Some New Equalities On The Intuitionistic Fuzzy Modal Operators

Transcription

1 SAKARYA ÜNİVERSİTESİ FEN BİLİMLERİ ENSTİTÜSÜ DERGİSİ SAKARYA UNIVERSITY JOURNAL OF SCIENCE e-issn: X Dergi sayfası: Geliş/Received Kabul/Accepted Doi Some New Equalities On The Intuitionistic Fuzzy Modal Operators Mehmet Çitil *1 Feride Tuğrul ABSTRACT In this study properties of the two modal operator ( ) defined on intuitionistic fuzzy sets were investigated. Afterwards some intuitionistic fuzzy operations $ # ) were researched with modal operators ( ). New equalities were obtained and proved. Keywords: intuitionistic fuzzy sets modal operators operations. Sezgisel Bulanık Modal Operatörlerde Bazı Yeni Eşitlikler ÖZ Bu çalışmada sezgisel bulanık kümeler üzerinde tanımlanmış iki modal operatörün ( ) özellikleri incelenmiştir. Ardından bazı sezgisel bulanık işlemler $ # ) modal operatörlerle ( ) birlikte araştırılmıştır. Yeni eşitlikler elde edilmiş ve ispatları yapılmıştır. Anahtar Kelimeler: sezgisel bulanık kümeler modal operatörler işlemler. * Sorumlu Yazar / Corresponding Author 1 Kahramanmaraş Sütçü İmam Üniversitesi Fen Edebiyat Fakültesi Matematik Bölümü Kahramanmaraş citil@ksu.edu.tr Kahramanmaraş Sütçü İmam Üniversitesi Fen Edebiyat Fakültesi Matematik Bölümü Kahramanmaraş feridetugrul@gmail.com 017 Sakarya Üniversitesi Fen Bilimleri Enstitüsü

2 1. INTRODUCTION The notion of fuzzy logic was firstly defined by L.A.Zadeh in Then intuitionistic fuzzy sets (shortly ifs) were defined by K.Atanassov in Intuitionistic fuzzy sets form a generalization of the notion of fuzzy set. In intuitionistic fuzzy set theory sum of the membership function and the non-membership function is a value between 0 and 1. The intuitionistic fuzzy set theory is useful in various application areas such as algebraic structures robotics control systems agriculture areas computer irrigation economy and various engineering fields. The knowledge and semantic representation of intuitionistic fuzzy set become more meaningful resourceful and applicable since it include the membership degree the nonmembership degree and the hesitation margin. The notion of Intuitionistic Fuzzy Operator (IFO) was defined firstly by K.Atanassov[3]. Several operators are defined in Intuitionistic Fuzzy Sets Theory.They are classified in three groups: modal topological and level operators.k.atanassov defined some first type modal operator ( ) on intuitionistic fuzzy sets [3]. α and α operators were defined by K.Dencheva in 004[5]. αβ and αβ operators were defined by K.Atanassov in 006[6]. αβγ and αβγ operators which are expansion of αβ and αβ operators respectively were defined K.Atanassov[7]. G.Cuvalcioglu was defined E αβ operator which is expansion of α and α operators [8]. K.Atanassov produced that it created a diagram of first type modal operators in 008. K.Atanassov defined αβγδετ operator in 009 and he produced that this operator is most general form of operators in the diagram. Z ωθ αβ operator was defined by G.Cuvalcioglu in 010 [9]. This operator is expansion ofe αβ αβ and αβ operators. Thisoperator settled in the diagram and expanded ωθ on the diagram [9]. G.Cuvalcioglu defined Z αβ operator which is expansion ofz ω αβ αβ and αβ operators in 01 [10]. So the diagram took its final state when this operator was defined. Some properties of first type modal operators were researched by many researchers[7][11][1]. Modal operators ( ) defined over the set of all IFS s transform every IFS into a FS. They are similar to the operators necessity and possibility defined in some modal logics. The notion of modal operator ( ) introduced on intuitionistic fuzzy sets were defined by K.Atanassov in 1986 []. The aim of this paper is to obtain new equalities by means of modal operators ( ) and some intuitionistic fuzzy operations. These equalities make it easy application areas of operators. Shorter equalities have obtained using features of some intuitionistic fuzzy operations with modal operators. In this paper; for every IFS M N in X we get M M x M x instead of M x M x M x x X and N N x N x instead of N x N x N x x X for simplicity in proofs. Some basic definitions that we build on our work are given as follow.. PRELIMINARIES Definition 1 [1] Let X be a nonempty set a fuzzy set A drawn from X is defined as A = x x X where : X [01] is the membership function of the fuzzy set A. Definition [][3] Let X be a nonempty set an intuitionistic fuzzy set A in X is an object having the form A = x x X where the function : X [01] define respectively the degree of membership and degree of nonmembership of the element x X to the set A which is a subset of X and for every element x X 0 1. Furthermore we have =1 called the intuitionistic fuzzy set index or hesitation on margin of x in A. is degree of indeterminacy of x X to the IFS A and [01] i.e. A : X [ 01] for every x X. expresses the lack of knowledge of whether x belongs to IFS A or not.

3 Definition 3 [][3] Let X be a nonempty set for every IFS M N in X. For every two IFS s M and N the following operations and relations are valid. M@N M x M x M x x X M c x M x M x x X x MxNx Mx Nx x X Theorem [][3] Let X be a nonempty set for every IFS M N in X. The following equality could be provided by means of definition operation. N M@N Theorem 3 [][3] Let X be a nonempty set for every IFS M N in X ; N M@N M N xmax M x N xmin M x N x x X M N xmin M x N xmax M x N x x X 3. MAIN RESULTS M N M N M N xmax M x N xmin M x N x x X x M x N x M x. N x M x. N x x X x M x. N x M x N x M x. N x x X M $ N x M x. N x M x. N x x X M # N M N x Mx. N x M x N x Mx. N x x X M x N x x M x B x M x B x 1 M x B x x X M x B x 1 Definition 4 [][3] Let X be a nonempty set for every IFS M N in X. and modal operators are defined as; M x M x x X x M x1 M x x X M x1 M x x X x1 M x M x x X 1 M c x1 M x M x x X M c x M x1 M x x X Theorem 1 [][3] Let X be a nonempty set for every IFS M N in X. The following equalities are ensured. (a) M M (b) M M (c) M M (d) M M In this section new equalities were obtained and proved by means of some intuitionistic fuzzy operations $ # ) and modal operators ( ). Theorem 4 [4] Let X be a nonempty set for M M N N Theorem 5 [4] Let X be a nonempty set for M M N N Theorem 6 [4] Let X be a nonempty set for M@N$M#N M$N Theorem 7 [4] Let X be a nonempty set for N $ M # N M $ N Theorem 8 [4] Let X be a nonempty set for N c N c Theorem 9 [4] Let X be a nonempty set for M N M c N M c M c Theorem 10 Let X be a nonempty set for every IFS M N in X. The following equality is holds for;

4 M M c N M M Theorem 1 Let X be a nonempty set for every IFS M N in X; M N M c N M c M c The above equality is obtained. M N 1 M x N x 1 M x N x M x1 N x 1 M x N x N x M x N x M x M x N x M N M x1 N x1 M x N x 1 M x N x M x M x N x1 M x N x N x M x N x 1 M x M x N x M x M x N x x X M N c 1 M x M x N x M x M x N x M c N M x N x1 N x 1 M x 1 M x1 N x M c N 1 M x 1 N x 1 M x1 N x M x N x M x N x1 N x 1 M x 1 M x N x M x N x M x N x1 M x N x 1 M x 1 N x 1 M x N x M x N x M x N x 1 M x N x M x N x M M c N 1Mx M x N x 1Mx M x N x M M c N M max 1Mx M x N x 1 M x min 1Mx M x N x M x 1 M x M x M M M c N M M Theorem 11 Let X be a nonempty set for every IFS M N in X. We get the following equality; M N M c N M c M N M N c M c N M x N x M c 1 M x N x 1 M x N x M x1 N x 1 M x N x N x M x N x M x M x N x 1 M x M x N x M x M x N x M x M x N x1 M x M x N x 1 N x 1 M x 1 M x1 N x M N M c N Mx M N M c N M c M N M c N M c 1 Mx max M x Mx M x1 M x M c M c min1 M x1 Mx M N M c N Mx M x M N M c N M c min Mx 1 M xmax M x M x M N M c N M c 1 M x M x M c M c Theorem 13 Let X be a nonempty set for every IFS M in X. We get; M M M M M M M x 1 M x M x1 M x1 M x M x M x 1 M x M x M x M x M x M x M x 1 M x M x M x M x M x M x M M M x1 M x1 M x M x M x1 M x M x M x M x1 M x M x M x M x M x M x M x M x1 M x M x M x M x N x1 N x 1 M x 1 M x N x M x N xm M M 1 Mx M x 1 Mx M x M x N x1 M x N x M M M M M Theorem 14 Let X be a nonempty set for every IFS M N in X. We obtain the following equality; M M N N

5 M N max M x1 N xmin1 M x N x M N min M x1 N xmax1 M x N x The proof is similar to proof of the Theorem 13. M N M x 1 N x M x1 N x1 M x N x M x 1 N x M x M x N x N x M x N x 1 N x M x N x N x M x N x M N M x1 N x1 M x N x N x1 M x M x M x N x1 M x N x N x M x N x M x M x N x1 M x M x N x M M N max M x1 N x min M x1 N x min1 M x N x max1 M x N x M x 1 N x N M M N N 1 M x N x M M N 1 Nx M x N M M N N 1 Nx M x Theorem 17 Let X be a nonempty set for every IFS M in X. We get the following equality; M $ M # M M $ M Theorem 15 Let X be a nonempty set for every IFS M in X; M M M M The above equality is holds for. M M max M x1 M xmin1 M x M x M M min M x1 M xmax1 M x M x M M M max M x1 M x min M x1 M x min1 M x M x max1 M x M x M x 1 M x M M M M M 1 M x M x Theorem 16 Let X be a nonempty set for every IFS M N in X ; M M N N It is easily seen similar to Theorem 15. M M x 1 M x M # M M $ M # M 1 M x M x M x1 M x M x 1 M x 1 M x M x 1 M x M x M x 1 M x 1 M x M x M x1 M x M $ M M $ M # M M $ M M x1 M x M x 1 M x 1 M x M x 1 M x M x 1 M x M x Theorem 18 Let X be a nonempty set for every IFS M N in X; N $ M # N M $ N It is easily seen similar to Theorem 17. N Mx 1 N x M # N 1 Mx N x Mx1 N x M x 1 N x 1 Mx N x 1 M x N x

6 N $ M # N M x 1 N x 1 M x N x M x1 N x M $ N N $ M # N M $ N M x1 N x M x 1 N x 1 M x N x 1 M x N x 1 M x N x Theorem 19 Let X be a nonempty set for every IFS M N in X. The equality is obtained. M N M N M N M N max1 M x1 N xmin M x N x M N min1 M x1 N xmax M x N x M N M N max1 M x1 N x min1 M x1 N x 1 M x1 N x 1 min M x N x M x N x 1 1 Mx1 N x Mx N x 1 M x1 N x1 M x N x1 M N M N M N M N M N max M x N xmin1 M x1 N x M N min M x N xmax1 M x1 N x M N M N max Mx N x min M x N x M x N x 1 min1 Mx1 N x max1 M x1 N x 1 M x1 N x 1 M x N x M x N x 1 1 M x 1 N x 1 M x1 N x 1 M N M N M N M N Theorem 0 Let X be a nonempty set for every IFS M N in X. We get the following equalities; M N M N M N M M N M N M N $ M N M N Conclusion 1 This conclusion is easy to see from Theorem 0. M N M N M M N M N $ M N M N Theorem Let X be a nonempty set for every IFS M N in X. The following equalities are hold for. M N M N M N $ M N M N M N M M N M N Conclusion This conclusion is easy to see from Theorem. M N M N M N $ M N M M N M N Theorem 3 Let X be a nonempty set for every IFS M N in X. We get; M $ N M $ N $ N M $ N c M $ N@M $ N c M $ N 1 M x1 N x M x N x M $ N c M x N x 1 M x1 N x M $ N M $ N c max 1 M x1 N x M x N x Theorem 1 Let X be a nonempty set for every IFS M N in X. We obtain; M N M N M N min 1 M x1 N x M x N x M $ N M $ N c min 1 M x1 N x M x N x max 1 M x1 N x M x N x

7 M $ N M $ N $ N M $ N c max 1 Mx1 Nx Mx Nx min 1 Mx1 Nx Mx Nx max 1 Mx1 Nx Mx Nx min 1 Mx1 Nx Mx Nx 1 Mx1 Nx Mx Nx 1 Mx1 Nx Mx Nx Theorem 4 Let X be a nonempty set for every IFS M N in X. The equality is obtained; M $ N M $ N $ N M $ N c M $ N@M $ N c The proof is similar to proof of the Theorem 3. Theorem 5 Let X be a nonempty set for every IFS M N in X. We get the following equality; M N # M N $ M M N M $ N The proof is similar to proof of the Theorem 5. Theorem 7 Let X be a nonempty set for every IFS M N in X. M N c M $ N M N M $ N c c M $ N M $ N The above equality is obtained. M N c M $ N M x N x M x N x 1 M x N x M x N x M N M $ N c c M x N x1 M x N x Theorem 8 Let X be a nonempty set for every IFS M N in X. We obtain the following equality; M N c M $ N M N M $ N c c M $ N M $ N M N max M x N xmin1 M x1 N x M N min M x N xmax1 M x1 N x The proof is similar to proof of the Theorem CONCLUSION M N # M N M M N max M x N xmin M x N x max M x N x min M x N x max1 M xmin1 N x max1 M x min1 N x M x N x M x N x 1 M x1 N x 1 M x 1 N x max M x N x min M x N x max1 M x min1 N x M x N x 1 M x 1 N x M N # M N $ M M N M x N x 1 M x1 N x M $ N M N # M N $ M M N M $ N In this paper properties of modal operators defined on intuitionistic fuzzy sets have been investigated. Then some intuitionistic fuzzy operations have been researched with modal operators. New equalities have been obtained. These equalities make it easy application areas of operators. Shorter equalities have been obtained using features of some intuitionistic fuzzy operations with modal operators. These equalities are more useful because they are shorter and more practical. These equalities could be made use of in many application areas of operators and these offer more practical solutions as they provide simplicity. Robotic economy control systems computer and algebraic structures are some of these application areas. Theorem 6 Let X be a nonempty set for every IFS M N in X. The following equality is holds for; M N # M N $ M M N M $ N REFERENCES [1] L.A.Zadeh Fuzzy Sets* Information and Controlvol.8 pp

8 [] K.T.Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol.0 no.1 pp [3] K.T.Atanassov Intuitionistic Fuzzy SetsTheory and Applications Physica-Verlag Heidelberg Germany pp [4] E. Eker F. Tuğrul and M. Çitil New equalities on the intuitionistic fuzzy operators and operations Notes on Intuitionistic Fuzzy Sets vol.1 no.4 pp [5] K.Dencheva Extension of intuitionistic fuzzy modal operators Proc.of the Second Int. IEEE Symp. Intelligent Systems vol.3 pp [6] K.T. Atanassov The most general form of one type of intuitionistic fuzzy modal operators Notes on Intuitionistic Fuzzy Sets vol.1 no. pp [7] K.T. Atanassov Some Properties of the operators from one type of intuitionistic fuzzy modal operators Advanced Studies on Contemporary Mathematics vol.15 no.1 pp [8] G. Çuvalcıoğlu Some Properties of E operator Advanced Studies on Contemporary Mathematics vol. 14 no. pp [9] G. Çuvalcıoğlu Expand the modal operator diagram with Z Jangjeon Math. Soc. vol.13 no.3 pp [10] G. Çuvalcıoğlu On the Diagram of One Type Modal Operators on Intuitionistic Fuzzy Sets: Last Expanding with Z Iranian J. of Fuzzy Systems vol.10 no.1 pp [11] G. Çuvalcıoğlu and S.Yılmaz Some properties of OTMOs on IFSs Advanced Studies in Contemporary Mathematics vol.0 no.4 pp [1] B. Doycheva Inequalities with intuitionistic fuzzy topological and Gökhan Çuvalcıoglu s operators Notes on Intuitionistic Fuzzy Sets vol.14 no.1 pp