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

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1 IFSCOM016 1 Proceeding Book No. 1 pp (016) ISBN: 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 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

2 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 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 [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 [8] Let X be a set A = { x, µ A (x), ν A (x) : x X} IF S(X), α, β [0, 1].

3 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 [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 [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 [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 [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 [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 [7]Let X be a universal, A IF S(X) α, β, ω [0, 1] then

4 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 [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;

5 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 )

6 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

7 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 , 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 [6] Çuvalcıoğlu G., New Tools Based on Intuitionistic Fuzzy Sets Generalized Nets, ISBN , Springer International Publishing Switzerl, 016, [7] Çuvalcıoğlu G., Yılmaz S. On New Intuitionistic Fuzzy Operators: S α,β T α,β,kasmera, 43(), 015, [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 [10] Zadeh L.A., Fuzzy Sets, Information Control, 8, (1965), p University of Mersin Department of Mathematics Mersin, Turkey. address: gcuvalcioglu@mersin.edu.tr Dept. of Bioinformatics Mathematical Modelling Institute of Biophysics Biomedical Engineering Bulgarian Academy of Sciences address: krat@bas.bg University of Mersin Department of Mathematics Mersin, Turkey. address: sinemyilmaz@mersin.edu.tr