A new modal operator over intuitionistic fuzzy sets

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1 1 st Int. Workshop on IFSs, Mersin, 14 Nov Notes on Intuitionistic Fuzzy Sets ISSN Vol. 20, 2014, No. 5, 1 8 A new modal operator over intuitionistic fuzzy sets Krassimir Atanassov 1, Gökhan Çuvalcıoğlu 2 and Vassia Atanassova 3 1 Department of Bioinformatics and Mathematical Modelling Institute of Biophysics and Biomedical Engineering Bulgarian Academy of Sciences 105 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria, and Intelligent Systems Laboratory Prof. Asen Zlatarov University, Burgas-8010, Bulgaria krat@bas.bg 2 Department Of Mathematics, University Of Mersin Mersin, Turkey gcuvalcioglu@gmail.com 3 Department of Bioinformatics and Mathematical Modelling Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences 105 Acad. G. Bonchev Str., Sofia 1113, Bulgaria vassia.atanassova@gmail.com Abstract: A new operator from modal type is introduced over the intuitionistic fuzzy sets. On one hand, this operator functions by reducing the degree of membership or non-membership, and, on the other hand, by simultaneously summing it with a part of the degree of non-membership or membership, respectively. Some of its properties are studied. Keywords: Intuitionistic fuzzy modal operator, Intuitionistic fuzzy operation. AMS Classification: 03E72. 1 Introduction In a series of papers of the two authors, a new type of intuitionistic fuzzy modal operators are introduced and some of their properties are studied. The definitions of these operattors are given in Section 2. In Section 3 a new operator from modal type is introduced and some of its bassic properties are studied. In the Conclusion an Open Problems are formulated. 1

2 2 Preliminary results Let a set E be fixed. The Intuitionistic Fuzzy Set (IFS) A in E is defined by (see, e.g., [1]): A = { x, µ A (x), ν A (x) x E}, where functions µ A : E [0, 1] and ν A : E [0, 1] define the degree of membership and the degree of non-membership of the element x E, respectively, and for every x E: 0 µ A (x) + ν A (x) 1. Different relations and operations are introduced over the IFSs. Some of them (see, e.g. [1, 4]) are the following A B iff ( x E)(µ A (x) µ B (x)&ν A (x) ν B (x)), A = B iff ( x E)(µ A (x) = µ B (x)&ν A (x) = ν B (x)), A = { x, ν A (x), µ A (x) x E}, A B = { x, min(µ A (x), µ B (x)), max(ν A (x), ν B (x)) x E}, A B = { x, max(µ A (x), µ B (x)), min(ν A (x), ν B (x)) x E}, A + B = { x, µ A (x) + µ B (x) µ A (x).µ B (x), ν A (x).ν B (x) x E}, A.B = { x, µ A (x).µ B (x), ν A (x) + ν B (x) ν A (x).ν B (x) x E}, A = { x, µ A (x), 1 µ A (x) x E}, A = { x, 1 ν A (x), ν A (x) x E}. In [1, 4] the following extended modal operators are defined: Let α, β [0, 1] and let: F α,β (A) = { x, µ A (x) + α.π A (x), ν A (x) + β.π A (x) x E}, where α + β 1 G α,β (A) = { x, α.µ A (x), β.ν A (x) x E}, H α,β (A) = { x, α.µ A (x), ν A (x) + β.π A (x) x E}, H α,β (A) = { x, α.µ A(x), ν A (x) + β.(1 α.µ A (x) ν A (x)) x E}, J α,β (A) = { x, µ A (x) + α.π A (x), β.ν A (x) x E}, J α,β (A) = { x, µ A(x) + α.(1 µ A (x) β.ν A (x)), β.ν A (x) x E}, In [1, 4] the following operator is defined X a,b,c,d,e,f (A) = { x, a.µ A (x) + b.(1 µ A (x) c.ν A (x)), d.ν A (x) + e.(1 f.µ A (x) ν A (x)) x E} where a, b, c, d, e, f [0, 1] and there, the following two conditions are given: a + e e.f 1, 2

3 b + d b.c 1. In addition, in [5] it is demonstrated that it is also necesary to add the following third condition: b + e 1. In [4] another type of modal operators are described. The following are the first two operators of modal type, which are similar to the standard modal operators and : +A = { x, µ A(x) 2, ν A(x) + 1 x E}, 2 A = { x, µ A(x) + 1, ν A(x) x E}. 2 2 For a given real number α [0, 1] and IFS A, the above operators are extended to the forms: + α A = { x, α.µ A (x), α.ν A (x) + 1 α x E}, α A = { x, α.µ A (x) + 1 α, α.ν A (x) x E}. The second extension was introduced by Katerina Dencheva in [8]. She extended the last two operators to the forms: + α,β A = { x, α.µ A (x), α.ν A (x) + β x E}, α,β A = { x, α.µ A (x) + β, α.ν A (x) x E}, where α, β, α + β [0, 1]. The third extension of the above operators gives the following operators (see [4]): + α,β,γ A = { x, α.µ A (x), β.ν A (x) + γ x E}, α,β,γ A = { x, α.µ A (x) + γ, β.ν A (x) x E}, where α, β, γ [0, 1] and max(α, β) + γ 1. In [2, 4], the idea for extending the last two operators naturally produced the operator α,β,γ,δ A = { x, α.µ A (x) + γ, β.ν A (x) + δ x E}, where α, β, γ, δ [0, 1] and max(α, β) + γ + δ 1. In 2007in [6], Gökhan Çuvalcıoğlu introduced operator E α,β by E α,β (A) = { x, β(α.µ A (x) + 1 α), α(β.ν A (x) + 1 β) x E}, where α, β [0, 1], and he studied some of its properties. Obviously, E α,β (A) = αβ,αβ,(1 α)β,(1 β)α A. In 2010, he extended the previous operator to the form: Z ω α,β(a) = { x, β(αµ A (x) + ω ω.α), α(βν A (x) + ω ω.β) x X}, 3

4 where ω, α, β [0, 1] (see [7]). A new (and probably final?) extension of the above operators is the operator where α, β, γ, δ, ε, ζ [0, 1] and (see [3, 4]). α,β,γ,δ,ε,ζ A = { x, α.µ A (x) ε.ν A (x) + γ, β.ν A (x) ζ.µ A (x) + δ x E}, max(α ζ, β ε) + γ + δ 1, min(α ζ, β ε) + γ + δ 0 Theorem [4]. Operators X a,b,c,d,e,f and α,β,γ,δ,ε,ζ are equivalent. Finally, we construct the following Figure 1 in which X Y denotes that operator X represents operator Y, while the reverse is not valid. α,β,γ,δ,ε,ζ X a,b,c,d,e,f α,β,γ,δ + α,β,γ Zα,β ω α,β,γ + α,β E α,β α,β G α,β F α,β H α,β Hα,β J α,β Jα,β + α α + D α Figure 1. 3 Main results Here, we introduce the following new operator from modal type, that is a modification of the above discussed operators. It has the form α,β,γ,δ A = { x, α.µ A (x) + γ.ν A (x), β.µ A (x) + δ.ν A (x) x E}, where α, β, γ, δ [0, 1] and α + β 1, γ + δ 1. According to this definition, on one hand, the operator reduces by α the degree of membership µ A (x) original IFS A s and sums it up with a part of the degree of non-membership (γ.ν A (x)), 4

5 and in the same time it reduces the original A s degree of non-membership (ν A (x)) by δ and sums it up with a part of the degree of membership (β.µ A (x)). It is easy to see that the operator 1,0,0,1 A = A, and 0,1,1,0 A = A. Therefore, this operator gives the possibility to express the operation identity and the operation classical negation. In this way, by varying the values of the variables α, β, γ, δ in the [0; 1] range, we can obtain the whole continuity of sets existing between a given set A and its classical negation A. Let us study the basic properties of the new operator. First, we check that the new set is an IFS. Really, 0 α.µ A (x) + γ.ν A (x) µ A (x) + ν A (x) 1, 0 β.µ A (x) + δ.ν A (x) µ A (x) + ν A (x) 1 and 0 α.µ A (x) + γ.ν A (x), β.µ A (x) + δ.ν A (x) = (α + β).µ A (x) + (γ + δ).ν A (x) µ A (x) + ν A (x) 1. Theorem 1: For every IFS A and for every four real numbers α, β, γ, δ [0, 1] such that α +β 1, γ + δ 1 α,β,γ,δ A = δ,γ,β,α A. Proof: We obtain sequentially that α,β,γ,δ A = α,β,γ,δ { x, ν A (x), µ A (x) x E} = { x, α.ν A (x) + γ.µ A (x), β.ν A (x) + δ.µ A (x) x E} = { x, β.ν A (x) + δ.µ A (x), α.ν A (x) + γ.µ A (x) x E} = δ,γ,β,α A. This completes the proof. Theorem 2: For every two IFSs A and B and for every four real numbers α, β, γ, δ [0, 1] such that α + β 1, γ + δ 1, it holds that (a) α,β,γ,δ (A B) = α,β,γ,δ A α,β,γ,δ B, (b) α,β,γ,δ (A B) = α,β,γ,δ A α,β,γ,δ B, (c) α,β,γ,δ (A + B) = α,β,γ,δ A + α,β,γ,δ B, (d) α,β,γ,δ (A.B) = α,β,γ,δ A. α,β,γ,δ B. 5

6 Proof: For (a), first, we obtain that Second, we obtain that Let α,β,γ,δ (A B) = α,β,γ,δ { x, max(µ A (x), µ B (x)), min(ν A (x), ν B (x)) x E} = { x, α. max(µ A (x), µ B (x)) + γ. min(ν A (x), ν B (x)), β. max(µ A (x), µ B (x)) + δ. min(ν A (x), ν B (x)) x E}. α,β,γ,δ A α,β,γ,δ B = { x, α.µ A (x) + γ.ν A (x), β.µ A (x) + δ.ν A (x) x E} { x, α.µ B (x) + γ.ν B (x), β.µ B (x) + δ.ν B (x) x E} = { x, max(α.µ A (x) + γ.ν A (x), α.µ B (x) + γ.ν B (x)), min(β.µ A (x) + δ.ν A (x), β.µ B (x) + δ.ν B (x)) x E} X max(α.µ A (x) + γ.ν A (x), α.µ B (x) + γ.ν B (x)) α. max(µ A (x), µ B (x)) γ. min(ν A (x), ν B (x)). Now, for µ A (x), µ B (x), ν A (x), ν B (x) we must study the following four cases. Case 1: µ A (x) µ B (x), ν A (x) ν B (x): X = max(α.µ A (x) + γ.ν A (x), α.µ B (x) + γ.ν B (x)) α.µ A (x) γ.ν B (x) α.µ A (x) + γ.ν A (x) α.µ A (x) γ.ν B (x) 0. Case 2: µ A (x) µ B (x), ν A (x) < ν B (x): X = max(α.µ A (x) + γ.ν A (x), α.µ B (x) + γ.ν B (x)) α.µ A (x) γ.ν A (x) 0. Case 3: µ A (x) < µ B (x), ν A (x) ν B (x): X = max(α.µ A (x) + γ.ν A (x), α.µ B (x) + γ.ν B (x)) α.µ B (x) γ.ν B (x) 0. Case 4: µ A (x) < µ B (x), ν A (x) < ν B (x): X = max(α.µ A (x) + γ.ν A (x), α.µ B (x) + γ.ν B (x)) α.µ B (x)) γ.ν A (x) α.µ B (x) + γ.ν B (x) α.µ B (x)) γ.ν A (x) 0. Asserions (b), (c) and (d) are proved analogously. The same is valid for the proofs of the next theorem. Theorem 3: For every IFS A and for every four real numbers α, β, γ, δ [0, 1] such that α +β 1, γ + δ 1 (a) α,β,γ,δ A α,β,γ,δ A, (b) α,β,γ,δ A α,β,γ,δ A. Now, Figure 1 from above is modified, as illustrated on Figure 2. 6

7 α,β,γ,δ α,β,γ,δ,ε,ζ X a,b,c,d,e,f α,β,γ,δ + α,β,γ Zα,β ω α,β,γ + α,β E α,β α,β G α,β F α,β H α,β Hα,β J α,β Jα,β + α α + D α Figure 2. 4 Conclusion In the present paper, a new modal operator is introduced. It is different from the rest modal operators, defined over IFSs. It arises some open problems, as the following ones. Open Problem 1: Can operator α,β,γ,δ be represented by the extended modal operators? Open Problem 2: Can operator α,β,γ,δ be represented by the modal operators from Section 2? Open Problem 3: Can operator α,β,γ,δ be used for representation of some type of modal operators? References [1] Atanassov, K., Intuitionistic Fuzzy Sets: Theory and Applications, Springer, Heidelberg, [2] Atanassov, K., The most general form of one type of intuitionistic fuzzy modal operators. Notes on Intuitionistic Fuzzy Sets, Vol. 12, 2006, No. 2, [3] Atanassov, K., The most general form of one type of intuitionistic fuzzy modal operators. Part 2. Proceedings of the Twelfth International Conference on Intuitionistic Fuzzy Sets (J. Kacprzyk and K. Atanassov, Eds), Sofia, May 2008, Vol. 1. In: Notes on Intuitionistic Fuzzy Sets, Vol. 14, 2008, No. 1, [4] Atanassov, K., On Intuitionistic Fuzzy Sets Theory, Springer, Berlin, [5] Atanassov, K., A short remark on intuitionistic fuzzy operators X a,b,c,d,e,f and x a,b,c,d,e,f, Notes on Intuitionistic Fuzzy Sets, Vol. 19, 2013, No. 1, [6] Çuvalcıoğlu, G., Some properties of E α,β operator. Advanced Studies on Contemporary Mathematics, Vol. 14, 2007, No. 2,

8 [7] Çuvalcıoğlu, G., Expand the model operator diagram with Zα,β ω. Proceedings of the Jangjeon Math. Society, Vol. 13, 2010, No. 3, [8] Dencheva, K., Extension of intuitionistic fuzzy modal operators + and. Proc. of the Second Int. IEEE Symposium Intelligent Systems, Varna, June , Vol. 3,