Extension Principle Based on Neutrosophic Multi-Fuzzy Sets and Algebraic Operations

Journal of Mathematical Extension Vol. 12, No. 1, (2018), 69-90 ISSN: 1735-8299 Journal of Mathematical Extension URL: http://www.ijmex.com Vol. 12, No. 1, (2018), 69-90 ISSN: 1735-8299 URL: http://www.ijmex.com

Extension Principle Based on Neutrosophic Multi-Fuzzy Sets and Algebraic Operations Extension Principle Based on Neutrosophic Multi-Fuzzy Sets and Algebraic Operations Mehmet Şahin

Gaziantep University

Mehmet Şahin Irfan Deli

Gaziantep University Kilis 7 Aralık University

Irfan Deli

Vakkas Uluçay Kilis 7 Aralık University Gaziantep University ∗

Vakkas Uluçay∗

Gaziantep University Abstract. In this paper, we first proposed the extension principles of neutrosophic multi-sets and cut sets which are a bridge between neutrosophic multi-sets and crisp sets. Then the representation theorem Abstract. In this paper, we first proposed the extension principles of of neutrosophic multi-sets based on cut sets are studied. Finally, the neutrosophic multi-sets and cut sets which are a bridge between neuaddition, subtraction, multiplication and division operations over neutrosophic multi-sets and crisp sets. Then the representation theorem trosophic multi-sets are defined based on the extension principle. of neutrosophic multi-sets based on cut sets are studied. Finally, the addition, subtraction, multiplication and03E99 division operations over neuAMS Subject Classification: 03E72; trosophic multi-sets are defined based on set, the multi-fuzzy extension principle.

Keywords and Phrases: Neutrosophic set, extension principle, representation theorem AMS Subject Classification: 03E72; 03E99 Keywords and Phrases: Neutrosophic set, multi-fuzzy set, extension principle, representation theorem

1.

Introduction

There are many mathematical tools that with uncertainties such as; 1. Introduction rough set theory [19], fuzzy set theory [43], intuitionistic fuzzy set theThere mathematical tools[26]. thatA with uncertainties such inas; ory [7] are and many neutrosophic set theory neutrosophic set theory rough set theory [19], fuzzy set [43], intuitionistic fuzzy set thedependently is characterized by theory a truth-membership, a indeterminacyory [7] and neutrosophic set theory [26].Therefore A neutrosophic set theory set inmembership and a falsity-membership. the neutrosophic dependently is characterized by a truth-membership, a indeterminacyReceived: May 2017; Accepted: September 2017 ∗ membership and a falsity-membership. Therefore the neutrosophic set Corresponding author ∗

Received: May 2017; Accepted: September 2017 69 Corresponding author

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MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY

theory has become a popular topic of investigation in the real problems which contain uncertainty. In recent years, the academic community has witnessed growing research interests in neutrosophic set theory [1, 2, 3, 4, 5, 6, 8, 13, 15, 27, 38, 41].

Since membership values are inadequate for providing complete information in some real problems which has different membership values for each element, different generalization of fuzzy sets, intuitionistic fuzzy sets and neutrosophic sets have been introduced is called multi fuzzy set [39], intuitionistic fuzzy multiset [36] and neutrosophic multiset [41, 42], respectively. In the multisets an element of a universe can be constructed more than once with possibly the same or different membership values. Some work on the multi fuzzy set [18, 28, 29, 30, 31, 32, 33, 34, 35], on intuitionistic fuzzy multiset [12, 16, 20, 21, 22, 23, 24, 25, 36] and on neutrosophic multiset [9, 10, 11, 37] have been studied.

As far as we know, however, there is less investigation on the cut sets and extension principle for neutrosophic set theory as well as algebraic operations. It is noted that the algebraic operations defined in this paper are remarkably different from those introduced by Chatterjee et al.

[11], Deli et al [14] and Ye [42]. The purpose of this paper is to develop a bridge between neutrosophic multi-fuzzy sets and crisp sets. To do this, in Section 2, some concepts relate to neutrosophic sets and neutrosophic multi-fuzzy sets are briefly reviewed. In Section 3, the cut sets and extension principles of neutrosophic multi-fuzzy sets are developed. In Section 4, based on the extension principle some arithmetic operations such as; addition, subtraction, multiplication and division operations over neutrosophic multi-fuzzy sets are defined. In last Section, conclusion is made.

2.

Definitions and Preliminary

In this section, we present the basic definitions and results of neutrosophic set theory [2, 38, 41] and neutrosophic multi (or refined) set theory [11, 14, 42] that are useful for subsequent discussions. See especially [2, 1, 9, 10, 11, 27, 37, 38, 41] for further details and background.

EXTENSION PRINCIPLE BASED ... ...

EXTENSION PRINCIPLE BASED ONON ...ON EXTENSION PRINCIPLE BASED

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Definition 2.1. ([38]) Let (objects), a generic Definition 2.1.

Let EEbe of of points (objects), withwith awith generic Definition 2.1.([38]) ([38]) Let Ebeabeaspace aspace space ofpoints points (objects), a generic element in E denoted by u.

A neutrosophic set (N-set) in U is characelement in u. u.

A neutrosophic set (N-set) in U in is characelement inEEdenoted denotedbyby A neutrosophic set (N-set) U is characterized by truth-membership function a, aindeterminacy-membership terized by function TNT,N terized byaaatruth-membership truth-membership function TaN,indeterminacy-membership indeterminacy-membership function and falsity-membership function T INI(x) and function IIINNN and function FNF. NF T.N (x), and function andaaafalsity-membership falsity-membership function . NT(x), N (x) NI(x), N (x) and − + − + − + FN (x) are real standard nonstandard subsets 1 1[. [.

N (x) oror nonstandard subsets of ]ofof 0,] 1]0,[.0, F (x) are arereal realstandard standard or nonstandard subsets It can be written as It can can be be written writtenasas N= = {< x, (T (x), IIN(x), (x),FF (x))>:>: xxE, ∈∈E, TN (u), INI(x), FNF(x) [0,[0, 1]}.1]}.

N IN (x)) x>:∈ TE, I(u), ⊆ [0,⊆1]}.

N = {< {<x, x,(T (TNNN(x), (x), TN ⊆ NF N (u), N (x), N (x) N (x), N (x)) NF(x), N (x)

−so There is no restriction the sum TT (u); (u) FNF(u), NN NIN There is onon the sum of of Tof IN I(u) and Fand 0 so  0−0  There isno norestriction restriction on the sum (u); (u)and (u), N (u); N (u), Nso + +3. + supT (u) supI (u) supF (u) supTNN (u) +++ supF (u)  3 supT (u)++ +supI supI (u) supF 3. .

NNN NN N(u) N (u) + + Here, = 1+ε, where standard part ε εits Here, 111+= isisis itsits standard part andand ε its non-standard Here, = 1+ε, 1+ε,where where1 11 its standard part and itsnon-standard non-standard −−0= 1+ε, where 0 is its standard part and ε its non− part.

Similarly, where 0 is0 its part part and εand its nonpart. Similarly, 0=1+ε, 1+ε, where is standard its standard ε its nonpart.

Similarly, 0= standard part. standard part. standard part. −

Definition 2.2.

[2] Let and neutrosophic onon E.E. Then, Definition 2.2.

NNN and MMbe neutrosophic set set onset E. Then, Definition 2.2.[2] [2]Let Let and Mbetwo betwo two neutrosophic Then, 1.

A neutrosophic set with (u) (u) 1 is NN NIN N (u) 1.

NNN with TNT(u) == 1,=1, IN1,I(u) =(u) 1=,F (u) = 1=is= 1. A A neutrosophic neutrosophicset set with T (u) =1N1,F ,F 1 is N (u) called normal neutrosophic set. called set.set. callednormal normalneutrosophic neutrosophic 2.

When the support set number following applies 2. set isisaisareal number setset and the following applies 2. When Whenthe thesupport support set areal real number setand andthe the following applies for all u ∈ [a, b] over any interval [a, b]; T (u)  T (a) ∧ T N M M for b] b]over anyany interval [a, b]; TM(a)TM ∧T for all alluu∈∈[a,[a, over interval [a,Tb]; TN(u) (a) ∧ T(b), N (u) M (b), M (b), (u)  I (a) ∧ I (b) and F (u)  F (a) ∧ F (b) M M N M M IIINN (u)  I (a) ∧ I (b) and F (u)  F (a) ∧ F (b) (u)  I (a) ∧ I (b) and F (u)  F (a) ∧ F (b) MM MM N N M M M N M A is said to be convex.

A A isissaid saidtotobebeconvex. convex.

3.

For neutrosophic set (T(T (u), INI(u), FNF(u)) u∈ NN 3. set NNN =={< u, u, (Tu,N (u), I(u), u>:∈>: ={< {< (u)) u∈ 3. For For aaaneutrosophic neutrosophic set N (u), N (u)) NF(u), N>: E} E} E} N = {u eitherT (u), αα orNor F 1α,− α,∈α, α∈ NN N N N eitherT INI(u) >> α> or F (u) 1<−< Nααα = ={u {u: ::uuu∈∈∈U,U, U, eitherT (u), IN(u) (u) F(u) 1α− α∈ N (u), N<(u) − + + + ]]]−−0, 0, 1 [} and 0,11 [}[}and and = {u eitherT (u), αα orNor F 1α,− α,∈α, α∈ N NN N N eitherT INI(u)  α or F (u) 1− N ={u {u: ::uuu∈∈∈U,U, U, eitherT (u), IN(u) (u) F(u) 1α− α∈ Nααα = N (u), N(u) + + −0, + 0, 1 [} are called the weak and strong cut respectively. ]]]−− thethe weak andand strong cut respectively.

0,11 [}[}are arecalled called weak strong cut respectively. : :X→ subset of of XX 4.

Extending the function f f:fX Y→,Ythe neutrosophic subset N ofNXN 4.

X→ Y, ,the theneutrosophic neutrosophic subset 4. Extending Extendingthe thefunction function is made correspond subset f (N ){T = {T , I)f, (N , Ff, (N } )} f,(N )(N) is totoneutrosophic subset f(N) Iff(N F ismade madetoto tocorrespond correspond toneutrosophic neutrosophic subset f= (N ) f= {T Iff)(N Ff)(N (N) (N))} of Y may the following ways of ways of Y Ymay maybebe bethe thefollowing following ways  −1−1 −1 (y) = ∅, −1f(y) ∨{T (x) ∨{T (x) : x: ∈ f −1 iffif =(y) ∅, = ∅, ∨{T :xx∈ ∈f (y)}, f (y)}, (y)}, if f −1 NN N (x) T (y) = (N ) TTff(N) (y) = (y) = f (N ) 0, otherwise 0, 0, otherwise otherwise

72

MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY

If (N ) (y) = Ff (N ) (y) =

3.





∧{IN (x) : x ∈ f −1 (y)}, if f −1 (y) = ∅, 0, otherwise ∧{FN (x) : x ∈ f −1 (y)}, if f −1 (y) = ∅, 0, otherwise.

Extension Principles and Cut Sets for Neutrosophic Multi-Sets

In this section, we present the extension principles of neutrosophic multisets and the representation theorem of neutrosophic multi-sets based on cut sets by extending the study are given in [2, 17].

Definition 3.1. Let E be a universe and Np = {1, 2, ...p}. A neutrosophic multi-fuzzy set(Nmfs) A on E is the set A = {x, (TA1 (x), TA2 (x), ..., TAp (x)), (IA1 (x), IA2 (x), ..., IAp (x)), (FA1 (x), FA2 (x), ..., FAp (x)) : x ∈ E}, where TAi (x), IAj (x), FAk (x) ∈ [0, 1] with 0  supTAi (x) + supIAj (x) + supFAk (x)  3 for i, j, k ∈ Np .

(TA1 (x), TA2 (x), ..., TAP (x)), (IA1 (x), IA2 (x), ..., IAP (x)) and (FA1 (x), FA2 (x), ..., FAP (x)) are the truth-membership sequence, indeterminacy-membership sequence and falsity-membership sequence of the element x, respectively.

Also, p is called the dimension (cardinality) of Nmfs A and it is denoted by d(A).

Definition 3.2. If a neutrosophic multi-sets A satisfies the following conditions, we call it a neutrosophic multi-numbers.

1. A neutrosophic multi-sets A with TA1 (u) = TA2 (u) = ... = TAP (u) = 1, IA1 (u) = IA2 (u) = ... = IAP (u) = 0, FA1 (u) = FA2 (u) = ... = FAP (u) = 0 is called normal neutrosophic multi-sets.

2. When the support set is a real number set and the following applies for all u ∈ [a, b] over any interval [a, b]; TAi (u)  TBi (a) ∧ TBi (b), i (a) ∨ I i (b) and F i (u)  F i (a) ∨ F i (b) IAi (u)  IB B A B B

A is said to be convex (TAi ; convex, IAi and FAi ; concave), (i ∈ {1, 2, ..., p}).

EXTENSIONPRINCIPLE PRINCIPLEBASED BASEDON ON EXTENSION ... ...

73 73

1 (u), T 2 P 1 2 1 (u), {<u,u,(T(T Definition 3.3. Let Let AA=={< TA2 (u), ...,..., TAPT(u)), (IA1(I(u), IA2 I(u), Definition 3.3.

A (u), A (u)), A (u), A (u), AA P (u)), (F11 (u), F22 (u), ..., FPP (u)) >: u ∈ U }, {T i i i P ..., IA (FAA(u), FAA(u), ..., FAA(u)) >: u ∈ U }, {TAi (u), IAi I(u), FAiF(u) ∈ ∈ A (u)), A (u), A (u), A (u) [0, 1]}, (i (i ∈ ∈ {1, {1,2, 2,..., ...,p}) p})bebeany anyneutrosophic neutrosophicmulti-sets multi-setsonon Then, U.U. Then, ex-extending the the function functionff : :XX→ →YY, ,the theneutrosophic neutrosophic multi-subsets A of X is multi-subsets A of X is i i i, I i iFi f (A) = {T , made to to correspond correspondto toneutrosophic neutrosophicmulti-subsets multi-subsets f (A) = {Tf (A) , If (A) , Ff (A)f (A) } } f (A) f (A) of Y may may be be the the following followingways ways



i (y)== TTfif(A) (A)(y)



i (y)== IIfif(A) (A)(y)



Ffifi(A) (y)== F (A)(y) 1, 2, 2,...p. ...p. for i = 1,

(x): :xx∈∈f f−1−1 (y)}, ifif f −1 ∨{TAiAi(x) (y)}, f −1 (y)(y) = = ∅, ∅, ∨{T 0, otherwise 0, otherwise i (x) : x ∈ f −1 (y)}, if−1 −1 ∧{IAi A (x) : x ∈ f −1 (y)}, if f f (y)(y) = = ∅, ∅, ∧{I otherwise 1,1, otherwise

∧{FAiAi(x) (y)}, ifif f −1 (x): :xx∈∈f −1 f −1 ∧{F (y)}, f −1 (y)(y) = = ∅, ∅, otherwise 1,1, otherwise

1 (u), T 2 P 1 2 1 (u), {<u,u,(T(T Definition 3.4. Let Let AA=={< TA2 (u), ...,..., TAPT(u)), (IA1(I(u), IA2 I(u), Definition 3.4.

A (u), A (u)), A (u), A (u), AA P (u)), (F11 (u), F22 (u), ..., FPP (u)) >: u ∈ U }, {T i i i P ..., IA (FAA(u), FAA(u), ..., FAA(u)) >: u ∈ U }, {TAi (u), IAi I(u), FAiF(u) ∈ ∈ A (u)), A (u), A (u), A (u) [0, 1]}, (i (i ∈ ∈ {1, {1,2, 2,..., ...,p}) p}) bebeany anyneutrosophic neutrosophicmulti-sets multi-setsononU.U. For For anyany ordered aa (α, (α,β, β,γ) γ) where whereα,α,β,β,γγ∈∈[0,[0,1],1],soso0 0α α Then ++ ββ ++ γ γ  3. 3.

Then (α, β, γ)-cut γ)-cut set set of of neutrosophic neutrosophicmulti-sets multi-setsAAis isdenoted denotedbyby A A(α,β,γ) , is, is (α,β,γ) defined by by i i i (u)  γ, u ∈ U }, A(α,β,γ) ={u {u: :TTAiAi(u) (u)α,α,IAiIA (u)β,β, A (u) FAF (u) (α,β,γ) = A  γ, u ∈ U },

That is; is; i i i (u) ∨ γ = γ, u ∈ U }.

A(α,β,γ) = {u {u :: TTAiAi(u) (u)∧∧αα==α,α,IAIi A (u)∨∨β β==β,β, (u) FAF (u) (α,β,γ) = A ∨ γ = γ, u ∈ U }.

(α,β, β,γ)-cut γ)-cut set setofofneutrosophic neutrosophicmulti-sets multi-setsA Ais is denoted The strong strong (α, denoted by by is defined defined by by A(α,β,γ),, is (α,β,γ) i i i (u) < γ, u ∈ U }, A(α,β,γ) = ={u {u: :TTAiAi(u) (u)>>α,α,IAiIA (u)<<β,β, A (u) FAF (u) A < γ, u ∈ U }, (α,β,γ) 1 (u), T 2 P 1 2 1 (u), Definition 3.5.Let LetAA=={< {<u,u,(T(T Definition 3.5.

TA2 (u), ...,..., TAPT(u)), (IA1(I(u), IA2 I(u), A (u), A (u)), A (u), A (u), AA

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MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY

P (u)), (F 1 (u), F 2 (u), ..., F P (u)) >: u ∈ U }, {T i (u), I i (u), F i (u) ∈ ..., IA A A A A A A [0, 1]}, (i ∈ {1, 2, ..., p}) be any neutrosophic multi-sets on U and A(α,β,γ) be any (α, β, γ)-cut set of neutrosophic multi-sets A. Then,

1. A((1,1,...,1),(0,0,...0),(0,0,...,0)) is called the core of neutrosophic multisets A. 2. A((0,0,...,0),(1,1,...1),(1,1,...,1)) is called the support of neutrosophic multisets A. Remark 3.6. For any ordered (α, β, γ), a crisp set on X can always be derived from the neutrosophic multi-sets A. Its an inverse image of the neutrosophic multi-sets A at the confidence cut (α, β, γ).

1 (u), I 2 (u), Theorem 3.7. Let A = {< u, (TA1 (u), TA2 (u), ..., TAP (u)), (IA A P (u)), (F 1 (u), F 2 (u), ..., F P (u)) >: u ∈ U }, {T i (u), I i (u), F i (u) ∈ ..., IA A A A A A A [0, 1]}, (i ∈ {1, 2, ..., p}) be any neutrosophic multi-sets on U. Then, (α1 , β1 , γ1 )  (α2 , β2 , γ2 )

1. A(α2 ,β2 ,γ2 ) ⊆ A(α1 ,β1 ,γ1 ).

2. A(α2 ,β2 ,γ2 ) ⊆ A(α1 ,β1 ,γ1 ).

3. A(α2 ,β2 ,γ2 ) ⊆ A(α1 ,β1 ,γ1 ).

Here, (α1 , β1 , γ1 )  (α2 , β2 , γ2 ) means that α1  α2 , β1  β2 and γ1  γ2

Proof. Here, (α1 , β1 , γ1 )  (α2 , β2 , γ2 ) means that α1  α2 , β1  β2 and γ1  γ2 i (u)  β , F i (u)  γ , u ∈ U } A(α2 ,β2 ,γ2 ) = {u : TAi (u)  α2 , IA 2 2 A i (u)  β , F i (u)  γ , u ∈ U }  {u : TAi (u)  α1 , IA 1 1 A ⊆ A(α1 ,β1 ,γ1 ).

Therefore A(α2 ,β2 ,γ2 ) ⊆ A(α1 ,β1 ,γ1 ).

2. i (u) < β , F i (u) < γ , u ∈ U } A(α2 ,β2 ,γ2 ) = {u : TAi (u) > α2 , IA 2 2 A i i i < {u : TA (u) > α1 , IA (u) < β1 , FA (u) < γ1 , u ∈ U } ⊆ A(α1 ,β1 ,γ1 ).

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A ⊆⊆AA 1 ,β1 ,γ1 ).

A(α Therefore (α2 ,β ,β22,γ ,γ22))⊆ A (α (α1 ,β1 ,γ1 ).

Therefore A (α22,β (α1 ,β1 ,γ1 ).

2 ,γ2 ) 3. It is proved similarly. proved similarly.

3. It is proved similarly.  3.8.

Suppose that {u 0.5, 0.4, 0.3), (0.3, 0.2, 0.4, 0.5), Example 3.8.Suppose Supposethat thatAAA==={u {u1:1:(0.6, :(0.6, (0.6, 0.3), Example 3.8.

0.5,0.5, 0.4,0.4, 0.3), (0.3,(0.3, 0.2, 0.2, 0.4, 0.4, 0.5),0.5), 1 (0.4, 0.6, 0.8, 0.7), u : (0.5, 0.8, 0.3, 0.4), (0.6, 0.4, 0.3, 0.5), (0.3, 0.5, 0.4, 0.6), 0.8,0.7), 0.7),uu22: :(0.5, (0.5,0.8, 0.8,0.3, 0.3, 0.4), (0.6, 0.5), (0.4, 0.6, 0.8, 0.4), (0.6, 0.4,0.4, 0.3,0.3, 0.5), (0.3,(0.3, 0.5, 0.5, 0.4, 0.4, 0.6),0.6), 2 u33 : (0.7, 0.2, 0.4, 0.5), (0.2, 0.6, 0.1, 0.4), (0.1, 0.3, 0.7, 0.5)}.

Let (α(α = 0.2,0.4, 0.4,0.5), 0.5),(0.2, (0.2,0.6, 0.6, 0.1, 0.4), (0.1, 0.5)}. u3 : (0.7, 0.2, 0.1, 0.4), (0.1, 0.3,0.3, 0.7,0.7, 0.5)}.

Let Let (α1 1=1 = = 0.4, = 0.3) [0, and (α(α β2β2==0.1, γ2γ= 0.2) ∈∈ 2 2==0.7, 0.5, 0.4,γγγ11 = = 0.3) 0.3)∈∈∈[0, [0,1]1]1]and and(α 0.7, 0.1, 0.2) 2 = 0.5, β β111 = 0.4, 1 2 = 0.7, β2 = 0.1, γ2 = 0.2) ∈ [0, 1].Then [0, 1].Then = {u , u , u } and A(α(α = {u3 }.3 }. It Itis isclear 1.A clearthat that (α11,β 2 ,β,β 2 ,γ,γ 2 ) ) = {u ,β1 ,γ ,γ1 )) = {u11 , u22 , u33 } and A 1.A(α (α1 ,β11,γ11) = {u1 , u2 , u3 } and A(α2 ,β22 ,γ22 ) 2= {u3 }. It is clear that A(α ⊆ A .

2 ,β ,β2 ,γ ,γ2 )) ⊆ A(α (α1 ,β ,β1 ,γ,γ1 )) .

A(α (α22 ,β22 ,γ22) ⊆ A(α11,β1 1,γ1 )1. = {u ,,uu2 ,,uu33}}and =={u{u A A2 ,β2 ,γ2 ) ⊆ ⊆ 1 2 }.Therefore 2.A = {u andAAA(α(α2 ,β 2 }.Therefore (α11,β 2 ,γ,γ 2 )= ,β11,γ ,γ11)) = {u11, u2 2 ,β ) {u2 }.Therefore (α2 ,β2⊆ ,γ2 ) 2.A(α , u3 } and A(α(α (α2 ,β22 ,γ22 ) 2 2 ,β2 ,γ2 ) A(α1(α,β11,β,γ11,γ) .1 ) A(α1 ,β1 ,γ1 ). (α1 ,β1 ,γ1 )

11 2 (u), P (u)), 1 1 I 2 (u), 2 P 2 Definition 3.9.

Let AA=={< u,u,(T (u), T2T ...,..., TPAT (I Definition 3.9.Let LetA {<u, (T (I(u), 1A(u), 1 (u), A(u), A AIA (u), A (u), A (u), A (u)), A (u), Definition 3.9. = {< (T T ..., T (u)), (I IA2i (u), A A A A P 1 2 P i i P (u)), (F P (u))>: i i i ∈ ..., IA },}, {Ti{T F (FA1A1(u), (u),FF2AA2(u), (u),..., ...,FF >: uu∈∈UU P PAA(u)) iIAI(u), i AF(u) A (u), A (u), A (u), A (u) ∈ ..., IA A (u)), (FA (u), FA (u), ..., FA (u)) >: u ∈ U }, {TA (u), IA (u), FA (u) ∈ (i ∈ {1, 2, ..., p}) be any neutrosophic multi-sets on U.U.Then α[0, ∈ {1, {1,2, 2,..., ...,p}) p})be beany anyneutrosophic neutrosophic multi-sets Then [0, 1]}, 1]}, (i ∈ multi-sets on on U. Then α- αcut, β-cut, γ-cut, (α, β)-cut, (α, γ)-cut and (β, γ)-cut set of neutrosophic γ-cut,(α, (α,β)-cut, β)-cut,(α, (α, γ)-cut and γ)-cut of neutrosophic cut, β-cut, γ-cut, γ)-cut and (β,(β, γ)-cut set set of neutrosophic A is denoted by AAαα, ,AAβ β, ,AA ), A ), A ). ). respecγ ,γ ,AA multi-sets A is denoted by ), A ), A). respec(α,β (α,γ (β,γ (β,γ multi-sets A is denoted by Aα , Aβ , Aγ , A(α,β(α,β ), A(α,γ(α,γ ), A(β,γ respectively, is defined by; defined by; by; tively, is defined

A = {u : T A(u)  α,α,u u∈∈UU }, (u) AAααα=={u{u: T: ATA(u)  α, u ∈ U }, }, A = {u : I A(u)  β,β,u u∈∈UU }, (u) AAβββ=={u{u: I:AIA(u)  β, u ∈ U }, }, A = {u : F A(u)  γ,γ,u u∈∈UU }, (u) AAγγ=={u{u: F: FA(u)  γ, u ∈ U }, }, γ

A

A(α,β) = {u α, IAIA (u) β,β, u ∈ U }, {u:: T :TTA(u) (u) A(u) (α,β)=={u AA(α,β) α,α, IA (u)(u)  β, u ∈uU∈},U }, A A (α,γ) = {u α, FF (u) γ,γ, u ∈ U }, {u:: T :TTA(u) (u) (u) A(u) A AA(α,γ) =={u α,α, F A(u)  γ, u ∈uU∈},U }, (α,γ)

A

A

A(β,γ) = {u β, FF (u) γ,γ, u ∈ U }. {u:: I:IIA(u) (u) (u) A(u) A (β,γ)=={u AA(β,γ) β,β, FAA(u)  γ, u ∈uU∈}.U }.

A

{u 0.5, 0.2), (0.5, 0.2, 0.4, 0.3), Example 3.10.

Suppose that 1 1: :(0.4, {u (0.4,0.3, 0.3, 0.5, 0.2), (0.5, 0.2, 0.4, 0.3), 3.10.Suppose Supposethat thatAAA==={u Example 3.10.

1 : (0.4, 0.3, 0.5, 0.2), (0.5, 0.2, 0.4, 0.3), 0.6, 0.5, 0.2), uu22 ::(0.8, 0.4, 0.3, 0.6), (0.4, 0.3, 0.6, 0.1), (0.7, 0.3, 0.4, 0.2), (0.3, 0.5, 0.2), (0.8, 0.4, 0.3, 0.6), (0.4, 0.3, 0.6, 0.1), (0.7, 0.3, 0.4, 0.2), (0.3, 0.6, 0.5, 0.2), u2 : (0.8, 0.4, 0.3, 0.6), (0.4, 0.3, 0.6, 0.1), (0.7, 0.3, 0.4, 0.2), u33 : (0.7, 0.6, 0.5, 0.8), (0.1, 0.6, 0.4, 0.2), (0.5, 0.4, 0.6, 0.8)}.

Let (α(α == 0.6, β= 0.6,0.5, 0.5,0.8), 0.8),(0.1, (0.1,0.6, 0.6,0.4, 0.4, 0.2), (0.5, 0.8)}. u3 : (0.7, 0.6, 0.2), (0.5, 0.4,0.4, 0.6,0.6, 0.8)}.

Let Let (α = 0.6, β0.6, =β = = 0.3) ∈ [0, 1] Then, 0.4, γ 0.3) ∈ [0, 1] Then, 0.4, γ = 0.3) ∈ [0, 1] Then,

76

MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY

Aα = {u2 , u3 }, Aβ = {u1 , u2 , u3 }, Aγ = {u1 , u2 }, A(α,β) = {u2 , u3 }, A(α,γ) = {u2 }, A(β,γ) = {u1 , u2 }.

1 (u), I 2 (u), Theorem 3.11. Let A = {< u, (TA1 (u), TA2 (u), ..., TAP (u)), (IA A P (u)), (F 1 (u), F 2 (u), ..., F P (u)) >: u ∈ U }, {T i (u), I i (u), F i (u) ∈ ..., IA A A A A A A [0, 1]}, (i ∈ {1, 2, ..., p}) be any neutrosophic multi-sets on U and A(α,β,γ) be a set of (α, β, γ)-cut. Then,

1. A(α,β) = Aα ∩ Aβ .

2. A(β,γ) = Aβ ∩ Aγ .

3. A(γ,γ) = Aγ ∩ Aγ .

4. A(α,β,γ) =Aα ∩ A(β,γ) .

5. A(α,β,γ) =A(α,γ) ∩ Aβ .

6. A(α,β,γ) =A(α,β) ∩ Aγ .

7. A(α,β,γ) =Aα ∩ Aβ ∩ Aγ .

Proof. It is trivial.  Theorem 3.12. Let A, B, Ai and Bi (i ∈ I) be any neutrosophic multisets on U. Then, 1. (A ∪ B)(α,β,γ) = A(α,β,γ) ∪ B(α,β,γ).

2. (A ∪ B)(α,β,γ) = A(α,β,γ) ∪ B(α,β,γ).

3.

4.

 

i (Ai )(α,β,γ) i (Ai )(α,β,γ)

 ⊆ ( i Ai )(α,β,γ).  = ( i Ai )(α,β,γ).

EXTENSION PRINCIPLE BASED ON ...

EXTENSION PRINCIPLE BASED ON ...

EXTENSION PRINCIPLE EXTENSION PRINCIPLE BASEDBASED ON ... ON ...

77 77

77 77

Proof.

Proof.

Proof.

1.Let LetA A B any be any neutrosophic multisets on U. Then, 1.Proof. andand B be neutrosophic multisets on U. Then, 1.

Let A and B be any neutrosophic multisets on 1. Let A and B be any neutrosophic multisets on U. Then,U. Then, i i i i : T(u) i α, i (u) B(α,β,γ)= {u =: {u β, FA∪B AA∪∪B(α,β,γ) TA∪B  (u) α, IA∪B (u)IA∪B  β,(u) FA∪B γ, u ∈(u) U} γ, u ∈ U } A∪B i i i i i i γ, (u) i i i i i i = {u : T (u)  α, I (u)  β, F (u)  Uγ} = γ, u ∈ U } ∧ α = α, I (u) ∨ β = β, AA∪∪BB = {u : T (u)  α, I (u)  β, F (u)  γ, u ∈ ∧ α = α, IA∪BA∪B (u) ∨A∪B β A∪B = β, FA∪B (u) ∨ F γU} = γ, u ∈∈∨ U} (α,β,γ)= {u : TA∪B (α,β,γ) A∪B A∪B A∪B A∪B A∪B(u) A∪B i i i i i i i i i i i i i i = ::{u : T(TA∪B αα(u)) I(Iα ∨A =∨(u) β, ∨=∧Tα, =β (u) α, (I (u) ∨ ∨uβ∨∈=γU} β= γ, u ∈ U } = {u {u TA∪B (u) α(u)) I= (u) = FβA∪B ∨ γ = (u) γ, = (T ∨ T∧B(u) ∧B = α, α,∧ ∨ Iβ, β I=F β(u)) A∪B A (u) BA∪B A∪B A (u) A∨(u) B (u)) i (u) i (u)) ∧ α = ii (u) ∨ I i (u)) ∨ β = β i (u) ii(u)) i (u) i∈ i (u)) = : (T ∨ T α, (I ,Ai (F ∨ F ∨ γ = γ, u U } :{u (T ∨ T (u)) ∧ α = α, (I (u) ∨ I (u)) ∨ β = β ,=(F{u (u) ∨ F ∨ γ = γ, u ∈ U} B A B A A A A AB A B i (u))i ∨ γ =iγ, u ∈ iU } i (u) i (u) ∨ γ = γ, u ∈ U }∨ i (u) ,= ∨ : FTAiA = α, I (u) ∨ β = β, F , (F ∨(u) (u)) ∨∧α,γαI= γ, u ∈ U} = {u : {u TAiAi(u) ∧ αFA = (u) ∨ β = β, F (u) ∨ γ = γ, u ∈ U}∨ A A A(F A A i (u) ∧ iα = i (u) ∨ i (u) ∨ γ = γ, u ∈ U }∨ iT(u) i I(u) iF(u) i:(u) i (u) i= i (u) α, β β, {u :(u) ∧ α = α, I ∨ β = β, F ∨ γ ∈=U}∨ γ, u ∈ U } = {u TT ∧ α = α, I ∨ β = β, F (u) ∨ γ = γ, u ∧ α = α, I (u) ∨ β = β, F ∨ γ = γ, u ∈ U} {u := TBi:{u BA AB A B A B A B A i (u) ∧ α =i α, I i (u) ∨ β i= β, F i (u) ∨ γ = γ, u ∈ U } i: (u) {u T = A ∪ B . ∧ α = α, I (u) ∨ β = β, F (u) ∨ γ = γ, u ∈ U} {u : T = A(α,β,γ) ∪ B(α,β,γ) .(α,β,γ) B B B B B B (α,β,γ) =(α,β,γ) A(α,β,γ) ∪ B(α,β,γ) . =A ∪ B(α,β,γ) .

LetA A B any be any neutrosophic multisets on U. Then, 2.2.Let andand B be neutrosophic multisets on U. Then, LetAAand and B be neutrosophic multisets on U. Then, 2.2.Let B be anyany neutrosophic multisets on U. Then, i i i i :T i> α, i <(u) β,<Fγ, (A∪ ∪ = {u=: {u TA∪B (u) > (u) α, IA∪B (u)IA∪B < β, (u) FA∪B u ∈(u) U}< γ, u ∈ U } (A B)B) A∪B A∪B (α,β,γ) (α,β,γ) i i i i i i i i i γ, (u) i i i =::{u : T(u) (u) F< < (A∪∪B)B) {u (u) α, (u) < Iβ, (u) γ, (A ∧A∪B α =IA∪B α, (u) βA∪B = u(u) β, ∨Uγ} = γ, u ∈ U } == {u TTA∪B ∧>α = I> α, Iα, (u) ∨(u) βFA∪B =< β,β, F∨A∪B (u) ∨∈γFU} = γ, uu∈∈U} A∪B A∪B (α,β,γ) (α,β,γ) A∪B A∪B A∪B A∪B i i i i i i i i i i i i i = {u : T (u) ∧ α = α, I (u) ∨ β = β, F (u) ∨ (T (u) ∨ T (u)) ∧ α = α, (I (u) ∨ I (u)) ∨ β {u::(T TA∪B = α, β= FAA∪B∨(u) γ = γ, u ∈=γ U}β= γ, u ∈ U } == {u ∨AT∧Bα(u)) ∧BαIA∪B = (u) (IA∪B ∨ Iβ,B (u)) β =∨BβA∪B A∪B A (u)(u) A∨(u) i i i i i i i i i i i =(F :F(u) (T ∨=TB ∧ (u) ∨ ∨ FT∨ ∈α, U∨(I }IA :{u (T ∧∨(u)) α,γ,α (IuA=(u) ∨ IβB=(u)) β ∨β =β ,=(F{u ∨(u) γ(u)) γ, uαγ∈==U} A∨(u) A A A B(u)) B (u)) A, (u) A (u)) ii (u) i∨ i (u))i ∨ γ =iγ, u ∈i U } i (u) i (u) ∨ γ = γ, u ∈ U }∨ i(F F ∧ α = α, I (u) ∨ β = β, F {u : T (u) ∨ F (u)) ∨ γ = γ, u ∈ U} =, (F {u,= : T (u) ∧ α = α, I (u) ∨ β = β, F (u) ∨ γ = γ, u ∈ U}∨ A A A A AA A A A A i (u) i∨ βi = β, iF i (u) ∨ γ = γ, u ∈ U }∨ iT(u) i I(u) i :(u) i (u) iα ∧Iα, = α, {u = α, I ∨ β = β, F (u) γ∈=U}∨ γ, u ∈ U } = {u T: AT I ∨ β = β, F (u) ∨ γ =u γ, {u := TBi:{u (u) ∧ αAi∧(u) =α∧α,=α (u) ∨ β = β, F (u) ∨ γ = γ, ∈∨uU} A A B B B B A B A i (u) ∧ α = i (u) ∨ β i = β, F i (u) ∨ γ = γ, u ∈ U } iA i (u) : T α, I ∪ B . := T (u) ∧ α = α, I ∨ β = β, F (u) ∨ γ = γ, u ∈ U} ∪ B . ={uA{u B (α,β,γ) (α,β,γ) B B B (α,β,γ) B B (α,β,γ) ∪ B(α,β,γ) .

A(α,β,γ) = A=(α,β,γ) ∪ B(α,β,γ) .

Theproofs proofs of (3) (4)becan be similarly. made similarly.  The of (3) and and (4) can made  Theproofs proofs (4) be canmade be made similarly.  The of of (3)(3) andand (4) can similarly.  Bibe (i any ∈ I)neutrosophic be any neutrosophic multiTheorem 3.13.

LetB,A,AiB, andAB (i ∈ I) multiTheorem 3.13.

Let A, i iand A, AB, Ai Band I) neutrosophic be any neutrosophic multiTheorem 3.13.

A, B, multiTheorem 3.13.

LetLet setsononU.U. Then, sets Then, i (ibe∈any i and i (i ∈BI) setsononU.U.Then, Then, sets ∩ (α,β,γ) B)(α,β,γ) = A(α,β,γ) ∩ B(α,β,γ).

1.1.(A(A ∩ B) = A(α,β,γ) ∩ B(α,β,γ). = A ∩ B(α,β,γ).

1.1.(A(A ∩∩ B)B) = A ∩ B (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ). = A(α,β,γ) ∩ B(α,β,γ). ∩ (α,β,γ) B)(α,β,γ) 2.2.(A(A ∩ B) = A(α,β,γ) ∩ B(α,β,γ).

A ∩ B(α,β,γ). =A= ∩ B(α,β,γ).

2.2. (A(A B)B) ∩ ∩ (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) ⊆i A( 3.3. ⊆( i )(α,β,γ) i )(α,β,γ) i )(α,β,γ). i (A i Ai )(α,β,γ). i (A Ai )(α,β,γ).

3.3. ⊆ (⊆i (A i )(α,β,γ) i )(α,β,γ) i )i(α,β,γ). i (A i (A =i A( 4.4. =( i )(α,β,γ) i )(α,β,γ) i )(α,β,γ). i (A i Ai )(α,β,γ). i (A Ai )(α,β,γ). = (=i (Ai )i(α,β,γ).

4.4. i (A i ) i )(α,β,γ) i (A Proof. (α,β,γ) Proof.

Proof.

1.We Wecan can 1.Proof. seesee thatthat Wecan canseesee that 1.1.We that

7878 78 78

MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY MEHMET MEHMET ŞAHIN, ŞAHIN, IRFAN IRFAN DELI AND DELI VAKKAS AND VAKKAS ULUÇAYULUÇAY

i i i (u) i α, I i i (u) i i(u) i α, i i i U} AA BB(α,β,γ) ==={u : :T:T{u (u) Fβ, γ, u ∈γ, AA ∩∩∩ ∩ {u (u) (u) α,α, IA∩B (u)  FA∩B (u) u∈ U}U} (α,β,γ) A∩B A∩B A∩B B(α,β,γ) {u = TA∩B :T (u) I Iβ, β, (u) FA∩B β, (u) F γ, u(u) ∈  γ, u ∈ U } (α,β,γ) A∩B A∩B A∩B A∩B A∩B i i i i i i i i i i i ==={u : :T:T{u (u) ∧∧ α∧α =(u) α,=α, I∧A∩B (u) ∨ β = β, F (u) ∨ γ = γ, ui= {u (u) = I (u) ∨ β = β, F (u) ∨ γ γ,U} u ∈u ∨ U} A∩B A∩B {u = TA∩B : T (u) α α, α I = α, (u) I ∨ β (u) = β, ∨ F β = (u) β, F ∨ γ∈= (u) γ, ∈ γU}= γ, u ∈ U } A∩B A∩B A∩B A∩B A∩B i A∩B i A∩B A∩B i i i i i i i i i i i i i i ==={u : (T (u) ∧ T (u)) ∧ α = α, (I (u) ∧ I (u)) ∨ β = β {u : (T (u) ∧ T (u)) ∧ α = α, (I (u) ∧ I (u)) ∨ β = β AA : (u) Bα,∧(I {u = :{u (T (TA∧B(u) TBB (u)) ∧ TB∧(u)) α = A∧ α,α(I =(u) (u))∧∨ IβB=(u)) β ∨β =β AA BIA B(u) i i i ∧iFAi (u)) ,,(F ∨Fγ∨i∨= γ, uγ, ∈uγU} FFAi Ai(u)) γ = ∈ U} AA(u) A ,(F (F ,A(u) (F (u)A∧∧(u) ∧(u)) (u)) γ = ∨ γ, u = ∈ U} γ, u ∈ U } A i i (u) i (u)i ∨i γ = γ, u i∈ U}∧ i i ∧∧ i ∨ α, iβ, ==={u : :T:T{u α,=∧ IAα, β∨=∨ F=β, {u α=α= α, IAiI= (u) FAFβ(u) γF γ, u∈ U}∧ AT Aβ, {u = :(u) TαAi∧(u) α(u) IβA β= (u) ∨ (u)∨β,∨ γ=A= (u) γ, u∨∈ γ U}∧ = γ, u ∈ U }∧ AA(u) A (u) A= i i i i i i i i i i i i (u) ∧ α = α, I (u) ∨ β = β, F (u) ∨ γ = γ, u ∈ U} {u : T ∧∧(u) αα==∧α,α, (u) ∨ β = β, F (u) ∨ γ = γ, u ∈ U} {u TBTBB(u) BαIB B I= (u) α, I ∨ β (u) = β, ∨ F β = (u) β, ∨ F γ = (u) γ, u ∨ ∈ γ U} = γ, u ∈ U } {u: :{u :(u) TB B B B B B ===AA(α,β,γ) ∩∩B∩B . B. . (α,β,γ) A= A B ∩ . (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ)(α,β,γ)(α,β,γ)

2.2.We We have 2.

2.

We We have have i i i (u) > i i (u) i (u) i i < i > i α, iF i U} ==={u :=:T:TA∩B α,>α, I(u) <(u) γ,< uF< ∈γ, (A(A {u (u) FA∩B u∈ U}U} (A ∩∩∩B) A∩B A∩B {u {u TA∩B : (u) T(u) α,IA∩B I> Iβ,< <β, β,(u) FA∩B <(u) β, (u) γ, u(u) ∈ < γ, u ∈ U } (A ∩ B)(α,β,γ) B) (α,β,γ) A∩B A∩B A∩B A∩B A∩B (α,β,γ) (α,β,γ) i i i i i i i i i i i ==={u : T (u) ∧ α = α, I (u) ∨ β = β, F (u) ∨ γ = γ, u {u : T (u) ∧ α = α, I (u) ∨ β = β, F (u) ∨ γ = γ,U} u ∈u ∨ U} A∩B A∩B {u =:A∩B {u TA∩B : T(u) ∧(u) α =∧ α,αA∩B IA∩B = α, (u) I ∨ β (u) = β,∨A∩B Fβ =(u) β, ∨ FA∩B γi ∈= (u) γ, ∈ γU}= γ, u ∈ U } A∩B A∩B i (u) i (u)) i (u)A∩B i (u))i ∨i A∩B i i i i i i i i i ==={u : (T ∧ T ∧ α = α, (I ∧ I β = β {u (u) ∧ I (u)) ∨ β = β A BT(u) B {u =: :(T {u (TAA(u) :(u) (T∧A∧ TBB(u)) (u)) ∧ T∧B ∧α(u)) α==α,Aα, ∧(Iα (I = (u) α, ∧ (I I (u)) (u) ∧ ∨ I β = (u)) β ∨ β =β AA BB A B i i(u) i (u)) i (u)) i i i i , ,(F ∧ F ∨ γ = γ, u ∈ U} (F (u) ∧ F ∨ γ = γ, u ∈ U} AAA AF , (F , (F (u)A∧(u) (u)) FA∨(u)) γ = ∨γ,γu = ∈ U} γ, u ∈ U } AA∧ i i i ∧∧ i∨(u) i∨(u) iβ= i γ== iγ,U}∧ ==={u :=:T:TAi{u = α,=∧ IAiα,(u) βα, =∨Iββ, F=β, γ, uγ=∈= {u α, IAiI= (u) ∨ F ∨β, γF u∈ U}∧ A (u) {u T(u) : TαAi∧α (u) (u) α= α β, ∨AFβ ∨ γ, u∨∈ γU}∧ = γ, u ∈ U }∧ AA(u) A A (u) A (u) A (u) i i i i i i i i i i i i (u) ∧ α = α, I (u) ∨ β = β, F (u) ∨ γ = γ, u ∈ U} {u : T (u) ∧ α = α, I (u) ∨ β = β, F (u) ∨ γ = γ, u ∈ U} {u BT B B (u) ∧ (u) α = ∧ α, α I = (u) α, ∨ I β (u) = β, ∨ F β = (u) β, ∨ F γ = (u) γ, u ∨ ∈ γ U} = γ, u ∈ U } {u: :T{u : T BB BB BB B B B ∩ B . ===AA(α,β,γ) ∩ B . ∩(α,β,γ) B(α,β,γ) ∩ B. .

A(α,β,γ) = A(α,β,γ) (α,β,γ) (α,β,γ)(α,β,γ)

The of can bebemade similarly.  The ofof(3) (3) and (4) can made similarly.  The Theproofs proofs proofs (3) ofand (3) and(4) and (4) can (4) be can made be made similarly. similarly.  Theorem 3.14.

Let bebebe any neutrosophic multi-sets onmulti-sets Uonand i for ∈on I,iU Theorem 3.14.

Let any multi-sets and ∈i and I, I, for i ∈ I, Theorem Theorem 3.14.

3.14.

LetAALet A Aany beneutrosophic neutrosophic any neutrosophic multi-sets onU Ufor and for ∈ (α , β , γ ) = max {(α , β , γ )} and (α , β , γ ) = min {(α , β , γ )}, i ii,i{(α i,ββ 2(α 2,2β, 2β 2,2(α i i , min iβ (α γ,2γ)2= , γi ,ii)}, 111,,1β,111β i {(α i ,iiγ 2and (α(α , 1γ11,1 )γ=1=)max max = imax ,iγi,)} )}iand , and γi )} (α ), β=min γ2ii{(α )i {(α = γ{(α i {(α iβ 2i ,min i ,iβ i )},i , βi , γi )},   ) = A(αA(α 1.

1. ,γ ,β11,β,γA ).,γ 1= i )i,β (Aiii(A )(α(αiii,β,β )i(α 1.1. iii(A ) ==A 11,γ 1 ). i ,γ i)(α ,β1 ,γ1 ).

1 ,β 1(α 11). i ,β i ,γ i i )i ,γi ) (α   ) ==AAA2= 2.

2.

2.2. iii(A (Aiii(A )(α(αiii,β,β )i(α ,γ ,γA ).,γ i )i,β i)(α ) =(α (α,β22,β 22,γ 2 ). i ,γ ,β2 ,γ2 ).

2 ,β 2(α 22). i ,β i ,γ i i )i ,γi ) (α

Proof.

Proof.

Proof.

Proof.

1.

Note that 1. that 1.1.Note Note Note that that  i     i   (A )  i (u) i (u) i i (u) ii ,α i,(u) i γi,, ==={u{u :=: : i{u Ti AT(u) T α I, Ai (u) β u∈ i, ii )(α , I βiiβ,F(u) (u) γiiF ,U} U}U} ,β ,γ ) i i i i i i (A (A ) {u T : (u) α α I , (u) F γuiAi, ∈ u(u) ∈  γi , u ∈ U } (α ,β ,γ ) iii(A i i Fi A A A i i i i i iA i  i   ,βii,γ,βi )i ,γi ) i (αi(α i  Ai i IA Aβ(u) ii i Ai A    i i (u) i ii (u) i i i i F (u) ∨ γ = γ , uiF ∈γiU} I (u) ∨ β = β , Ti AT(u) ∧ α = α , ==={u{u :=: : i{u i 1 i 1 F ∨ γ u1 ,∈u∨U} I (u) ∨ β = β , (u) ∧ α = α , i i A A 1 ,γ i 1 i 1 F ∨ (u) ∈γU} I (u) ∨ I β (u) = β ∨ , β = β , T (u) T ∧ (u) α = ∧ α α , = α , {u : AA i A i i = γ1 , u ∈ U i 1 i iA A 1 1i γ= 1 i i iA A1 i = i i A A i i i i i i i i i i i i i i i i ==={u{u : (T (u) ∧ T (u)) ∧ α = α , (I (u) ∧ I (u)) ∨ β = , 1 1 1 1 : (T (u) ∧ T (u)) ∧ α = α , (I (u) ∧ I (u)) ∨ β = β , A B (u) A 1α,1(I Bα1∧ 1α1 =∧ 1α 1 β1 ,∨ β1 = β1 , {u=: {u (TA :(u) (TA∧ TBB (u)) ∧ TB ∧ (u)) , B(I IBA(u)) (u) ∨ ∧1βI1B= (u)) A= A (u) i i i ∧ iF iA(u)) i (u)) i∧ i1∨(u)) (F ∨ γ = γ , u ∈ U} 1 (F (u) ∧ F ∨ γ = γ , u ∈ U} AA(u) A 1 1 (F (F (u) ∧ (u) F (u)) F γ = ∨ γ γ , u = ∈ γ U} , u ∈ U } 11 1 A A AA A 1 ===AA(α ,β1 ,β ,γ11,γ ).1 ).,β 1(α 1A A(α = (α 1 ,β 1 ,γ 11 ). 1 ,γ1 ).

EXTENSION PRINCIPLE BASED ��������� ��������� ��������� ��������� ����� �� ��� ����� ON �� ... �����

79 ��

2. ����  �� � �> α ,  �� i � < β ,  � F i�(u) < γ , u ∈ U } � {u� :  � T i� (u) (A )(αi ,βi ,γ = i� ��� �i I�A��(u) i�� �� �i �� � ��� � � ��� � � ��� � � ��� ��� � � � � � �i� � � � � � � �� � i� �A ) �� ��� � ����i������ � � i � �� �i�� � � � � �A� ���  � � � � � �� �� � �� �   � �� � � � � � � � � i�(u) ∨ γ = γ , u ∈ U } i i � � � � � = {u :� �i �T (u) = β�i�� = � �� �� �� ��� �� �∧ � �α ��i�� ��� �α � �2��,�� �i��I��A��(u) ��� ��∨� ��� � �� β��2��,�� � �i��F�A ����� � �i � � �2� � � � � ��� �i��A � ����� i� (u)) ∧� α = �α , (I i� (u) ∧ I i� (u)) ∨�β = β , � � � = {u : (T (u) ∧ T 2 2 � �� �� ����� ���� ����A ��� ����� B������ �� � ������ ����������� �A� ����� � �� � � �� � � �B� ���� � �2� � �2� � ����� � i� (u) i� (u)) ∨ γ = � � � (F ∧ F γ , u ∈ U } ��� ����� �A ���� �A ���� �� � ����2�� �� � � �2� � � � � � � � ������� � =A .  � � � � � ����� � ��� ���(α �� � 2�,β ��2�,γ ��2�)� Definition 3.15.

A be on ��������� ��������� ���������� ��� � Let ��� �� ��� � ��� �� any ������������ ��� two ��� neutrosophic ������������ ���������� �� �multi-sets ���������� ��� �� U� and ��� for (α, β,��� γ) ∈ [0, 1]��3���� (α, γ)A ������ �� ���� ���� ������ ����� �����β, �������� ��� is �� ���defined ������ �� � �� as; ��� ∀u �� ∈� U�,� ��� ��� i i (α, β, γ)A(u) =� {u : TAi� (u) ∧ α, IA � � (u) ∨ β, FA� (u) ∨ γ, u ∈ U }. ������ �� ������ �� ������ � �� � � �� ��� �� �� � ������� � �� ������� ��� ���� � �� �������� � ���� � ��� � �� � � � ��

Even though A is a crisp set, (α, β, γ)A(u) is still a neutrosophic multi� ������������ �� ����� ������� ������������ ������ ���� ���� ������ ������ � �� ������� �� ����� ����� ��� ������ ������ ��� ��������� ������ sets defined by ���� ���� ������ ������ �� ��  � � (α, β, γ), u ∈ A (α, β, γ)A(u) ��� ��=��� �(0, ��� � 0, ���0), ��� u� ∈ � /� A. ��� �� ������ ��� �� ������ � � ��� �� ��� ���� �� �� �� ��� � � � �� Theorem 3.16. Let A, B be any neutrosophic multi-sets on U and for ������� ������� ���������� ��� ����� � ���� ���������������� �� ��� ������������ ���������� �� ����������� ��� ��� �� � ��� ��� (αi , βi , γi ) ∈ [0, 1]3 (i = 1, 2). We have ���� �� ��� � ������������� �� �� ������������ ��� �� ���� �� � � �� ���� �� � 1. If (α1 , β1 , γ1 )  (α2 , β2 , γ2 ), then (α1 , β1 , γ1 )A ⊆ (α2 , β2 , γ2 )A ���� �� ������ ���� �� � � �� ��� �� ��� � ����� � �� ��� ������� �� � ��� � �� � �� �� �� ���� �� ��� ��� � � ����� � ����� � � �� �� �� � � ��� � 2. If A ⊆ B, then (α1 , β1 , γ1 )A ⊆ (α1 , β1 , γ1 )B �� ���� ��� � �� � ��� ���� � �� ����� ����� � �� � �� �� �� ���� �������� ���� � �� �� � � � ���

Proof. Let A and B be any neutrosophic multisets on U and for (αi , βi , γi ) ������ ������ ∈ [0, 1]3 (i = 1, 2). We have ���� �β��1��,��γ�1�� �) �� � ���� � �2��,� β � �2�,� γ�2�)� ⇒ � 1.���� (α  (α � � ��1��,� � i� � ∧ α , I i� (u) ∨ β , F i� (u) ∨ γ , u ∈ U } (α β��1�� )A�� � = ����� � ��1��,�� �, �γ�1��� � ��� {u �� ��� �:� T �A ��� �� �������1��� �A� ��������� 1�� ����� ���(u) A � ��� � ��� � �1� � � � � � i� (u) i� (u) ∨ β , F i� (u) ∨ γ , u ∈ U } � � ��� ∧ α , I � �� �  � ��{u �� ��� �:� T �A � ��� � � � � � � � ��� ��� � � � � � � � � � �� 2�� A � 2� A �� � � � � ��� � �2� � � � � � � ��� �= � �� �� �(α �� ��� �2�,� γ�2�)A. ��� 2�,� β

Therefore (α β �� ��1��,�� ���1 �,�γ �1���)A �� � �� �2�,� γ�2�)A. ��� ��������� ��������� ��� � �� 2�,� β � � �⊆ � � �(α � ��� 2.�� �� If A �⊆ B TB (u), IA�(u) � IB�(u) ,F�� � ���� �� � � ���� � then ���� �� � (A �����⊆ ���B ��� �if� ��T��A��(u) ���� ��� ������ �� ���� �� ��� �� � A (u) � � ��� � ��� � ��� � ���

80 80

MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY

80

MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY

FB (u), ∀u ∈ U ) FB (u), ∀u ∈ U ) i (u) ∨ β , F i (u) ∨ γ , u ∈ U } (α1 , β1 , γ1 )A = {u : TAii (u) ∧ α1 , IA i (u) ∨ β1 , FA i F ∈ U )= {u : TAi (u) ∧ α1 , IA (α , β1 ,∀u γ1 )A ∨ γ11 , u ∈ U } B1(u), i (u) ∨ β1 , FAi (u)  {u : TBi (u) ∧ α1 , IB (u) γ , u ∈ U} 1 i i (u) ∨ B  {u {u :: T TBi (u) (u) ∧ ∧α α1,, IIB ∨ γγ11,, uu ∈ ∈U U}} i (u) ∨ β1 , FB i (u) (α1 , β1 , γ1 )A = ∨ (α1 , β1A , γ1 )B. 1 A (u) ∨ β1 , FA 1 = (α , β , γ )B. i i i 1 ∧ α , I (u) ∨ β , F (u) ∨ γ , u ∈ U }  {u1: T1B (u) 1 B 1 1 B Therefore (α1 , β1 , γ1 )A ⊆ (α 1 , β1 , γ1 )B.  (α1⊆, β(α )B. Therefore (α1 , β1 , γ=1 )A 1 , 1γ,1β 1 , γ1 )B.  Theorem 3.17. Let A be any a neutrosophic multi-sets on U. Then, Theorem(α 3.17.

Let A be any, βa ,neutrosophic Therefore γ )B.  multi-sets on U. Then, 1 , β1 , γ1 )A ⊆ (α1  1 1 (α, β, γ A A = ).

Theorem 3.17. LetAA=be any a neutrosophic multi-sets on U. Then, (α, β, γ)) A(α,β,γ (α,β,γ) . (α,β,γ)∈[0,1]  33 A = (α,β,γ)∈[0,1] (α, β, γ) A(α,β,γ) .

Proof. Let A be any neutrosophic multisets on U and for (α, β, γ) ∈ Proof.

Let A be any neutrosophic (α,β,γ)∈[0,1]3 multisets on U and for (α, β, γ) ∈ 3 [0, 1]3 . We have [0, have Proof.

Let A be any neutrosophic 1] . We  multisets on U and for (α, β, γ) ∈ (α,β,γ)∈[0,1] = (α,β,γ)∈[0,1]3 {(α, β, γ) ∧ A(α,β,γ) } 3 (α, β, γ)A(α,β,γ 3 . We have ) [0, 1](α,β,γ)∈[0,1] (α, β, γ)A =  3 (α,β,γ) (α,β,γ)∈[0,1]3 {(α, β, γ) ∧ Ai (α,β,γ) } (α,β,γ)∈[0,1] = (u)  α, 3 {(α, β, γ) : TA  =i (α,β,γ)∈[0,1] {(α, β, β, γ) γ) ∧ : TAAi (u) }α, 3 {(α, (α, β, γ)A = i 3 3 (α,β,γ) I (u) (α,β,γ)∈[0,1] (α,β,γ)∈[0,1]  β, F i (u)  γ, u ∈ U }(α,β,γ) i IA (u)i  β, Fi A  β, γ, γ) u ∈: T Ui}(u)  α, A (u) = 3 {(α, =A (T(α,β,γ)∈[0,1] (u), FAii (u)) A i (u), I i (u), A A = (T I (u), F (u)) i i A (u)  A γ, u ∈ U } I=A A. (u)A  β, F A = (T A. i (u), I i (u), F i (u)) = A A multi-sets A on U. Then, Theorem 3.18. Let A be any a neutrosophic A. multi-sets on U. Then, Theorem 3.18. Let A be any a =neutrosophic  . = (α, β, γ) A(α,β,γ multi-sets on U. Then, Theorem 3.18. LetA ).

AA=be any a neutrosophic (α, β, γ) A(α,β,γ ) (α,β,γ)∈[0,1]  33 A = (α,β,γ)∈[0,1] (α, β, γ) A(α,β,γ .

Proof. Let A be any neutrosophic multisets on )U and for (α, β, γ) ∈ Proof. Let A be any neutrosophic (α,β,γ)∈[0,1]3 multisets on U and for (α, β, γ) ∈ [0, 1]33 . We have [0, have Proof.

Let A be any neutrosophic 1] . We  multisets on U and for (α, β, γ) ∈ (α,β,γ)∈[0,1] = (α,β,γ)∈[0,1]3 {(α, β, γ) ∧ A(α,β,γ } 3 (α, β, γ)A 3 (α,β,γ ) [0, 1](α,β,γ)∈[0,1] . We have = (α,β,γ)∈[0,1]3 {(α, β, γ) ∧ A(α,β,γ) } 3 (α, β, γ)A (α,β,γ) = (α,β,γ)∈[0,1]3 {(α, β, γ) : TAii (u) >) α,  = (α,β,γ)∈[0,1] : TA (u) >}α, 3 {(α, β, γ) ∧ 3 (α,β,γ)∈[0,1] (α,β,γ)∈[0,1]3 (α, β, γ)A(α,β,γ) I i (u) < β, FAii (u) < γ, u ∈ UA}(α,β,γ)  i A IA (u)i < β, Fi A (u) < β, γ, γ) u ∈: T Ui}(u) > α, = 3 {(α, = (T(α,β,γ)∈[0,1] IA FAii (u)) A i (u), i (u), A = (T (u), I (u), F (u)) i (u) A i (u) < A γ, u ∈ U } I=A A.A < β, F A = (T A. i (u), I i (u), F i (u)) = A A A Theorem 3.19. Let A, B be any = A.two neutrosophic multi-sets on U. Theorem 3.19. Let A, B be any two neutrosophic multi-sets on U. Then, Then, Theorem 3.19.

B β, be γ) any two A = BLet⇔A, ∀(α, ∈ [0, 1]3neutrosophic ,A = B multi-sets on U. ) = B(α,β,γ) A = B ⇔ ∀(α, β, γ) ∈ [0, 1]33 , A(α,β,γ (α,β,γ (α,β,γ ) Then, ⇔ ∀(α, β, γ) ∈ [0, 1]3 , A(α,β,γ = B(α,β,γ) ) = B ) 3 ⇔ ∀(α, β, γ) ∈ [0, 1] , A A = B ⇔ ∀(α, β, γ) ∈ [0, 1] , A(α,β,γ (α,β,γ)) = B(α,β,γ (α,β,γ)) ⇔ ∀(α, β, γ) ∈ [0, 1]3 , A(α,β,γ = B(α,β,γ )

)

EXTENSION PRINCIPLE BASED ON ...

81

Proof. Let A and B be any neutrosophic multisets on U and for (α, β, γ) ∈ [0, 1]3 . We hav A = B(A = B if TA (u) = TB (u), IA (u) = IB (u) ,FA (u) = FB (u), ∀u ∈ U )⇔ A(α,β,γ)

i (u)  β, F i (u)  γ, u ∈ U } = {u : TAi (u)  α, IA A i i = {u : TA (u) ∧ α, IA (u) ∨ β, FAi (u) ∨ γ, u ∈ U } i (u) ∨ β, F i (u) ∨ γ, u ∈ U } = {u : TBi (u) ∧ α, IB B = B(α,β,γ) .

Therefore A(α,β,γ) = B(α,β,γ) .

A = B(A = B if TA (u) = TB (u), IA (u) = IB (u) ,FA (u) = FB (u), ∀u ∈ U )⇔ A(α,β,γ

)

i (u) < β, F i (u) < γ, u ∈ U } = {u : TAi (u) > α, IA A i (u) ∨ β, F i (u) ∨ γ, u ∈ U } = {u : TAi (u) ∧ α, IA A i (u) ∨ β, F i (u) ∨ γ, u ∈ U } = {u : TBi (u) ∧ α, IB B = B(α,β,γ . )

Therefore A(α,β,γ = B(α,β,γ . )

)

Theorem 3.20. Let A be any a neutrosophic multi-sets on U and for δ = (α, β, γ) ∈ [0, 1]3 . If H : [0, 1]3 → P (X) satisfies that ∀δ ∈ [0, 1]3 , Aδ ⊆ H(δ) ⊆ Aδ , then  A= δH(δ) δ∈[0,1]3

and the following properties are fulfilled: 1. H(δ1 ) ⊇ H(δ2 ) whenever δ1 < δ2 .  2. ∀δ ∈ [0, 1]3 , Aδ = θ<δ H(θ).  3. ∀δ ∈ [0, 1]3 , Aδ = θ>δ H(θ).

Proof.

1. It is obvious.  2. When θ < δ , we have H(θ) ⊃ Aθ ⊃ Aδ . Thus θ<δ H(θ) ⊃ Aδ . On    the other hand θ<δ H(θ) ⊂ θ<δ Aθ = Aδ . Hence Aδ = θ<δ H(θ).

82 82

MEHMET ŞAHIN, IRFAN DELI DELI AND VAKKAS ULUÇAYULUÇAY MEHMET ŞAHIN, IRFAN AND VAKKAS

  On 3. δ, δ, we we havehave H(θ)H(θ) ⊂ Aθ⊂⊂ AAθδ .⊂Thus Aδ ⊃ A H(θ).

3. When Whenθ θ> > ⊃ θ>δ H(θ). On θ>δ Aδ . Thus   δ = A . Therefore, the other hand,H(θ) ⊂ A H ⊃ A and the other ⊃ θ θ>δ A θ>δ θθ>δ Hθθ>δ δ θ = Aδ . Therefore,  hand,H(θ) ⊂θ Aθ and H(θ).

A H(θ).

Aδδ == θ>δ θ>δ f : Xf → Y .→ weYrecall (direct)image of a Let thethe function :X . we the recall the (direct)image of a Let extending extending function f is defined as the neutrosophic multi-subsets A of X under the neutrosophic multi-subsets A of X under f is defined as i i f(A) = {T=f(A) , Fi i ,}F⊆i Y, } ⊆ Y, f (A) {T, fIi f(A) (A) , Iff(A) (A) f (A)

and image of aofneutrosophic multi-subsets B of Y isBdefined andthe theinverse inverse image a neutrosophic multi-subsets of Y is defined as as i i −1 = {T i f −1f(A) (A) =f(A) {T, Iif(A),, IFif(A) }, F⊆i X. } ⊆ X. f (A)

f (A)

f (A)

f : Xf → induces two newtwo mappings In thethe mapping : XY → Y induces new mappings In other otherwords, words, mapping −1 −1 ff :: PP(X) ) and f f: P (Y: )P→ mappings mappings (X)→→P (Y P (Y ) and (YP) (X). → PThese (X). induced These induced possess properties. possessthe thefollowing following properties.

Theorem f :fX: → B,A, AiB, ∈P ∈ J), then the induced Theorem3.21.

3.21.

X Y,→A,Y, A(X)(i i ∈ P (X)(i ∈ J), then the induced mappings thatthat mappingsf satisfies f satisfies f(A) ⊆ f(B).

1.1. AA⊆⊆BBimplies implies f (A) ⊆ f (B).    2.2. f(f ( i (A(A = (= i(f(Afi )). i ))i )) i i (Ai )).    ⊆ (⊆ i(f(Afi )).

3.3. f(f ( i (A(A i ))i )) i i (Ai )).     A(x),   A(x), A(x), 4.4. ∀y = = i i i ∀y∈∈Y (f(A))(y) Y (f (A))(y) Tf(A) (x)=y I (x)=y (x)=y A(x). i T (x)=y Ii (x)=yFA(x), Fi (x)=y A(x). f(A) f(A) f (A)

f (A)

f (A)

3 [0, 1]3 , f(A

γ)γ) ∈ ∈ [0, 1] , f(α,β,γ) ) ⊆ (f(A)) . (α,β,γ) .

5.5. ∀(α, (α,β,γ) ∀(α,β,β, (A(α,β,γ) ) ⊆ (f (A))    (α, β, γ)f(A γ)(f(A)) 6.6. f(A) ). (A(α,β,γ) ).

3 (α,β,γ) f (A)= = (α,β,γ)∈[0,1]3 (α, 3β,(α, = β, γ)(f(α,β,γ) (A))= β, γ)f (α,β,γ)∈.[0,1] 3 (α, (α,β,γ) (α,β,γ)∈[0,1]

(α,β,γ)∈.[0,1]

Proof.

Proof.

1.

1. ItItisisTrivial.

Trivial.

Then 2.

Then 2. Let Letyy∈∈Y.Y.    i y, F i = i yy ∈∈f( ∃x ∈ {T i), {T (x)i = y, I i = (x) i ))i ))⇔ ⇔ f ( i (A (A ∃x ∈ y, I= y,=Ffy} i (Ai ), f(A) (x) i i (Aif(A) f (A) (x)f(A) f (A) (x) (A) (x) = y} i i i i i ⇔⇔ ∃i ∈∃iJ,∈x J, ∈ (A )and{T (x) = y, I (x) = y, F (x) =Fy}i (x) = y} i (Ai )and{T x∈ = y, If (A) (x) f(A) f (A) (x) f(A) f(A)= y, f (A) ⇔⇔ ∃i ∈∃i ∈ f(A y ∈i )f (Ai ) J,∈yJ, ⇔⇔ y ∈y ∈i f(Aif).(Ai ). i

EXTENSION PRINCIPLE BASED...ON ...

EXTENSION PRINCIPLE BASED EXTENSION PRINCIPLE BASED ON ...ON EXTENSION PRINCIPLE BASED ON83...

83

83

83

3. It issimilar similartotothethe proof of (2) proof of (2) 3.3. to theto proof (2) 3.ItItItisissimilar is similar theof proof of (2) 4. Let Letyy∈∈Y.Y.Then Then 4.

4.4.Let y ∈yY.∈Then Let Y. Then (f(A))(y)=={1,{1, 0, 0} (f(A))(y) (f(A))(y) = {1,=0, 0, 0}0}0, 0} (f (A))(y) {1, ⇔∈yf(A) ∈ f(A) ⇔⇔ y ∈yf(A) ⇔ y ∈ if (A) i i i i y, (x) i y,(x) ⇔ ∃x ∈i {T A, {T(x) = If(A) Ff(A) (x) i= y} ∈{T A, =(x) I(x) = y, = y, y} ii y,(x) i (x) iF= f(A) f(A) f(A) f(A) ⇔ ∃x ∈ A, {T = y, I (x) = F (x) = y}i ⇔⇔ ∃x ∃x ∈ A, (x) = y, I = y, F (x) = y} f (A) f (A) f(A) f(A) f(A) i i f (A) i (x) = (x) i= y, i = (x) ⇔ ∃x ∈A(x) A, A(x) =0,{1, 0,i 0}and{T Ii f(A) (x) F=f(A) = y} ⇔ ∃x ∈ A, = {1, 0}and{T y, I (x) = y, F y}=(x) i i y, (x) i f(A) f(A) f(A) ⇔ ∃x =∈{1, A,0,A(x) =f(A) {1, Iy} y, Ffi (A) (x ⇔ ∃x  ∈ A, 0}and{T =0}and{T y, If(A) (x)f =(A) y,(x) Ff(A) (x)y,=f(A) A(x) (x)0, =  f (A)  = 0.  ⇔ i T   A(x) =i 0, F i A(x) = 1, A(x) = 0. i A(x)A(x) = 1, (x)=y Ifi1, (x)=y Ii = (x)=y (x)=y 0,(x)=y A(x) = f(x)=y (A) (A) f 0.

(A) i = 1, i ⇔ ⇔ i T⇔ A(x) A(x) =A(x) 0,i =i 0, F A(x) A(x) == i A(x) f(A) (x)=y Tf (A) (x)=yIf(A) (x)=y Tf(A)f(A)

If (A) (x)=y Ff(A) (x)=yf(A)

Ff (A) (x)=y

Then 5. Let Letyy∈∈Y.Y.Then Y. Then 5.Let Let y ∈yY.∈Then 5.5. i i i T i (y)i (y) i I(y) iβ, F y ∈(f(A)) (f(A)) >I α, (y) ii β, < (α,β,γ) ⇔i⇔ Tf(A) >I iα, < F (y) < (y) γ i < γ(y) <γ f(A) f(A) f(A) (α,β,γ) (f (A)) ⇔ T (y) > α, I (y) < β, F f(A) f(A) yyy∈∈∈ (f(A)) ⇔ Tf(A) (y) > α, (y) < β, F (y) < γ   (α,β,γ)(α,β,γ) f (A) f (A) f (A)    f(A) f(A)   < A(x) ⇔ i T i (x)=y  α,> α,i I i (x)=y < A(x) A(x) <γ i A(x) ⇔ A(x) > A(x) β, <FA(x) γ i β, (x)=y F (x)=y ⇔ A(x) > α, < β, i i T (x)=y I (x)=y f (A) f (A)<Iβ, f (A) ⇔ T i f(A) A(x) > α, A(x) A(x) < γ i i T (x)=y (x)=y Ffi (A) (x)=y A(x) < γ f(A) f(A) (x)=y f (A) i If(A) (x)=y i f (A)Ff(A) i(x)=y f(A) i T (x) = i y, I(x) =(x) ⇔⇔ ∃x ∈i TX, y, F= (x) A(x) >γ)(α, β, γ) i i y,= i F=i (x) i= y, ∈ T∃x X, If(A) =y,y, f(A) f(A) ∈f(A) X, T (x) (x) =f(A) y,i Iy, (x) A(x) > (α, β, γ) ⇔⇔ ∃x∃x ∈ X, (x)if(A) = y,(x) A(x) >FA(x) (α, β,> γ) (α,=β,y, f (A) f (A) f(A) (x) = y,fIi (A) f(A)i(x) = y, Fif(A) i i y,(x) ⇔ ∃x ∈ XA , T(x) (x) y,(x) I= (x) Ff(A) (x)F = iy, I=(x) iF= i y (x) = y (α,β,γ) ⇔ ∃x ∈ XA , T = = y, = y i i i f(A) f(A) ⇔ ∃x ∈ XA , T y, I (x) = y, (α,β,γ) f(A)= y, I f (A) f(A) ⇔ ∃x ∈ XA(α,β,γ) , Tf(A)(α,β,γ) (x) (x)f(A) = y, Ff(A) (x) =y f (A) f (A) ⇔ y f(A ∈y f(A ). ). f(A) (α,β,γ) ⇔ y ∈ ). ⇔ ∈ f (A (α,β,γ) ⇔ y ∈ f(A(α,β,γ) ). (α,β,γ)   =A= (α,β,γ)∈[0,1] A ∈ P (X), β, γ)A(α,β,γ) and (5) 6. By By for every 3 (α, A ∈ P (X), A γ)A 6. for every 3 (α, β, A ∈ P (X), A = 3 (α, (α, γ)A(5) 6.

By for every 3(α,β,γ) (α,β,γ)∈[0,1] (α,β,γ) and (5) β, γ)A(α,β,γ) and β, (5)and 6. By for every A ∈ P (X), A = (α,β,γ)∈[0,1] (α,β,γ)∈[0,1]      (α, β, γ)(f(A)) =  f(A) = β, γ)f(A ). (α,β,γ) f(A) = (α, β, γ)(f(A)) = β,(α, γ)f(A f (A) (α, β, (α,β,γ) γ)(f (A)) = β, (α, (α, β,).(α,β,γ) γ)f (A (α,β,γ) (α,β,γ) (α,β,γ) ). f(A) == (α, β, γ)(f(A)) = (α,β,γ)(α, γ)f(A (α,β,γ) ). (α,β,γ)∈[0,1] 3 3 3 (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] 3 (α,β,γ)∈[0,1]

3 (α,β,γ)∈[0,1] 3 3 (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] (α,β,γ)∈[0,1]3

Forevery every y ∈ Y, we we have For For every Y,have , have we have y y∈ ∈Yy,Y∈we For every    (α,β,γ)∈[0,1] = = = β, γ)(f(A)) β,3(f(A γ) ∧(α,β,γ) (f(A ))(y) 3 (α, 3 (α, (α,β,γ) (α,β,γ) β, γ)(f(A)) β, γ)(f (A)) (α, β, γ)))(y) ∧ (f (A (α,β,γ)∈[0,1] 3 (α, 3 (α, β, 3 (α, (α,β,γ) (α,β,γ) (α,β,γ) ))(y) (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] γ) ∧(α,β,γ)   3(α, γ) ∧ (f(A β,  ))(y) (α,β,γ)∈[0,1]3 (α, β, γ)(f(A))(α,β,γ) = (α,β,γ)∈[0,1] = = (α ∧ A (x), i α = (α ∧ A (x), (α ∧ A (x), i i α∈[0,1] Tf (A) (x)=y αT (x)=y α α∈[0,1] T (x)=y =  (α ∧ i α∈[0,1] f (A) f(A)Aα (x),  α∈[0,1] T (x)=y  f(A)   (β ∨ A (x)), i β Aβ (x)), (β i∨β∈[0,1] A(x)=y (x)), i i (β β∈[0,1] I∨ β∈[0,1] If(A) (A) βIf (A) (x)=y βf(x)), A(x)=y ∨ If(A) β∈[0,1] (β  (x)=y    γ∈[0,1] (γ ∨i (γF ∨ i γ (x))Aγ (x)) i A A(x)=y γ (x)) (x)=y γ∈[0,1] Ff(A)A(x)=y (A) F (γ ∨(γF∨iγ∈[0,1] f (A) γ f(x))  γ∈[0,1] (x)=y    f(A)  =i = i (x)=y α ∧ A (x), i α(x), ∧ Aα∈[0,1] α (x), αα ∧ Aα (x), Tf(A)T(x)=y f (A) Tf α∈[0,1] = = α ∧(x)=y Aα∈[0,1] i  (A) α  Tf(A) (x)=y α∈[0,1]      i ∨ Aββ(x), ∨ Aβ (x), β ∨ Aββ (x), i Ifi (A) (x)=y β∈[0,1] If (A) If(A) β∈[0,1] β ∨(x)=y i  (x)=y Aβ (x),β∈[0,1] If(A) (x)=y β∈[0,1]      γ ∨ A (x) i i γ(x) ∨ Aγ∈[0,1] i F(x)=y (x)=y γ (x) γγ ∨ Aγ (x) f (A)Ff (A) γ ∨(x)=y Aγ∈[0,1] i F γ f(A)   Ff(A) (x)=y γ∈[0,1]  γ∈[0,1]  i A(x), = =i = A(x), i A(x), i A(x), A(x), i I (x)=y T (x)=y Ifi (A) (x)=y A(x), T (x)=y If(A) (x)=y Tf(A) (x)=y f (A) f (A) f (A) A(x), A(x), =  i i   If(A) (x)=y Tf(A) (x)=y  A(x) A(x) i i A(x) i F(x)=y (x)=y f (A)Ff (A) (x)=y i F f(A) A(x) Ff(A) (x)=y = (f(A))(y). = (f (A))(y). = (f(A))(y). = (f(A))(y).  f (A)== (α,β,γ)∈[0,1] (α, β,(α,β,γ) γ)f(α,β,γ) (A ). ).

Thus f(A) β, γ)f(A Thusf(A) 3 (α, β,3γ)f(A Thus (α,β,γ)∈[0,1] 3 (α, (α,β,γ)∈[0,1] f(A) == (α,β,γ)∈[0,1] ). ). (α,β,γ) Thus 3 (α, β, γ)f(A(α,β,γ)

84 8484 84 84

MEHMET DELI AND VAKKAS ULUÇAY MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY

Theorem 3.22.

B, B )(i ∈∈then J), then the induced Theorem 3.22.

3.22.

YYA, BA, ∈iB, P∈(Y ∈(Y J), then the induced Theorem 3.22. :→ X→ → ∈ P )(i ∈ J), then induced Theorem 3.22. :X YB,Y A, P)(i (Y )(i J), then thethe induced iiB Theorem f ff: fX X → A, B, P∈)(i (Y ∈(Y J), the induced iB i P −1 −1 −1 −1 −1 mappings f satisfies that mappingsffff satisfies satisfies that mappings satisfies mappings satisfies mappings thatthat −1−1 −1−1f −1−1 −1 −1 ⊆⊆BB BB implies 1.

A⊆ implies ff−1 ⊆(A) implies f(A) ⊆(B). f (B).

(B).

1.1.AAA ⊆ implies f(A) (A) 1. implies ⊆f f⊆(B).

−1 −1 −1 −1 c .c c c . −1(B −1 (B)) −1 −1 ccc)c)))c= −1 (B =(f (f 2. =)= (B)) (B =(f (f 2.2.fff−1 (B (B)) 2.

(B (B)) . .         −1 −1−1 −1 −1 −1 −1 (B )) = fi (B (B 3. ((=(i if(f−1 ))= fi )).

(B 3.3.fff−1 (B 3. i )). ((( (ii(B i= i )). ii)) ii )) i )). i (B i (B ii(B    −1−1  −1 −1 −1 −1 −1 (B (B )) = 4. ((=(i if(f−1 ))= (B 4.4.fff−1 (B (B 4. i )). ((( (ii(B i= i )). ii)) ii )) i )). i (B i fi )). ii(B i f(B

−1 33, f1] 3(B −1 (B −1. (B)) . −1(B)) 5. ∀(α, β, γ) ∈[0, [0, (f ∀(α,β, β, γ) 1] ) (α,β,γ) =) (f 5.5.∀(α, ∀(α, β, γ) ∈[0, [0, f(α,β,γ) )−1= (f 5. ∀(α, β, γ) 1], 3f3−1 ,−1 f,−1 (B (f (B)) . . γ) ∈∈∈ 1] (B =)−1 (f=(B)) (B)) . (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ)          −1 −1 −1 −1 −1(B) −1 (B)) −1 −1(B)) −1 −1 −1 (B) ==(α,β,γ)∈[0,1] (α, β,−1 γ)(f = (α, γ)f (B ).).

6.

(B)) =(α,β,γ) (α, β, ).γ)f (B) (α, β, γ)(f = (α, β, γ)f (B(α,β,γ) 6.6.fff−1 (B) (B)) = (α, β,(B (B 6.

3 (α, 3(α,β,γ)∈[0,1] 3γ)(f 3β, 3β, 3 (B) === β, γ)(f (B)) = (α,−1 β,3(B γ)f ).−1 (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) 3 (α, 3γ)f (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] (α,β,γ)∈[0,1] (α,β,γ) (α,β,γ) (α,β,γ)∈[0,1]

Proof.

We proof (4) examples.

Proof.We We proof (4)(4) and (5) asas examples.

Proof.

We proof (4) and (5) as examples.

Proof.

We proof and as examples.

Proof. proof (4) and (5)(5) examples.

4.

4.

4.4.4.      −1 −1 −1 ∀x X, (B (x)) ∀x∈∈∈∈X, X, ))(x) = ∀x X, f−1 (B ))(x) = (x))  ∀x X, (B ))(x) (x)) i i )(f ∀x fff−1 (((( (ii(B ))(x) = )(f(x)) i i )(f(x)) i i )(f i i i i(B i (B i (B i i i (B =  ii i i   i i i     ii i (x)), i (B i (B i (x)), i i(B i i )(F (B )(T (x)), (x)) == = (B )(T (Bi(x)), )(I (x)), (B (x)) = )(T (x)), (x)), )(I (x)), (B )(Ffi )(F i (B )(T (x)), (Bf(A) )(I (x)), (x)) i if)(F i ifi)(I ii f(A) ii (B i (B ff(A) (A) i i )(F i i i i i)(I ii(B if(A) f(A) f (A) (A) (A) (x)) (A) f(A) (A) f (A)f(x)) i i i (B f(A) f(A)  −1 −1−1 −1 f(B == =f = (B fi(B (B )(x) ii)(x) )(x) i )(x) i )(x)   f−1 −1−1 −1 (B ))(x). f (B ))(x). ==( = f (B = ( f ( ii ))(x). ( i i f i (B ii i ))(x). i ))(x).

5.

5.5.

5.

5.

−1 ii i (x)} ∈ B −1 i i {T i (x), i(x), −1(B −1 i i I(x), i (x), i(x), Ifi(x), FF(x)} x∈∈∈fff−1 (B {T Ffi f(A) (x)} B (B )⇔ ⇔ I(A) F (x)} ∈(α,β,γ) B(α,β,γ) xxxx∈ (B {T (x), (x)} ∈B (B ) )) ⇔ ⇔ {T (x), If(A) Ff(A) ∈B (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ) f(x), (A) (A) ff∈ (A) (α,β,γ) (α,β,γ) f(A) f(A) f (A) (A) f (A) (A) f(A) i ii i (x)) i β,iB(F i i (x)) i i i i i ⇔ B(T (x))  α, B(I γγ γ ⇔  α, B(IB(I (x)) β, B(F (x)) γff(A) ⇔f(A) B(T (x))  B(I (x)) f(A) β, B(F α,α, B(I (x))  (x)) β, B(F ⇔ B(T B(T (x)) α,  β, B(F fi(x)) γ (x)) ff (A) f(A) f(A) f (A) f (A) (A) f(x)) (A) (A) (A) f(A) f(A) (x)) −1 −1 −1 −1(B))(x)  γ) (α, β, ⇔ (f(f  (α,(α, β, ⇔(f (f (B))(x)  (α, β, (B))(x) (α,γ) β,γ) γ) γ) ⇔⇔ (B))(x) β, −1 −1 −1 −1 (B)) .. . ⇔ xx⇔ ∈∈ (B)) .(α,β,γ) (f (B)) x(fx(f ∈ ∈(f ⇔⇔ (B)) .(α,β,γ) (α,β,γ) (α,β,γ) (α,β,γ)

The proofs (1),(2),(3) bebe made similarly. similarly.

Theproofs proofsofof ofofof (1),(2),(3) and (6) becan made similarly.

The proofs (1),(2),(3) and (6) can made The (1),(2),(3) and (6) made similarly.    The proofs (1),(2),(3) and (6)can can be made similarly.

4.

Algebraic Over Neutrosophic MultiAlgebraic Operations Over Neutrosophic Multi4.4. Algebraic Algebraic Operations Over Neutrosophic Multi4.

Algebraic Operations Over Neutrosophic Multi4.

Operations Over Neutrosophic MultiSets Based Extension Principle Sets Based on Extension Principle SetsBased Based on Extension Principle Sets Based Principle Sets onon Extension Principle Let and BB be two multi-sets on the universal XX X LetAA Aand and be two neutrosophic multi-sets on the universal sets Xsets Xsets Let A andBB be two neutrosophic multi-sets universal Let and B be two neutrosophic multi-sets onon thethe universal setssets Let be two neutrosophic multi-sets on the universal and Y respectively. to the extension principle of the neutroand Y respectively.

According to the extension principle of the neutroand Y respectively.

According to the extension principle of the neutroand Y respectively.

According extension principle of the neutroand respectively. According to the extension principle of the neutrosophic multi-sets.

We algebraic operations over the neutrosophicmulti-sets. multi-sets.

We can define algebraic operations over over the neutrosophic multi-sets.

We can define algebraic operations over neutrosophic multi-sets.

We can define algebraic operations over thethe neutrosophic We can define algebraic operations the neutrosophic multi-sets A and addition, subtraction, multiplication, sophic multi-sets A and B, i.e.,the addition, subtraction, multiplication, sophic multi-sets A and B, i.e.,the addition, subtraction, multiplication, sophic multi-sets A and B, i.e.,the addition, subtraction, multiplication, sophic multi-sets and B, i.e.,the addition, subtraction, multiplication,

EXTENSION PRINCIPLE BASED ONON ...ON EXTENSION PRINCIPLE BASED ... ...

EXTENSION PRINCIPLE BASED

85 85 85

and division division are defined asasfollows: and operations are defined and divisionoperations operations are defined asfollows: follows: ii i i i(x)}, ∧ i (x) i ∨ I i i(x)}, A ∗∗∗B {T (x) ∧∧T∧fTi (B) {Ifi{I A {z, {T (x) (x)}, ∧z=x∗y (x)(x) I∨ (x)}, B =={z, {z,∨∨∨ {T (x) (x)}, ∧z=x∗y Ifi (B) (x)}, z=x∗y z=x∗y z=x∗y z=x∗y ff(A) (A) f∨(B) (A) fT(B) f{I (A) f (B) f (A) f (B) f (A) ii i i i i ∧∧∧ {F (x) ∨∨F∨F (x)}|(x, y) y) ∈ y) X × {F (x) (x)}|(x, ∈X ×},Y {F (x) (x)}|(x, ∈ YX ×},Y }, z=x∗y z=x∗y z=x∗y ff(A) f (B) (A) fF(B) f (A) f (B)

represent one ofofthe algebraic operations “+”, “− where the represent one the algebraic operations “+”, “−“− where symbol “∗” represent one of the algebraic operations “+”, where thesymbol symbol“∗” “∗” division operation “÷ ”“ is to satisfy ”, and ÷÷”.”.”.The The division operation “÷ ” required is required to to satisfy ”,““ × ×”, ”, and and“““÷ The division operation ÷ ” is required satisfy i i i(y)  0, I i i (y) i i(y)i  condition: =={y|T (y) 1, the supp(B) (y)(y)  0,f (B) If (B) (y) F1, F1,f (B) (y)(y)   the condition: condition:000∈∈∈supp(B) supp(B) ={y|T {y|T 0, Ifi (B)  Ff (B) f (B) f (B) f (B) f (B) 1, y ∈ Y. 1, y ∈ Y. +++ B, AA− AB, ÷A B÷ and λand ×λB defined as as as More specifically, specifically, B, A×A ×B, B, A ÷ B and × B are defined More B, A−B, −B,A B, × B λ are ×B are defined More specifically,AAA follows: follows: follows: i i i (x) ∧ T i i i(y)}, ∧ i i (x) i(y)}, i (x) A+ + {T {Ifi{I ∨ I(x) A {z, {T (x) (y)}, ∧z=x+y I∨ +BB == {z, {z,∨∨∨ {T (x)∧ ∧ (y)}, ∧z=x+y Ifi (y)}, z=x+y z=x+y z=x+y z=x+y f (A) fT(B) (A) f∨(B) f (A) fT(B) f{I (A) f (B) f (A) f (B) f (A) (B) (y)}, i i i i i i ∧ {F (x) ∨∨F∨F (y)}|(x, y) y) ∈ y) X × ∧∧z=x+y {F (x) (y)}|(x, ∈X ×}Y {F (x) (y)}|(x, ∈ YX ×}Y } z=x+y z=x+y ff(A) f (B) (A) fF(B) f (A) f (B) i i i (x) ∧ T i i i(y)}, ∧ i i (x) i(y)}, i (x) A− − {T {Ifi{I ∨ I(x) A {z, {T (x) (y)}, ∧z=x−y I∨ −BB == {z, {z,∨∨∨ {T (x)∧ ∧ (y)}, ∧z=x−y Ifi (y)}, z=x−y z=x−y z=x−y z=x−y f (A) fT(B) (A) f∨(B) f (A) fT(B) f{I (A) f (B) f (A) f (B) f (A) (B) (y)}, i i i i i i ∧ {F (x) ∨∨F∨F (y)}|(x, y) y) ∈ y) X × ∧∧z=x−y {F (x) (y)}|(x, ∈X ×}Y {F (x) (y)}|(x, ∈ YX ×}Y } z=x−y z=x−y ff(A) f (B) (A) fF(B) f (A) f (B) i i i (x) ∧ T i i i(y)}, ∧ i i (x) i(y)}, i (x) A× × {T {Ifi{I ∨ I(x) A {z, {T (x) (y)}, ∧z=x×y I∨ ×BB == {z, {z,∨∨∨ {T (x)∧ ∧ (y)}, ∧z=x×y Ifi (y)}, z=x×y z=x×y z=x×y z=x×y f (A) fT(B) (A) f∨(B) f (A) fT(B) f{I (A) f (B) f (A) f (B) f (A) (B) (y)}, i i i i i i ∧ {F (x) ∨ F (y)}|(x, y) ∈ X × Y } ∧∧z=x×y {F (x) (y)}|(x, y)y) ∈X ×Y {F (x)∨∨F (y)}|(x, ∈X ×}Y } z=x×y z=x×y ff(A) f (B) (A) fF(B) f (A) f (B) i i i (x) ∧ T i i i(y)}, ∧ i i (x) i(y)}, i (x) A÷ ÷ {T {Ifi{I ∨ I(x) A {z, {T (x) (y)}, ∧z=x÷y I∨ ÷B B == {z, {z,∨∨∨ {T (x)∧ ∧ (y)}, ∧z=x÷y Ifi (y)}, z=x÷y z=x÷y z=x÷y z=x÷y f (A) fT(B) (A) f∨(B) f (A) fT(B) f{I (A) f (B) f (A) f (B) f (A) (B) (y)}, i i i i i i ∧ {F (x) ∨ F (y)}|(x, y) ∈ X × Y } ∧∧z=x÷y {F (x) (y)}|(x, y)y) ∈X ×Y {F (x)∨∨F (y)}|(x, ∈X ×}Y } z=x÷y z=x÷y ff(A) f (B) (A) fF(B) f (A) f (B) i i i(y)}, ∧ i (y)}, i (y)}, ∨∨z=λ×y {T{T λB {Ifi{I == {z, {z, (y)}, ∧z=λ×y λB {z, ∨ (y)}, ∧z=λ×y λB= (y)}, z=λ×y z=λ×y z=λ×y f{T (B) (B) f (B) f{I (B) f (B) f (B) i i i ∧∧z=λ×y {F (y)}|(x, y) ∈ Y }, {F (y)}|(x, y) ∈ Y }, ∧z=λ×y {F (y)}|(x, y) ∈ Y }, z=λ×y f (B) f (B) f (B)

respectively, where === 00is0isany real number. respectively, where real number. respectively, whereλλλ isany any real number.

Example 4.1. and BBbe two neutrosophic multi-set as follows; Example 4.1.

Let and two neutrosophic multi-set as as follows; Example 4.1.Let LetAAA and Bbebe two neutrosophic multi-set follows; A= = 0.8, 0.7), (0.2, 0.1, 0.3, 0.5), (0.3, 0.4,0.4, 0.1, 0.5), 2, (0.6, 0.4, 0.7,0.4, 0.9), A (0.5, 0.6, 0.8, 0.7), (0.2, 0.1, 0.3, 0.5), (0.3, 0.1, 0.5), 2, 2, (0.6, 0.4, 0.7,0.7, 0.9),0.9), = {1, {1,(0.5, (0.5,0.6, 0.6, 0.8, 0.7), (0.2, 0.1, 0.3, 0.5), (0.3, 0.4, 0.1, 0.5), (0.6, 0.2), (0.2, 0.1, 0.4, 0.3), 3,3, (1, 0.8, 0.5, 0.7), (0.1, 0.2, 0.4,0.2, 0.3), 0.3, 0.5, 0.2), (0.2, 0.1, 0.4, 0.3), (1, 0.8, 0.5, 0.7), (0.1, 0.2, 0.4, 0.3),0.3), (0.4, 0.3,0.5, 0.5, 0.2), (0.2, 0.1, 0.4, 0.3), 3, (1, 0.8, 0.5, 0.7), (0.1, 0.4, (0.4,0.3, 0.1)}.

(0.4, 0.3, 0.2, 0.1)}.

0.3,0.2, 0.2, 0.1)}.

(0.4,0.3,

and and and B= = 0.6, 0.7), (0.3, 0.1, 0.2, 0.4), (0.6, 0.4,0.4, 0.5, 0.7), 2, (0.7, 0.6, 0.4,0.6, 0.8), B (0.8, 0.5, 0.6, 0.7), (0.3, 0.1, 0.2, 0.4), (0.6, 0.5, 0.7), 2, 2, (0.7, 0.6, 0.4,0.4, 0.8),0.8), = {4, {4,(0.8, (0.8,0.5, 0.5, 0.6, 0.7), (0.3, 0.1, 0.2, 0.4), (0.6, 0.4, 0.5, 0.7), (0.7, 0.2), (0.3, 0.7, 0.4, 0.6)}.

0.3, 0.4, 0.2), (0.3, 0.7, 0.4, 0.6)}.

(0.5, (0.5,0.3, 0.3,0.4, 0.4, 0.2), (0.3, 0.7, 0.4, 0.6)}.

868686

MEHMET ŞAHIN, IRFAN DELI AND VAKKAS ULUÇAY ULUÇAY MEHMET MEHMET ŞAHIN, ŞAHIN, IRFAN IRFAN DELI AND DELI VAKKAS ANDULUÇAY VAKKAS

respectively.

Then, respectively. respectively.

Then, Then, ={3,{3, 0.4, 0.8), (0.5, 0.3, 0.5), (0.3, 0.7, 0.4, 0.6), 4, 0.4, 0.4, 0.8), 0.4, 0.4, 0.8), AA +A+ B+= BB = {3, (0.5,(0.5, (0.5, 0.6,0.6, 0.4, 0.6, 0.8), 0.4, (0.5, 0.8), 0.3,(0.5, 0.4,0.4, 0.5), 0.3, (0.3, 0.4, 0.7, 0.5), 0.4,(0.3, 0.6), 0.7, 4, (0.6, 0.4,(0.6, 0.4, 0.6), 0.4,4, 0.8), (0.6, 0.5, 0.2), (0.3, 0.7, 5, (0.7, 0.6, 0.4, 0.4, 0.3),0.3, 0.4, 0.3), (0.5,(0.5, (0.5, 0.3,0.3, 0.5, 0.3, 0.2), 0.5, (0.3, 0.2), 0.7,(0.3, 0.4,0.4, 0.6), 0.7,0.6), 0.4, 5, (0.7, 0.6), 0.6,5, 0.4,(0.7, 0.7),0.7), 0.6, (0.5,(0.5, 0.4, 0.3,0.3, 0.4, 0.7), 0.3), (0.5, 0.4, 0.6), 6, (0.6, 0.4, 0.7), (0.4, 0.3, 0.5, 0.5, 0.7),0.4, 0.5, 0.7), (0.4,(0.4, (0.4, 0.7,0.7, 0.4, 0.7, 0.6), 0.4, 6, 0.6), (0.6, 6, 0.4,(0.6, 0.6,0.6, 0.7), 0.4, (0.4, 0.6, 0.3, 0.7), 0.5,(0.4, 0.4),0.4), 0.3, (0.6,(0.6, 0.5, 0.4,0.4, 0.5, 0.4), 0.7), (0.6, 7, 0.5, 0.7), (0.3, 0.2, 0.4), (0.6, 0.4, 0.5, 0.7)}. 7, (0.8, 7,(0.8, (0.8, 0.5,0.5, 0.5, 0.5, 0.7), 0.5, (0.3, 0.7), 0.2,(0.3, 0.4,0.4, 0.4), 0.2, (0.6, 0.4, 0.4, 0.4), 0.5,(0.6, 0.7)}.

0.4, 0.5, 0.7)}.

={−3, {−3, 0.5, 0.6, 0.7), (0.3, 0.4), (0.6, 0.4, 0.5, 0.7), 0.6, AA −A− B−= BB = {−3, (0.5,(0.5, 0.5, (0.5, 0.6, 0.5, 0.7), 0.6, (0.3, 0.7), 0.1,0.1, (0.3, 0.3,0.3, 0.4), 0.1, (0.6, 0.3, 0.4, 0.4), 0.5,(0.6, 0.7), 0.4, −2,−2, 0.5, (0.6,(0.6, 0.7), 0.4,0.4, 0.6, −2, 0.7),0.7), (0.6, 0.4, 0.6, 0.7), 0.5, 0.4), (0.6, 0.4, 0.5, 0.5, 0.4, 0.4), 0.2, 0.4, 0.4), 0.3,0.3, 0.5, 0.3, 0.4), 0.5, (0.6, 0.4), 0.4,(0.6, 0.5,0.5, 0.7), 0.4,0.7), 0.5, −1,−1, 0.7), (0.8,(0.8, 0.5, −1, 0.5,(0.8, 0.7),0.7), (0.3, 0.5,(0.3, 0.5, 0.2,0.2, 0.4, 0.7), 0.4), (0.3, (0.4,(0.4, (0.4, 0.5, 0.7), 0, (0.6, 0.4, 0.8), (0.5, 0.3, 0.5, 0.4, 0.6),0.7, 0.4, 0.6), (0.6,(0.6, (0.6, 0.4,0.4, 0.5, 0.4, 0.7), 0.5, 0, 0.7), (0.6, 0, 0.4,(0.6, 0.4,0.4, 0.8), 0.4, (0.5, 0.4, 0.3, 0.8), 0.5,(0.5, 0.2),0.2), 0.3, (0.3,(0.3, 0.5, 0.7,0.7, 0.4, 0.2), 0.6), (0.3, 1, 0.4, 0.7), (0.5, 0.3, 0.3), (0.4, 0.7, 0.4, 0.6)}. 1, (0.7, 1,(0.7, (0.7, 0.6,0.6, 0.4, 0.6, 0.7), 0.4, (0.5, 0.7), 0.3,(0.5, 0.4,0.4, 0.3), 0.3, (0.4, 0.4, 0.7, 0.3), 0.4,(0.4, 0.6)}.

0.7, 0.4, 0.6)}. ={2,{2, 0.4, 0.8), (0.5, 0.3, 0.5), (0.3, 0.7, 0.4, 0.6), 4, 0.4, 0.4, 0.8), 0.4, 0.4, 0.8), AA ×A× B×= BB = {2, (0.5,(0.5, (0.5, 0.6,0.6, 0.4, 0.6, 0.8), 0.4, (0.5, 0.8), 0.3,(0.5, 0.4,0.4, 0.5), 0.3, (0.3, 0.4, 0.7, 0.5), 0.4,(0.3, 0.6), 0.7, 4, (0.6, 0.4,(0.6, 0.4, 0.6), 0.4,4, 0.8), (0.6, 0.5, 0.2), (0.3, 0.7, 6, (0.7, 0.6, 0.4, 0.4, 0.3),0.3, 0.4, 0.3), (0.5,(0.5, (0.5, 0.3,0.3, 0.5, 0.3, 0.2), 0.5, (0.3, 0.2), 0.7,(0.3, 0.4,0.4, 0.6), 0.7,0.6), 0.4, 6, (0.7, 0.6), 0.6,6, 0.4,(0.7, 0.7),0.7), 0.6, (0.5,(0.5, 0.4, 0.3,0.3, 0.4, 0.7), 0.3), (0.5, 0.4, 0.6), 8, (0.6, 0.4, 0.7), (0.4, 0.3, 0.5, 0.5, 0.7),0.4, 0.5, 0.7), 0.7,0.7, 0.4, 0.7, 0.6), 0.4, 8, 0.6), (0.6, 8, 0.4,(0.6, 0.6,0.6, 0.7), 0.4, (0.4, 0.6, 0.3, 0.7), 0.5,(0.4, 0.4),0.4), 0.3, (0.6,(0.6, 0.5, 0.4,0.4, 0.5, 0.4), 0.7), (0.6, (0.4,(0.4, (0.4, 0.5, 0.7), (0.3, 0.2, 0.4), (0.6, 0.4, 0.5, 0.7)}. 12,12, 12, (0.8,(0.8, (0.8, 0.5,0.5, 0.5, 0.5, 0.7), 0.5, (0.3, 0.7), 0.2,(0.3, 0.4,0.4, 0.4), 0.2, (0.6, 0.4, 0.4, 0.4), 0.5,(0.6, 0.7)}.

0.4, 0.5, 0.7)}.

={1/4, {1/4, 0.5, 0.6, 0.7), (0.3, 0.4), (0.6, 0.4, 0.5, 0.7), (0.5, 0.4, AA ÷A÷ B÷= BB = {1/4, (0.5,(0.5, 0.5, (0.5, 0.6, 0.5, 0.7), 0.6, (0.3, 0.7), 0.1,0.1, (0.3, 0.3,0.3, 0.4), 0.1, (0.6, 0.3, 0.4, 0.4), 0.5,(0.6, 0.7), 0.4, 1/2,1/2, 0.5, (0.5,0.7), 0.6,0.6, 0.4, 1/2, 0.8),0.8), (0.5, 0.6, 0.4, 0.8), 0.4, 0.5), (0.3, 0.7, 0.5, 0.5, 0.4, 0.4), 0.2, 0.4, 0.4), 0.3,0.3, 0.4, 0.3, 0.5), 0.4, (0.3, 0.5), 0.7,(0.3, 0.4,0.4, 0.6), 0.7,0.6), 0.4, 3/4,3/4, 0.6), (0.8,(0.8, 0.5, 3/4, 0.5,(0.8, 0.7),0.7), (0.3, 0.5,(0.3, 0.2, 0.5,0.2, 0.4, 0.7), 0.4), (0.3, (0.5,(0.5, (0.5, 0.5, 0.7), 1, (0.6, 0.4, 0.8), (0.5, 0.3, 0.5, 0.4, 0.6),0.7, 0.4, 0.6), (0.6,(0.6, (0.6, 0.4,0.4, 0.5, 0.4, 0.7), 0.5, 1, 0.7), (0.6, 1, 0.4,(0.6, 0.4,0.4, 0.8), 0.4, (0.5, 0.4, 0.3, 0.8), 0.5,(0.5, 0.2),0.2), 0.3, (0.3,(0.3, 0.5, 0.7,0.7, 0.4, 0.2), 0.6), (0.3, 0.6, 0.4, 0.7), (0.5, 0.3, 0.3), (0.4, 0.7, 0.4, 0.6)}. 3/2,3/2, 3/2, (0.7,(0.7, 0.6, (0.7, 0.4, 0.6, 0.7), 0.4, (0.5, 0.7), 0.3,(0.5, 0.4,0.4, 0.3), 0.3, (0.4, 0.4, 0.7, 0.3), 0.4,(0.4, 0.6)}.

0.7, 0.4, 0.6)}. λB {4λ, 0.6, 0.7), (0.3, 0.1, 0.4), 0.4, 0.5, 0.7), 2λ, (0.7, 0.4, 0.8), 0.6, 0.4, 0.8), λB λB = =={4λ, {4λ, (0.8,(0.8, (0.8, 0.5,0.5, 0.6, 0.5, 0.7), 0.6, (0.3, 0.7), 0.1,(0.3, 0.2,0.2, 0.4), 0.1, (0.6, 0.2,(0.6, 0.4, 0.4), 0.5,(0.6, 0.7), 0.4, 2λ,0.5, (0.7,0.7), 0.6,0.6, 0.4, 2λ, 0.8), (0.7, 0.2), (0.3, 0.7, 0.6)}.

(0.5, 0.3,0.3, 0.4, 0.3,0.4, 0.2), 0.4, (0.3, 0.2), 0.7, (0.3, 0.4,0.4, 0.7, 0.6)}.

0.4, 0.6)}.

(0.5,(0.5,

Note that algebraic operations defined in this section remarkably Note Note that that the the the algebraic algebraic operations operations defined in defined this section in this are are section remarkably are remarkably different from those introduced reason isreason thatYe different different from from those those introduced introduced by by Ye Ye [42]. by[42].

Ye TheThe [42]. reason The is thatYe [42]is[42] thatYe [42] defined the operators neutrosophic multiset on the basis of algebraic defined defined thethe operators operators overover neutrosophic over neutrosophic multiset on multiset the basis on of the algebraic basis of algebraic sum and product which is t-norm t-conorm. sum sum and and product product which which is t-norm is t-norm andand t-conorm. and t-conorm.

Conclusion 5.5.5.Conclusion Conclusion Using the concept ofofcut sets of neutrosophic multi-sets, proposed Using Using thethe concept concept of cut sets cut ofsets neutrosophic of neutrosophic multi-sets, multi-sets, we we proposed we proposed the representation theorem neutrosophic multi-sets.

Based the on the the the representation representation theorem theorem for for neutrosophic for neutrosophic multi-sets. multi-sets.

Based on on theBased extension principle, defined algebraic operations as the addiextension extension principle, principle, we we defined we defined algebraic algebraic operations operations suchsuch as the such addias the addition, subtraction, multiplication division, which extensions of tion, tion, subtraction, subtraction, multiplication multiplication andand division, and division, which werewere which extensions wereofextensions of the operations intuitionistic fuzzy sets in sets [17].

The algebraic the the operations operations overover over the the intuitionistic the intuitionistic fuzzy setsfuzzy in [17].

Thein algebraic [17].

The algebraic operations proposed in in this paper were remarkably different from thosefrom those operations operations proposed proposed in this paper this were paper remarkably were remarkably different from different those defined by which not always rational from a logical defined defined by by [11,[11, [11, 14, 14, 42], 14,42], which 42], which werewere notwere always notrational always from rational a logical from a logical point of view view practical aspects.

Different applications of the point point of of view andand and practical practical aspects. aspects.

Different Different applications applications of the rep-rep-of the representation theorem extension principles of neutrosophic multi-sets multi-sets resentation resentation theorem theorem andand extension and extension principles principles of neutrosophic of neutrosophic multi-sets will be examined examined ininnear future. will will bebe examined in near future. near future.

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Mehmet Şahin Associate Professor of Mathematics Department of Mathematics Gaziantep University Gaziantep,Turkey E-mail: mesahin@gantep.edu.tr Irfan Deli Assistant Professor of Mathematics Muallim Rıfat Faculty of Education Kilis 7 Aralık University Kilis, Turkey E-mail: irfandeli@kilis.edu.tr Vakkas Uluçay Doctor of Mathematics Department of Mathematics Gaziantep University Gaziantep,Turkey E-mail: vulucay27@gmail.com