SOME OPERATIONS ON TYPE-2 INTUITIONISTIC FUZZY SETS

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1 Journal of Computer Science and Cybernetics, V.28, N.3 (2012), SOME OPEATIONS ON TYPE-2 INTUITIONISTIC FUZZY SETS BUI CONG CUONG 1, TONG HOANG ANH, BUI DUONG HAI 1 Institute of Information Technology, Vietnam Academy of Science and Technology Tóm tắt. Trong nhúng thªp ni n g n y, mët sè mð rëng cõa kh i ni»m tªp mí ñc xu t. Tªp mí lo i hai v tªp mí trüc c m l hai kh i ni»m mîi thu hót ñc nhi u sü quan t m cõa c c nh nghi n cùu v¼ sü phong phó cõa c c ùng döng. B i b o giîi thi»u mët kh i ni»m mîi - tªp mí trüc c m lo i hai v chùng minh mët sè t½nh ch t tr n â. Abstract. In the last decades, there have been some extensions of fuzzy sets and their applications. ecently, type-2 fuzzy set and intuitionistic fuzzy set are two of them, drawing a great deal of scientist's attention because of their widespread range of applications. In this paper, we introduce a new concept - type-2 intuitionistic fuzzy set and propose some properties of their operations. 1. INTODUCTION Type-2 fuzzy sets - that is, fuzzy sets with fuzzy sets as truth values - seem destined to play an increasingly important role in applications. They were introduced by Zadeh [13], extending the notion of ordinary fuzzy sets. The Mendel's book [5] on Uncertain ule-based Fuzzy ogic Systems and other researches [6, 7, 8, 11] are discussions of both theoretical and practical aspects of type-2 fuzzy sets. In [1] Atanassov K. introduced the concept of intuitionistic fuzzy set characterized by a membership function and a non-membership function, which is a generalization of fuzzy set. In [1] Atanassov K. also defined some operators of IFSs. ecently, intuitionistic fuzzy set (IFS) theory have been applied to many different fields, such as decision making, medical diagnosis, pattern recognition. In this paper, we introduce concept of type-2 intuitionistic fuzzy sets. We define some basic operations and derive some their properties. The paper is organized as follow: Section 2 gives a briefly review some basic definitions of fuzzy sets, type-2 fuzzy sets and intuitionistic fuzzy sets. Section 3 is devoted to the new main definitions and some their properties. Section 4 is a first discussion of a subclass of the type-2 IFS Definition of fuzzy sets 2. BASIC DEFINITION et T, S be nonempty sets. The Map(S, T ) be the set of all function from S into T. A This paper was supported by NAFOSTED grant

2 SOME OPEATIONS ON TYPE-2 INTUITIONISTIC FUZZY SETS 275 fuzzy set A of a set S is a maping A : S [0, 1]. The set S has no operations on it. So operations on the set Map(S, [0, 1]) of all fuzzy subsets of S come from operations on [0, 1]. Common operations on [0, 1] of interest in fuzzy theory are,, and given by x y = min{x, y}, x y = max{x, y}, x = 1 x The constant 0 and 1 are generally considered as part of the algebraic structure. So the algebra basic to fuzzy set theory is [0, 1],,,, 0, 1. The corresponding operations on the set Map(S, [0, 1]) of all fuzzy subsets of S are given pointwise by the following formulas (A B)(s) = A(s) B(s), (A B)(s) = A(s) B(s), A (s) = (A(s)) and the two nullary operations are given by 1(s) = 1 and 0(s) = 0 for all s S. We use the same symbols for the pointwise operations on the elements of Map(S, [0, 1]). There are many properties hold in the algebra I = ([0, 1],,,.0.1) (see [9, 11]). Thus, I is a bounded distributive lattice with an involution that satisfies De Morgan's laws and the Kleene inequality. Thus is, I is a Kleene algebra. Thus Map(S, [0, 1]) is also a Kleene algebra. Basic knowledge of fuzzy sets has been presented in [3, 9] Definition of type-2 fuzzy sets et S be a universe of discourse, the a type-2 fuzzy set (T2 FS) is defined as following. Definition [5] A type-2 fuzzy set, denoted by A, is characterized by a type-2 membership function µ A (x, u), where x S and u J x [0.1], i.e. A = {((x, u), µ A (x, u)) x X, u J x [0.1]}, in which 0 µ A (x, u) 1. A can be express as A = µ A (x, u)/(x, u), x X u J x J x [0, 1] 2.3. Basic definition and some properties of IFS et Y be a universe of discourse, then a fuzzy set A = {< y, µ A (y) > y Y } defined by Zadeh [13] is characterized by a membership function µ A : Y [0.1], where µ A (y) denotes the degree of membership of element y to the set A. Definition [1] An intuitionistic fuzzy set (IFS) A = {< y, µ A (y), ν A (y) > y Y } is characterized by a membership function µ A : Y [0.1], and a non-membership function ν A : Y [0, 1] with the condition 0 µ A (y) + ν A (y) 1 for all y Y, where the numbers µ A (y) and ν A (y) represent the degree of membership and the degree of non-membership of the element y to the set A, respectively. Definition [1, 4] If A and B are two IFS of the set Y, then A B iff y Y, µ A (y) µ B (y) and ν A ν B (y), A B iff B A, A = B iff y Y, µ A (y) = µ B (y) and ν A = ν B (y), A B = {< y, min(µ A (y), µ B (y)), max(ν A (y), ν B (y)) > y Y }, A B = {< y, max(µ A (y), µ B (y)), min(ν A (y), ν B (y)) > y Y },

3 276 BUI CONG CUONG, TONG HOANG ANH, BUI DUONG HAI Definition [1, 4] If A 1 and A 2 are two intuitionistic fuzzy sets, then A 1 = {< y, ν A1 (y), µ A1 (y) > y Y } A 1 + A 2 = {< y, µ A1 (y) + µ A2 (y) µ A1 (y).µ A2 (y), ν A1 (y).ν A2 (y) > y Y } A 1.A 2 = {< y, µ A1 (y).µ A2 (y), ν A1 (y) + ν A2 (y) ν A1 (y).ν A2 (y) > y Y } λa 1 = {< y, 1 (1 µ A1 (y)) λ, (ν A1 (y)) λ > y Y } A 1 λ = {< y, (µ A1 (y)) λ, 1 (1 ν A1 (y)) λ > y Y } 3. DEFINITION OF OPEATIONS ON TYPE-2 IFS Now we introduce the notion of a type-2 intuitionistic fuzzy set Definition Definition et S be a nonempty set. A is a type-2 intuitionistic fuzzy set (T2IFS) of S. A is defined by: A : S Map(D, [0, 1]) Map(D, [0, 1]), where D = {(u, v) [0, 1] [0, 1] : u + v 1}. For convenience in description, the binary operations betweenf(u, v) and g(u, v) in Map(D, [0, 1]) are written in the forms (f g) = (f g)(u, v) = f(u, v) g(u, v), (f g) = (f g)(u, v) = f(u, v) g(u, v) and (f, g) = (f(u, v), g(u, v)). Now we give the definitions of main operations in T2IFS theory The operation AND Definition et (f 1, g 1 ) and (f 2, g 2 ) be in Map(D, [0, 1]) Map(D, [0, 1]). We define the intersection operation denoted and it is defined by: (f 1, g 1 ) (f 2, g 2 ) = (f, g) where for any (u, v) D The operation O f(u, v) = f 1(u 1, v 1 ) f 2 (u 2, v 2 ) u 1 u 2 =u,v 1 v 2 =v g(u, v) = g 1(u 1, v 1 ) g 2 (u 2, v 2 ) u 1 u 2 =u,v 1 v 2 =v Definition et (f 1, g 1 ) and (f 2, g 2 ) be in Map(D, [0, 1]) Map(D, [0, 1]). We define the union operation denoted and it is defined by: (f 1, g 1 ) (f 2, g 2 ) = (f, g) where for any (u, v) D The operation NEGATION f(u, v) = f 1(u 1, v 1 ) f 2 (u 2, v 2 ), u 1 u 2 =u,v 1 v 2 =v g(u, v) = g 1(u 1, v 1 ) g 2 (u 2, v 2 ) u 1 u 2 =u,v 1 v 2 =v (f(u, v), g(u, v)) = (f(v, u), g(v, u)) The followings are definitions of identities. They are 1 = (1 1, 1 0 ) and 0 = (1 0, 1 1 ).

4 SOME OPEATIONS ON TYPE-2 INTUITIONISTIC FUZZY SETS (u, v) = { { 1 if u = 1, v = 0 0 if u 1, v if u = 0, v = 1 0(u, v) = 0 if u 0, v 1 Thus, we defined the algebra M = (Map(D, [0, 1]) Map(D, [0, 1]),,,, 1, 0) for T2IFS with operations,,. Our aim in this paper is to examine some properties on the algebra of T2IFS such as idempotent, involution, commutative laws, associative laws or distributive laws Some properties of these operations. In this section, we are going to demonstrate some properties of the operations on T2 IFS. We start with the below theorem which clarifies the first properties such as idempotent, commutative, absorption laws with identities, involution, and De Morgan's laws. Theorem For every (f, g), (f 1, g 1 ), (f 2, g 2 ) M, we have 1.(f, g) (f, g) = (f, g) and (f, g) (f, g) = (f, g) 2.(f 1, g 1 ) (f 2, g 2 ) = (f 2, g 2 ) (f 1, g 1 ) and (f 1, g 1 ) (f 2, g 2 ) = (f 2, g 2 ) (f 1, g 1 ) 3.(11, 10) (f, g) = (f, g) and (10, 11) (f, g) = (f, g) 4.(f, g) = (f, g) 5.{(f 1, g 1 ) (f 2, g 2 )} = (f 1, g 1 ) (f 2, g 2 ) and {(f 1, g 1 ) (f 2, g 2 )} = (f 1, g 1 ) (f 2, g 2 ) Proof We omit properties 1,2 and 4. The proof of property 3. First, we handle the absorption law of identity (1 1, 1 0 ). et (f, g) be in M.Suppose that (1 1, 1 0 ) (f, g) = (f, g ), we have f (u, v) = 1 1(u 1, v 1 ) f(u 2, v 2 ) u 1 u 2 =u,v 1 v 2 =v To find f (u, v) we look for all values of 1 1 (u 1, v 1 ) f(u 2, v 2 ), where u 1 u 2 = u, v 1 v 2 = v and then find their sup. In the first case, if (u 1, v 1 ) = (1, 0) then 1 1 (u 1, v 1 ) = 1 and (u 2, v 2 ) must be (u, v). We have 1 1 (u 1, v 1 ) f(u 2, v 2 ) = 1 f(u, v) = f(u, v). In the other case, if (u 1, v 1 ) (1, 0), then 1 1 (u 1, v 1 ) = 0, we have 1 1 (u 1, v 1 ) f(u 2, v 2 ) = 0 f(u 2, v 2 ) = 0. Hence, f (u, v) is the sup of two results for the two cases or f (u, v) = 0 f(u, v) = f(u, v). In similar way, we prove that g (u, v) = g(u, v). With the same arguments, the absorption law of the other identity can be proven. For property 5,we prove only the first formula. The second formula automatically has the analogous proof. et (f 1, g 1 ), (f 2, g 2 ) M. Suppose that (f, g ) = (f 1, g 1 ) (f 2, g 2 ), we have f (u, v) = f 1(u 1, v 1 ) f 2 (u 2, v 2 ), u 1 u 2 =u,v 1 v 2 =v g (u, v) = g 1(u 1, v 1 ) g 2 (u 2, v 2 ). u 1 u 2 =u,v 1 v 2 =v then (f (u, v), g (u, v)) = (f (v, u), g (v, u)), where f (v, u) = f 1(v 1, u 1 ) f 2 (v 2, u 2 ), u 1 u 2 =u,v 1 v 2 =v g (v, u) = g 1(v 1, u 1 ) g 2 (v 2, u 2 ). u 1 u 2 =u,v 1 v 2 =v

5 278 BUI CONG CUONG, TONG HOANG ANH, BUI DUONG HAI Otherwise, (f 1, g 1 ) (f 2, g 2 ) = (f 1 (u, v), g 1 (u, v)) (f 2 (u, v), g 2 (u, v)) = (f 1 (v, u), g 1 (v, u)) (f 2 (v, u), g 2 (v, u) = (f (v, u), g (v, u)), where f (v, u) = f 1(v 1, u 1 ) f 2 (v 2, u 2 ), v 1 v 2 =v,u 1 u 2 =u g (v, u) = g 1(v 1, u 1 ) g 2 (v 2, u 2 ). v 1 v 2 =v,u 1 u 2 =u Comparing (f, g ) and (f, g ), we can imply (f, g ) = (f, g ). Thus, we have {(f 1, g 1 ) (f 2, g 2 )} = (f 1, g 1 ) (f 2, g 2 ) is proved. Definition For f Map(D, [0, 1]). et f, f, f and f be elements of Map(D, [0, 1]) defined by f (u, v) = u u f(u, v), f (u, v) = u u f(u, v), f (u, v) = v v f(u, v ), f (u, v) = v v f(u, v ). The below figures visualize our definitions. (a) f (b) f (c) f (d) f (e) f Figure 3.1: Geometrical interpretation of f, f, f, f, and f Theorem The following properties hold for all (f 1, g 1 )(f 2, g 2 ) M: 1.(f 1, g 1 ) (f 2, g 2 ) = (f, g) provided f = (f 1 f 2 ) (f 1 f 2 ) (f 1 f 2 ) (f 1 f 2 ), g = (g 1 g 2 ) (g 1 g 2 ) (g 1 g 2 ) (g 1 g 2 ). 2.(f 1, g 1 ) (f 2, g 2 ) = (f, g) provided f = (f 1 f 2 ) (f 1 f 2 ) (f 1 f 2 ) (f 1 f 2 ), g = (g 1 g 2 ) (g 1 g 2 ) (g 1 g 2 ) (g 1 g 2 ). Proof et (f 1, g 1 ), (f 2, g 2 ) M. We have (f 1, g 1 ) (f 2, g 2 ) = (f, g) such that f(u, v) = g(u, v) = u 1 u 2 =u,v 1 v 2 =v u 1 u 2 =u,v 1 v 2 =v (f 1 (u 1, v 1 ) f 2 (u 2, v 2 )), (g 1 (u 1, v 1 ) g 2 (u 2, v 2 ))

6 SOME OPEATIONS ON TYPE-2 INTUITIONISTIC FUZZY SETS 279 First, f(u, v) = (f 1 (u 1, v 1 ) f 2 (u 2, v 2 )) u 1 u 2 =u,v 1 v 2 =v = [ (f 1 (u 1, v 1 ) f 2 (u 2, v 2 ))] u 1 u 2 =u v 1 v 2 =v = {[ (f 1 (u 1, v 1 ) f 2 (u 2, v))] [ (f 1 (u 1 ) f 2 (u 2, v 2 ))]} u 1 u 2 =u v 1 v=v vv 2 =v = {[( f 1 (u 1, v 1 )) f 2 (u 2, v)] [f 1 (u 1, v) ( f 2 (u 2, v 2 ))]} u 1 u 2 =u v 1 v v2 v = ((f 1 (u 1, v) f 2 (u 2, v)) (f 1 (u 1, v) f 2 (u 2, v))) u 1 u 2 =u = (f 1 (u 1, v) f 2 (u 2, v)) (f 1 (u 1, v) f 2 (u 2, v)) u 1 u 2 =u u 1 u 2 =u = (f 1 (u, v) f 2 (u 2, v)) (f 1 (u 1, v) f 2 (u, v)) u u 2 =u u 1 u=u (f 1 (u, v) f 2 (u 2, v)) (f 1 (u 1, v) f 2 (u, v)) u u 2 =u u 1 u=u = (f 1 (u, v) f 2 (u 2, v)) ( f 1 (u 1, v) f 2 (u, v)) u 2 u u 1 u (f 1 (u, v) f 2 (u 2, v)) ( f 1 (u 1, v) f 2 (u, v)) u 2 u u 1 u Analogously, = (f 1 (u, v) f 2 (u, v)) (f 1 (u 1, v) f 2 (u, v)) (f 1 (u, v) f 2 (u, v)) (f 1 (u, v) f 2 (u, v)) = (f 1 f 2 ) (f 1 f 2 ) (f 1 f 2 ) (f 1 f 2 ) we can easily prove the fomulas stated for g. It is no doubt that the operation between (f 1, g 1 ) and (f 2, g 2 ) has the similar proof. Note.To make convenient for further descriptions and proofs, let denote operations between (f 1, g 1 ) and (f 2, g 2 ) as the followings: 1.(f 1, g 1 ) (f 2, g 2 ) = (f 1 f 2, g 1 g 2 ), where f 1 f 2 = (f 1 f 2 ) (f 1 f 2 ) (f 1 f 2 ) (f 1 f 2 ) g 1 g 2 = (g 1 g 2 ) (g 1 g 2 ) (g 1 g 2 ) (g 1 g 2 ) (f 1, g 1 ) (f 2, g 2 ) = (f 1 f 2, g 1 g 2 )where f 1 f 2 = (f 1 f 2 ) (f 1 f 2 ) (f 1 f 2 ) (f 1 f 2 ) g 1 g 2 = (g 1 g 2 ) (g 1 g 2 ) (g 1 g 2 ) (g 1 g 2 ) 2.f g = (f g) (f g ) and f g = (f g) (f g ) such that f g = (f g ) (f g) and f g = (f g ) (f g) Corollary et f, g Map(D, [0, 1]). We have 1.(f g) = (f g ) (f g ) and (f g) = (f g ) (f g ). 2. (f g) = (f g ) (f g )and (f g) = (f g ) (f g ). 3.(f g) = f g. Proof 1.We have

7 280 BUI CONG CUONG, TONG HOANG ANH, BUI DUONG HAI (f g) (u, v) = u u = ( u 1 u 2 u = u 1 u 2 u = (f g)(u, v) = v 1 v 2 =v ( u u u 1 u 2 =u,v 1 v 2 =v (f(u 1, v 1 ) g(u 2, v 2 ))) (f(u 1, v 1 ) g(u 2, v 2 ))) ((f (u 1, v) g(u 2, v)) (f(u 1, v) g (u 2, v))) (f (u 1, v) g(u 2, v)) (f(u 1, v) g (u 2, v)) u 1 u 2 u u 1 u 2 u = (f (u 1, v) g(u 2, v)) (f(u 1, v) g (u 2, v)) u 1 u,u 2 u u 1 u,u 2 u = ( f (u 1, v) g(u 2, v)) ( f(u 1, v) g (u 2, v)) u 1 u u 2 u = (f (u, v) g (u, v)) (f (u, v) g (u, v)). no difficulty to prove the other formula by using the same arguments used in the above proof. 2.We have (f g) (u, v) = (f g)(u, v ) = f(u 1, v 1 ) g(u 2, v 2 ) v v v v u 1 u 2 =u,v 1 v 2 =v = f(u 1, v 1 ) g(u 2, v 2 ) u 1 u 2 =u,v 1 v 2 v = f(u 1, v 1 ) g(u 2, v 2 ) v 1 v 2 v u 1 u 2 =u = [( f(u, v 1 ) g(u 2, v 2 )) ( f(u 1, v 1 ) g(u, v 2 ))] v 1 v 2 v u u 2 =u u 1 u=u = [(f(u, v 1 ) g(u 2, v 2 )) (( f(u 1, v 1 )) g(u, v 2 ))] v 1 v 2 v u 2 u u 1 u = [(f(u, v 1 ) g (u, v 2 )) (f (u, v 1 ) g(u, v 2 ))] Thus, v 1 v 2 v = [(f(u, v 1 ) g (u, v 2 )) (f (u, v 1 ) g(u, v 2 ))] v 1 v,v 2 [(f(u, v 1 ) g (u, v 2 )) (f (u, v 1 ) g(u, v 2 ))] v 2 v,v 1 = [(f (u, v) g (u, v)) (f (u, v) g (u, v))] [(f (u, v) g (u, v)) (f (u, v) g (u, v))] = (f g ) (f g ) (f g ) (f g ) = (f g ) (f g ). (f g) = (f g ) (f g ). With the same arguments, it becomes easy to prove the following formula: u 1 u (f g) = (f g ) (f g ). u 2 u There is 3. We have (f g) = (f g)(u, v ) = ( (f(u 1, v 1 ) g(u 2, v 2 ))) u u,v v u u,v v u 1 u 2 =u,v 1 v 2 =v = (f(u 1, v 1 ) g(u 2, v 2 ) = ( (f(u 1, v 1 ) g(u 2, v 2 )) u 1 u 2 u,v 1 v 2 v u 1 u 2 u v 1 v 2 v = ( f(u 1, v 1 ) g(u 2, v 2 ) = (f (u 1, v) g (u 2, v) u 1 u 2 u v 1 v v 2 v u 1 u 2 u = ( (f (u 1, v)) ( g (u 2, v)) = f (u, v) g (u, v) u 1 u = f g u 2 u Theorem The following associative laws hold for and. (f 1, f 2 ) [(g 1, g 2 ) (h 1, h 2 )] = [(f 1, f 2 ) (g 1, g 2 )] (h 1, h 2 )

8 SOME OPEATIONS ON TYPE-2 INTUITIONISTIC FUZZY SETS 281 (f 1, f 2 ) [(g 1, g 2 ) (h 1, h 2 )] = [(f 1, f 2 ) (g 1, g 2 )] (h 1, h 2 ) Proof We are going to prove the following formula. For f, g, h Map(D, [0, 1]) f (g h) = (f g) h (3.1) From the right side of (1), we have (f g) h = [(f g) h ] [(f g) h ] [(f g) h] [(f g) h ] = {[(f g ) (f g )] h } {[(f g ) (f g )] h } [(f g ) h] {[(f g ) (f g ) (f g) (f g )] h } = [f g h ] [f g h ] [f g h ] [f g h ] [f g h] [f g h ] [f g h ] [f g h ] [f g h ] the left side of (1), we have f (g h) = [f (g h) ] [f (g h) ] [f (g h)] [f (g h) ] = {f [(g h ) (g h )]} {f [(g h ) (g h )]} {f [(g h ) (g h ) (g h) (g h )]} [f (g h )] = [f g h ] [f g h ] [f g h ] [f g h ] [f g h ] [f g h ] [f g h] [f g h ] [f g h ] Hence, f (g h) = (f g) h is proved. Because of this property, we imply From (f 1, f 2 ) [(g 1, g 2 ) (h 1, h 2 )] = (f 1 (g 1 h 1 ), f 2 (g 2 h 2 )) To prove the absorption law of, we have the similar proof. = ((f 1 g 1 ) h 1, (f 2 g 2 ) h 2 ) = [(f 1, f 2 ) (g 1, g 2 )] (h 1, h 2 ) Corollary The following distributive laws hold for operations, and : f (g h) = (f g) (f h) f (g h) = (f g) (f h) This is one of the results which has been proved for type-2 fuzzy sets in [11]. The following results will show that the operations of type-2 intuitionistic fuzzy sets do not have distributive laws. Theorem et f, g, h be in M We have 1.f (g h) = [f (g h)] [f (g h )] [f (g h )] [f (g h )]. 2.(f g) (f h) = [(f g ) (f h)] [(f g ) (f h )] [(f g ) (f h)] [(f g ) (f h )] [(f g) (f h )] [(f g) (f h )] [(f g ) (f h )] [(f g ) (f h )].

9 282 BUI CONG CUONG, TONG HOANG ANH, BUI DUONG HAI Proof 1.From the left side, we have. f (g h) = [f (g h)] [f (g h) ] = {f [(g h) (g h )]} {f [(g h ) (g h )]} = [f (g h)] [f (g h )] [f (g h )] [f (g h )] 2.We have (f g) (f h) = [(f g) (f h)] [(f g) (f h) ] the first expression, we have (f g) (f h) = [(f g ) (f g )] [(f h) (f h )] Deal with the second expression, we have Hence, (f g) (f h) = = [(f g ) (f h)] [(f g ) (f h )] [(f g ) (f h)] [(f g ) (f h )] = [(f g) (f h )] [(f g) (f h )] [(f g ) (f h )] [(f g ) (f h )] (f g) (f h) = [(f g ) (f h)] [(f g ) (f h )] [(f g ) (f h)] [(f g ) (f h )] [(f g) (f h )] [(f g) (f h )] [(f g) (f h )] [(f g) (f h )] [(f g) (f h )] [(f g) (f h )] [(f g ) (f h )] [(f g ) (f h )] In short, this theorem shows that the distributive laws do not hold. 4. CONCUSION In this paper, we have introduced a new concept of type-2 intuitionistic fuzzy set and their operations. It is hopefully more general and applicable than the ordinary intuitionistic fuzzy set. There would be an overwhelmingly large amount of applications on many different fields that derive from it. EFEENCES [1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (2006) [2] K. Atanassov, New operations defined on intuitionistic fuzzy sets, Fuzzy Sets and Systems 61 (1994)

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