2.8. (see [18]) Let ${\tilde{A}}_{n},$ n = 1, 2, . . . , N be N complex fuzzy sets on X, and $μ_{{\tilde{A}}_{n}} (x) = r_{{\tilde{A}}_{n}} (x) . e^{i \arg_{{\tilde{A}}_{n}} (x)}$ their membership functions, respectively. The complex fuzzy Cartesian product of ${\tilde{A}}_{n}, n = 1, \dots, N,$ denoted ${\tilde{A}}_{1} \times \dots \times {\tilde{A}}_{N},$ is specified by a function $\begin{matrix} μ_{{\tilde{A}}_{1} \times \dots \times {\tilde{A}}_{N}} & = & r_{{\tilde{A}}_{1} \times \dots \times {\tilde{A}}_{N}} (x) {. e}^{i \arg_{{\tilde{A}}_{1} \times \dots \times {\tilde{A}}_{N}} (x)} \\ = & min (r_{{\tilde{A}}_{1}} (x_{1}), \dots, r_{{\tilde{A}}_{N}} (x_{N})) \\ {. e}^{i min (\arg_{{\tilde{A}}_{1}} (x_{1}), \dots, \arg_{{\tilde{A}}_{N}} (x_{N}))}, \end{matrix}$ where $x = (x_{1}, \dots, x_{N}) \in \underset{N}{\underset{︸}{X \times \dots \times X}}$ .

Zhang et al. [18] proposed a distance measure for complex fuzzy sets by identifying the difference between the amplitude terms and the difference between the phase terms as follows:

Definition 2.9. (see [18]) A distance of complex fuzzy sets is a function ρ : CFS (X) ×CFS (X) → [0, 1] with the properties: For any $\tilde{A}, \tilde{B}, \tilde{C} \in CFS (X)$

$ρ (\tilde{A}, \tilde{B}) \geq 0,$

$ρ (\tilde{A}, \tilde{B}) = 0$ if and only if $\tilde{A} = \tilde{B},$

$ρ (\tilde{A}, \tilde{B}) = ρ (\tilde{B}, \tilde{A}),$

$ρ (\tilde{A}, \tilde{B}) \leq ρ (\tilde{A}, \tilde{C}) + ρ (\tilde{C}, \tilde{B}) .$

Zhang et al. [18] also introduced a function ρ as a distance between two complex fuzzy sets as follows: $\begin{matrix} ρ (\tilde{A}, \tilde{B}) & = & max (\sup_{x \in X} | r_{\tilde{A}} (x) - r_{\tilde{B}} (x) |, \\ \frac{1}{2 π} \sup_{x \in X} | \arg_{\tilde{A}} (x) - \arg_{\tilde{B}} (x) |) . \end{matrix}$

Now, we introduce the definition of δ-Equalities of complex fuzzy sets.

Definition 2.10. (see [18]) Let $\tilde{A}$ and $\tilde{B}$ be two complex fuzzy sets on X, and $μ_{\tilde{A}} (x) = r_{\tilde{A}} (x) . e^{i \arg_{\tilde{A}} (x)}$ and $μ_{\tilde{B}} (x) = r_{\tilde{B}} (x) . e^{i \arg_{\tilde{B}} (x)}$ their membership functions, respectively. Then $\tilde{A}$ and $\tilde{B}$ are said to be δ-equal, denoted $\tilde{A} = (δ) \tilde{B},$ if and only if $d (\tilde{A}, \tilde{B}) \leq 1 - δ$ , where 0 ≤ δ ≤ 1 .

3 Motivation for complex multi-fuzzy set

The concept of classical set is extended to fuzzy set [1], whose membership grades are within the real-valued interval [0,1]. Fuzzy set is an influential approach that deals with different types of uncertainties. Further, on the basis of the fuzzy set, intuitionistic fuzzy set [5] was proposed by adding a non-membership, followed by neutrosophic set [6] by adding an independent indeterminacy-membership on the basis of intuitionistic fuzzy set. Neutrosophic set was defined as a new mathematical tool for dealing with problems involving incomplete, indeterminacy and inconsistent knowledge. However, many real life problems cannot completely be characterized by a single membership function of Zadeh’s fuzzy sets like characterization problems which include complete color characterization of color images, taste recognition of food items, decision making problems with multi aspects and others. The notion of multi-fuzzy sets [7] provides a new method to represent these problems.

Research on combining complex numbers with fuzzy set has been earlier proposed by Buckley [19] on fuzzy complex number and by Ramot et al. [10] on complex fuzzy set. Unlike Buckley’s proposals, in which complex numbers are used as elements in a universe of discourse, the complex numbers in a complex fuzzy set are used as the membership degrees. In other words, Ramot’s complex fuzzy sets is characterized by a membership function whose range is not limited to [0, 1] but extends to the unit circle in the complex plane, which includes the amplitude and phase terms, while the range of Buckley’s complex fuzzy numbers are within the closed interval [0, 1]. Dey and Pal [20] extended fuzzy complex numbers and sets to multi-fuzzy complex numbers and multi-fuzzy complex sets by using the ordered sequences of membership functions where each membership function represents a mapping from the complex numbers to the interval [0,1]. Our proposed complex multi-fuzzy set has the added advantages of both complex fuzzy set and multi-fuzzy complex set. On the one hand, complex multi-fuzzy set has the phase term which gives it the wavelike properties that could be used to describe constructive and destructive interference depending on the phase value of an element that involve periodicity and varies with time, whereas multi- fuzzy complex set does not have this feature. On the other hand, complex multi- fuzzy set uses complex fuzzy sets as building blocks where its membership function is an ordered sequence of complex fuzzy membership functions. Thus, complex multi- fuzzy set effectively handles the uncertainty and periodicity in the information of multiple periodic factor prediction problems (MPFP) such as bushfire danger rating prediction [21].

Prediction of a bushfire danger rating in advance can reduce costs of damage and save people’s lives. Australian fire authorities currently use a six-level fire danger rating system that is based on the fire danger Index (FDI). The FDI is determined using the data (observations and/or predictions) of four primary meteorological indicators(factors), that is, “maximum temperature,” “efficient precipitation,” “wind speed,” and “relative humidity”. It is well-known that the four indicators change seasonally. Hence, the same data for an indicator mean different things at different times; for instance, a 20° temperature might mean a cool day in summer, a warm day in winter or a fair day in spring. This phenomenon indicates that data are of semantic uncertainty and periodicity.

Complex multi-fuzzy set consists of multi-membership functions such that each membership function is composed of amplitude term and phase term that handle the uncertainty and periodicity, simultaneously. Thus, in this example we have a complex multi-fuzzy set with four complex-valued membership functions. Multi-fuzzy complex set cannot handle this situation since it does not have the wave-like feature provided by the phase terms.

The comparisons, advantages and drawbacks of the various hybrid structures of fuzzy sets, and complex numbers discussed above, served as the motivation for this paper.

4 Complex multi-fuzzy sets

In this section, we introduce the definition of complex multi-fuzzy sets which are a generalization of multi-fuzzy sets [7], by extending the range of multi-membership functions from the interval [0, 1] to the unit circle in the complex plane.

We begin by proposing the definition of complex multi-fuzzy sets based on multi-fuzzy sets and complex fuzzy sets, and give an illustrative example of it.

Definition 4.1. (see [22]) Let X be a non-empty set, N the set of natural numbers and {L_j : j ∈ N} a family of complete lattices. A complex multi-fuzzy set (CMFS) $\tilde{S}$ , defined on a universe of discourse X, is characterised by a multi-membership function $μ_{\tilde{s}} (x) = {(μ_{{\tilde{s}}_{j}} (x))}_{j \in N}$ , that assigns to any element x ∈ X a complex-valued grade of multi- membership in $\tilde{S}$ . By definition, the values of $μ_{\tilde{s}} (x)$ may receive all lying within the unit circle in the complex plane, and are thus of the form $μ_{\tilde{s}} (x) = {(r_{{\tilde{s}}_{j}} (x) {. e}^{{iArg}_{{\tilde{s}}_{j}} (x)})}_{j \in N}$ , $(i = \sqrt{- 1})$ , ${(r_{{\tilde{s}}_{j}} (x))}_{j \in N}$ are real-valued function such that ${(r_{{\tilde{s}}_{j}} (x))}_{j \in N} \in [0, 1]$ and ${(e^{{iArg}_{{\tilde{s}}_{j}} (x)})}_{j \in N}$ are periodic function whose periodic law and principal period are, respectively, 2π and $0 < {(\arg_{{\tilde{s}}_{j}} (x))}_{j \in N} \leq 2 π$ , i.e., ${({Arg}_{{\tilde{s}}_{j}} (x))}_{j \in N} = \arg_{{\tilde{s}}_{j}} (x) + 2 k π, K = 0, \pm 1, . . .,$ where $\arg_{{\tilde{s}}_{j}} (x)$ is the principal argument. The principal argument ${(\arg_{{\tilde{s}}_{j}} (x))}_{j \in N}$ will be used on the following text. The CMFS $\tilde{S}$ may be represented as the set of ordered sequence $\begin{matrix} \tilde{S} & = & {(x {(μ_{{\tilde{s}}_{j}} (x) = a_{j})}_{j \in N}) : x \in X} \\ = & {x, ({(r_{{\tilde{s}}_{j}} (x) {. e}^{{iarg}_{{\tilde{s}}_{j}} (x)})}_{j \in N}) : x \in X} . \end{matrix}$

Where $μ_{{\tilde{s}}_{j}} : X \to L_{j}$ = {a_j : a_j ∈ C, |a_j|≤1} for all j.

The function $(μ_{\tilde{S}} (x) = r_{{\tilde{s}}_{j}} (x) {. e}^{{iarg}_{{\tilde{s}}_{j}} (x)})_{j \in N}$ is called the complex multi-membership function of complex multi-fuzzy set $\tilde{S}$ . If |N| = k, then k is called the dimension of $\tilde{S}$ . The set of all complex multi-fuzzy sets of dimension k in X is denoted by CM ^k FS(X). In this paper L_j = {a_j : a_j ∈ C, |a_j|≤1} and we will study some properties of complex multi-fuzzy sets of dimension k.

Remark 4.2.

A complex multi-fuzzy set is a generalization of multi-fuzzy set. Assume the multi-fuzzy set $\tilde{S}$ is characterized by the real-valued multi- membership function (γ_j (x)) _j∈N. Transforming $\tilde{S}$ into complex multi-fuzzy sets is achieved by setting the amplitude terms ${(r_{{\tilde{s}}_{j}} (x))}_{j \in N}$ to be equal to (γ_j (x)) _j∈N and the phase terms ${(\arg_{{\tilde{s}}_{j}} (x))}_{j \in N}$ to equal zero for all x. In other words, without a phase term, the complex multi-fuzzy sets effectively reduces to conventional multi-fuzzy set. This interpretation is supported by the fact that the range of ${(r_{{\tilde{s}}_{j}} (x))}_{j \in N}$ is [0, 1], as for real-valued grade of multi-membership function.

A complex multi-fuzzy set of dimension one is reduced to a complex fuzzy set, while a complex Atanassov’s intuitionistic fuzzy set can be considered as a complex multi-fuzzy set of dimension two in terms of multi-dimensional membership functions. A complex multi-fuzzy set of dimension two encompasses two complex- valued membership functions whilst a complex Atanassov’s intuitionistic fuzzy set is characterized by a complex-valued membership function and complex-valued nonmembershipfunction.

If $\sum_{j = 1}^{k} | μ_{{\tilde{s}}_{j}} (x) | \leq 1$ , where $μ_{{\tilde{s}}_{j}} (x) = r_{{\tilde{s}}_{j}} (x)$ $. e^{i \arg_{{\tilde{s}}_{j}} (x)}$ for all x ∈ X, then the complex multi-fuzzy set of dimension k is called a normalized complex multi-fuzzy set. If $\sum_{j = 1}^{k} | μ_{{\tilde{s}}_{j}} (x) | \leq l > 1,$ for some x ∈ X, we redefine the complex multi-membership degree $(| μ_{{\tilde{s}}_{1}} (x), |, . . ., | μ_{{\tilde{s}}_{k}} (x) |)$ as $\frac{1}{l} (| μ_{{\tilde{s}}_{1}} (x) |, . . ., | μ_{{\tilde{s}}_{k}} (x) |)$ . Hence the non-normalized complex multi-fuzzy set can be changed into a normalized multi-fuzzy set.

In real life, the decision information is often multi-valued, uncertain and varies from time to time and how to express the decision information is the first task of decision making. The following example reveals how the complex multi-fuzzy set can express these information effectively.

Example 4.3. Suppose, we want to discuss the influence of modern methods in education to student’s performance. Let X be the universal set of indicators that measure the student’s achievement and his interaction, where X = {x₁ = academicachievement, x₂ = emotionalinteraction, x₃ = socialinteraction}. Suppose the modern methods are “using technology in the classroom”, “using gamification in the classroom” and “social media in the classroom”. For any x ∈ X the influence of these three modern methods on student performance can be characterized by the complex multi-membership function $(μ_{\tilde{s_{1}}} (x), μ_{\tilde{s_{2}}} (x), μ_{\tilde{s_{3}}} (x))$ , where $μ_{\tilde{s_{1}}} (x)$ represents the influence of using technology on the student’s performance, $μ_{\tilde{s_{2}}} (x)$ represents the influence of using gamification on the student’s performance and $μ_{\tilde{s_{3}}} (x)$ represents in the influence of the social media in the classroom. The complex multi-membership function $(μ_{\tilde{s_{1}}} (x), μ_{\tilde{s_{2}}} (x), μ_{\tilde{s_{3}}} (x))$ can be represented by the complex multi-fuzzy set $\tilde{S}$ of dimension 3 over X as follows.

$\begin{array}{l} \tilde{S} \\ = {〈 x_{1}, (0.1) e^{i π (11 / 12)}, (0.8) e^{i π (7 / 12)}, (0.8) e^{i π (7 / 12)} 〉, \\ 〈 x_{2}, (0.3) e^{i π (4 / 12)}, (0.6) e^{i π (4 / 12)}, (0.1) e^{i π (4 / 12)} 〉, \\ x_{3}, (0.8) e^{i π (1 / 12)}, (0.6) e^{i π (4 / 12)}, (0.1) e^{i π (4 / 12)} 〉} . \end{array}$

In this example, the amplitude terms of the membership values represent the degree of the influence of the modern methods in education on the student’s performance, whereas the phase terms represent the time that elapses before the influence of modern methods in education evident on the student’s performance, considering that this study lasts 12 months. Both the amplitude and phase terms lie in [0, 1], where an amplitude term of value close to zero implies that a particular modern method has no or very little influence on a student’s performance whereas an amplitude term with value close to one implies that a particular modern method greatly affects a particular student. Similarly, a phase term with value close to zero (one) implies that a very short (long) time elapses before the influence of a modern method becomes evident in a student’s performance.

This example shows how the complex multi-fuzzy set can characterize a multi-valued module and shows its ability to describe accurately the problem parameters that varies with time.

Now, we present the concept of the subset and equality operations on two complex multi-fuzzy sets in the following definition.

Definition 4.4. Let k be a positive integer and let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets in a universe of discourse X of dimension k, given as follows: $\begin{matrix} \tilde{A} & = & {〈 x, r_{{\tilde{A}}_{1}} (x) . e^{{iarg}_{{\tilde{A}}_{1}} (x)}, . . ., r_{{\tilde{A}}_{k}} (x) {. e}^{i \arg_{{\tilde{A}}_{k}} (x)} 〉 \\ : x \in X}, \\ \tilde{B} & = & {〈 x, r_{{\tilde{B}}_{1}} (x) {. e}^{{iarg}_{{\tilde{B}}_{1}} (x)}, . . ., r_{{\tilde{B}}_{k}} (x) . e^{i \arg_{{\tilde{B}}_{k}} (x)} 〉 \\ : x \in X}, \end{matrix}$

Then

$\tilde{A} \subset \tilde{B}$ if and only if $r_{{\tilde{A}}_{j}} (x) \leq r_{{\tilde{B}}_{j}} (x)$ and $\arg_{{\tilde{A}}_{j}} (x) \leq \arg_{{\tilde{B}}_{j}} (x),$ for all x ∈ X and j = 1, 2, . . . , k .

$\tilde{A} = \tilde{B}$ if and only if $r_{{\tilde{A}}_{j}} (x) = r_{{\tilde{B}}_{j}} (x)$ and $\arg_{{\tilde{A}}_{j}} (x) = \arg_{{\tilde{B}}_{j}} (x)$ for all x ∈ X and j = 1, 2, . . . , k .

5 Basic operations on complex multi-fuzzy sets

The operations on complex multi-fuzzy sets are not intuitive compared with those of conventional fuzzy sets. Ramot et al. [10, 11] presented some set-theoretical operations. These operations are mainly defined on the modulus part of the complex-valued membership degrees, rather than the phase part. In this section we introduce some basic theoretic operations on complex multi-fuzzy sets, which are complement, union and intersection and show that the commutative, associative, distributive, De Morgan’s law and other pertaining laws also hold in complex multi-fuzzy sets.

Definition 5.1. Let k be a positive integer and let $\tilde{A}$ be a complex multi-fuzzy set in a universe of discourse X of dimension k, given by $\begin{matrix} \tilde{A} & = & {〈 x, r_{{\tilde{A}}_{1}} (x) . e^{{iarg}_{{\tilde{A}}_{1}} (x)}, . . ., r_{{\tilde{A}}_{k}} (x) {. e}^{i \arg_{{\tilde{A}}_{k}} (x)} 〉 \\ : x \in X}, \end{matrix}$

The complement of $\tilde{A}$ is denoted by ${\tilde{A}}^{c}$ , and is defined by $\begin{matrix} {\tilde{A}}^{c} & = & {〈 x, r_{{\tilde{A}}_{1}^{c}} (x) {. e}^{{iarg}_{{\tilde{A}}_{1}^{c}} (x)}, \dots, r_{{\tilde{A}}_{k}^{c}} (x) {. e}^{{iarg}_{{\tilde{A}}_{k}^{c}} (x)} 〉 \\ : x \in X}, \\ = & {〈 x, (1 - r_{{\tilde{A}}_{1}} (x)) . e^{i (2 π - \arg_{{\tilde{A}}_{1}} (x))}, \dots \\ (1 - r_{{\tilde{A}}_{k}} (x)) . e^{i (2 π - \arg_{{\tilde{A}}_{k}} (x))} 〉 : x \in X}, \end{matrix}$

Example 5.2. Consider Example 4.3. By using the basic complex multi-fuzzy complement, we have $\begin{matrix} {\tilde{S}}^{c} \\ = {〈 x_{1}, (0.9) . e^{i π (13 / 12)}, (0.2) e^{i π (17 / 12)}, (0.2) \\ {. e}^{i π (17 / 12)} 〉, 〈 x_{2}, (0.7) . e^{i π (20 / 12)}, (0.4) {. e}^{i π (20 / 12)}, \\ (0.9) . e^{i π (20 / 12)} 〉, 〈 x_{3}, (0.2) {. e}^{i π (23 / 12)} (0.4) \\ {. e}^{i π (20 / 12)}, (0.9) . e^{i π (20 / 12)} 〉} . \end{matrix}$

Proposition 5.3. Let $\tilde{A}$ be a complex multi-fuzzy set in a universe of discourse X of dimension k. Then ${({\tilde{A}}^{c})}^{c} = \tilde{A} .$

Proof. By Definition 5.1, we have $\begin{matrix} {({\tilde{A}}^{c})}^{c} \\ = {〈 x, r_{{({\tilde{A_{1}}}^{c})}^{c}} (x) {. e}^{{iarg}_{{({\tilde{A_{1}}}^{c})}^{c}} (x)}, \dots, \\ r_{{({\tilde{A_{k}}}^{c})}^{c}} (x) {. e}^{{iarg}_{{({\tilde{A_{k}}}^{c})}^{c}} (x)} 〉 : x \in X} \\ = {〈 x, (1 - r_{{\tilde{A}}_{1}^{c}} (x)) {. e}^{i (2 π - \arg_{{\tilde{A}}_{1}^{c}} (x))}, \dots, \\ (1 - r_{{\tilde{A}}_{k}^{c}} (x)) . e^{i (2 π - \arg_{{\tilde{A}}_{k}^{c}} (x))} 〉 : x \in X} \\ = {〈 x, [1 - (1 - r_{{\tilde{A}}_{1}} (x))] . e^{i [2 π - (2 π - \arg_{{\tilde{A}}_{1}} (x))]}, \\ \dots, [1 - (1 - r_{{\tilde{A}}_{k}} (x))] . e^{i [2 π - (2 π - \arg_{{\tilde{A}}_{k}} (x))]} 〉 \\ : x \in X} \\ = {〈 x, r_{{\tilde{A}}_{1}} (x) . e^{i \arg_{{\tilde{A}}_{1}} (x)}, \dots, r_{{\tilde{A}}_{k}} (x) {. e}^{i \arg_{{\tilde{A}}_{k}} (x)} 〉 \\ : x \in X} \\ = \tilde{A} . \end{matrix}$

Thus ${({\tilde{A}}^{c})}^{c} = \tilde{A} .$

Definition 5.4. Let k be a positive integer and let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets in a universe of discourse X of dimension k, given as follows. $\begin{matrix} \tilde{A} & = & {〈 x, r_{{\tilde{A}}_{1}} (x) . e^{{iarg}_{{\tilde{A}}_{1}} (x)}, . . ., r_{{\tilde{A}}_{k}} (x) {. e}^{i \arg_{{\tilde{A}}_{k}} (x)} 〉 \\ : x \in X}, \\ \tilde{B} & = & {〈 x, r_{{\tilde{B}}_{1}} (x) {. e}^{{iarg}_{{\tilde{B}}_{1}} (x)}, . . ., r_{{\tilde{B}}_{k}} (x) . e^{i \arg_{{\tilde{B}}_{k}} (x)} 〉 \\ : x \in X}, \end{matrix}$

The union of $\tilde{A}$ and $\tilde{B}$ , denoted by $\tilde{A} \cup \tilde{B}$ , is defined as $\begin{matrix} \tilde{A} \cup \tilde{B} & = & {〈 x, r_{{\tilde{A}}_{1} \cup {\tilde{B}}_{1}} (x) . e^{i \arg_{{\tilde{A}}_{1} \cup {\tilde{B}}_{1}} (x)}, . . ., \\ r_{{\tilde{A}}_{k} \cup {\tilde{B}}_{k}} (x) . e^{i \arg_{{\tilde{A}}_{k} \cup {\tilde{B}}_{k}} (x)} 〉 : x \in X} \\ = & {〈 x, \lor (r_{{\tilde{A}}_{1}} (x), r_{{\tilde{B}}_{1}} (x)) . e^{i \arg_{{\tilde{A}}_{1} \cup {\tilde{B}}_{1}} (x)}, \dots, \\ \lor (r_{{\tilde{A}}_{k}} (x), r_{{\tilde{B}}_{k}} (x)) . e^{{iarg}_{{\tilde{A}}_{k} \cup {\tilde{B}}_{k}} (x)} 〉 : x \in X} . \end{matrix}$ where ∨ denote the max operator.

The following definition of complex multi-fuzzy union is required to calculate the phase term $(e^{{iarg}_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x)})$ (for j = 1, . . . , k).

Definition 5.5. Let k be a positive integer and let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets in a universe of discourse X of dimension k, with the complex-valued multi membership functions $μ_{\tilde{A}} (x)$ and $μ_{\tilde{B}} (x)$ . The complex multi-fuzzy union of $\tilde{A}$ and $\tilde{B}$ , denoted by $\tilde{A} \cup \tilde{B}$ , is specified by the function $\begin{matrix} \tilde{s} : {(a_{1}, . . ., a_{j}) : a_{1}, . . ., a_{j} \in C, | a_{1} |, \dots, | a_{j} | \leq 1, \\ \forall j = 1, 2, . . ., k} \\ \times {(b_{1}, \dots, b_{j}) : b_{1}, \dots, b_{j} \in C, \\ | b_{1} |, \dots, | b_{j} | \leq 1, \forall j = 1, 2, . . ., k} \\ \to {(d_{1}, \dots, d_{j}) : d_{1}, \dots, d_{j} \in C, \\ | d_{1} |, \dots, | d_{j} | \leq 1, \forall j = 1, 2, . . ., k}, \end{matrix}$ where (a₁, …, a_j) , (b₁, …, b_j) and (d₁, … d_j) are the complex multi-membership functions of $\tilde{A}$ , $\tilde{B}$ and $\tilde{A} \cup \tilde{B}$ , respectively. $\tilde{s}$ assigns a complex value, for all x ∈ X and j = 1, 2, …, k . $\begin{matrix} \tilde{s} ((a_{1}, \dots, a_{j}), \dots, (b_{1}, \dots, b_{j})) \\ = ({\tilde{s}}_{μ} (a_{1}, b_{1}), \dots, {\tilde{s}}_{μ} (a_{j}, b_{j})) \\ = (μ_{{\tilde{A}}_{1} \cup {\tilde{B}}_{1}} (x), \dots, μ_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x)) \\ = (d_{1}, \dots, d_{j}) \end{matrix}$

The complex multi-fuzzy union functions $\tilde{s}$ must satisfy at least the following axiomatic requirements, for any (a₁, …, a_j) , (b₁, …, b_j) , (c₁, …, c_j) and

Axiom 1: $| {\tilde{s}}_{μ} (a_{j}, 0) | = | a_{j} |,$ ∀j = 1, 2, . . . , k (boundary condition).

Axiom 2: ${\tilde{s}}_{μ} (a_{j}, b_{j}) = {\tilde{s}}_{μ} (b_{j}, a_{j}),$ ∀j = 1, 2, . . . , k (commutative condition).

Axiom 3: if |b_j| ≤ |d_j| then $| {\tilde{s}}_{μ} (a_{j}, b_{j}) | \leq | {\tilde{s}}_{μ} (a_{j}, d_{j}) |,$ ∀j = 1, 2, . . . , k (monotonic condition).

Axiom 4: ${\tilde{s}}_{μ} (a_{j}, {\tilde{s}}_{μ} (b_{j}, c_{j})) = {\tilde{s}}_{μ} ({\tilde{s}}_{μ} (a_{j}, b_{j}), c_{j}), \forall j = 1, 2, . . ., k,$ (associative condition).

In some cases, it may be desirable that

\tilde{s}

also satisfy the following requirements:

Axiom 5: $\tilde{s}$ is a continuous function (continuity).

Axiom 6: $| {\tilde{s}}_{μ} (a_{j}, a_{j}) | \geq | a_{j} |,$ ∀j = 1, 2, . . . , k (superidempotency).

Axiom 7: if: if |a_j| ≤ |c_j| and |b_j| ≤ |d_j| then $| {\tilde{s}}_{μ} (a_{j}, b_{j}) | \leq | {\tilde{s}}_{μ} (c_{j}, d_{j}) |,$ ∀j = 1, 2, . . . , k (strict monotonicity).

The phase term for complex multi-membership functions belongs to (0, 2π] . To define the phase terms $\arg_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x)$ (for j = 1, 2, . . . , k .), we take those forms which Ramot et al.[10] presented to calculate $\arg_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x)$ as follows:

Sum: $\arg_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x) = \arg_{{\tilde{A}}_{j}} (x) + \arg_{{\tilde{B}}_{j}} (x),$ for all j = 1, 2, . . . , k .

Max: $\arg_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x) = max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x)),$ for all j = 1, 2, . . . , k .

Min: $\arg_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x) = min (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x)),$ for all j = 1, 2, . . . , k .

“Winner Takes All”: $\begin{matrix} \arg_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x) & = & {\begin{matrix} \arg_{{\tilde{A}}_{j}} (x), & r_{{\tilde{A}}_{j}} > r_{{\tilde{B}}_{j}} \\ \arg_{{\tilde{B}}_{j}} (x), & r_{{\tilde{A}}_{j}} < r_{{\tilde{B}}_{j}} \end{matrix}, \\ for all j = 1, 2, . . ., k . \end{matrix}$

Example 5.6. Let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets in X ={ x₁, x₂ } of dimension four, given by $\begin{array}{l} \tilde{A} = {〈 x_{1} (0.4) e^{i (π)}, (0.3) e^{i (0.4 π)}, (0.2) e^{i (1.5 π)}, \\ (0.1) . e^{i (0.1 π)} 〉, x_{2}, (0.1) . e^{i (π)}, (0.2) . e^{i (0.7 π)}, 〉 \\ (0.2) . e^{i (0.5 π)}, (0.5) . e^{i (π)} 〉}, \\ \tilde{B} = {〈 x_{1} (0.3) . e^{i (0.1 π)}, (0.2) . e^{i (π)}, (0.1) . e^{i (1.1 π)}, \\ (0.3) . e^{i (0.2 π)} 〉, x_{2}, (0.5) . e^{i (π)}, (0.3) . e^{i (0.2 π)}, 〉 \\ (0.0) . e^{i (0.8 π)}, (0.1) . e^{i (2 π)} 〉 .} . \end{array}$

Using the max union function for calculating $(r_{{\tilde{A}}_{k} \cup {\tilde{B}}_{k}} (x))$ (for k = 1, . . . , 4) and the “Winner Takes All” for determining $(\arg_{{\tilde{A}}_{k} \cup {\tilde{B}}_{k}} (x))$ (for k = 1, . . . , 4), the following results are obtained for $\tilde{A} \cup \tilde{B} :$ $\begin{matrix} \tilde{A} \cup \tilde{B} \\ = {〈 x_{1}, (0.4) . e^{i (π)}, (0.3) {. e}^{i (0.4 π)}, \\ (0.2) . e^{i (1.5 π)}, (0.3) . e^{i (0.2 π)} 〉, 〈 x_{2}, (0.5) . e^{i (π)}, \\ (0.3) . e^{i (0.2 π)}, (0.2) . e^{i (0.5 π)}, (0.5) . e^{i (π)} 〉} . \end{matrix}$

We introduce the definition of the intersection of two complex multi-fuzzy sets with an illustrative example as follows.

Definition 5.7. Let k be a positive integer and let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets in a universe of discourse X of dimension k, given as follows. $\begin{matrix} \tilde{A} & = & {〈 x, r_{{\tilde{A}}_{1}} (x) . e^{{iarg}_{{\tilde{A}}_{1}} (x)}, . . ., r_{{\tilde{A}}_{k}} (x) {. e}^{i \arg_{{\tilde{A}}_{k}} (x)} 〉 \\ : x \in X}, \\ \tilde{B} & = & {〈 x, r_{{\tilde{B}}_{1}} (x) {. e}^{{iarg}_{{\tilde{B}}_{1}} (x)}, . . ., r_{{\tilde{B}}_{k}} (x) . e^{i \arg_{{\tilde{B}}_{k}} (x)} 〉 \\ : x \in X}, \end{matrix}$

The intersection of $\tilde{A}$ and $\tilde{B}$ , denoted by $\tilde{A} \cap \tilde{B}$ , is defined as

$\begin{matrix} \tilde{A} \cap \tilde{B} \\ = {〈 x, r_{{\tilde{A}}_{1} \cap {\tilde{B}}_{1}} (x) . e^{i \arg_{{\tilde{A}}_{1} \cap {\tilde{B}}_{1}} (x)}, . . ., \\ r_{{\tilde{A}}_{k} \cap {\tilde{B}}_{k}} (x) . e^{i \arg_{{\tilde{A}}_{k} \cap {\tilde{B}}_{k}} (x)} 〉 : x \in X} \\ = {〈 x, \land (r_{{\tilde{A}}_{1}} (x), r_{{\tilde{B}}_{1}} (x)) . e^{i \arg_{{\tilde{A}}_{1} \cap {\tilde{B}}_{1}} (x)}, \dots, \\ \land (r_{{\tilde{A}}_{k}} (x), r_{{\tilde{B}}_{k}} (x)) . e^{{iarg}_{{\tilde{A}}_{k} \cap {\tilde{B}}_{k}} (x)} 〉 : x \in X} . \end{matrix}$ where ∧ denote the max operator.

The following definition of complex multi-fuzzy intersection is required to calculate the phase term $(e^{{iarg}_{{\tilde{A}}_{j} \cap {\tilde{B}}_{j}} (x)})$ (for j = 1, 2, . . . , k).

Definition 5.8. Let k be a positive integer and let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets in a universe of discourse X of dimension k, with the complex-valued multi membership functions $μ_{\tilde{A}} (x)$ and $μ_{\tilde{B}} (x)$ . The complex multi-fuzzy intersection of $\tilde{A}$ and $\tilde{B}$ , denoted by $\tilde{A} \cap \tilde{B}$ , is specified by the function

$\begin{matrix} \tilde{t} : {(a_{1}, . . ., a_{j}) : a_{1}, . . ., a_{j} \in C, \\ | a_{1} |, \dots, | a_{j} | \leq 1, \forall j = 1, \dots, k} \\ \times {(b_{1}, \dots, b_{j}) : b_{1}, \dots, b_{j} \in C, \\ | b_{1} |, \dots, | b_{j} | \leq 1, \forall j = 1, \dots, k} \\ \to {(d_{1}, \dots, d_{j}) : d_{1}, \dots, d_{j} \in C, \\ | d_{1} |, \dots, | d_{j} | \leq 1, \forall j = 1, \dots, k}, \end{matrix}$ where (a₁, …, a_j) , (b₁, …, b_j) and (d₁, … d_j) are the complex multi-membership functions of $\tilde{A}$ , $\tilde{B}$ and $\tilde{A} \cap \tilde{B}$ , respectively. $\tilde{t}$ assigns a complex value, for all x ∈ X and j = 1, 2, …, k . $\begin{matrix} \tilde{t} ((a_{1}, \dots, a_{j}), \dots, (b_{1}, \dots, b_{j})) \\ = ({\tilde{t}}_{μ} (a_{1}, b_{1}), \dots, {\tilde{t}}_{μ} (a_{j}, b_{j})) \\ = (μ_{{\tilde{A}}_{1} \cap {\tilde{B}}_{1}} (x), \dots, μ_{{\tilde{A}}_{j} \cap {\tilde{B}}_{j}} (x)) \\ = (d_{1}, \dots, d_{j}) \end{matrix}$

The complex multi-fuzzy intersection functions $\tilde{t}$ must satisfy at least the following axiomatic requirements, for any (a₁, …, a_j) , (b₁, …, b_j) , (c₁, …, c_j) and (d₁, … d_j) ∈ {x : x ∈ C, |x| ≤ 1} .

Axiom 1: if $| b_{j} | = 1, | {\tilde{t}}_{μ} (a_{j}, b_{j}) | = | a_{j} |,$ ∀j = 1, 2, . . . , k (boundary condition).

Axiom 2: ${\tilde{t}}_{μ} (a_{j}, b_{j}) = {\tilde{t}}_{μ} (b_{j}, a_{j}),$ ∀j = 1, 2, . . . , k (commutative condition).

Axiom 3: if |b_j| ≤ |d_j| then $| {\tilde{t}}_{μ} (a_{j}, b_{j}) | \leq | {\tilde{t}}_{μ} (a_{j}, d_{j}) |,$ ∀j = 1, 2, . . . , k (monotonic condition).

Axiom 4: ${\tilde{t}}_{μ} (a_{j}, {\tilde{t}}_{μ} (b_{j}, c_{j})) = {\tilde{t}}_{μ} ({\tilde{t}}_{μ} (a_{j}, b_{j}), c_{j}),$ fo ∀j = 1, 2, . . . , k, (associative condition).

In some cases, it may be desirable that $\tilde{t}$ also satisfy the following requirements:

Axiom 5: $\tilde{t}$ is a continuous function (continuity).

Axiom 6: $| {\tilde{t}}_{μ} (a_{j}, a_{j}) | \geq | a_{j} |,$ ∀j = 1, 2, . . . , k (superidempotency).

Axiom 7: if |a_j| ≤ |c_j| and |b_j| ≤ |d_j| then $| {\tilde{t}}_{μ} (a_{j}, b_{j}) | \leq | {\tilde{t}}_{μ} (c_{j}, d_{j}) |,$ for all j = 1, 2, . . . , k (strict monotonicity).

We consider some forms to calculate the phase term $\arg_{{\tilde{A}}_{j} \cap {\tilde{B}}_{j}} (x)$ (for j = 1, . . . , k .), such that the same possible choices are given to calculate the $\arg_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x)$ .

We will now give some theorems and propositions on the union, intersection,complement of complex multi fuzzy sets. These theorems and propositions illustrate the relationship between the set theoretic operations that have been mentioned above.

Proposition 5.9. Let $\tilde{A}$ , $\tilde{B}$ and $\tilde{C}$ be any three complex multi-fuzzy sets on X of dimension k. Then the following commutative and associative properties hold true.

$\tilde{A} \cup \tilde{A} = \tilde{A},$ $\tilde{A} \cap \tilde{A} = \tilde{A}$ .

$\tilde{A} \cup \tilde{B} = \tilde{B} \cup \tilde{A},$ $\tilde{A} \cap \tilde{B} = \tilde{B} \cap \tilde{A} .$

$\tilde{A} \cup (\tilde{B} \cup \tilde{C}) = (\tilde{A} \cup \tilde{B}) \cup \tilde{C},$ $\tilde{A} \cap (\tilde{B} \cap \tilde{C}) = (\tilde{A} \cap \tilde{B}) \cap \tilde{C} .$

Proof. The proof is straightforward by using Definitions 5.4 and 5.7.

Theorem 5.10. Let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets on X of dimension k. Then

$\tilde{A} \subseteq \tilde{B}$ if and only if $\tilde{A} \cup \tilde{B} = \tilde{B} .$

$\tilde{A} \subseteq \tilde{B}$ if and only if $\tilde{A} \cap \tilde{B} = \tilde{A} .$

$\tilde{A} = \tilde{B}$ if and only if $\tilde{A} \cup \tilde{B} = \tilde{A} \cap \tilde{B} .$

Proof. We shall prove assertion 1. Assertions 2 and 3 can be similarly proven.

Let $\tilde{A} = {〈 x, [r_{{\tilde{A}}_{j}} (x) . e^{i {arg}_{{\tilde{A}}_{j}} (x)}]_{j \in k} 〉 : x \in X}$ and $\tilde{B} = {〈 x, [r_{{\tilde{B}}_{j}} (x) . e^{i {arg}_{{\tilde{B}}_{j}} (x)}]_{j \in k} 〉 : x \in X}$ be two complex multi-fuzzy sets in a universe of discourse X of dimension k.

(⇒) From the Definition 4.4, $\tilde{A} \subset \tilde{B}$ implies that $r_{{\tilde{A}}_{j}} (x) \leq r_{{\tilde{B}}_{j}} (x)$ and $\arg_{{\tilde{A}}_{j}} (x) \leq \arg_{{\tilde{B}}_{j}} (x),$ for all x ∈ X and j = 1, 2, . . . , k .

Therefore $max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x)) = r_{{\tilde{B}}_{j}} (x)$ and $max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x)) = \arg_{{\tilde{B}}_{j}} (x)$ , for all x ∈ X and j = 1, 2, . . . , k .

Consequently, by Definition 5.4 we have

\begin{matrix} \tilde{A} \cup \tilde{B} & = & {〈 x, [\max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x)) \\ . e^{i max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x))}]_{j \in k} 〉 : x \in X} \\ = & {〈 x, [r_{{\tilde{B}}_{j}} (x) . e^{i {arg}_{{\tilde{B}}_{j}} (x)}]_{j \in k} 〉 : x \in X} \\ = & \tilde{B} \end{matrix}

Thus $\tilde{A} \cup \tilde{B} = \tilde{B} .$

(⇐) Let $\tilde{A} \cup \tilde{B} = \tilde{B}$ , From Definition 5.4 we have $\begin{matrix} \tilde{A} \cup \tilde{B} & = & {〈 x, [\max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x)) \\ . e^{i max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x))}]_{j \in k} 〉 : x \in X} \\ = & {〈 x, [r_{{\tilde{B}}_{j}} (x) . e^{i {arg}_{{\tilde{B}}_{j}} (x)}]_{j \in k} 〉 : x \in X} \\ = & \tilde{B} . \end{matrix}$

This implies that $max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x)) = r_{{\tilde{B}}_{j}} (x)$ and $max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x)) = \arg_{{\tilde{B}}_{j}} (x),$ for all x ∈ X and j = 1, 2, . . . , k .

Therefore $r_{{\tilde{A}}_{j}} (x) \leq r_{{\tilde{B}}_{j}} (x)$ and $\arg_{{\tilde{A}}_{j}} (x) \leq \arg_{{\tilde{B}}_{j}} (x)$ for all x ∈ X, than $\tilde{A} \subseteq \tilde{B} .$

Hence $\tilde{A} \subseteq \tilde{B}$ if and only if $\tilde{A} \cup \tilde{B} = \tilde{B} .$

Theorem 5.11. Let $\tilde{A}$ , $\tilde{B}$ and $\tilde{C}$ be any three complex multi-fuzzy sets on X of dimension k. Then the following distributive properties hold true.

$\tilde{A} \cup (\tilde{B} \cap \tilde{C}) = (\tilde{A} \cup \tilde{B}) \cap (\tilde{A} \cup \tilde{C}) .$

$\tilde{A} \cap (\tilde{B} \cup \tilde{C}) = (\tilde{A} \cap \tilde{B}) \cup (\tilde{A} \cap \tilde{C}) .$

Proof. The proof is straightforward by using Definitions 5.4 and 5.7.

Theorem 5.12. Let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets on X of dimension k. Then

$(\tilde{A} \cup \tilde{B}) \cap \tilde{A} = \tilde{A},$

$(\tilde{A} \cap \tilde{B}) \cup \tilde{A} = \tilde{A} .$

Proof. (1) Let $\tilde{A} = {〈 x, [r_{{\tilde{A}}_{j}} (x) . e^{i {arg}_{{\tilde{A}}_{j}} (x)}]_{j \in k} 〉 : x \in X}$ and $\tilde{B} = {〈 x, [r_{{\tilde{B}}_{j}} (x) . e^{i {arg}_{{\tilde{B}}_{j}} (x)}]_{j \in k} 〉 : x \in X}$ be two complex multi-fuzzy sets in a universe of discourse X of dimension k. By Definitions 5.4 and 5.7, we have $\begin{matrix} (\tilde{A} \cup \tilde{B}) \cap \tilde{A} \\ = {〈 x, [min [r_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x), r_{{\tilde{A}}_{j}} (x)] . e^{i min [\arg_{{\tilde{A}}_{j} \cup {\tilde{B}}_{j}} (x), \arg_{{\tilde{A}}_{j}} (x)]}]_{j \in k} 〉 : x \in X} \\ = {〈 x, [min [max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x)), r_{{\tilde{A}}_{j}} (x)] . e^{i min [max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x)), \arg_{{\tilde{A}}_{j}} (x)]}]_{j \in k} 〉 : x \in X} \\ = {〈 x, [r_{{\tilde{A}}_{j}} (x) . e^{i {arg}_{{\tilde{A}}_{j}} (x)}]_{j \in k} 〉 : x \in X} \\ = \tilde{A} . \end{matrix}$ (2) The proof is similar to that in part (1) and therefore is omitted.

Proposition 5.13. Let A and $\tilde{B}$ be two complex multi-fuzzy sets on X of dimension k. Then the following De Morgans’s laws hold true.

${(\tilde{A} \cup \tilde{B})}^{c} = {\tilde{A}}^{c} \cap {\tilde{B}}^{c} .$

${(\tilde{A} \cap \tilde{B})}^{c} = {\tilde{A}}^{c} \cup {\tilde{B}}^{c} .$

Proof. (1) Let the union of $\tilde{A} = {〈 x, [r_{{\tilde{A}}_{j}} (x) . e^{i {arg}_{{\tilde{A}}_{j}} (x)}]_{j \in k} 〉 : x \in X}$ and $\tilde{B} = {〈 x, [r_{{\tilde{B}}_{j}} (x) . e^{i {arg}_{{\tilde{B}}_{j}} (x)}]_{j \in k} 〉 : x \in X}$ be the basic max operator. Then $\begin{matrix} \tilde{A} \cup \tilde{B} & = & {〈 x, [\max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x)) . \\ e^{i max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x))}]_{j \in k} 〉 : x \in X} \\ {\tilde{A}}^{c} & = & {〈 x, [(1 - r_{{\tilde{A}}_{j}} (x)) . e^{i (2 π - \arg_{{\tilde{A}}_{j}} (x))}]_{j \in k} 〉 \\ : x \in X}, \\ {\tilde{B}}^{c} & = & {〈 x, [(1 - r_{{\tilde{B}}_{j}} (x)) . e^{i (2 π - \arg_{{\tilde{B}}_{j}} (x))}]_{j \in k} 〉 \\ : x \in X}, \\ (\tilde{A} \cup \tilde{B})^{c} & = & {〈 x, [(1 - \max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x))) . \\ e^{i (2 π - max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x)))}]_{j \in k} 〉 : x \in X} \\ {\tilde{A}}^{c} \cap {\tilde{B}}^{c} & = & {〈 x, [(min (1 - r_{{\tilde{A}}_{j}} (x), 1 - r_{{\tilde{B}}_{j}} (x))) . \\ e^{i (min (2 π - \arg_{{\tilde{A}}_{j}} (x), 2 π - \arg_{{\tilde{B}}_{j}} (x)))}]_{j \in k} 〉 : x \in X} . \end{matrix}$

To proof $(\tilde{A} \cup \tilde{B})^{c} = {\tilde{A}}^{c} \cap {\tilde{B}}^{c}$ we need to show that the amplitude term $1 - max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x)) = min (1 - r_{{\tilde{A}}_{j}} (x), 1 - r_{{\tilde{B}}_{j}} (x))$ and that the phaseterm $(2 π - max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x)) = min (2 π - \arg_{{\tilde{A}}_{j}} (x)$ $, 2 π - \arg_{{\tilde{B}}_{j}} (x))$ , for j = 1, . . . , k.

To show the amplitude term, we consider the two possible cases:

Case 1: If $r_{{\tilde{A}}_{j}} (x) \geq r_{{\tilde{B}}_{j}} (x),$ then $1 - r_{{\tilde{A}}_{j}} (x) \leq 1 - r_{{\tilde{B}}_{j}} (x) .$ Thus, $1 - max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x)) = 1 - r_{{\tilde{A}}_{j}} (x)$ and $min (1 - r_{{\tilde{A}}_{j}} (x), 1 - r_{{\tilde{B}}_{j}} (x)) = 1 - r_{{\tilde{A}}_{j}} (x) .$

Case 2: If $r_{{\tilde{A}}_{j}} (x) < r_{{\tilde{B}}_{j}} (x)$ , then $1 - r_{{\tilde{A}}_{j}} (x) > 1 - r_{{\tilde{B}}_{j}} (x)$ . Thus, $1 - max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x)) = 1 - r_{{\tilde{B}}_{j}} (x)$ and $min (1 - r_{{\tilde{A}}_{j}} (x), 1 - r_{{\tilde{B}}_{j}} (x)) = 1 - r_{{\tilde{B}}_{j}} (x) .$

Therefore, by Case 1 and Case 2: $\begin{matrix} 1 - \max (r_{{\tilde{A}}_{j}} (x), r_{{\tilde{B}}_{j}} (x)) \\ = \min (1 - r_{{\tilde{A}}_{j}} (x), 1 - r_{{\tilde{B}}_{j}} (x)), \forall j = 1, . . ., k . \end{matrix}$

To show the phase term, we consider the two possible cases:

Case 1: If $\arg_{{\tilde{A}}_{j}} (x) \geq \arg_{{\tilde{B}}_{j}} (x),$ then $2 π - \arg_{{\tilde{A}}_{j}} (x) \leq 2 π - \arg_{{\tilde{B}}_{j}} (x) .$ Thus, $2 π - \max (\arg_{A_{j}} (x), \arg_{{\tilde{B}}_{j}} (x)) = 2 π - \arg_{A_{j}} (x)$ and $\min (2 π - \arg_{{\tilde{A}}_{j}} (x), 2 π - \arg_{{\tilde{B}}_{j}} (x)) = 2 π - \arg_{{\tilde{A}}_{j}} (x) .$

Case 2: If $\arg_{{\tilde{A}}_{j}} (x) < \arg_{{\tilde{B}}_{j}} (x),$ then $2 π - \arg_{{\tilde{A}}_{j}} (x) > 2 π - \arg_{{\tilde{B}}_{j}} (x) .$ Thus, $2 π - \max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x)) = 2 π - \arg_{{\tilde{B}}_{j}} (x)$ and $\min (2 π - \arg_{{\tilde{A}}_{j}} (x), 2 π - \arg_{{\tilde{B}}_{j}} (x)) = 2 π - \arg_{{\tilde{B}}_{j}} (x) .$

Therefore, by Case 1 and Case 2: $\begin{matrix} 2 π - \max (\arg_{{\tilde{A}}_{j}} (x), \arg_{{\tilde{B}}_{j}} (x)) \\ = \min (2 π - \arg_{{\tilde{A}}_{j}} (x), 2 π - \arg_{{\tilde{B}}_{j}} (x)), \\ \forall j = 1, \dots, k . \end{matrix}$

(2) The proof is similar to that in part (1) and therefore is omitted.

In the following, we introduce the concept of the product of two multi-fuzzy sets and the Cartesian product of multi-fuzzy sets.

Definition 5.14. Let k be a positive integer and let $\tilde{A}$ and $\tilde{B}$ be two multi-fuzzy sets in a universe of discourse X of dimension k, given as follows:

$\tilde{A} = {〈 x, μ_{{\tilde{A}}_{1}} (x), . . ., μ_{{\tilde{A}}_{k}} (x) 〉 : x \in X}$ and $\tilde{B} = {〈 x, μ_{{\tilde{B}}_{1}} (x), . . ., μ_{{\tilde{B}}_{k}} (x) 〉 : x \in X} .$ The multi-fuzzy product of $\tilde{A}$ and $\tilde{B}$ , denoted by $\tilde{A} \circ \tilde{B}$ , is defined as $\begin{matrix} \tilde{A} \circ \tilde{B} & = & {〈 x, r_{{\tilde{A}}_{1} \circ {\tilde{B}}_{1}} (x), . . ., r_{{\tilde{A}}_{k} \circ {\tilde{B}}_{k}} (x) 〉 : x \in X} \\ = & {〈 x, (r_{{\tilde{A}}_{1}} (x) \cdot r_{{\tilde{B}}_{1}} (x)), \dots, \\ (r_{{\tilde{A}}_{k}} (x) \cdot r_{{\tilde{B}}_{k}} (x)) 〉 : x \in X} . \end{matrix}$

Definition 5.15. Let k be a positive integer and let ${\tilde{A}}_{1}, \dots, {\tilde{A}}_{n}$ be multi-fuzzy sets in X₁, …, X_n of dimension k. The Cartesian product ${\tilde{A}}_{1} \times \dots \times {\tilde{A}}_{n}$ is a multi-fuzzy set defined by $\begin{matrix} {\tilde{A}}_{1} \times \dots \times {\tilde{A}}_{n} \\ = {〈 (x_{1}, \dots, x_{n}), μ_{{\tilde{A}}_{1} \times . . . \times {\tilde{A}}_{n}}^{1} (x_{1}, \dots, x_{n}), \\ \dots, μ_{{\tilde{A}}_{1} \times . . . \times {\tilde{A}}_{n}}^{k} (x_{1}, . . ., x_{n}) 〉 : (x_{1}, . . ., x_{n}) \\ \in X_{1} \times . . . \times X_{n}} \end{matrix}$

Where $\begin{matrix} μ_{{\tilde{A}}_{1} \times . . . \times {\tilde{A}}_{n}}^{j} (x_{1}, . . ., x_{n}) \\ = min [μ_{{\tilde{A}}_{1}}^{j} (x_{1}), . . ., μ_{{\tilde{A}}_{n}}^{j} (x_{n})], \forall j = 1, . . ., k . \end{matrix}$

In the following, we introduce the concept of the product of two complex multi-fuzzy sets and the Cartesian product of complex multi-fuzzy sets with an illustrative example of the product operation.

Definition 5.16. Let k be a positive integer and let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets in a universe of discourse X of dimension k, given as follows: $\begin{matrix} \tilde{A} & = & {〈 x, r_{{\tilde{A}}_{1}} (x) . e^{{iarg}_{{\tilde{A}}_{1}} (x)}, . . ., r_{{\tilde{A}}_{k}} (x) {. e}^{i \arg_{{\tilde{A}}_{k}} (x)} 〉 \\ : x \in X}, \\ \tilde{B} & = & {〈 x, r_{{\tilde{B}}_{1}} (x) {. e}^{{iarg}_{{\tilde{B}}_{1}} (x)}, . . ., r_{{\tilde{B}}_{k}} (x) . e^{i \arg_{{\tilde{B}}_{k}} (x)} 〉 \\ : x \in X}, \end{matrix}$

The complex multi-fuzzy product of $\tilde{A}$ and $\tilde{B}$ , denoted by $\tilde{A} \circ \tilde{B}$ , is defined as $\begin{matrix} \tilde{A} \circ \tilde{B} \\ = {〈 x, r_{{\tilde{A}}_{1} \circ {\tilde{B}}_{1}} (x) . e^{{iarg}_{{\tilde{A}}_{1} \circ {\tilde{B}}_{1}} (x)}, . . ., \\ r_{{\tilde{A}}_{k} \circ {\tilde{B}}_{k}} (x) . e^{i \arg_{{\tilde{A}}_{k} \circ {\tilde{B}}_{k}} (x)} 〉 : x \in X .}, \\ = {〈 x, (r_{{\tilde{A}}_{1}} (x) \cdot r_{{\tilde{B}}_{1}} (x)) . e^{i 2 π (\frac{\arg_{{\tilde{A}}_{1}} (x)}{2 π} \cdot \frac{\arg_{{\tilde{B}}_{1}} (x)}{2 π})}, \\ \dots, (r_{{\tilde{A}}_{k}} (x) \cdot r_{{\tilde{B}}_{k}} (x)) . e^{i 2 π (\frac{\arg_{{\tilde{A}}_{k}} (x)}{2 π} \cdot \frac{\arg_{{\tilde{B}}_{k}} (x)}{2 π})} 〉 \\ : x \in X} . \end{matrix}$

Example 5.17. Consider Example 5.6. By using the basic complex multi-fuzzy product, $\begin{matrix} \tilde{A} \circ \tilde{B} \\ = {〈 x_{1}, (0.12) e^{i (0.05 π)}, (0.06) e^{i (0.2 π)}, (0.02) \\ e^{i (0.83 π)}, (0.03) e^{i (0.01 π)} 〉, 〈 x_{2}, (0.05) e^{i (0.1 π)}, \\ (0.09) e^{i (0.35 π)}, (0.0) e^{i (0.2 π)}, (0.05) e^{i (π)} 〉} . \end{matrix}$

Definition 5.18. Let k be a positive integer and let ${\tilde{A}}_{1}, \dots, {\tilde{A}}_{n}$ be complex multi-fuzzy sets in X₁, …, X_n of dimension k, respectively. The Cartesian product ${\tilde{A}}_{1} \times \dots \times {\tilde{A}}_{n}$ is a complex multi-fuzzy set defined by $\begin{matrix} {\tilde{A}}_{1} \times . . . \times {\tilde{A}}_{n} \\ = {〈 (x_{1}, . . ., x_{n}), μ_{{\tilde{A}}_{1} \times . . . \times {\tilde{A}}_{n}}^{1} (x_{1}, . . ., x_{n}), \\ . . ., μ_{{\tilde{A}}_{1} \times . . . \times {\tilde{A}}_{n}}^{k} (x_{1}, . . ., x_{n}) 〉 : (x_{1}, . . ., x_{n}) \\ \in X_{1} \times . . . \times . X_{n}} \end{matrix}$ where $\begin{matrix} μ_{{\tilde{A}}_{1} \times \dots \times {\tilde{A}}_{n}}^{j} (x_{1}, \dots, x_{n}) \\ = \min [μ_{{\tilde{A}}_{1}}^{j} (x_{1}), . . ., μ_{{\tilde{A}}_{n}}^{j} (x_{n}) . \\ e^{i min [\arg_{{\tilde{A}}_{1}}^{j} (x_{1}), \dots, \arg_{{\tilde{A}}_{n}}^{j} (x_{n})]}], \forall j = 1, . . . ., k . \end{matrix}$

6 Distance measure and δ-equalities of complex multi-fuzz sets

In this section we give the structure of distance measure on complex multi-fuzzy sets by extending the structure of distance measure of complex fuzzy sets [18].

We will first introduce the axiom definition of distance measure between complex multi-fuzzy sets and give an illustrative example.

Definition 6.1. The distance between complex multi-fuzzy sets is a function $\hat{d} : C M^{k} F S (X) \times {CM}^{k} FS (X) \to [0, 1]$ for any $\tilde{A}, \tilde{B}, \tilde{C} \in C M^{k} F S (X)$ , satisfying the following properties:

$0 \leq \hat{d} (\tilde{A}, \tilde{B}) \leq 1,$

$\hat{d} (\tilde{A}, \tilde{B}) = 0$ if and only if $\tilde{A} = \tilde{B},$

$\hat{d} (\tilde{A}, \tilde{B}) = \hat{d} (\tilde{B}, \tilde{A}),$

$\hat{d} (\tilde{A}, \tilde{B}) \leq \hat{d} (\tilde{A}, \tilde{C}) + \hat{d} (\tilde{C}, \tilde{B}) .$

We define $\begin{matrix} \hat{d} (\tilde{A}, \tilde{B}) \\ = max (\begin{matrix} max (\begin{matrix} sup_{x \in X} | r_{{\tilde{A}}_{1}} (x) - r_{{\tilde{B}}_{1}} (x) |, \dots, \\ sup_{x \in X} | r_{{\tilde{A}}_{k}} (x) - r_{{\tilde{B}}_{k}} (x) | \end{matrix}) \\ max (\begin{matrix} \frac{1}{2 π} sup_{x \in X} | \arg_{{\tilde{A}}_{1}} (x) - \arg_{{\tilde{B}}_{1}} (x) | \\ , \dots, \\ \frac{1}{2 π} sup_{x \in X} | \arg_{{\tilde{A}}_{k}} (x) - \arg_{{\tilde{B}}_{k}} (x) | \end{matrix}) \end{matrix}) \end{matrix}$

Example 6.2. Let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets in X ={ x₁, x₂ } of dimension four, given by

\begin{array}{l} \tilde{A} \\ = {〈 x_{1}, (0.2) . e^{i (π)}, (0.3) . e^{i (0.4 π)}, (0.0) . e^{i (0.1 π)}, \\ (0.3) . e^{i (2 π)} 〉, 〈 x_{2}, (0.2) . e^{i (2 π)}, (0.1) . e^{i (0.5 π)}, \\ (0.3) . e^{i (0.2 π)}, (0.4) . e^{i (0.2 π)} 〉}, \\ \tilde{B} \\ = {〈 x_{1}, (0.6) . e^{i (0.1 π)}, (0.0) . e^{i (π)}, (0.3) . e^{i (1.1 π)}, \\ (0.1) . e^{i (0.7 π)} 〉, 〈 x_{2}, (0.5) . e^{i (π)}, (0.3) . e^{i (0.8 π)}, \\ (0.1) . e^{i (0.6 π)}, (0.2) . e^{i (2 π)} 〉} . \end{array}

we see

\begin{matrix} \sup_{x \in X} | r_{{\tilde{A}}_{1}} (x) - r_{{\tilde{B}}_{1}} (x) | \\ = \sup (| - 0.4 |, | - 0.3 |) = 0.4, \\ \sup_{x \in X} | r_{{\tilde{A}}_{2}} (x) - r_{{\tilde{B}}_{2}} (x) | = 0.3, \\ \sup_{x \in X} | r_{{\tilde{A}}_{3}} (x) - r_{{\tilde{B}}_{3}} (x) | = 0.3, \\ \sup_{x \in X} | r_{{\tilde{A}}_{4}} (x) - r_{{\tilde{B}}_{4}} (x) | = 0.2, \end{matrix}

and

\begin{matrix} \frac{1}{2 π} \sup_{x \in X} | \arg_{{\tilde{A}}_{1}} (x) - \arg_{{\tilde{B}}_{1}} (x) | \\ = \frac{1}{2 π} (| 0.9 π |, | π |) = \frac{1}{2 π} (π) = 0.5, \\ \frac{1}{2 π} \sup_{x \in X} | \arg_{{\tilde{A}}_{2}} (x) - \arg_{{\tilde{B}}_{2}} (x) | = 0.5, \\ \frac{1}{2 π} \sup_{x \in X} | \arg_{{\tilde{A}}_{3}} (x) - \arg_{{\tilde{B}}_{3}} (x) | = 1.8, \\ \frac{1}{2 π} \sup_{x \in X} | \arg_{{\tilde{A}}_{4}} (x) - \arg_{{\tilde{B}}_{4}} (x) | = 0.9, \end{matrix}

Therefore

\begin{matrix} \hat{d} (\tilde{A}, \tilde{B}) \\ = \max [\max (0.4, 0.3, 0.3, 0.2), \\ \max (0.5, 0.3, 0.5, 0.9)] \\ = \max [0.4.0.9] = 0.9 . \end{matrix}

Now, we propose the definition of δ-equalities of complex multi-fuzzy sets.

Definition 6.3. Let k be a positive integer and let $\tilde{A}$ and $\tilde{B}$ be two complex multi-fuzzy sets in a universe of discourse X of dimension k. Then $\tilde{A}$ and $\tilde{B}$ are said to be δ-equal, denoted by $\tilde{A} = (δ) \tilde{B}$ , if and only if $\hat{d} (\tilde{A}, \tilde{B}) \leq 1 - δ,$ where 0 ≤ δ ≤ 1 .

Given the definition above, we can easily obtain the following proposition.

Proposition 6.4. Let $\tilde{A}$ , $\tilde{B}$ and $\tilde{C}$ be any three complex multi-fuzzy sets in X of dimension k. Then

$\tilde{A} = (0) \tilde{B},$

$\tilde{A} = (1) \tilde{B}$ if and only if $\tilde{A} = \tilde{B},$

if $\tilde{A} = (δ) \tilde{B}$ if and only if $\tilde{B} = (δ) \tilde{A},$

$\tilde{A} = (δ_{1}) \tilde{B}$ and δ₂ ≤ δ₁, then $\tilde{A} = (δ_{2}) \tilde{B},$

if for all α ∈ I, $\tilde{A} = (δ_{α}) \tilde{B},$ where I is an index set, then $\tilde{A} = (\sup_{α \in I} δ_{α}) \tilde{B},$

for all $\tilde{A}$ and $\tilde{B}$ , there exists a unique δ such that $\tilde{A} = (δ) \tilde{B}$ and if $\tilde{A} = (δ^{'}) \tilde{B}$ then δ′ ≤ δ .

Proof. Properties 1–4 can be easily proven using Definitions 6.1 and 6.3. Here we only prove Properties 5 and 6.

5. Since for all α ∈ I, $\tilde{A} = (δ_{α})$ , we have $\begin{matrix} \hat{d} (\tilde{A}, \tilde{B}) \\ = max (\begin{matrix} max (\begin{matrix} sup_{x \in X} | r_{{\tilde{A}}_{1}} (x) - r_{{\tilde{B}}_{1}} (x) |, \dots, \\ sup_{x \in X} | r_{{\tilde{A}}_{k}} (x) - r_{{\tilde{B}}_{k}} (x) | \end{matrix}) \\ max (\begin{matrix} \frac{1}{2 π} sup_{x \in X} | \arg_{{\tilde{A}}_{1}} (x) - \arg_{{\tilde{B}}_{1}} (x) | \\ , \dots, \\ \frac{1}{2 π} sup_{x \in X} | \arg_{{\tilde{A}}_{k}} (x) - \arg_{{\tilde{B}}_{k}} (x) | \end{matrix}) \end{matrix}) \\ \leq 1 - δ_{α} . \end{matrix}$

Therefore $\sup_{x \in X} | r_{{\tilde{A}}_{j}} (x) - r_{{\tilde{B}}_{j}} (x) | \leq 1 - \sup_{α \in I} δ_{α},$ ∀j = 1, 2, …, k. and $\frac{1}{2 π} \sup_{x \in X} | \arg_{{\tilde{A}}_{j}} (x) - \arg_{{\tilde{B}}_{j}} (x) | \leq 1 - \sup_{α \in I} δ_{α},$ ∀j = 1, 2, …, k .

Thus

\begin{matrix} \hat{d} (\tilde{A}, \tilde{B}) \\ = max (\begin{matrix} max (\begin{matrix} sup_{x \in X} | r_{{\tilde{A}}_{1}} (x) - r_{{\tilde{B}}_{1}} (x) |, \dots, \\ sup_{x \in X} | r_{{\tilde{A}}_{k}} (x) - r_{{\tilde{B}}_{k}} (x) | \end{matrix}) \\ max (\begin{matrix} \frac{1}{2 π} sup_{x \in X} | \arg_{{\tilde{A}}_{1}} (x) - \arg_{{\tilde{B}}_{1}} (x) | \\ , \dots, \\ \frac{1}{2 π} sup_{x \in X} | \arg_{{\tilde{A}}_{k}} (x) - \arg_{{\tilde{B}}_{k}} (x) | \end{matrix}) \end{matrix}) \\ \leq 1 - \sup_{α \in I} δ_{α} . \end{matrix}

Hence $\tilde{A} = (\sup_{α \in I} δ_{α}) \tilde{B} .$

6. Let $δ = 1 - \hat{d} (\tilde{A}, \tilde{B})$ . Then $\tilde{A} = (δ) \tilde{B} .$ If $\tilde{A} = (δ^{'}) \tilde{B}$ , we have $1 - δ = \hat{d} (\tilde{A}, \tilde{B}) \leq 1 - δ^{'}$ . Consequently δ′ ≤ δ . Now assume there exist two constants δ₁ and δ₂ which simultaneously satisfy the required properties, then δ₁ ≤ δ₂ and δ₂ ≤ δ₁. This implies δ₁ = δ₂. Hence the desired δ is unique.

Now, we give a theorem of the complement of δ-equalities of complex multi-fuzzy sets.

Theorem 6.5. If $\tilde{A} = (δ) \tilde{B},$ then ${\tilde{A}}^{c} = (δ) {\tilde{B}}^{c},$ where ${\tilde{A}}^{c}$ and ${\tilde{B}}^{c}$ are the complement of the complex multi-fuzzy sets $\tilde{A}$ and $\tilde{B}$ .

Proof. Since

\begin{matrix} \hat{d} ({\tilde{A}}^{c}, {\tilde{B}}^{c}) & = & \max (\begin{matrix} \max (\begin{matrix} \sup_{x \in X} | r_{{\tilde{A}}^{c}_{1}} (x) - r_{{\tilde{B}}^{c}_{1}} (x) |, \dots, \sup_{x \in X} | r_{{\tilde{A}}^{c}_{k}} (x) - r_{{\tilde{B}}^{c}_{k}} (x) | \end{matrix}), \\ \max (\begin{matrix} \frac{1}{2 π} \sup_{x \in X} | \arg_{{\tilde{A}}^{c}_{1}} (x) - \arg_{{\tilde{B}}^{c}_{1}} (x) |, \dots, \\ \underset{x \in X}{\frac{1}{2 π} \sup} | \arg_{{\tilde{A}}^{c}_{k}} (x) - \arg_{{\tilde{B}}^{c}_{k}} (x) | \end{matrix}) \end{matrix}) \\ = & \max (\begin{matrix} \max (\begin{matrix} \sup_{x \in X} | (1 - r_{{\tilde{A}}_{1}} (x)) - ({1 - r}_{{\tilde{B}}_{1}} (x)) |, \dots, \\ \sup_{x \in X} | (1 - r_{{\tilde{A}}_{k}} (x)) - (1 - r_{{\tilde{B}}_{k}} (x)) | \end{matrix}), \\ \max (\begin{matrix} \frac{1}{2 π} \sup_{x \in X} | (2 π - \arg_{{\tilde{A}}_{1}} (x)) - (2 π - \arg_{{\tilde{B}}_{1}} (x)) |, \dots, \\ \underset{x \in X}{\frac{1}{2 π} \sup} | (2 π - \arg_{{\tilde{A}}_{k}} (x)) - {(2 π - \arg)}_{{\tilde{B}}_{k}} (x) | \end{matrix}) \end{matrix}) \\ = & \max (\begin{matrix} \max (\sup_{x \in X} | r_{{\tilde{A}}_{1}} (x) - r_{{\tilde{B}}_{1}} (x) |, \dots, \sup_{x \in X} | r_{{\tilde{A}}_{k}} (x) - r_{{\tilde{B}}_{k}} (x) |), \\ \max (\begin{matrix} \frac{1}{2 π} \sup_{x \in X} | \arg_{{\tilde{A}}_{1}} (x) - \arg_{{\tilde{B}}_{1}} (x) |, \dots, \\ \underset{x \in X}{\frac{1}{2 π} \sup} | \arg_{{\tilde{A}}_{k}} (x) - \arg_{{\tilde{B}}_{k}} (x) | \end{matrix}) \end{matrix}) \\ = & \hat{d} (\tilde{A}, \tilde{B}) \leq 1 - δ, \end{matrix}

Hence $\hat{d} ({\tilde{A}}^{c}, {\tilde{B}}^{c}) \leq 1 - δ,$ thus ${\tilde{A}}^{c} = (δ) {\tilde{B}}^{c} .$

7 Conclusion

Complex multi-fuzzy set theory is an extension of complex fuzzy set and complex Atanassov intuitionistic fuzzy set theory. In this paper, we have introduced the novel concept of complex multi-fuzzy set and studied the basic theoretic operations of this new concept which are complement, union and intersection on complex multi-fuzzy sets. We also presented some properties of these basic theoretic operations and other relevant laws pertaining to the concept of complex multi-fuzzy sets. Finally, we present the distance measure between two complex multi-fuzzy sets. This distance measure is used to define δ-equalities of complex multi-fuzzy sets.

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