TWO NEW OPERATOR DEFINED OVER INTERVAL VALUED INTUITIONISTIC FUZZY SETS S. Sudharsan 1 2 and D. Ezhilmaran 3 1 Research Scholar Bharathiar University Coimbatore -641046 India. 2 Department of Mathematics C. Abdul Hakeem College of Engineering & Technology MelvisharamVellore 632 509 Tamilnadu India. 3 School of Advanced Sciences VIT University Vellore 632 014 Tamilnadu India. ABSTRACT In this paper two new Operator defined over IVIFSs were introduced which will be Multiplication of an IVIFS with and Multiplication of an IVIFS with the natural number are proved. Key Words: Intuitionistic fuzzy set Interval valued Intuitionistic fuzzy sets Operations over interval valued intuitionistic fuzzy sets. AMS CLASSIFICATION: 03E72 1. INTRODUCTION In 1965 Fuzzy sets theory was proposed by L. A. Zadeh[18]. In 1986 the concept of intuitionistic fuzzy sets (IFSs) as a generalization of fuzzy set were introduced by K. Atanassov[1]. After the introduction of IFS many researchers have shown interest in the IFS theory and applied in numerous fields such as pattern recognition machine learning image processing decision making and etc... In 1994 new operations defined over the intuitionistic fuzzy sets was proposed by K. Atanassov[3]. In 2000 Some operations on intuitionistic fuzzy sets were proposed by Supriya Kumar De Ranjit Biswas and Akhil Ranjan Roy[12]. In 2001 an application of intuitionistic fuzzy sets in medical diagnosis were proposed by Supriya Kumar De Ranjit Biswas and Akhil Ranjan Roy [13]. In 2006 n-extraction operation over intuitionistic fuzzy sets were proposed by B. Riecan and K. Atanassov[9]. In 2010 Operation division by n over intuitionistic fuzzy sets were proposed by B. Riecan and K. Atanassov [10]. In 2010 Remarks on equalities between intuitionistic fuzzy sets was K. Atanassov[4]. In 2008 properties of some IFS operators and operations were proposed by Liu Q Ma C and Zhou X [7]. In 2008 Four equalities connected with intuitionistic fuzzy sets was proposed by T. Vasilev [14]. In 2011 Intuitionistic fuzzy sets: Some new results were proposed by R. K. Verma and B. D. Sharma[15]. In 1989 the notion of Interval-Valued Intuitionistic Fuzzy Sets which is a generalization of both DOI : 10.5121/ijfls.2014.4401 1
Intuitionistic Fuzzy Sets and Interval-Valued Fuzzy Sets were proposed by K. Atanassov and G. Gargov [5]. After the introduction of IVIFS many researchers have shown interest in the IVIFS theory and applied it to the various field. In 1994 Operators over interval-valued intuitionistic fuzzy sets was proposed by K. Atanassov[6]. In 2007 methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making was proposed by Z.S. Xu [17]. In 2007 Some geometric aggregation operators based on interval-valued intuitionistic fuzzy sets and their application to group decision making were proposed by G.W.Wei and X.R.Wang [16]. In 2012 Some Results on Generalized Interval-Valued Intuitionistic Fuzzy Sets were proposed by Monoranjan Bhowmik and Madhumangal Pal[8]. In 2013 Interval-Valued Intuitionistic Hesitant Fuzzy Aggregation Operators and Their Application in Group Decision- Making were proposed by Zhiming Zhang [19]. In 2014 new Operations over Interval Valued Intuitionistic Hesitant Fuzzy Set were proposed by Broumi and Florentin Smarandache [11]. This paper proceeds as follows: In section 2 some basic definitions related to intuitionistic fuzzy sets interval valued intuitionistic fuzzy sets and set operations are introduced over the IVIFSs are presented. In section 3 two new operators and defined over IVIFSs are introduced and proved. In section 4 and 5 Conclusion and Acknowledgments are given. 2. PRELIMINARIES Definition 2.1: Intuitionistic Fuzzy Set [12]: An intuitionistic fuzzy set A in the finite universe X is defined as where : 01 and : 01 with the condition 0 sup sup 1 for any. The intervals and denote the degree of membership function and the degree of non-membership of the element x to the set A. Definition 2.2: Interval valued Intuitionistic Fuzzy Set [5]: An Interval valued intuitionistic fuzzy set A in the finite universe X is defined as A. The intervals and denote the degree of membership function and the degree of non-membership of the element x to the set A. For every and are closed intervals and their Left and Right end points are denoted by and.let us denote Where 0 1 0 0. Especially if and then the given IVIFS A is reduced to an ordinary IFS. Let us define the empty IVIFS the totally uncertain IVIFS and the unit IVIFS by: 0011 0011 1100. Definition 2.2. Set operations on IVIFSs [5]: Let A and B be two IVIFSs on the universe X where 2
Here we define some set operations for IVIFSs: 1 1 1 1 11 11 11 11 1 1 1 1 1 1 1 1 Where 1 is natural number. 3. TWO NEW OPERATOR AND DEFINED OVER IVIFS ARE INTRODUCED AND PROVED Two new Operators defined over IVIFS are introduced which will be an analogous as of Operations extraction as well as of operation Multiplication of an IVIFS with and Multiplication of an IVIFS with. It has the form for every IVIFS and for every natural number 1 1 1 1 1 1 1 1 1 1 11 11 11 11 3
Theorem 3.1. For any two IVIFSs A and B and for every natural number 1:... 1 1 1. 1 1 1. Proof:. 11 11 11 11 11 11 11 11 11 11 11 11 11 11 1 1 1 1 1 1 11 11 11 11 1 1 1 1 1 1 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 4
5 Hence is proved and similarly (b) is proved by analogy. Proof:. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Hence is proved and similarly (d) is proved by analogy.
Theorem 3.2. For every IVIFS A and for every natural number 1:. ;. ;. ;.. Proof: (a). 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 11 1 1 1 1 1 1 1 1 1 (3.1) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (3.2) From (3.1) and (3.2) we get 1 1 Hence (a) is proved and similarly (b) is proved by analogy. Proof:. 11 11 11 11 11 11 7
11 11 111 111 11 11 1 1 (3.3) 11 11 1 1 11 11 1 1 (3.4) From (3.3) and (3.4) we get Hence is proved and similarly (d) is proved by analogy. Theorem 3.3. For any two IVIFSs A and B and for every natural number 1:. 1 1 1. 1 1 1... Proof: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8
1 1 1 1 1 1 1 1 1 1 1 1 Hence (a) is proved and similarly (b) is proved by analogy. Proof: 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 Hence (c) is proved and similarly (d) is proved by analogy. Theorem 3.4. For every IVIFS A and for every natural number 1:. 1. 1 Proof.. 1 1 1 1 1 1 9
1 1 1 1 1 1 1 1 111 1 111 1 11 11 1 1 1 1 1 1 1 1 11 11 1 1 1 1 1 1 1 1 Hence (a) is proved. Proof.. 1 1 11 11 1 11 11 11 11 10
1 111 11 1 111 11 1 1 11 11 11 11 11 11 11 11 11 Hence (b) is proved. Theorem 3.5: For every IVIFS A and for every natural number n 1: (a). (b). Proof. (a). 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11
1 1 1 1 1 1 1 1 1 Hence (a) is proved. Proof. (b). 11 11 11 11 11 11 11 11 11 11 Hence (b) is proved. 4. CONCLUSION In this paper two new operators based IVIFS were introduced and few theorems were proved. In future the application of this operator will be proposed and another two operators based on IVIFS are to be introduced. 5. ACKNOWLEDGMENTS The authors are highly grateful to the Editor-in-Chief and Reviewer Professor Ayad Ghany Ismaeel and the referees for their valuable comments and suggestions. REFERENCES [1] K. Atanassov (1986) Intuitionistic fuzzy sets Fuzzy Sets & Systems. vol. 20 pp. 87-96. [2] K. Atanassov (1999) Intuitionistic Fuzzy Sets Springer Physica-Verlag Berlin. [3] K. Atanassov (1994) New operations defined over the intuitionistic fuzzy sets Fuzzy Sets &Systems vol.61 pp 37-42. [4] K. Atanassov (2010) Remarks on equalities between intuitionistic fuzzy sets Notes on Intuitionistic Fuzzy Sets vol.16 no.3 pp. 40-41. 12
[5] K. Atanassov and G. Gargov(1989) Interval-valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol. 31 pp. 343-349. [6] K. Atanassov (1994) Operators over interval-valued intuitionistic fuzzy sets Fuzzy Sets and Systems. vol. 64 no.2 pp. 159-174. [7] Q.Liu C. Ma and X. Zhou (2008) On properties of some IFS operators and operations Notes on Intuitionistic Fuzzy Sets Vol. 14 no.3 pp. 17-24. [8] Monoranjan Bhowmik and Madhumangal Pal(2012) Some Results on Generalized Interval-Valued Intuitionistic Fuzzy Sets International Journal of Fuzzy Systems vol. 14 no.2. [9] B. Riecan and K. Atanassov(2006) n-extraction operation over intuitionistic fuzzy sets Notes in Intuitionistic Fuzzy Sets vol 12 no.4 pp. 9 11. [10] B. Riecan and K. Atanassov(2010) Operation division by n over intuitionistic fuzzy sets Notes in Intuitionistic Fuzzy Sets vol. 16 no.4 pp. 1 4. [11] Said Broumi and Florentin Smarandache(2014) New Operations over Interval Valued Intuitionistic Hesitant Fuzzy Set Mathematics and Statistics vol. 2 no. 2 pp. 62-71. [12] Supriya Kumar De Ranjit Biswas and Akhil Ranjan Roy (2000) Some operations on intuitionistic fuzzy sets Fuzzy Sets and Systems vol. 114 no. 4 pp. 477 484. [13] Supriya Kumar De Ranjit Biswas and Akhil Ranjan Roy (2001) An application of intuitionistic fuzzy sets in medical diagnosis Fuzzy Sets & System vol. 117 no. 2 pp. 209-213. [14] T. Vasilev (2008) Four equalities connected with intuitionistic fuzzy sets Notes on Intuitionistic Fuzzy Sets vol. 14 no. 3 pp. 1-4. [15] R. K. Verma and B. D. Sharma (2011) Intuitionistic fuzzy sets: Some new results Notes on Intuitionistic Fuzzy Sets vol. 17 no. 3 pp.1-10. [16] G. W. Wei and X. R. Wang (2007) Some geometric aggregation operators based on interval-valued intuitionistic fuzzy sets and their application to group decision making In Proceedings of the international conference on computational intelligence and security pp. 495-499. [17] Z. S. Xu (2007) Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making Control Decis. vol. 22 no.2 pp. 215-219. [18] L. A. Zadeh (1965) Fuzzy Sets Information & Control Vol.8 pp. 338-353. [19] Zhiming Zhang (2013) Interval-Valued Intuitionistic Hesitant Fuzzy Aggregation Operators and their application in Group Decision Making Journal of Applied Mathematics. 13