OPERATIONS ON COMPLEX MULTI-FUZZY SETS

Transcription

1 OPERATIONS ON COMPLEX MULTI-FUZZY SETS Yousef Al-Qudah and Nasruddin Hassan School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, UKM Bangi Selangor DE, MALAYSIA 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,] 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. Introduction The concept of fuzzy set FS was used for the first time by Zadeh [] to handle uncertainty in many fields of everyday life. Fuzzy set theory generalised the range values of classical set theory from the integer 0 and to the interval [0, ]. In addition, this concept has proven to be very useful in many different fields [2-4]. The theory of multi-fuzzy sets [5,6] is a generalisation of Zadeh s fuzzy sets [] and Atanassov s intuitionistic fuzzy sets [7] in terms of multi-dimensional membership functions and multi-level fuzziness. Ramot et al. [8] 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,] 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 s xe iω sx i=, where r s x and ω s x are both real-valued functions and r s xe iω sx the range are in a complex unit circle. They also added an additional term Corresponding author. Nasruddin Hassan, School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia UKM Bangi Selangor DE, Malaysia. Tel.: ; nas@ukm.edu.my

2 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. [8,9] 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 [0] 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 [] 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 [2], with an attempt to show that precise membership values should normally be of no practical significance. Hong and Hwang [3] then proposed an important generalization. Moreover, Cai [4,5] 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.[6] introduced the concept of δ-equalities of complex fuzzy set and applied it to signal processing. We will extend the discussion on multi-fuzzy sets further by proposing a new concept of complex multi-fuzzy sets CM k FSs by adding a second dimension phase term to multi-membership function of multi-fuzzy sets. Complete characterization of many real life problems that have a periodic nature can then be handled by complex multifuzzy membership functions of the objects involved in these periodic problems. Then, we define its basic operations, namely complement, union, intersection, complex multifuzzy product, Cartesian product, distance measure and study their properties. Finally, we introduce the δ-equalities of complex multi-fuzzy sets. 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 965 [] is as follows. DEFINITION 2.. see [] A fuzzy set S in a universe of discourse X is characterised by a membership function µ S x that takes values in the interval [0,] for all x X. Sebastian and Ramakrishnan [6] introduced the following definition of multi-fuzzy sets.

3 DEFINITION 2.2. Let k be a positive integer and X be a non-empty set. A multifuzzy set S in X is a set of ordered sequences S = { x, µ x,..., µ k x : x X}, where µ i : X L i = [0,], i =,2,...,k. The function µ S x = µ x,..., µ k x is called the multi-membership function of multi-fuzzy sets S, k is called a dimension of S. The set of all multi-fuzzy sets of dimension k in X is denoted by M k FSx. In the following, we give some basic definitions and set theoretic operations of complex fuzzy sets. DEFINITION 2.3. see [8] A complex fuzzy set CFS S, defined on a universe of discourse X, is characterised by a membership function µ s x that assigns to any element x X a complex-valued grade of membership in S. By definition, the values of µ s x may receive all lying within the unit circle in the complex plane and are thus of the form µ s x = r s x.e iarg sx, where i =, r s x is a real-valued function such that r s x [0,] and e iarg sx is a periodic function whose periodic law and principal period are, respectively, 2π and 0 <arg s x 2π, i.e., Arg s x = arg s x + 2kπ, k = 0,±,±2,..., where arg s x is the principal argument. The principal argument arg S x will be used in the following text. The CFS S may be represented as the set of ordered pairs S = {x, µ s x : x X} = {x,r s x.e iarg sx : x X }. DEFINITION 2.4. see [8] Let à and B be two complex fuzzy sets on X and, let µã x = rã x.e iarg à x and µ B x = r B x.eiarg B x be their membership functions, respectively. athe complex fuzzy complement of Ã, denoted by à c, is specified by a function à c = {x,rãc x.e i arg à c x : x X } = {x, rãx.e i2π arg à cx : x X }, b The complex fuzzy union of à and B, denoted by à B, is specified by a function µã B x = r à B x.ei arg à B x = maxrãx,r B x.ei maxarg à x,arg B x, c The complex fuzzy intersection of à and B, denoted by à B, is specified by a function µã B x = r à B x.ei arg à B x = minrãx,r B x.ei minarg à x,arg B x, d We say that à is greater than B, denoted by à B or B Ã, if for any x X, rãx r B x and arg à x arg B x. DEFINITION 2.5. see [6] Let à and B be two complex fuzzy sets on X and, let µã x = rã x.e iarg à x and µ B x = r B x.eiarg B x be their membership functions, respectively. The complex fuzzy product of à and B, denoted à B, is specified by a function

4 µã B = r à B x.ei arg à B x = rãx r B x.ei2π argãx 2π arg B x 2π. DEFINITION 2.6. see [6] Let à n be N complex fuzzy sets on µãx = rãx.e iarg à x their membership functions, respectively. The complex fuzzy Cartesian product of à n, n =,...,N, denoted Ã... à N, is specified by a function µã... à N = rã... à N x.e iarg Ã... à N x = min rã x,...,rãn x N.e i min argã x,...,argãn x N, Where x = x,...,x N X... X. }{{} N DEFINITION 2.7. see [6] Let à and B be two complex fuzzy sets on X and, let µã x = rã x.e iarg à x and µ B x = r B x.eiarg B x be their membership functions, respectively. Then à and B are said to be δ-equal, denoted à = δ B, if and only if dã, B δ, where 0 δ. 3. Complex multi-fuzzy sets In this section, we introduce the definition of complex multi-fuzzy sets which are a generalisation of multi-fuzzy sets [5], by extending the range of multi-membership functions from the interval [0,] to the unit circle in the complex plane. We begin by proposing the definition of complex multi-fuzzy sets based on multifuzzy sets and complex fuzzy sets, and give an illustrative example of it. DEFINITION 3.. see [7] 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 multifuzzy set CMFS S, defined on a universe of discourse X, is characterised by a multi-membership function µ s x = µ s j x, that assigns to any element x X j N a complex-valued grade of membership in S. By definition, the values of µ s x may receive all lying within the unit circle in the complex plane, and are thus of the form µ s x = r s j x.e iarg s j x j N, i =, r s j x are real-valued function j N such that r s j x j N [0,] and eiarg s j x j N are periodic function whose periodic law and principal period are, respectively, 2π and 0 < arg s j x 2π, i.e., j N Arg s j x j N = arg s j x + 2kπ, K = 0,±,...,where arg s j x is the principal argument. The principal argument arg s j x will be used on the following text. The j N CMFS S may be represented as the set of ordered sequence S = {x, µ s j x = a j j N : x X } = {x, r s j x.e iarg s j x j N : x X }. where µ s j : X L j = {a j : a j C, a j } for all j.

5 The function µ S x = r s j x.e iarg s j x j N is called the complex multi-membership function of complex multi-fuzzy set S. If N = k, then k is called the dimension of S. The set of all complex multi-fuzzy sets of dimension k in X is denoted by CM k FSX. In this paper L j = {a j : a j C, a j } and we will study some properties of complex multi-fuzzy sets of dimension k. Remark 3.2. i. A complex multi-fuzzy set is a generalization of multi-fuzzy set. Assume the multi-fuzzy set S is characterized by the real-valued multi- membership function γ j x j N. Transforming S into complex multi-fuzzy sets is achieved by setting the amplitude terms r s j x j N to be equal to γ jx j N and the phase terms arg s j x to equal zero for all x. In other words, without a phase term, the j N complex multi-fuzzy sets effectively reduces to conventional multi-fuzzy set.this interpretation is supported by the fact that the range of r s j x is [0,], as j N for real-valued grade of multi- membership function. ii. A complex multi-fuzzy set of dimension one is reduced to a complex fuzzy set, while a complex multi-fuzzy set of dimension two with µ s x + µ s2 x is reduced to a complex Atanassov s intuitionistic fuzzy set. iii. If k j= µ s j x, where µ s j x = r s j x.e iarg 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 k j= µ s j x l >, for some x X, we redefine the complex multimembership degree µ s x,..., µ sk x as l µ s x,..., µ sk x. Hence the non-normalized complex multi-fuzzy set can be changed into a normalized multi-fuzzy set. Example 3.3. Let X ={x,x 2,x 3 } be a universe of discourse. Then, S is a complex multi-fuzzy set in Xof dimension three, as given below: S = { x,0..e iπ,0.3e i0.4π,0.2.e i.5π, x 2,0.0.e i0.2π,0.3.e i0.7π, 0.2.e i0.5π, x 3, 0.3.e i0.8π,0.2.e i0.π,0.5.e i.2π }. Now, we present the concept of the subset and equality operations on two complex multifuzzy sets in the following definition. DEFINITION 3.4. Let k be a positive integer and let à and B be two complex multifuzzy sets in a universe of discourse X of dimension k, given as follows: Then à = { x,rã x.e iarg à x,...,rãk x.e iarg à k x : x X }, B = { x,r B x.e iarg B x,...,r B k x.e iarg B k x : x X },. à B if and only if rã j x r B j x and argã j x arg B j x, for all x X and j =,2,...,k. 2. à = B if and only if rã j x = r B j x and argã j x = arg B j x for all x X and j =,2,...,k.

6 4. Basic operations on complex multi-fuzzy sets In this section, we introduce some basic theoretic operations on complex multi-fuzzy sets, which are complement, union and intersection, derive their properties and give some examples. DEFINITION 4. Let k be a positive integer and let à be a complex multi-fuzzy set in a universe of discourse X of dimension k, given by à = { x,rã x.e iarg à x,...,rãk x.e iarg à k x : x X}, The complement of à is denoted by à c, and is defined by à c = { x,rãc x.e iarg à c x,...,rãc k x.e iarg à c k x : x X } = { x, rã x.e i 2π arg à x,..., rãk x.e i 2π arg à k x : x X }. Example 4.2. Consider Example 3.3.By using the basic complex multi-fuzzy complement, we have à c = { x, 0..e i2π π, 0.3.e i2π 0.4π, 0.2.e i2π.5π, x 2, 0.0.e i2π 0.2π, 0.3.e i2π 0.7π, 0.2.e i2π 0.5π, x 3, 0.3.e i2π 0.8π, 0.2.e i2π 0.π, 0.5.e i2π.2π }, = { x, 0.9.e iπ, 0.7.e i.6π,0.8.e i0.5π, x 2,.0.e i.8π,0.7.e i.3π,0.8.e i.5π, x 3, 0.7.e i.2π,0.8.e i.9π, 0.5.e i0.8π }. Proposition 4.3. Let à be a complex multi-fuzzy set in a universe of discourse X of dimension k. Then à c c = Ã. Proof. By Definition 4., we have Ãc c = { x,r A c c x.eiarg A c c x,...,r A c k c x.eiarg A c k c x : x X } = { x, rãc x.e i 2π arg à c x,..., rãc k x.e i 2π arg à c k x : x X }. = { x, [ rã x].e i [2π 2π arg à x ],..., [ rãk x]. e i [2π 2π arg à k x ] : x X } = { x,rã x.e iarg à x,..., rãk x.e iarg à k x : x X } = Ã. Thus à c c = Ã. In the following, we introduce the definition of the union of two complex multifuzzy sets with an illustrative example.

7 DEFINITION 4.4 Let k be a positive integer and let à and B be two complex multifuzzy sets in a universe of discourse X of dimension k, given as follows: à = { x,rã x.e iarg à x,...,rãk x.e iarg à k x : x X }, B = { x,r B x.e iarg B x,...,r B k x.e iarg B k x : x X }, The union of à and B, denoted by à B, is defined as à B = { x,rã B 2 x.e iarg à B x,...,rãk B k x.e i arg à k B k x : x X } = { x, rã x,r B x.e iarg à B x,..., rãk x,r B k x.e iarg à k B k x : x X }. where denote the max operator. The following definition of complex multi-fuzzy union is required to calculate the phase term e iarg à j B j x for j =,...,k. DEFINITION 4.5 Let k be a positive integer and let à and B be two complex multifuzzy sets in a universe of discourse X of dimension k, with the complex-valued multi membership functions µã x and µ B x. The complex multi-fuzzy union of à and B, denoted by à B, is specified by the function s : {a,...,a j : a,...,a j C, a,..., a j, j =,...,k } {b,...,b j : b,...,b j C, b,..., b j, j =,...,k } { d,...,d j : d,...,d j C, d,..., d j, j =,...,k }, where a,...,a j,b,...,b j and d,...,d j are the complex multi-membership functions of Ã, B and à B, respectively. s assigns a complex value, for all x X and j =,2,...,k. sa,...,a j,...,b,...,b j = s µ a,b,..., s µ a j,b j = µã B x,..., µã j B j x = d,...,d j The complex multi-fuzzy union functions s must satisfy at least the following axiomatic requirements, for any a,...,a j,b,...,b j,c,...,c j and d,...,d j {x : x C, x }.. Axiom : s µ a j,0 = a j, j =,2,...,k boundary condition. 2. Axiom 2: s µ a j,b j = s µ b j,a j, j =,2,...,k commutative condition. 3. Axiom 3: if b j d j then s µ a j,b j s µ a j,,d j, j =,2,...,k monotonic condition.

8 4. Axiom 4: s µ a j, s µ b j,c j = s µ sµ a j,b j,c j, j =,2,...,k, associative condition. In some cases, it may be desirable that s also satisfy the following requirements: 5. Axiom 5: s is a continuous function continuity. 6. Axiom 6: sµ a j,a j a j, j =,2,...,k superidempotency. 7. Axiom 7: if : if a j c j and b j d j then sµ a j,b j sµ c j,d j, j =,2,...,k strict monotonicity. The phase term for complex multi-membership functions belongs to 0, 2π]. To define the phase terms argã j B j x for j =,...,k., we take those forms which Ramot et al.[8] presented to calculate argã j B j x as follows:. Sum: argã j B j x = argã j x + arg B j x, for all j =,2,...,k. 2. Max: argã j B j x = maxargã j x,arg B j x, for all j =,2,...,k. 3. Min: argã j B j x = minargã j x,arg B j x, for all j =,2,...,k. 4. Winner Takes All : argã j B j x = { argã j x rã j > r B j arg B j x rã j < r B j, for all j =,2,...,k. Example 4.6. Let à and B be two complex multi-fuzz sets in X = {x,x 2 } of dimension four, given by à = { x, 0.4.e iπ,0.3.e i0.4π,0.2.e i.5π,0..e i0.π, x 2, 0..e iπ, 0.2.e i0.7π,0.2.e i0.5π,0.5.e iπ }, B = { x, 0.3.e i0.π,0.2.e iπ,0..e i.π,0.3.e i0.2π, x 2, 0.5.e iπ, 0.3.e i0.2π,0.0.e i0.8π,0..e i2π }. Using the max union function for calculating rãk B k x for k =,...,4. and the Winner Takes All for determining argãk B k x for k =,...,4., the following results are obtained for à B : à B = { x, 0.4.e iπ,0.3.e i0.4π,0.2.e i.5π,0.3.e i0.2π, x 2, 0.5.e iπ, 0.3.e i0.2π,0.2.e i0.5π,0.5.e iπ }. We introduce the definition of the intersection of two complex multi-fuzzy sets with an illustrative example as follows. DEFINITION 4.7 Let k be a positive integer and let à and B be two complex multifuzzy sets in a universe of discourse X of dimension k, given as follows: à = { x,rã x.e iarg à x,...,rãk x.e iarg à k x : x X }, B = { x,r B x.e iarg B x,...,r B k x.e iarg B k x : x X }, The intersection of à and B, denoted by à B, is defined as

9 à B = { x,rã B 2 x.e iarg à B x,...,rãk B k x.e i arg à k B k x : x X } = { x, rã x,r B x.e iarg à B x,..., rãk x,r B k x.e iarg à k B k x : x X }. where denote the max operator. The following definition of complex multi-fuzzy intersection is required to calculate the phase term e iarg à j B j x for j =,...,k. DEFINITION 4.8 Let k be a positive integer and let à and B be two complex multifuzzy sets in a universe of discourse X of dimension k, with the complex-valued multi membership functions µã x and µ B x. The complex multi-fuzzy intersection of à and B, denoted by à B, is specified by the function t : {a,...,a j : a,...,a j C, a,..., a j, j =,...,k } {b,...,b j : b,...,b j C, b,..., b j, j =,...,k } {d,...,d j : d,...,d j C, d,..., d j, j =,...,k }, where a,...,a j,b,...,b j and d,...,d j are the complex multi-membership functions of Ã, B and à B, respectively. t assigns a complex value, for all x X and j =,2,...,k. t a,...,a j,...,b,...,b j = t µ a,b,..., t µ a j,b j = µã B x,..., µã j B j x = d,...,d j The complex multi-fuzzy intersection functions t must satisfy at least the following axiomatic requirements, for any a,...,a j,b,...,b j,c,...,c j and d,...,d j {x : x C, x }.. Axiom : if b j =, t µ a j,b j = a j, j =,2,...,k boundary condition. 2. Axiom 2: t µ a j, b j = t µ b j, a j, j =,2,...,k commutative condition. 3. Axiom 3: if b j d j then t µ a j,b j t µ a j,d j, j =,2,...,k monotonic condition. 4. Axiom 4: t µ a j, t µ b j,c j = t µ tµ a j,b j,c j, fo j =,2,...,k, associative condition. In some cases, it may be desirable that t also satisfy the following requirements: 5. Axiom 5: t is a continuous function continuity. 6. Axiom 6: t µ a j,a j a j, j =,2,...,k superidempotency. 7. Axiom 7: if a j c j and b j d j then t µ a j,b j t µ c j,d j, for all j =,2,...,k strict monotonicity. We consider some forms to calculate the phase term argã j B j xfor j =,...,k., such that the same possible choices are given to calculate the argã j B j x.

10 Example 4.9. Consider Example 4.6. By using the basic complex multi-fuzzy intersection, we have à B = { x, 0.3.e i0.π,0.2.e i0.4π,0.e i.π,0..e i0.π, x 2, 0.. e iπ, 0.3.e i0.2π,0.0.e i0.5π,0..e iπ }. We will now give some propositions on the union, intersection,complement of complex multi fuzzy sets. These propositions illustrate the relationship between the set theoretic operations that have been mentioned above. Proposition 4.0 Let Ã, B and C be any three complex multi-fuzzy sets on X of dimension k. Then the following properties hold.. à à = Ã, A à = Ã. 2. à B = B Ã, à B = B Ã. 3. à B C = à B C, A B C = à B C. 4. à B C = à B à C, à B C = à B à C. Proof. The proof is straightforward by using Definitions 4.4 and 4.7. Proposition 4. Let A and B be two complex multi-fuzzy sets on X of dimension k. Then the following properties hold.. 2. à B c = à c B c. à B c = à c B c. Proof. Let the union of à = { x,rã x.e iarg à x,...,rãk x.e iarg à k x : x X } and B = { x,r B x.e iarg B x,..., r B k x.e iarg B k x : x X }, be the basic max operator. Then à B = { x,max rã x,r B x.e i maxarg à x,arg B x,...,max rãk x,r B k x.e i maxarg à k x,arg B k x : x X }, à c = { x, rã x.e i 2π arg à x,..., rãk x.e i 2π arg à k x : x X }, B c = { x, r B x.e i 2π arg B x,..., r B k x.e i 2π arg B x k : x X }, à B c = { x, max rã x,r B x.e i2π maxarg à x,arg B x,..., max rãk x,r B k x.e i2π maxarg à x,arg k B x k : x X }, à c B c = { x, min rã x, r B x.e i min 2π arg à x,2π arg B x,..., min rãk x, r B k x.e i min 2π arg à k x,2π arg B k x : x X }. To proof à B c = à c B c we need to show that the amplitude term max rã j x,r B j x = min rã j x, r B j x and that the phase term 2π maxargã j x,arg B j x = min2π argã j x, 2π arg B j x, for j =,...,k. To show the amplitude term, we consider the two possible cases:

11 Case : If rã j x r B j x, then rã j x r B j x. Thus, max rã j x,r B j x = rã j x and min rã j x, r B j x = rã j x. Case 2: If rã j x < r B j x, then rã j x > r B j x. Thus, max rã j x,r B j x = r B j x and min rã j x, r B j x = r B j x. Therefore, by Case and Case 2: max rã j x,r B j x = min rã j x, r B j x, j =,...,k. To show the phase term, we consider the two possible cases: Case : If argã j x arg B j x, then 2π argã j x 2π arg B j x. Thus, 2π max arg A j x,arg B j x = 2π arg A j x and min 2π argã j x, 2π arg B j x = 2π argã j x. Case 2 : If argã j x < arg B j x, then 2π argã j x > 2π arg B j x. Thus, 2π max argã j x,arg B j x = 2π arg B j x and min 2π argã j x, 2π arg B j x = 2π arg B j x. Therefore, by Case and Case 2: 2π max argã j x,arg B j x = min 2π argã j x,2π arg B j x, j =,...,k. 2 The proof is similar to that in part 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 4.2 Let k be a positive integer and let à and B be two multi-fuzzy sets in a universe of discourse X of dimension k, given as follows: à = { x, µã x,..., µãk x : x X } and B = { x, µ B x,..., µ B k x : x X }. The multi-fuzzy product of à and B, denoted by à B, is defined as à B = { x,rã B x,...,rãk B k x : x X } = { x, rã x r B x,..., rãk x r B k x : x X }. DEFINITION 4.3. Let k be a positive integer and let Ã,...,à n be multi-fuzzy sets in X,...,X n of dimension k. The Cartesian product Ã... à n is a multi-fuzzy set defined by Ã... à n = { x,...,x n, µ Ã... à n x,...,x n,..., µ k Ã... à n x,...,x n : x,...,x n X... X n } Where µ j Ã... à n x,...,x n = min [µ j à x,..., µ j à n x n ], j =,2,...,k In the following, we introduce the concept of the product of two complex multifuzzy sets and the Cartesian product of complex multi-fuzzy sets with an illustrative ex-

12 ample of the product operation. DEFINITION 4.4 Let k be a positive integer and let à and B be two complex multifuzzy sets in a universe of discourse X of dimension k, given as follows: à = { x,rã x.e iarg à x,...,rãk x.e iarg à k x : x X }, B = { x,r B x.e iarg B x,...,r B k x.e iarg B k x : x X }, The complex multi-fuzzy product of à and B, denoted by à B, is defined as à B = { x,rã B x.e iarg à B x,...,rãk B k x.e iarg à k B k x : x X }, = { x, rã x r B x.e i2π arg x à 2π argãk x. e i2π 2π arg B x k 2π : x X }. arg B x 2π,..., rãk x r B k x. Example 4.5. product, Consider Example 4.6. By using the basic complex multi-fuzzy à B = { x,0.2.e i0.05π,0.06.e i0.2π,0.02.e i0.83π,0.03.e i0.0π, x 2, 0.05.e i0.π,0.09.e i0.35π,0.0e i0.2π,0.05e iπ }. DEFINITION 4.6. Let k be a positive integer and let Ã,...,à n be complex multifuzzy sets in X,...,X n of dimension k, respectively. The Cartesian product Ã... à n is a complex multi-fuzzy set defined by where Ã... à n = { x,...,x n, µ Ã... à n x,...,x n,..., µ k Ã... à n x,...,x n : x,...,x n X... X n } µ j Ã... à n x,...,x n = min [µ j à x,..., µ j à n x n e imin [arg j à x,...,arg j Ãn x n ] ], j =,2,...,k. 5. 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 [6]. We will first introduce the axiom definition of distance measure between complex multi-fuzzy sets and give an illustrative example. DEFINITION 5.. The distance between complex multi-fuzzy sets is a function d ˆ: CM k FSX CM k FSX [0,] for any Ã, B, C CM k FSX, satisfying the following properties:

13 . 0 d ˆ Ã, B, 2. d ˆ Ã, B = 0 if and only if à = B, 3. d ˆ Ã, B = d ˆ B,Ã, 4. d ˆ Ã, B d ˆ Ã, C + d ˆ C, B. We define d ˆ Ã, B max sup x X rã x r B x,...,sup x X rãk x r B k x, = max max 2π sup x X argã x arg B x,..., 2π sup x X arg à k x arg B k x Example 5.2. Let à and B be two complex multi-fuzzy sets in X = {x,x 2 } of dimension four, given by à = { x, 0.2.e iπ,0.3.e i0.4π,0.0.e i0.π,0.3.e i2π, x 2, 0.2.e i2π, 0..e i0.5π,0.3.e i0.2π,0.4.e i0.2π }, B = { x, 0.6.e i0.π,0.0.e iπ,0.3.e i.π,0..e i0.7π, x 2, 0.5.e iπ, 0.3.e i0.8π,0..e i0.6π,0.2.e i2π }. we see sup x X rã x r B x = 0.4, 0.3 = 0.4, sup x X rã2 x r B 2 x = 0.3, sup x X rã3 x r B 3 x = 0.3, sup x X rã4 x r B 4 x = 0.2 and 2π sup x X argã x arg B x = 2π 0.9π, π = 2π π = 0.5, 2π sup x X argã x arg B x = 0.5, 2π sup x X argã x arg B x =.8, 2π sup x X argã x arg B x = 0.9 Therefore ˆ dã, B = max[max0.4, 0.3, 0.3, 0.2, max0.5, 0.3, 0.5, 0.9] = max[ ] = 0.9. Now, we propose the definition of δ-equalities of complex multi-fuzzy sets. DEFINITION 5.3. Let k be a positive integer and let à and B be two complex multifuzzy sets in a universe of discourse X of dimension k, given as follows: à = { x,rã x.e iarg à x,...,rãk x.e iarg à k x : x X }, B = { x,r B x.e iarg B x,...,r B k x.e iarg B k x : x X }, Then à and B are said to be δ-equal, denoted by à = δ B, if and only if ˆ d Ã, B δ, where 0 δ. Given the definition above, we can easily obtain the following proposition.

14 Proposition 5.4 Let Ã, B and C be any three complex multi-fuzzy sets in X of dimension k. Then. à = 0 B, 2. à = B if and only if à = B, 3. if à = δ B if and only if B = δã, 4. à = δ B and δ 2 δ, then à = δ 2 B, 5. if for all α I, à = δ α B, where I is an index set, then à = sup α I δ α B, 6. for all à and B, there exists a unique δ such that à = δ B and if à = δ B then δ δ, 7. if à = δ B and à = δ 2 B, then à = δ C, where δ = δ δ 2. Proof. Properties - 4 can be easily proven using Definitions 5. and 5.3. Here we only prove properties 5, 6 and Since for all α I, à = δ α, we have Therefore d ˆ Ã, B max sup x X rã x r B x,...,sup x X rãk x r B k x, = max max 2π sup x X argã x arg B x,..., 2π sup x X arg à k x arg B k x δ α. sup x X rã j x r B j x sup α I δ α, j =,2,...,k. and 2π sup x X argã j x arg B j x sup α I δ α, j =,2,...,k. Thus d ˆ Ã, B max sup x X rã x r B x,...,sup x X rãk x r B k x, = max max 2π sup x X argã x arg B x,..., 2π sup x X arg à k x arg B k x sup α I δ α. Hence à = sup α I δ α B. 6. Let δ = d ˆ Ã, B. Then à = δ B. If à = δ B, we have δ = d ˆ Ã, B δ. Consequently δ δ. Now assume there exist two constants δ and δ 2 which simultaneously satisfy the required properties, then δ δ 2 and δ 2 δ. This implies δ = δ 2. Hence the desired δ is unique. 7. Since à = δ B, we have d ˆ Ã, B max sup x X rã x r B x,...,sup x X rãk x r B k x, = max max 2π sup x X argã x arg B x,..., 2π sup x X arg à k x arg B k x δ,

15 which implies sup x X rã j x r B j x δ and 2π sup x X argã j x arg B j x δ, j =,2,...,k. Also we have B = δ C, thus d ˆ B, C max sup x X r B x r C x,...,sup x X r B k x r C k x, = max max 2π sup x X arg B x arg C x,..., 2π sup x X arg B k x arg C k x δ 2, which implies sup x X r B j x r C j x δ 2 and 2π sup x X arg B j x arg C j x δ 2, j =,2,...,k. Now, we have d ˆ Ã, C max sup x X rã x r C x,...,sup x X rãk x r C k x, = max max 2π sup x X argã x arg C x,..., 2π sup x X arg à k x arg C k x supx X rã x r max B x + sup x X r B x r C x,...,, sup x X rãk x r B k x + sup x X r B k x r C k x 2π sup x X argã x arg B max x + 2π max sup, x X arg B x arg C x..., 2π sup argãk x X x arg B k x + 2π sup x X arg B k x arg C k x max max δ + δ 2,..., δ + δ 2, max δ + δ 2,..., δ + δ 2 max δ + δ 2, δ + δ 2 = δ + δ 2 = δ + δ 2 From definition 5., since ˆ d Ã, C. Therefore d Ã, C δ δ 2 = δ, which implies à = δ C. Now, we give a theorem of the complement of δ-equalities of complex multi-fuzzy sets. THEOREM 5.6. If à = δ B, then à c = δ B c, where à c and B c are the complement of the complex multi-fuzzy sets à and B. Proof. Since max supx X d ˆ rãc à c, B c x r B c x,...,sup x X rãc x r k B cx, k = max max 2π sup x X argãc x arg B c x,..., 2π sup x X arg à cx arg k B cx k

16 max = max max supx X rã x r B x,...,, sup x X rãk x r B k x 2π sup x X 2π argã x 2π arg B x,..., 2π sup x X 2π arg à k x 2π arg B k x max sup x X rã x r B x,..., sup x X rãk x r B k x, = max max 2π sup x X argã x arg B x,..., 2π sup x X arg à k x arg B k x = ˆ d Ã, B δ Hence ˆ d à c, B c δ, thus à c = δ B c. 6. 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. References [] L.A. Zadeh, Fuzzy set, Information and Control , [2] F.J. Cabrerizo, F. Chiclana, R. Al-Hmouz, A. Morfeq, A.S. Balamash and E. Herrera-Viedma, Fuzzy decision making and consensus: Challenges, Journal of Intelligent and Fuzzy Systems , [3] T. Sarkodie-Gyan, H. Yu, M. Alaqtash, M.A. Bogale, J. Moody and R. Brower, Application of fuzzy sets for assisting the physician s model of functional impairments in human locomotion, Journal of Intelligent and Fuzzy Systems , [4] L.W. Lee and S.M. Chen, Fuzzy decision making and fuzzy group decision making based on likelihoodbased comparison relations of hesitant fuzzy linguistic term sets, Journal of Intelligent and Fuzzy Systems , [5] S. Sebastian and T.V. Ramakrishnan, Multi-fuzzy sets, International Mathematical Forum , [6] S.Sebastian and T.V. Ramakrishnan, Multi-fuzzy sets: An extension of fuzzy sets, Fuzzy Information and Engineering 3 20, [7] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems , [8] D. Ramot, R. Milo, M. Friedman and A. Kandel, Complex fuzzy sets, IEEE Transactions on Fuzzy Systems , [9] D. Ramot, M. Friedman, G. Langholz, and A. Kandel, Complex fuzzy logic, IEEE Transactions on Fuzzy Systems ,

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