Int. J. Nonlinear Anal. Appl. 8 (2017) No. 1, 11-21
ISSN: 2008-6822 (electronic)
http://dx.doi.org/10.22075/ijnaa.2017.11099.1542
The operators over the generalized intuitionistic
fuzzy sets
Ezzatallah Baloui Jamkhaneh
Department of Statistics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran
(Communicated by M. Eshaghi)
Abstract
In this paper, newly defined level operators and modal-like operators over extensional generalized
intuitionistic fuzzy sets (GIF SB ) are proposed. Some of the basic properties of the new operators
are discussed.
Keywords: Generalized intuitionistic fuzzy sets; intuitionistic fuzzy sets; modal-like operators;
level operators.
2010 MSC: 03E72.
1. Introduction
Modal operators, topological operators, level operators, negation operators are different groups of
operators over the intuitionistic fuzzy sets (IFS) due to Atanassov [2]. Atanassov [3] defined intuitionistic fuzzy modal operator of α A and α A. Then Dencheva [10] defined an extension of these
operators as α,β A and α,β A. Atanassov [4] defined another extension of these operators as α,β,γ A
and α,β,γ A. Atanassov [5] defined an operator as α,β,γ,η which is an extension of all the operators
defined. In 2007, Cuvaleglu defined operator of Eα,β over IF Ss and studied some of its properties.
Eα,β is a generalization of Atanassov’s operator α A. Cuvalcoglu et al. [9] examined intuitionistic fuzzy operators with matrices and they examined algebraic structures of them. Atanassov [1]
defined level operators Pα,β , Qα,β . In 2008, he studied some relations between intuitionistic fuzzy
negations and intuitionistic fuzzy level operators Pα,β , Qα,β . In 2009, Parvathi and Geetha defined
some level operators, max-min implication operators and Pα,β , Qα,β operators on temporal intuitionistic fuzzy sets. Lupianez [11] show relations between topological operators and intuitionistic fuzzy
topology. The intuitionistic fuzzy operators are important from the point of view application. The
Email address: e [email protected] (Ezzatallah Baloui Jamkhaneh)
Received: February 2016
Revised: October 2016
12
Baloui Jamkhaneh
intuitionistic fuzzy operators applied in contracting a classifier recognizing imbalanced classes, image
recognition, image processing, multi-criteria decision making, deriving the similarity measure, sales
analysis, new product marketing, medical diagnosis, financial services, solving optimization problems
and etc. Baloui Jamkhaneh and Nadarajah [7] considered a new generalized intuitionistic fuzzy sets
(GIF SB ) and introduced some operators over GIF SB . By analogy we shall introduce the some of
operators over GIF SB and we will discuss their properties.
2. Preliminaries
In this section we recall some of the basic definition of IFS which will be helpful in further study of
the paper. Let X be a non empty set.
Definition 2.1. (Atanassov, [1]) An IFS A in X is defined as an object of the form
A = {hx, µA (x), νA (x)i : x ∈ X}
where the functions µA : X → [0, 1] and νA : X → [0, 1] denote the degree of membership and
non-membership functions of A respectively and 0 ≤ µA (x) + νA (x) ≤ 1 for each x ∈ X.
Definition 2.2. (Baloui Jamkhaneh and Nadarajah, [7]) Generalized intuitionistic fuzzy sets (GIF SB )
A in X, is defined as an object of the form A = {hx, µA (x), νA (x)i : x ∈ X} where the functions µA : X → [0, 1] and νA : X → [0, 1] denote the degree of membership and degree of nonmembership functions of A respectively, and 0 ≤ µA (x)δ + νA (x)δ ≤ 1 for each x ∈ X where
δ = nor n1 , n = 1, 2, . . . , N . The collection of all our generalized intuitionistic fuzzy sets is denoted by
GIF SB (δ, X).
Definition 2.3. (Baloui Jamkhaneh and Nadarajah, [7]) The degree of non-determinacy (uncertainty) of an element x ∈ X to the GIF SB A is defined by
1
πA (x) = (1 − µA (x)δ − νA (x)δ ) δ
Definition 2.4. (Baloui Jamkhaneh and Nadarajah, [7]) Let A and B be two GIF SB s such that
A = {hx, µA (x), νA (x)i : x ∈ X},
B = {hx, µB (x), νB (x)i : x ∈ X},
define the following relations and operations on A and B
i. A ⊂ B if and only if µA (x) ≤ µB (x) and νA (x) ≥ νB (x), ∀x ∈ X,
ii. A = B if and only if µA (x) = µB (x) and νA (x) = νB (x), ∀x ∈ X,
S
iii. A B = {hx, max(µA (x), µB (x)), min(νA (x), νB (x))i : x ∈ X},
T
iv. A B = {hx, min(µA (x), µB (x)), max(νA (x), νB (x))i : x ∈ X},
v. A + B = {hx, µA (x)δ + µB (x)δ − µA (x)δ µB (x)δ , νA (x)δ νB (x)δ i : x ∈ X},
vi. A.B = {hx, µA (x)δ .µB (x)δ , νA (x)δ + νB (x)δ − νA (x)δ νB (x)δ i : x ∈ X},
vii. Ā = {hx, νA (x), µA (x)i : x ∈ X}.
Let X is a non-empty finite set, and A ∈ GIF SB , as A = {hx, µA (x), νA (x)i : x ∈ X}. Baloui
Jamkhaneh and Nadarajah [7] introduced following operators of GIF SB and investigated some their
properties
1
viii. A = {hx, µA (x), (1 − µA (x)δ ) δ i : x ∈ X}, (modal logic: the necessity measure).
1
ix. ♦A = {hx, (1 − νA (x)δ ) δ , νA (x)i : x ∈ X}, (modal logic: the possibility measure).
The operators over the generalized intuitionistic fuzzy sets 8 (2017) No. 1, 11-21
13
3. Main results
Here, we will introduce new operators over the GIF SB which extend some operators in the research
literature related to IFS. Let X is a non-empty finite set.
Definition 3.1. Letting α, β ∈ [0, 1]. For every GIF SB as A = {hx, µA (x), νA (x)i : x ∈ X}, we
define the level operators as follows
1
1
1
1
i. Pα,β (A) = {hx, max(α δ , µA (x)), min(β δ , νA (x)) : x ∈ X}, where α + β ≤ 1,
ii. Qα,β (A) = {hx, min(α δ , µA (x)), max(β δ , νA (x)) : x ∈ X}, where α + β ≤ 1.
Theorem 3.2. For every A ∈ GIF SB and α, β, γ, η ∈ [0, 1], where α + β ≤ 1, γ + η ≤ 1, we have
i. Pα,β (A) ∈ GIF SB ,
ii. Qα,β (A) ∈ GIF SB ,
iii. Pα,β (Ā) = Qβ,α (A),
iv. Qα,β (Pγ,η (A)) = Pmin(α,γ),max(β,η) (Qα,β (A)),
v. Pα,β (Qγ,η (A)) = Qmax(α,γ),min(β,η) (Pα,β (A)),
vi. Pα,β (Pγ,η (A)) = Pmax(α,γ),min(β,η) (Pα,β (A)),
vii. Qα,β (Qγ,η (A)) = Qmin(α,γ),max(β,η) (Qα,β (A)),
1
1
viii. Pα,β (Qα,β (A)) = Qα,β (Pα,β (A)) = {hx, α δ , β δ i : x ∈ X},
ix. Qα,β (A) ⊂ A ⊂ Pα,β (A).
Proof .
(i)
1
1
µPα,β (A) (x)δ + νPα,β (A) (x)δ = (max(α δ , µA (x)))δ + (min(β δ , νA (x)))δ ,
= max(α, µA (x)δ ) + min(β, νA (x)δ ) = I,
(1) if max(α, µA (x)δ ) = α and min(β, νA (x)δ ) = β then I = α + β ≤ 1.
(2) if max(α, µA (x)δ ) = α and min(β, νA (x)δ ) = νA (x)δ then I = α + νA (x)δ ≤ α + β ≤ 1.
(3) if max(α, µA (x)δ ) = µA (x)δ and min(β, νA (x)δ ) = β then I = µA (x)δ + β ≤ µA (x)δ +
νA (x)δ ≤ 1.
(4) if max(α, µA (x)δ ) = µA (x)δ and min(β, νA (x)δ ) = νA (x)δ then I = µA (x)δ + νA (x)δ ≤ 1.
The proof is completed. The Proof of (ii) is similar to that of (i). Proof of (iii) is obvious.
14
Baloui Jamkhaneh
(iv)
1
1
,
Qα,β (Pγ,η (A)) = Qα,β
hx, max γ δ , µA (x) , min η δ , νA (x) i : x ∈ X
1
1
1
1
= hx, min α δ , max(γ δ , µA (x)) , max β δ , min(η δ , νA (x)) i : x ∈ X ,
1
1
1
= hx, max min(α δ , γ δ ), min(α δ , µA (x))
1
1
1
, min max(β δ , η δ ), max(β δ , νA (x)) i : x ∈ X ,
1
1
= Pmin(α,γ),max(β,η) hx, min(α δ , µA (x)), max(β δ , νA (x))i : x ∈ X ,
= Pmin(α,γ),max(β,η) (Qα,β (A)).
Since α + β ≤ 1, γ + η ≤ 1 then min(α, γ) + max(β, η) ≤ 1.
(v)
1
1
Pα,β (Qγ,η (A)) = Pα,β
hx, min γ δ , µA (x) , max η δ , νA (x) i : x ∈ X
,
1
1
1
1
= hx, max α δ , min(γ δ , µA (x)) , min β δ , max(η δ , νA (x)) i : x ∈ X ,
1
1
1
= hx, min max(α δ , γ δ ), max(α δ , µA (x))
1
1
1
, max min(β δ , η δ ), min(β δ , νA (x)) i : x ∈ X ,
1
1
δ
δ
= Qmax(α,γ),min(β,η) hx, max(α , µA (x)), min(β , νA (x))i : x ∈ X ,
= Qmax(α,γ),min(β,η) (Pα,β (A)).
Since α + β ≤ 1, γ + η ≤ 1 then max(α, γ) + min(β, η) ≤ 1. The proof is completed. The
Proofs of (vi), (vii) and (viii) are similar to that of (iv). The Proof of (ix) is obvious.
Theorem 3.3. For every A ∈ GIF SB and α, β ∈ [0, 1], where α + β ≤ 1 we have
i. Pα,β (A ∩ B) = Pα,β (A) ∩ Pα,β (B),
ii. Pα,β (A ∪ B) = Pα,β (A) ∪ Pα,β (B),
iii. Qα,β (A ∩ B) = Qα,β (A) ∩ Qα,β (B),
iv. Qα,β (A ∪ B) = Qα,β (A) ∪ Qα,β (B).
Proof .
The operators over the generalized intuitionistic fuzzy sets 8 (2017) No. 1, 11-21
15
(i)
1
1
Pα,β (A ∩ B) = hx, max(α δ , min(µA (x), µB (x))), min(β δ , max(νA (x), νB (x)))i : x ∈ X ,
1
1
1
1
= hx, min max(α δ , µA (x)), max(α δ , µB (x)) , max min(β δ , νA (x)), min(β δ , νB (x)) i :
1
1
x ∈ X = hx, max(α δ , µA (x)), min(β δ , νA (x))i : x ∈ X ∩
1
1
hx, max(α δ , µB (x)), min(β δ , νB (x))i : x ∈ X = Pα,β (A) ∩ Pα,β (B).
(ii)
1
1
Pα,β (A ∪ B) = hx, max(α δ , max(µA (x), µB (x))), min(β δ , min(νA (x), νB (x)))i : x ∈ X ,
1
1
1
1
= hx, max max(α δ , µA (x)), max(α δ , µB (x)) , min min(β δ , νA (x)), min(β δ , νB (x)) i :
1
1
δ
δ
x ∈ X = hx, max(α , µA (x)), min(β , νA (x))i : x ∈ X ∪
1
1
δ
δ
hx, max(α , µB (x)), min(β , νB (x))i : x ∈ X = Pα,β (A) ∪ Pα,β (B)
The proofs of (iii) and (iv) are similar to that of (i). Definition 3.4. Let α ∈ [0, 1] and A ∈ GIF SB , we define modal-like operators as follows
1
1
i. α A = {hx, α δ µA (x), (ανA (x)δ + 1 − α) δ i : x ∈ X},
1
1
ii. α A = {hx, (αµA (x)δ + 1 − α) δ , α δ νA (x)i : x ∈ X}.
Definition 3.5. Let α, β ∈ [0, 1] and A ∈ GIF SB , we define the first extension modal-like operators
as follows
1
1
i. α,β A = {hx, α δ µA (x), (ανA (x)δ + β) δ i : x ∈ X}, where α + β ≤ 1,
1
1
ii. α,β A = {hx, (αµA (x)δ + β) δ , α δ νA (x)i : x ∈ X}, where α + β ≤ 1.
Definition 3.6. Let α, β ∈ [0, 1] and A ∈ GIF SB , we define the operator Eα,β A as follows
1
1
Eα,β A = {hx, (β(αµA (x)δ + 1 − α)) δ , (α(βνA (x)δ + 1 − β)) δ i : x ∈ X}.
Definition 3.7. Let α, β, γ, η ∈ [0, 1] and A ∈ GIF SB , where αβ + βγ + αγ ≤ 1 we define the
operator Eα,β,γ,η A as follows
1
1
Eα,β,γ,η A = {hx, (β(αµA (x)δ + γ)) δ , (α(βνA (x)δ + η)) δ i : x ∈ X}.
Theorem 3.8. For every A ∈ GIF SB and α, β, γ, η ∈ [0, 1], where αβ + βγ + αη ≤ 1, we have
16
Baloui Jamkhaneh
i. Eα,β,γ,η A ∈ GIF SB ,
ii. Eα,1,γ,0 A = α,γ A,
iii. E1,β,0,η A = β,η A,
iv. Eα,1,1−α,0 A = α A,
v. E1,β,0,1−β A = β A.
Proof . (i)
1
1
µEα,β,γ,η A (x)δ + νEα,β,γ,η A (x)δ = ((β(αµA (x)δ + γ)) δ )δ + ((α(βνA (x)δ + η)) δ )δ ,
= (αβµA (x)δ + βγ) + (αβνA (x)δ + αη) = αβ(µA (x)δ + νA (x)δ ) + βγ + αη,
≤ αβ + βγ + αη ≤ 1.
The proof is completed. The Proofs of (ii), (iii), (iv) and (v) are obvious. Definition 3.9. Let α, β ∈ [0, 1] and A ∈ GIF SB , we define the second extension modal-like
operators as follows
1
1
δ
i. α,β,γ A = hx, α δ µA (x), (βνA (x) + γ) δ i : x ∈ X , where max(α, β) + γ ≤ 1.
1
1
δ
δ
δ
ii. α,β,γ A = hx, (αµA (x) + γ) , β νA (x)i : x ∈ X , where max(α, β) + γ ≤ 1.
Theorem 3.10. For every A ∈ GIF SB and α, β, γ, ε, η, λ ∈ [0, 1], where max(α, β) + γ ≤ 1, and
max(ε, η) + λ ≤ 1, we have
i. α,β,γ A ∈ GIF SB ,
ii. α,β,γ A ∈ GIF SB ,
iii. α,β,γ Ā = β,α,γ A,
iv. α,β,γ ε,η,λ A = αε,βη,βλ+γ A,
v. α,β,γ (A ∩ B) = α,β,γ A ∩ α,β,γ B,
vi. α,β,γ (A ∪ B) = α,β,γ A ∪ α,β,γ B,
vii. α,β,γ (A ∩ B) = α,β,γ A ∩ α,β,γ B,
viii. α,β,γ (A ∪ B) = α,β,γ A ∪ α,β,γ B,
ix. α,β,γ Pε,η (A) = Pαε,βη+γ (α,β,γ A), if αε + βη + γ ≤ 1,
x. α,β,γ Qε,η (A) = Qαε,βη+γ (α,β,γ A), if αε + βη + γ ≤ 1,
xi. α,β,γ Pε,η (A) = Pαε,βη+γ (α,β,γ A), if αε + βη + γ ≤ 1,
xii. α,β,γ Qε,η (A) = Qαε,βη+γ (α,β,γ A), if αε + βη + γ ≤ 1,
The operators over the generalized intuitionistic fuzzy sets 8 (2017) No. 1, 11-21
17
Proof . (i)
1
1
µα,β,γ A (x)δ + να,β,γ A (x)δ = (α δ µA (x))δ + ((βνA (x)δ + γ) δ )δ ,
= αµA (x)δ + βνA (x)δ + γ ≤ max(α, β)µA (x)δ + max(α, β)νA (x)δ + γ,
= max(α, β)(µA (x)δ + νA (x)δ ) + γ,
≤ max(α, β) + γ ≤ 1.
The proof is completed. The Proof of (ii) is similar to that of (i).
(iii) Since Ā = {hx, νA (x), µA (x)i : x ∈ X}, we have
1
1
δ
α,β,γ Ā = hx, α δ νA (x), (βµA (x) + γ) δ i : x ∈ X ,
then
1
1
δ
δ
δ
α,β,γ Ā = hx, (βµA (x) + γ) , α νA (x)i : x ∈ X = β,α,γ A.
The proof is completed.
(iv) Since
1
1
δ
α,β,γ A = hx, α δ µA (x), (βνA (x) + γ) δ i : x ∈ X ,
1
1
δ
δ
δ
ε,η,λ A = hx, ε µA (x), (ηνA (x) + λ) i : x ∈ X ,
we have
α,β,γ ε,η,λ A = hx, α ε µA (x), (β((ηνA (x) + λ) ) + γ) i : x ∈ X ,
1 1
1
δ
= hx, α δ ε δ µA (x), (βηνA (x) + βλ + γ) δ i : x ∈ X ,
1
δ
1
δ
δ
1
δ
δ
1
δ
= αε,βη,βλ+γ A.
Since max(α, β) + γ ≤ 1, and max(ε, η) + λ ≤ 1, then max(αε, βη) + βλ + γ ≤ 1.
(v) Since
A ∩ B = {hx, min(µA (x), µB (x)), max(νA (x), νB (x))i : x ∈ X},
we have
1
δ
δ
1
δ
α,β,γ (A ∩ B) = hx, α min(µA (x), µB (x)), (β(max(νA (x), νB (x))) + γ) i : x ∈ X ,
1
1
1
1
δ
δ
δ
δ
δ
δ
= hx, min(α µA (x), α µB (x)), (max((βνA (x) + γ) , (βνB (x) + γ) i : x ∈ X ,
= α,β,γ A ∩ α,β,γ B.
The proof is completed. The Proofs of (vi), (vii) and (viii) are similar to that of (v)
(ix) Since
1
1
δ
δ
Pε,η (A) = hx, max(ε , µA (x)), min(η , νA (x))i : x ∈ X ,
18
Baloui Jamkhaneh
we have
1
1
1
1
δ
α,β,γ Pε,η (A) = hx, α δ max ε δ , µA (x) , (β(min(η δ , νA (x))) + γ) δ i : x ∈ X ,
1 1
1
1
δ
= hx, max α δ ε δ , α δ µA (x) , (min(βη + γ, βνA (x) + γ)) δ i : x ∈ X ,
= Pαε,βη+γ (α,β,γ A).
The proof is completed. Proofs (x), (xi) and (xii) are similar to that of (ix). If in operators α,β,γ A and α,β,γ A, we choose β = α and γ = β, then they are converted to the
operators α,β A and α,β A, respectively. Therefore operators α,β,γ A and α,β,γ A are the extension
of the α,β A and α,β A, respectively. If in operators α,β A and α,β A. we chooseβ = α and
β = 1 − α, then they are converted to the operators α A and α A, respectively. Therefore operators
α,β A and α,β A are the extension of the α A and α A, respectively. That is
i. α,α,β A = α,β A, α,α,1−α A = α,1−α A = α A,
ii. α,α,β A = α,β A, α,α,1−α A = α,1−α A = α A.
Therefore using Theorem 3.10, we have the following proposition:
Proposition 3.11. For every A ∈ GIF SB and α, β, ε, η ∈ [0, 1], where α + β ≤ 1, and ε + η ≤ 1,
we have
i. α,β A ∈ GIF SB ,
ii. α,β A ∈ GIF SB ,
iii. α,β Ā = β,α A,
iv. α,β ε,η A = αε,αη+β A,
v. α,β (A ∩ B) = α,β A ∩ α,β B,
vi. α,β (A ∪ B) = α,β A ∪ α,β B,
vii. α,β (A ∩ B) = α,β A ∩ α,β B,
viii. α,β (A ∪ B) = α,β A ∪ α,β B,
ix. α,β Pε,η (A) = Pαε,αη+β (α,β A), if αε + αη + β ≤ 1,
x. α,β Qε,η (A) = Qαε,αη+β (α,β A), if αε + αη + β ≤ 1,
xi. α,β Pε,η (A) = Pαε,αη+β (α,β A), if αε + αη + β ≤ 1,
xii. α,β Qε,η (A) = Qαε,αη+β (α,β A), if αε + αη + β ≤ 1.
Proposition 3.12. For every A ∈ GIF SB and α, β, ε, η ∈ [0, 1], where ε + η ≤ 1, we have
i. α A ∈ GIF SB ,
ii. α A ∈ GIF SB ,
The operators over the generalized intuitionistic fuzzy sets 8 (2017) No. 1, 11-21
19
iii. α Ā = 1−α A,
iv. α ε A = 1−αε A,
v. α (A ∩ B) = α A ∩ α B,
vi. α (A ∪ B) = α A ∪ α B,
vii. α (A ∩ B) = α A ∩ α B,
viii. α (A ∪ B) = α A ∪ α B,
ix. α Pε,η (A) = Pαε,αη+1−α (α A),
x. α Qε,η (A) = Qαε,αη+1−α (α A),
xi. α Pε,η (A) = Pαε,αη+1−α (α A),
xii. α Qε,η (A) = Qαε,αη+1−α (α A).
Definition 3.13. Let α, β, γ, η ∈ [0, 1] and A ∈ GIF SB , we define the third extension modal-like
operator as follows
1
1
δ
δ
hx, (αµA (x) + γ) δ , (βνA (x) + η) δ i : x ∈ X , max(α, β) + γ + η ≤ 1.
α,β,γ,η A =
Theorem 3.14. For every A ∈ GIF SB and α, β, γ, η, ε, θ, λ, τ ∈ [0, 1], max(α, β) + γ + η ≤ 1,
max(ε, θ) + λ + τ ≤ 1 and ε + θ ≤ 1, we have
i.
α,β,γ,η A
∈ GF ISB ,
ii.
α,β,0,η A
= α,β,γ A,
iii.
α,β,γ,0 A
= α,β,γ A,
iv.
α,β,γ,η Ā
=
α,β,γ,η
ε,θ,λ,τ
v.
β,α,η,γ A,
A=
αε,βθ,αλ+γ,βτ +η A,
vi.
α,β,γ,η (A
∩ B) =
α,β,γ,η A
∩
α,β,γ,η B,
vii.
α,β,γ,η (A
∪ B) =
α,β,γ,η A
∪
α,β,γ,η B,
viii.
α,β,γ,η Pε,θ (A)
= Pαε+γ,βθ+η (
ix.
α,β,γ,η Qε,θ (A)
= Qαε+γ,βθ+η (
α,β,γ,η A),
α,β,γ,η A),
Proof . (i)
µ
α,β,γ,η A
(x)δ + ν
1
α,β,γ,η A
1
(x)δ = ((αµA (x)δ + γ) δ )δ + ((βνA (x)δ + η) δ )δ ,
= αµA (x)δ + γ + βνA (x)δ + η ≤ max(α, β)µA (x)δ + max(α, β)νA (x)δ + γ + η,
= max(α, β)(µA (x)δ + νA (x)δ ) + γ + η ≤ max(α, β) + γ + η ≤ 1.
20
Baloui Jamkhaneh
The proof is completed. The Proofs of (ii) and (iii) are obvious.
(iv) Since Ā = {hx, νA (x), µA (x)i : x ∈ X} we have
1
1
δ
δ
hx, (ανA (x) + γ) δ , (βµA (x) + η) δ i : x ∈ X ,
α,β,γ,η Ā =
then
1
1
δ
δ
hx, (βµA (x) + η) δ , (ανA (x) + γ) δ , i : x ∈ X =
α,β,γ,η Ā =
β,α,η,γ A.
The proof is completed.
(v) Since
hx, (αµA (x) + γ) , (βνA (x) + η) i : x ∈ X ,
α,β,γ,η A =
1
1
δ
δ
hx, (εµA (x) + λ) δ , (θνA (x) + τ ) δ i : x ∈ X ,
ε,θ,λ,τ A =
1
δ
δ
1
δ
δ
hence
α,β,γ,η
αε,θ,λ,τ
A=
=
hx, (εµA (x) + αλ + γ) , (βθνA (x) + βτ + η) i : x ∈ X ,
1
δ
δ
1
δ
δ
αε,βθ,αλ+γ,βτ +η A
The proof is completed.
(vi) Since A ∩ B = {hx, min(µA (x), µB (x)), max(νA (x), νA (x))i : x ∈ X}, we have
1
1
δ
δ
δ
δ
hx, (α min(µA (x), µB (x)) + γ) , (β(max(νA (x), νB (x))) + η) i : x ∈ X ,
α,β,γ,η (A ∩ B) =
1
1
1
= hx, min((αµA (x)δ + γ) δ , (αµB (x)δ + γ) δ ), max((βνA (x)δ + η) δ , (βνB (x)δ
1
+ γ) δ i : x ∈ X ,
=
α,β,γ,η A
∩
α,β,γ,η B,
The proof is completed. The Proof of (vii) is similar to that of (vi)
(viii) Since
1
1
Pε,θ (A) = hx, max(ε δ , µA (x)), min(θ δ , νA (x))i : x ∈ X ,
we have
α,β,γ,η Pε,θ (A)
δ
1
1
1
δ
δ
δ
δ
hx, (α max ε , µA (x) + γ) , (β min(θ , νA (x)) + η) i : x ∈ X ,
hx, max(αε + γ, αµA (x) + γ) , (min(βθ + η, βνA (x) + η) i : x ∈ X ,
=
=
= Pαε+γ,βθ+η
1
δ
δ
α,β,γ,η
1
δ
A.
The proof is completed. The Proof of (ix) is similar to that of (viii). δ
1
δ
The operators over the generalized intuitionistic fuzzy sets 8 (2017) No. 1, 11-21
21
4. Conclusions
We have introduced level operators and modal-like operators over GIF SB and proved their relationships. An open problem is as follows: definition of negation operator and other types over GIF SB
and study their properties.
References
[1] K.T. Atanassov, Intuitionistic fuzzy sets, Springer Physica-Verlag, Heidelberg, 1999.
[2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96.
[3] K.T. Atanassov, Remark on two operations over intuitionistic fuzzy sets, Int. J. of Uncertainty, Fuzziness Knowledge Syst. 9 (2001) 71–75.
[4] K.T. Atanassov, The most general form of one type of intuitionistic fuzzy modal operators, NIFS. 12 (2006) 36–38.
[5] K.T. Atanassov, Some properties of the operators from one type of intuitionistic fuzzy modal operators, Adv.
Studies Contemporary Math. 15 (2007) 13–20.
[6] K.T. Atanassov, On intuitionistic fuzzy negation and intuitionistic fuzzy level operators, NIFS. 14 (2008) 15–19.
[7] E. Baloui Jamkhaneh and S. Nadarajah, A new generalized intuitionistic fuzzy sets, Hacettepe J. Math. Stat. 44
(2015) 1537–1551.
[8] G. Cuvalcoglu, Some properties of Eα,β operator, Adv. Studies Contemporary Math. 14 (2007) 305–310.
[9] G. Cuvalcoglu., S. Ylmaz and K.T. Atanassov, Matrix representation of the second type of intuitionistic fuzzy
modal operators, Notes Intuitionistic Fuzzy Set. 20 (2014) 9–16.
[10] K. Dencheva, Extension of intuitionistic fuzzy modal operators and , In Intelligent Systems, 2004. Proceedings.
2004 2nd International IEEE Conference (Vol. 3, pp. 21–22). IEEE.
[11] F.G. Lupianez, Intuitionistic fuzzy topology operators and topology, IEEE Symp. Int. J. Pure Appl. Math. 17
(2004) 35–40.
[12] R. Parvathi and S.P. Geetha, A note on properties of temporal intuitionistic fuzzy sets, Notes Intuitionistic Fuzzy
Set. 15 (2009) 42–48.