Fuzzy Sets Systems 15 (2005) 151 155 www.elsevier.com/locate/fss On properties of four IFS operators Deng-Feng Li a,b,c,, Feng Shan a, Chun-Tian Cheng c a Department of Sciences, Shenyang Institute of Aeronautical Engineering, Shenyang 11003, Liaoning, China b Department Five, Dalian Naval Academy, Dalian 116018, Liaoning, China c Department of Civil Engineering, Dalian University of Technology, Dalian 11602, Liaoning, China Received 20 February 2005; received in revised form 2 February 2005; accepted 8 March 2005 Available online 30 March 2005 Abstract The aim of this paper is to point out correct some errors of properties of the four operators defined over the IFSs in Atanassov (Intuitionistic Fuzzy Sets, Springer, Heidelberg, 1999). 2005 Elsevier B.V. All rights reserved. Keywords: Fuzzy sets; IFS; Operator; Operation Mathematical objects introduced by Atanassov [1,2] studied under the name intuitionistic fuzzy set (IFS) have become a popular topic of investigation in the fuzzy set community [6]. There exists a large amount of literature involving IFS theory applications [3 5]. The aim of this paper is to point out correct some errors in [1]. Briefly we mention the definition, operators properties given in [1]. Definition 1 (See Atanassov [1, Definition 1.1, p. 1]). Let a (non-fuzzy) set E be fixed. An IFS A in E is defined as an object of the following form: A =x,μ A (x), ν A (x) Corresponding author. Department Five, Dalian Naval Academy, No. 1, Xiaolong Street, Dalian 116018, Liaoning, China. Tel.: +86 011 85856357; fax: +86 011 85856357. E-mail addresses: lidengfeng65@hotmail.com, dengfengli@sina.com (D.-F. Li). 0165-011/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.03.00
152 D.-F. Li et al. / Fuzzy Sets Systems 15 (2005) 151 155 where the functions μ A : E [0, 1] ν A : E [0, 1] define the degree of membership the degree of non-membershipof the element x E to the set A E, respectively, for every x E, 0 μ A (x) + ν A (x) 1. Definition 2 (See Atanassov [1, Definition 1., p. 9]). Let A = x,μ A (x), ν A (x) B = x,μ B (x), ν B (x) be two IFSs. We define the following relations: (1) A B iff for every x E, μ A (x) μ B (x) ν A (x) ν B (x); (2) A = B iff for every x E, μ A (x) = μ B (x) ν A (x) = ν B (x). Atanassov [1] introduced the following two IFS operators. Definition 3 (See Atanassov [1, Definition 1.11, p. 121]). Let A be an IFS. We define the following operators: A = A = x, μ A(x), ν A(x) + 1 2 2 x, μ A(x) + 1, ν A(x) 2 2 In general, we have the following Theorem 1. } 1 x E (1). (2) Theorem 1. For every IFS A, we have A A. (3) Proof. Using Eqs. (1) (2), we have A = x, μ A(x) + 2, ν A(x) + 1 () 1 In MS word, there are not the symbols given in [1]. Here they are replaced with the corresponding symbols with, respectively.
A = D.-F. Li et al. / Fuzzy Sets Systems 15 (2005) 151 155 153 x, μ A(x) + 1, ν A(x) + 2. (5) Obviously, the following two inequalities are always valid simultaneously: μ A (x) + 2 > μ A(x) + 1 ν A (x) + 1 < ν A(x) + 2. According to Definition 2 Eqs. () (5), it follows that A A. Atanassov [1] (See [1, Eq. (1.503) of Theorem 1.115, p. 122]) stated that for every IFS A, the following equality is always valid: A = A. However, Eq. (8) is not correct. In fact, the following inequality is not valid: A A, according to Eqs. (6) (7) Definition 2. Hence, A = A does not hold, i.e., A = A. The two operators introduced in Definition 3 were further generalized in the following: Definition (See Atanassov [1, Definition 1.118, p. 123]). Let α [0, 1] A be an IFS. We define the following two operators: α A =x,αμ A (x), αν A (x) + 1 α (10) α A =x,αμ A (x) + 1 α, αν A (x). (11) (6) (7) (8) (9) According to Definitions 3, we have 0.5 A = A 0.5 A = A. Moreover, we have 1 A = 1 A = A. Therefore, the operators α α are generalizations of.
15 D.-F. Li et al. / Fuzzy Sets Systems 15 (2005) 151 155 In general, we have the following Theorem 2. Theorem 2. For every IFS A for every α [0, 1), we have (12) Proof. Using Eqs. (10) (11), we have α α A =x,α[αμ A (x)]+1 α, α[αν A (x) + 1 α] =x,α 2 μ A (x) + 1 α, α 2 ν A (x) + α(1 α) (13) α α A =x,α[αμ A (x) + 1 α], α[αν A (x)]+1 α =x,α 2 μ A (x) + α(1 α), α 2 ν A (x) + 1 α. (1) Obviously, for any α [0, 1), the following two inequalities are always valid simultaneously: α 2 μ A (x) + α(1 α) <α 2 μ A (x) + 1 α (15) α 2 ν A (x) + 1 α > α 2 ν A (x) + α(1 α). (16) Hence, using Definition 2 Eqs. (13) (1), we obtain It is easily seen that for α = 1, 1 1 A = 1 1 A = A. (17) Atanassov [1] (See [1, Eq. (1.532) of Theorem 1.119, p. 123]) stated that for every IFS A every α [0, 1], the following equality is always valid: α α A = α α A. (18) However, Eq. (18) is wrong. In fact, the following inequality is not valid: α α A α α A, according to Eqs. (15) (17) Definition 2. Hence, from Eq. (12) the following equality is not valid: α α A = α α A. In other words, the following inequality is always valid:
D.-F. Li et al. / Fuzzy Sets Systems 15 (2005) 151 155 155 References [1] K. Atanassov, Intuitionistic Fuzzy Sets, Springer, Heidelberg, 1999. [2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Systems 20 (1986) 87 96. [3] Dengfeng Li, Some measures of dissimilarity in intuitionistic fuzzy structures, J. Comput. Syst. Sci. 68 (200) 115 122. [] Dengfeng Li, An approach to multiattribute decision makings in intuitionistic fuzzy sets, J. Comput. Syst. Sci. 70 (1) (2005) 73 85. [5] Dengfeng Li, Chuntian Cheng, New similarity measures of intuitionistic fuzzy sets application to pattern recognitons, Pattern Recognition Lett. 23 (1 3) (2002) 221 225. [6] D. Dubois, S. Gottwald, P. Hajek, J. Kacpryzk, H. Prade, Are (Atanassov s) intuitionalistic fuzzy sets intuitionalistic, Fuzzy Sets Systems, to appear.