GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)

IFSCOM016 1 Proceeding Book No. 1 pp. 155-161 (016) ISBN: 978-975-6900-54-3 SOME RESULTS ON S α,β AND T α,β INTUITIONISTIC FUZZY MODAL OPERATORS GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ) Abstract. In 1999, first Intuitionistic Fuzzy Modal Operators introduced in[]. Expansion of these operators new operators defined by different authors[3, 5, 6, 7, 8, 9]. Characteristics of these operators has been studied by several researchers. In this study, we obtained new results on modal operators which are called S α,β T α,β. Received: 5 July 016 Accepted: 9 August 016 1. Introduction The concept of Intuitionistic fuzzy sets was introduced by Atanassov in 1986 [1], form an extension of fuzzy sets[10] by exping the truth value set to the lattice [0, 1] [0, 1]. Intuitionistic fuzzy modal operators defined firstly by Atanassov[1, ]. Then severel extensions of these operators introduced by different authors[, 8, 5, 6]. Some algebraic characteristic properties of these operators were studied by several authors. Definition 1.1. [1] An intuitionistic fuzzy set (shortly IFS) on a set X is an object of the form A = { x, µ A (x), ν A (x) : x X} where µ A (x), (µ A : X [0, 1]) is called the degree of membership of x in A, ν A (x), (ν A : X [0, 1])is called the degree of non- membership of x in A, where µ A ν A satisfy the following condition: µ A (x) + ν A (x) 1, for all x X. The class of intuitionistic fuzzy sets on X is denoted by IF S(X). The hesitation degree of x is defined by π A (x) = 1 µ A (x) ν A (x) Definition 1.. [1]An IFS A is said to be contained in an IFS B (notation A B) if only if, for all x X : µ A (x) µ B (x) ν A (x) ν B (x). It is clear that A = B if only if A B B A. 1 3 rd International Intuitionistic Fuzzy Sets Contemporary Mathemathics Conference 010 Mathematics Subject Classification. 03E7,47S40. Key words phrases. Intuitionistic fuzzy sets, Intuitionistic fuzzy modal operators. 155

156GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ) Definition 1.3. [1]Let A IF S let A = { x, µ A (x), ν A (x) : x X} then the above set is callede the complement of A A c = { x, ν A (x), µ A (x) : x X} The intersection the union of two IFSs A B on X is defined by A B = { x, µ A (x) µ B (x), ν A (x) ν B (x) : x X} A B = { x, µ A (x) µ B (x), ν A (x) ν B (x) : x X} The notion of Second Type Intuitionistic Fuzzy Modal Operators was firstly introduced by Atanassov as following: Definition 1.4. [1]Let X be universal A IF S(X) then (1) (A) = { x, µ A (x), 1 µ A (x) : x X} () (A) = { x, 1 ν A (x), ν A (x) : x X} Definition 1.5. []Let X be universal A IF S(X), α [0, 1] then D α (A) = { x, µ A (x) + απ A (x), ν A (x) + (1 α)π A (x) : x X} Definition 1.6. []Let X be universal A IF S(X), α, β [0, 1] α+β 1 then F α,β (A) = { x, µ A (x) + απ A (x), ν A (x) + βπ A (x) : x X} Definition 1.7. []Let X be universal A IF S(X), α, β [0, 1] then G α,β (A) = { x, αµ A (x), βν A (x) : x X} Definition 1.8. []Let X be universal A IF S(X), α, β [0, 1] then (1) H α,β (A) = { x, αµ A (x), ν A (x) + βπ A (x) : x X} () H α,β (A) = { x, αµ A(x), ν A (x) + β(1 αµ A (x) ν A (x)) : x X} (3) J α,β (A) = { x, µ A (x) + απ A (x), βν A (x) : x X} (4) J α,β (A) = { x, µ A(x) + α(1 µ A (x) βν A (x)), βν A (x) : x X} The simplest One Type Intuitionistic Fuzzy Modal Operators defined in 1999. Definition 1.9. [] Let X be a set A = { x, µ A (x), ν A (x) : x X} IF S(X), α, β [0, 1]. { } (1) A = x, µ A(x), ν A(x)+1 : x X { } () A = x, µ A(x)+1, ν A(x) : x X After this definition, in 001, the extension of these operators were defined as following: Definition 1.10. [3] Let X be a set A = { x, µ A (x), ν A (x) : x X} IF S(X), α, β [0, 1]. (1) α A = { x, αµ A (x), αν A (x) + 1 α : x X} () α A = { x, αµ A (x) + 1 α, αν A (x) : x X} The operators α α are the extensions of the operators,, resp. In 004, Dencheva introduced the second extension of α α. Definition 1.11. [8] Let X be a set A = { x, µ A (x), ν A (x) : x X} IF S(X), α, β [0, 1].

SOME RESULTS ON S α,β AND T α,β INTUITIONISTIC FUZZY MODAL OPERATORS 157 (1) α,β A = { x, αµ A (x), αν A (x) + β : x X} where α + β [0, 1]. () α,β A = { x, αµ A (x) + β, αν A (x) : x X}where α + β [0, 1]. In 006, the third extension of the above operators was studied by author. He defined the following operators; Definition 1.1. [3]Let X be a set A = { x, µ A (x), ν A (x) : x X} IF S(X). (1) α,β,γ (A) = { x, αµ A (x), βν A (x) + γ : x X} where α, β, γ [0, 1], max{α, β} + γ 1. () α,β,γ (A) = { x, αµ A (x) + γ, βν A (x) : x X} where α, β, γ [0, 1], max{α, β} + γ 1. In 007, author[5] defined a new operator named E α,β studied some of its properties. This operator as following: Definition 1.13. [5]Let X be a set A = { x, µ A (x), ν A (x) : x X} IF S(X), α, β [0, 1]. We define the following operator: E α,β (A) = { x, β(αµ A (x) + 1 α), α(βν A (x) + 1 β) : x X} At the same year, Atanassov introduced the operator α,β,γ,δ which is a natural extension of all these operators in [3]. Definition 1.14. [3]Let Xbe a set, A IF S(X), α, β, γ, δ [0, 1] such that then the operator α,β,γ,δ defined by max(α, β) + γ + δ 1 α,β,γ,δ (A) = { x, αµ A (x) + γ, βν A (x) + δ : x X} In 008, most general operator α,β,γ,δ,ε,ζ defined as following: Definition 1.15. [3]Let X be a set,a IF S(X), α, β, γ, δ, ε, ζ [0, 1] such that max(α ζ, β ε) + γ + δ 1 min(α ζ, β ε) + γ + δ 0 then the operator α,β,γ,δ,ε,ζ defined by α,β,γ,δ,ε,ζ (A) = { x, αµ A (x) εν A (x) + γ, βν A (x) ζµ A (x) + δ : x X} In 010, Çuvalcıoğlu[6] defined a new operator which is a generalization of E α,β. Definition 1.16. [6]Let X be a set A = { x, µ A (x), ν A (x) : x X} IF S(X), α, β, ω [0, 1] then Z ω α,β(a) = { x, β(αµ A (x) + ω ω.α), α(βν A (x) + ω ω.β) : x X} Definition 1.17. [6]Let X be a set A = { x, µ A (x), ν A (x) : x X} IF S(X), α, β, ω, θ [0, 1] then Z ω,θ α,β (A) = { x, β(αµ A(x) + ω ω.α), α(βν A (x) + θ θ.β) : x X} The operator Z ω,θ α,β is a generalization of Zω α,β, also, E α,β, α,β, α,β. Uni-type intuitionistic fuzzy modal operators introduced by author as following; Definition 1.18. [7]Let X be a universal, A IF S(X) α, β, ω [0, 1] then

158GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ) (1) ω α,β (A) = { x, β(µ A(x) + (1 α)ν A (x)), α(βν A (x) + ω ωβ) : x X} () ω α,β (A) = { x, β(αµ A(x) + ω ωα), α((1 β)µ A (x) + ν A (x)) : x X} Definition 1.19. [7]Let X be a set A IF S(X), α, β, ω, θ [0, 1] then { } x, β((1 (1 α)(1 θ))µa (x) + (1 α)θν A (x) + (1 α)(1 θ)ω), E ω,θ α,β (A) = : x X α((1 β)θµ A (x) + (1 (1 β)(1 θ))ν A (x) + (1 β)(1 θ)ω) Definition 1.0. [7]Let X be a set, A IF S(X) α, β [0, 1] then (1) B α,β (A) = { x, β(µ A (x) + (1 α)ν A (x)), α((1 β)µ A (x) + ν A (x)) : x X} () α,β (A) = { x, β(µ A (x) + (1 β)ν A (x)), α((1 α)µ A (x) + ν A (x)) : x X} In 014, new one type intuitionistic fuzzy modal operators were defined in [9]. Definition 1.1. [9]Let X be a set A IF S(X), α, β, ω [0, 1] α+β 1 (1) L ω α,β (A) = { x, αµ A(x) + ω(1 α), α(1 β)ν A (x) + αβ(1 ω) x X} () K ω α,β (A) = { x, α(1 β)µ A(x) + αβ(1 ω), αν A (x) + ω(1 α) x X} As above, we get the following diagram;

SOME RESULTS ON S α,β AND T α,β INTUITIONISTIC FUZZY MODAL OPERATORS 159 Figure 1 The intuitionistic fuzzy modal operator, represented by α,β,γ,δ, introduced in 014 as following; Definition 1.. [4]Let X be a set A IF S(X), α, β, γ, δ [0, 1] α+β 1, γ + δ 1 then α,β,γ,δ (A) = { x, αµ A (x) + γν A (x), βµ A (x) + δν A (x) }. Some Properties of New Intuitionistic Fuzzy Modal Operators Definition.1. Let X be a set A IF S(X), α, β, α + β [0, 1]. (1) T α,β (A) = {< x, β(µ A (x) + (1 α)ν A (x) + α), α(ν A (x) + (1 β)µ A (x)) >: x X} where α + β [0, 1]. () S α,β (A) = {< x, α(µ A (x) + (1 β)ν A (x)), β(ν A (x) + (1 α)µ A (x) + α) >: x X} where α + β [0, 1]. It is clear that; β(µ A (x) + (1 α)ν A (x) + α) + α(ν A (x) + (1 β)µ A (x)) = (µ A (x) + ν A (x))(α + β αβ) + αβ α + β αβ + αβ 1 Theorem.1. Let X be a set A IF S(X). If α, β, α + β [0, 1] then T α,β (A) c = S α,β (A c ). Proof. It is clear from definition. Proposition.1. Let X be a set A IF S(X). If α, β, α + β [0, 1] then (1) T β,α (A) c T α,β (A c )

160GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ) Figure () S α,β (A c ) S β,α (A) c Proof. (1)From definition of this operators complement of an intuitionistic fuzzy set we get that, β(ν A (x) + (1 α)µ A (x)) β(ν A (x) + (1 α)µ A (x) + α) α(µ A (x) + (1 β)ν A (x) + β) α(µ A (x) + (1 β)ν A (x)) So, we can say T β,α (A) c T α,β (A c ). ()We can show this inclusion same way. Theorem.. Let X be a set A IF S(X). If α, β, α + β [0, 1] β α then (1) T α,β (A) T β,α (A) () S β,α (A) S α,β (A) Proof. It is clear. Theorem.3. Let X be a set A, B IF S(X). If α, β, α + β [0, 1] then (1) T α,β (A) T α,β (B) T α,β (A B) () T α,β (A B) T α,β (A) T α,β (B) Proof. (1) Let α, β [0, 1], β(1 α) min(ν A (x)) β(1 α) max(ν A (x)) β (min (µ A (x)) + (1 α) min (ν A β (min (µ A (x)) + (1 α) max (ν A

SOME RESULTS ON S α,β AND T α,β INTUITIONISTIC FUZZY MODAL OPERATORS 161 α(1 β) max(µ A (x)) α(1 β) min(µ A (x)) α (max (ν A (x)) + (1 β) max (µ A (x))) α (max(ν A (x)) + (1 β) min(µ A (x))) It is appear from here that T α,β (A) T α,β (B) T α,β (A B). () It can be shown easily. Theorem.4. Let X be a set A, B IF S(X). If α, β, α + β [0, 1] then (1) S α,β (A B) S α,β (A) S α,β (B) () S α,β (A) S α,β (B) S α,β (A B) Proof. (1) Let α, β [0, 1], α(1 β) min(ν A (x)) α(1 β) max(ν A (x)) α (max (µ A (x)) + (1 β) min (ν A (x))) α (max (µ A (x)) + (1 β) max (ν A (x))) β(1 α) max(µ A (x)) β(1 α) min(µ A (x)) β (min (ν A (x)) + (1 α) max (µ A β (min (ν A (x)) + (1 α) min (µ A Thus, S α,β (A B) S α,β (A) S α,β (B). () Can be proved similarly. References [1] Atanassov K.T., Intuitionistic Fuzzy Sets, VII ITKR s Session, Sofia, June (1983). [] Atanassov K.T., Intuitionistic Fuzzy Sets, Phiysica-Verlag, Heidelberg, NewYork, (1999). [3] Atanassov K.T., Studies in Fuzziness Soft Computing-On Intuitionistic Fuzzy Sets Theory, ISBN 978-3-64-916-5, Springer Heidelberg, New York, 01. [4] Atanassov K.T., Çuvalcıoğlu, G., Atanassova V. K., A new modal operator over intuitionistic fuzzy sets, Notes on IFS, 0(5), 014, 1-8. [5] Çuvalcıoğlu, G., On the diagram of One Type Modal Operators on Intuitionistic Fuzzy Sets: Last Exping with Z ω,θ α,β, Iranian Journal of Fuzzy Systems Vol. 10, No. 1, (013) pp. 89-106. [6] Çuvalcıoğlu G., New Tools Based on Intuitionistic Fuzzy Sets Generalized Nets, ISBN 978-3-319-6301-4, Springer International Publishing Switzerl, 016, 55-71. [7] Çuvalcıoğlu G., Yılmaz S. On New Intuitionistic Fuzzy Operators: S α,β T α,β,kasmera, 43(), 015, 317-37. [8] Dencheva K., Extension of intuitionistic fuzzy modal operators,proc.of the Second Int. IEEE Symp. Intelligent systems, Varna, June -4, (004), Vol. 3, 1-. [9] Yılmaz, S., Bal, A., Extentsion of Intuitionistic Fuzzy Modal Operators Diagram with New Operators, Notes on IFS, Vol. 0, 014, Number 5, pp. 6-35. [10] Zadeh L.A., Fuzzy Sets, Information Control, 8, (1965), p. 338-353. University of Mersin Department of Mathematics Mersin, Turkey. E-mail address: gcuvalcioglu@mersin.edu.tr Dept. of Bioinformatics Mathematical Modelling Institute of Biophysics Biomedical Engineering Bulgarian Academy of Sciences E-mail address: krat@bas.bg University of Mersin Department of Mathematics Mersin, Turkey. E-mail address: sinemyilmaz@mersin.edu.tr