Mapping on complex neutrosophic soft expert sets

Mapping on complex neutrosophic soft expert sets Ashraf Al-Quran, and Nasruddin Hassan

Citation: AIP Conference Proceedings 1940, 020111 (2018); doi: 10.1063/1.5028026 View online: https://doi.org/10.1063/1.5028026 View Table of Contents: http://aip.scitation.org/toc/apc/1940/1 Published by the American Institute of Physics

Articles you may be interested in Preface: The 2017 UKM FST Postgraduate Colloquium AIP Conference Proceedings 1940, 010001 (2018); 10.1063/1.5027915

Mapping on Complex Neutrosophic Soft Expert Sets Ashraf Al-Quran1, a) and Nasruddin Hassan2, b) 1,2

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia 43600 UKM Bangi Selangor DE, Malaysia a)

b)

aquraan @siswa.ukm.edu.my Corresponding author: nas@ukm.edu.my

Abstract. We introduce the mapping on complex neutrosophic soft expert sets. Further, we investigated the basic operations and other related properties of complex neutrosophic soft expert image and complex neutrosophic soft expert inverse image of complex neutrosophic soft expert sets.

INTRODUCTION Smarandache [1] firstly proposed the theory of neutrosophic set as a generalization of fuzzy set [2] and intuitionistic fuzzy set [3]. Unlike goal programming [4-8], neutrosophic set can deal with uncertain, indeterminate and incongruous information where the indeterminacy is quantified explicitly and truth-membership, indeterminacymembership and falsity-membership are completely independent. Molodtsov [9] introduced the notion of soft set which is free from the inadequacy of the parameterized tools of the former theories. It was extended to many variants including vague soft sets [10-17], Q-fuzzy soft sets [18-24] and genetic algorithms [25-27]. Maji [28] introduced the notion of neutrosophic soft set to combine the strengths of both of the soft sets and neutrosophic sets. Soft set and neutrosophic soft set are then extended to soft expert set [29] and neutrosophic soft expert set [30], where the user can know the opinions of all the experts in these models without any operations. The uncertainty sets that have been mentioned above are further extended to the complex field. The introduction of fuzzy sets was followed by their extension to the complex fuzzy set [31], intuitionistic fuzzy set also extended to complex intuitionistic fuzzy set [32]. Following in this direction, Ali and Smarandache [33] introduced complex neutrosophic set as a generalization of neutrosophic set. Complex neutrosophic set is characterized by complex-valued truth membership function, complex-valued indeterminate membership function, and complexvalued falsehood membership function such that each membership function associates with a phase term. The main purpose of this paper is to continue investigating complex neutrosophic set, soft expert set and their combinations. Kharal and Ahmad [34] introduced the notions of a mapping on the classes of soft sets and studied the properties of soft images and soft inverse images. They [35] also presented the concept of a mapping on classes of fuzzy soft sets. In the neutrosophic environment, Alkhazaleh and Marei [36] studied the notion of mapping on neutrosophic soft classes, followed by a mapping on neutrosophic soft expert sets by Broumi et al.[37]. We define the notion of complex neutrosophic soft expert sets (CNSES) and its basic operations, which will then lead to the concept of mapping on classes of complex neutrosophic soft expert sets. We also provide verifications of some fundamental operations and properties of this concept. This will further extend studies on neutrosophic soft expert set [38-40]. It enriches current studies on fuzzy sets [41-46] compared to those in certain environments [47-50].

PRELIMINARIES In this section, we recapitulate the concepts of neutrosophic and complex neutrosophic sets and present an overview of the operations structures of the complex neutrosophic model that are relevant to the work in this paper. The complex neutrosophic soft expert set (CNSES) and its basic operations are also introduced.

The 2017 UKM FST Postgraduate Colloquium AIP Conf. Proc. 1940, 020111-1–020111-10; https://doi.org/10.1063/1.5028026 Published by AIP Publishing. 978-0-7354-1632-1/$30.00

020111-1

We begin by recalling the definition of neutrosophic set. Definition 1. (See [1]) Let U be a universe of discourse. A neutrosophic set N in U is defined as N { u ;T N (u ); I N (u ); FN (u ) !; u Â? U }, where T N (u ), I N (u ) and FN (u ) are the truth membership function, the indeterminacy membership function and the falsity membership function, respectively, such that T ; I ; F : X o] 0;1 [ and 0 d T N (u ) I N (u ) FN (u ) d 3 . Ali and Smarandache [33] conceptualized complex neutrosophic set and gave the basic operations in the following two definitions. Definition 2. (See [33]) Let a universe of discourse U. A complex neutrosophic set S in U is characterized by a truth membership function T S (u ) , an indeterminacy membership function I S (u ) and a falsity membership function FS (u ) that assigns a complex-valued grade for each of these membership functions in S for any u Â?U . The values T S (u ) , I S (u ) , FS (u ) and their sum may all be within the unit circle in the complex plane and are of the form, TS (u ) and

pS (u ).e

PS (u ),Q S (u ), ZS (u )

j PS (u )

, I S (u )

are,

qS (u ).e

respectively,

jQS (u )

and FS (u )

real

valued

rS (u ).e

and

j ZS (u )

, each of p S (u ), q S (u ), rS (u )

p S (u ), q S (u ), rS (u ) Â? [0,1]

such

that

0 d pS (u ) qS (u ) rS (u ) d 3 . Definition 3. (See [33]) Let A and B be two complex neutrosophic sets on the universe U, where A is characterized by a truth membership function T A (u )

I A (u )

q A (u ).e

jQ A (u )

p A (u ).e

j PA (u )

and a falsity membership function FA (u )

truth membership function T B (u )

p B (u ).e

and a falsity membership function FB (u )

j PB (u )

, an indeterminacy membership function

rA (u ).e

j ZA (u )

and B is characterized by a

, an indeterminacy membership function I B (u )

rB (u ).e

j ZB (u )

q B (u ).e

jQ B (u )

.

We define the subset, union and intersection operations as follows. 1.

A set A is said to be complex neutrosophic subset of B ( A ÂŽ B ) if and only if the following conditions are satisfied: (a) T A (u ) d T B (u ) such that pA (u )d pB (u ) and P A (u ) d P B (u ) . (b) I A (u ) t I B (u ) such that q A (u ) t q B (u ) and Q A (u ) t Q B (u ) . (c) FA (u ) t FB (u ) such that rA (u ) t rB (u ) and ZA (u ) t ZB (u ) .

2.

The union(intersection) of

A and B , denoted as A ‰ (ˆ)B and the truth membership function

T A ‰( ˆ) B (u ) , the indeterminacy membership function I A ‰( ˆ) B (u ) and the falsity membership function FA ‰(ˆ) B (u ) are defined as follows: T A ‰(ˆ) B (u )

[( p A (u ) › (š) p B (u ))].e

j ( PA (u )›( š) PB (u ))

,

I A‰(ˆ) B (u) [(qA (u) š (›)qB (u))].e j (Q A (u )š(›)Q B (u )) and FA‰(ˆ) B (u) [(rA (u) š (›)rB (u))].e j (ZA (u )š(›)ZB (u )) , where

› = max and š = min .

020111-2

We will now give a definition of complex neutrosophic soft expert set below, by extending the concept of complex neutrosophic set defined earlier and give the definitions of empty and absolute CNSES. Definition 4. Let U be a universe, E a set of parameters, X a set of experts (agents) and O {1 agree , 0 disagree } a set of opinions. Let Z E u X uO and A Z . A pair (H , A ) is called a complex neutrosophic soft expert set (CNSES) over U , where H is a mapping given by

H : A o CN where CN

U

U

,

denotes the power complex neutrosophic set of U.

Definition 5. Let ( H, A ) be a CNSES over U. Then, 1.

(H , A ) is said to be empty CNSES, denoted by ) if T H (D ) (u )

0, I H (D ) (u ) 1, FH (D ) (u ) 1, D A

and u U . 2.

(H , A ) is said to be absolute CNSES, denoted by < if T H (D ) (u ) 1, I H (D ) (u )

0, FH (D ) (u )

0,

D A and u U . In the following, we define the subset, union and intersection operations of two CNSESs. Definition 6. For two CNSESs (H , A ) and (G , B ) over U, (H , A ) is called a CNSE subset of (G , B ) if 1. A B , 2. A , H ( ) is complex neutrosophic subset of G ( ) . Definition 7. The union of two CNSESs (H , A ) and (G , B ) over a universe U, is a CNSES (K , C ) , denoted by

(H , A ) (G , B ) , such that C

C , u U , A B and C H (H ), ­ ° (K , C ) = ® G (H ), °̄ H (H ) G (H ),

if H A B ; if H B A ; if H A B ;

where denotes the complex neutrosophic union. Definition 8. The intersection of two CNSESs denoted by (H , A ) (G , B ) , such that C

(H , A ) and (G , B ) over a universe U, is a CNSES (K , C ) ,

C , u U , A B and C

(K , C ) =

H (H ), ­ ° (H ), G ® °̄ H (H ) G (H ),

if H A B ; if H B A ; if H A B ;

where denotes the complex neutrosophic intersection.

MAPPING ON COMPLEX NEUTROSOPHIC SOFT EXPERT SETS In this section, we introduce the notion of a mapping on complex neutrosophic soft expert classes. We will define the complex neutrosophic soft expert images and complex neutrosophic soft expert inverse images of

020111-3

complex neutrosophic soft expert sets, and support them with an illustrative example. We will give also some operations and properties related with this concept. Now we propose the definition of a complex neutrosophic soft expert class, followed by the definitions of complex neutrosophic soft expert image and inverse image of the complex neutrosophic soft expert set. Definition 9. Let U be a universe, E a set of parameters, X a set of experts and O {1 agree , 0 disagree } a set of opinions. Let Z E u X uO . Then the collection of all complex neutrosophic soft expert sets over U with parameters from Z is called a complex neutrosophic soft expert class and is denoted as (U , Z ) . Definition 10 . Let (U , Z ) and (V , Z c ) be complex neutrosophic soft expert classes. Let r : U oV c c s : Z o Z be mappings. Then a mapping F : (U , Z ) o (V , Z ) is defined as follows: For a complex neutrosophic soft expert set (H , A ) in (U , Z ) , (F (H , A ), M ) , where M

s (Z ) Z

and

c

is a

c

complex neutrosophic soft expert set in (V , Z ) obtained as follows: 1 1 ­ H (D ))) (u ) if r (v ) and s ( E ) A z ; ° 1 ( 1 F ( H , A )( E )(v ) ®u r v D s ( E ) A °̄(0, 1, 1 ) otherwise. 1 c 1 For u r (v ) , E M Z , v V and D s ( E ) A , (F (H , A ), M ) is called a complex neutrosophic soft c expert image of the complex neutrosophic soft expert set H , A . If M Z , then we shall write (F (H , A ), M )

as F (H , A ) . Definition 11.

c

Let (U , Z ) and (V , Z ) be two complex neutrosophic soft expert classes. Let r : U oV and

c

s : Z o Z be mappings. Then a mapping F

1

c

: (V , Z ) o (U , Z ) is defined as follows: c

For a complex neutrosophic soft expert set (G , B ) in (V , Z ) , ( F

1

(G , B ), N ), where N

s

1

is a

( B ),

complex neutrosophic soft expert set in (U , Z ) obtained as follows:

­G (s (D ))(r (u )) 1 F (G , B )(D )(u ) ® ¯(0, 1, 1) for D N Z and u U . ( F

1

if s (D ) B , otherwise,

(G , B ), N ) is called a complex neutrosophic soft expert inverse image of the

complex neutrosophic soft expert set G , B . If N

1 1 Z , we shall write ( F (G , B ), N ) as F (G , B ) .

The Definitions 10 and 11 are illustrated as follows. Example 1. Let U E u X uO

and Z

Z

c

c

c

E u X uO

{u 1 , u 2 , u 3 } , V

{v 1 ,v 2 } and let E

{e 1 , e 2 } , E c

c

c

{e1 , e 2 } , X

{p , q } , X

c

{(e1 , p ,1), (e1 , q ,1),(e 2 , p ,1),(e 2 , q ,1),(e1 , p , 0), (e1 , q , 0), (e 2 , p , 0), (e 2 , q , 0)} and

{(e1c , p c ,1), (e1c , q c ,1), (e 2c , p c ,1), (e 2c , q c ,1), (e1c , p c , 0), (e1c , q c , 0), (e 2c , p c , 0), (e 2c , q c , 0)}.

020111-4

c

c

{p , q }

c

Suppose that (U , Z ) and (V , Z ) are complex neutrosophic soft expert classes. Define r : U oV and c

s : Z o Z as follows: r (u 1 ) v 2 , r (u 2 ) v 1 , r (u 3 ) c

c

c

s (e 2 , p ,1)

(e1 , q , 0), s (e 2 , q ,1)

s (e 2 , p , 0)

(e 2 , p ,1), s (e 2 , q , 0)

c

Let A Z

c

c

(e 2 , p ,1),s (e1, p , 0)

c

c

v 1 , s (e1 , p ,1)

c

c

(e 2 , p ,1), s (e1 , q ,1)

c

c

(e1, q , 0), s (e1, q , 0)

c

(e1 , p ,1),

c

(e1, p ,1),

(e1c , p c ,1) . c

B Z

{(e1 , p ,1), (e1 , q ,1),(e 2 , p , 0),(e 2 , q , 0)} ,

c

c

c

c

c

c

c

c

{(e1 , p ,1), (e1 , q , 0),(e 2 , p , 0),(e 2 , q ,1)}

and (H , A ) , (G , B ) be two complex neutrosophic soft expert sets over U and V respectively such that

(H , A ) =

{{(e1 , p ,1),{ ¢ 0.1e

j 2 3 (0.2)

j 2 3 (0.5)

¢ 0.1e

, 0.4e

j 2 3 (0.4)

, 0.3e

j 2 3 (0.6)

j 2 3 (0.4)

, 0.5e

j 2 3 (0.6)

²

, 0.3e

j 2 3 (0.7)

j 2 3 (0.3)

, 0.7e

u2 ¢ 0.1e

j 2 3 (0.8)

j 2 3 (0.7)

, 0.9e

, 0.9e

j 2 3 (0.8)

u1 ¢ 0.3e

j 2 3 (0.4)

, 0.5e

j 2 3 (0.6)

j 2 3 (0.4)

, 0.7e

u3 ¢ 0.3e

j 2 3 (0.4)

j 2 3 (0.7)

u1

u3 ¢ 0.3e

, 0.8e

, 0.6e

j 2 3 (0.7)

, 0.1e

j 2 3 (0.2)

u2

² ¢0.5e ,

j 2 3 (0.4)

¢0.9e

j 2 3 (0.4)

}}, {(e1 , q ,1),{

² ¢ 0.7e ,

j 2 3 (0.8)

² ¢ 0.6e ,

j 2 3 (0.4)

²

, 0.8e

j 2 3 (0)

, 0.8e

j 2 3 (0.3)

u2 j 2 3 (0.6)

, 0.5e

j 2 3 (0)

, 0.1e

j 2 3 (0.2)

²

u3

,

²

,

, 0.5e

j 2 3 (0.5)

, 0.4e

j 2 3 (0.1)

}},{(e 2 , p , 0), ²

,

u2

j 2 3 (0.5)

²

u1

, 0.9e

}},{(e 2 , q , 0),{

² ¢ 0.1e ,

j 2 3 (0.7)

, 0.2e

¢ 0.8e

j 2 3 (0.5)

, 0.4e

j 2 3 (0.1)

, 0.8e

j 2 3 (0.9)

²

u1

, 0.9e

j 2 3 (0.6)

, 0.4e

j 2 3 (0.1)

u3

²

,

}}}

(G , B ) =

c

c

{{(e1 , p ,1),{ {(e1c , q c , 0),{

j 2 3 (0.4)

, 0.8e

j 2 3 (0.5)

, 0.6e

j 2 3 (0.2)

v1

¢ 0.3e

{(e 2c , p c , 0),{

{(e 2c , q c ,1),{

¢ 0.5e

j 2 3 (0.5)

, 0.6e

j 2 3 (0.7)

j 2 3 (0.9)

, 0.8e

v1 ¢ 0.6e

¢ 0.4e

j 2 3 (0.8)

, 0.3e

j 2 3 (0.5)

, 0.4e

j 2 3 (0.1)

v1 j 2 3 (0.3)

, 0.5e

j 2 3 (0.7)

, 0.9e

j 2 3 (0.4)

v1

j 2 3 (0.7)

² ¢0.5e ,

j 2 3 (0.1)

, 0.8e

j 2 3 (0.2)

²

v2

² ¢ 0.4e ,

j 2 3 (0.5)

² ¢0.2e ,

j 2 3 (0.1)

² ¢ 0.5e ,

, 0.2e

, 0.2e

j 2 3 (0.6)

, 0.1e

j 2 3 (0.3)

, 0.9e

j 2 3 (0.4)

²

v2

j 2 3 (0.6)

, 0.2e

j 2 3 (0.8)

v2

, 0.7e

j 2 3 (0.5)

v2

, 0.4e

j 2 3 (0.6)

²

}},

}},

²

}},

}}}.

c

Then we define a mapping F : (U , Z ) o (V , Z ) as follows. For a complex neutrosophic soft expert set (H , A ) in (U , Z ) , (F (H , A ), M ) , where M c c c c c (e1 , p , 0) , (e1 , q , 0), (e 2 ,

{

c

c

c

s (Z )

p ,1)}, is a complex neutrosophic soft expert set in (V , Z ) and is obtained as follows:

c

F (H , A )(e1 , p ,1)(v 1 )

020111-5

(

u r 1 v 1 D s 1 e1c , p c ,1 1 A

u {u 2 ,u 3 }

¢ 0.7e

({

j 2 3 (0.8)

H (D ))(u )

j 2 3 (0.5)

¢ 0.9e ¢0

, 0.4e

(

u {u 2 ,u 3 } D ^ e 2 ,q ,0 0 ,((e1 ,q ,1) 1)`

j 2 3 (0.1)

¢ 0.3e

j 2 3 (0.3)

u1 j 2 3 (0)

, 0.9e

, 0.1e

j 2 3 (0.1)

²

j 2 3 (0.6)

, 0.3e

j 2 3 (0.7)

, 0.1e

² ¢0.3e ,

j 2 3 (0.6)

(H (e 2 , q , 0) H (e1 , q ,1))(u )

u {u 2 ,u 3 }

, 0.3e

j 2 3 (0.7)

, 0.1e

j 2 3 (0.2)

u2

²

,

})(u )

u3

(

, 0.8e

H (D ))(u )

j 2 3 (0.2)

u2

²

) (

¢0.7 0 7e

j 2 3 (0.8)

j 2 3 (0)

, 0.9e

, 0.1e

j 2 3 (0.1)

u3

²

)

¢ 0.7e j 23 (0.8) , 0.3e j 23 (0) , 0.1e j 23 (0.1) ² . c

c

(

F (H , A )(e1 , p ,1)(v 2 )

u r 1 v 2 D s 1 e1c , p c ,1 1 A

H (D ))(u )

(

u {u1 } D ^ e 2 ,q ,0 0 ,((e1 ,q ,1) 1)`

H (D )) (u )

¢ 0.9e j 23 (0.5) , 0.4e j 23 (0.1) , 0.8e j 23 (0.3) ² . c

c

Thus F (H , A )(e1 , p ,1) c

c

j 2 3 (0.8)

j 2 3 (0)

, 0.3e

, 0.1e

j 2 3 (0.1)

v1 c

c

(0,1,1) and F (H , A )(e1 , q , 0)(v 2 )

F (H , A )(e1 , q , 0)(v 1 ) c

¢ 0.7e

{

c

Thus F ( H , A )(e1 , q , 0)

j 2 3 (0.5)

² ¢0.9e ,

, 0.4e

j 2 3 (0.1)

, 0.8e

j 2 3 (0.3)

²

v2 1

c

c

(0,1,1) , since s (e1 , q , 0) A

},

I.

­ ° ¢ 0, 1, 1² ¢ 0, 1, 1² ½ ° , ® ¾. v v ° 2 ° ¯ 1 ¿

By similar calculations, consequently, we get c

c

¢0.6e j 23(0.4) , 0.2e j 23(0.4) , 0.4e j 23(0) ²

c

c

¢0.1e j 23(0.8) , 0.4e j 23(0.4) , 0.8e j 23(0.7) ² .

F (H , A )(e 2 , p ,1)(v 1 )

F (H , A )(e 2 , p ,1)(v 2 ) c

c

Thus F ( H , A )(e 2 , p ,1)

{

¢ 0.6e

j 2 3 (0.4)

, 0.2e

j 2 3 (0.4)

j 2 3 (0)

, 0.4e

v1

² ¢ 0.1e ,

j 2 3 (0.8)

, 0.4e

j 2 3 (0.4)

v2

Hence,

(F (H , A ), M ) =

{{(e1c , p c ,1),{ {(e1c , q c , 0),{ ¢ 0.1e

¢ 0.7e

j 2 3 (0.8)

, 0.3e

j 2 3 (0)

, 0.1e

j 2 3 (0.1)

v1

¢ 0, 1, 1² ¢ 0, 1, 1² ¢ 0.6e c c , }}, {(e 2 , p ,1),{ v1 v2

j 2 3 (0.8)

, 0.4e

j 2 3 (0.4)

v2

, 0.8e

j 2 3 (0.7)

²

² ¢ 0.9e ,

j 2 3 (0.4)

j 2 3 (0.5)

, 0.4e

j 2 3 (0.1)

, 0.8e

j 2 3 (0.3)

v2

, 0.2e

j 2 3 (0.4)

v1

, 0.4e

j 2 3 (0)

²

,

}} } .

Next for the complex neutrosophic soft expert inverse images, we have the following:

020111-6

²

}},

, 0.8e

j 2 3 (0.7)

²

}.

1 c For a complex neutrosophic soft expert set (G , B ) in (V , Z ) , ( F (G , B ), N ) , where N

s

1

(B )

{(e 2 , q , 0), (e1 , q , 0), (e1 , p , 0), (e 2 , p ,1), (e1 , q ,1)} is a complex neutrosophic soft expert set in (U , Z ) , where 1

c

c

F (G , B ) (e 2 , q , 0) (u1 ) G (s (e 2 , q , 0))r (u1 ) G (e1, p ,1)(v 2 )

({

¢ 0.5e

j 2 3 (0.4)

, 0.8e

j 2 3 (0.5)

, 0.6e

j 2 3 (0.2)

v1

² ¢0.5e ,

j 2 3 (0.7)

, 0.2e

j 2 3 (0.1)

, 0.8e

j 2 3 (0.2)

²

v2

})(v 2 )

¢ 0.5e j 23 (0.7) , 0.2e j 23 (0.1) , 0.8e j 23 (0.2) ². 1

c

c

F (G , B ) (e 2 , q , 0) (u 2 ) G (s (e 2 , q , 0))r (u 2 ) G (e1, p ,1)(v 1 )

({

¢ 0.5e

j 2 3 (0.4)

, 0.8e

j 2 3 (0.5)

, 0.6e

j 2 3 (0.2)

v1

² ¢0.5e ,

j 2 3 (0.7)

, 0.2e

j 2 3 (0.1)

, 0.8e

j 2 3 (0.2)

v2

²

})(v 1 )

¢ 0.5e j 23 (0.4) , 0.8e j 23 (0.5) , 0.6e j 2 3 (0.2) ². In the similar manner, 1

F (G , B ) (e 2 , q , 0) (u 3 )

¢0.5e

j 23(0.4)

, 0.8e

j 23(0.5)

, 0.6e

j 23(0.2)

².

Thus, 1

F (G , B ) (e 2 , q , 0) {

¢ 0.5e

j 2 3 (0.4)

, 0.8e

¢ 0.5e

j 2 3 (0.5)

j 2 3 (0.7)

, 0.2e

j 2 3 (0.1)

, 0.8e

j 2 3 (0.2)

u1

, 0.6e

j 2 3 (0.2)

² ¢ 0.5e ,

j 2 3 (0.4)

, 0.8e

j 2 3 (0.5)

, 0.6e

u2

²

}. u3 By similar calculations, consequently, we get

(F

1

(G , B ), N )

{{(e 2 , q , 0),{¢0.5e

j 23 (0.7)

, 0.2e j 23 (0.1) , 0.8e j 23 (0.2) ² ¢ 0.5e j 23 (0.4) , 0.8e j 23(0.5) , 0.6e j 23(0.2) ² , , u1 u2

¢0.5e j 23 (0.4) , 0.8e j 23 (0.5) , 0.6e j 23 (0.2) ² ¢ 0.5e j 23 (0.7) , 0.2e j 23 (0.1) , 0.8e j 23(0.2) ² }},{(e1, q , 0),{ , u3 u1 ¢0.5e j 23 (0.4) , 0.8e j 23 (0.5) , 0.6e j 23 (0.2) ² ¢0.5e j 23 (0.4) , 0.8e j 23 (0.5) , 0.6e j 23 (0.2) ² , }},{(e1, p , 0), u2 u3

{

¢0.4e j 23 (0.5) , 0.2e j 23 (0.6) , 0.1e j 23 (0.3) ² ¢0.3e j 23 (0.5) , 0.6e j 23 (0.7) , 0.8e j 23 (0.9) ² , , u1 u2

¢0.3e j 23 (0.5) , 0.6e j 23 (0.7) , 0.8e j 23 (0.9) ² ¢0.4e j 23 (0.5) , 0.2e j 23 (0.6) , 0.1e j 23 (0.3) ² }}, {(e 2 , p ,1),{ , u3 u1

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j 2 3 (0.2)

²

,

¢0.3e j 23 (0.5) , 0.6e j 23 (0.7) , 0.8e j 23 (0.9) ² ¢0.3e j 23 (0.5) , 0.6e j 23 (0.7) , 0.8e j 23 (0.9) ² , }},{(e1, q ,1), u2 u3

{

¢0.5e j 23 (0.7) , 0.2e j 23 (0.1) , 0.8e j 23 (0.2) ² ¢0.5e j 23 (0.4) , 0.8e j 23 (0.5) , 0.6e j 23 (0.2) ² , , u1 u2

¢0.5e j 23 (0.4) , 0.8e j 23 (0.5) , 0.6e j 23 (0.2) ² }}}. u3 c

Definition 12. Let F : (U , Z ) o (V , Z ) be a mapping and (H , A ) , (G , B ) complex neutrosophic soft expert c

sets in (U , Z ) . Then for E Z , Q V , the union and intersection of the complex neutrosophic soft expert images F (H , A ) and F (G , B ) are defined as follows.

F H , A ˆ F G , B

E Q

F H , A ˆ F G , B

E Q

F H , A E Q F G , B E Q , F H , A E Q F G , B E Q ,

where and denote the complex neutrosophic soft expert union and intersection, respectively. c

Definition 13. Let F : (U , Z ) o (V , Z ) be a mapping and (H , A ) , (G , B ) complex neutrosophic soft expert c

sets in (V , Z ) . Then for D Z , u U , the union and intersection of the complex neutrosophic soft expert inverse 1

images F ( H , A ) and F

1

(G , B ) are defined as follows.

F H , A ˆ F G , B

D u

F H , A ˆ F G , B

D u

where

1

1

1

1

H , A D u F 1 G , B D u , 1 1 F H , A D u F G , B D u ,

F

1

and denote the complex neutrosophic soft expert union and intersection, respectively.

Proposition 1. Let F : U , Z o (V , Z ) be a mapping. Then for complex neutrosophic soft expert sets H , A

c

and

G , B

in the complex neutrosophic soft expert class U , Z ,

1.

F () )

).

2.

F (< )

<.

3.

F

4. 5.

we have:

H , A G , B

F H , A F G , B . F H , A G , B

F H , A F G , B .

If H , A G , B , then F H , A F G , B .

Proof: The proof is straightforward by Definitions 10 and 12. Proposition 2.

H , A

and

Let F : U , Z o (V , Z ) be a mapping. Then for complex neutrosophic soft expert sets c

G , B

F

1

() )

).

2.

F

1

(< )

<.

3.

F

1.

1

c

in the complex neutrosophic soft expert class (V , Z ) ,

H , A G , B

F

1

H , A F 1 G , B .

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we have:

4. 5.

H , A G , B

H , A F 1 G , B . 1 1 If H , A G , B , then F H , A F G , B .

F

1

F

1

Proof: The proof is straightforward by Definitions 11 and 13.

CONCLUSION We established the concept of mapping on complex neutrosophic soft expert sets. The basic operations and properties of mapping on complex neutrosophic soft classes have been studied. We hope these fundamental results will help researchers to enhance and promote the research on complex neutrosophic soft expert theory.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

F. Smarandache, Int. J. Pure Appl. Math. 24 (3), 287-297 (2005). L.A. Zadeh, Inf. Control. 8 (3), 338-353 (1965). K. Atanassov, Fuzzy. Sets. Syst. 20 (1), 87-96 (1986). N. Hassan and Z. Ayop, Advances in Environmental Biology 6 (2), 510-513 (2012). N. Hassan and B.A. Halim, Sains Malaysiana 41 (9), 1155-1161 (2012). N. Hassan and L.L. Loon, Appl. Math. Sci. 6 (110), 5487-5493 (2012). N. Hassan and S. Sahrin, Advances in Environmental Biology 6 (5), 1868-1872 (2012). N. Hassan, L.W. Siew and S.Y. Shen, Appl. Math. Sci. 6 (110), 5483-5486 (2012). D. Molodtsov, Comput. Math. App. 37 (2), 19-31 (1999). K. Alhazaymeh and N. Hassan, Appl. Math. Sci. 7 (140), 6983- 6988; ibid 6989-6994 (2013). K. Alhazaymeh and N. Hassan, Int. J. Pure Appl. Math. 93 (4), 511-523 (2014). K. Alhazaymeh and N. Hassan, Adv. Fuzzy Syst. Vol 2012, Article ID 208489 K. Alhazaymeh, S. Abdul Halim, A. R. Salleh and N. Hassan, Appl. Math. Sci. 6 (54), 2669-2680 (2012). K. Alhazaymeh and N. Hassan, Int. J. Pure Appl. Math. 77 (3), 391-401; ibid (4), 549-563 (2012). K. Alhazaymeh and N. Hassan, AIP Conference Proceedings 1571, 970-974 (2013). K. Alhazaymeh and N. Hassan, Int. J. Pure Appl. Math. 93 (3), 351-360; ibid 361-367; ibid 369-376 (2014). K. Alhazaymeh, and N. Hassan, J. Intell. Fuzzy Systems 28 (3), 1205-1212 (2015). F. Adam and N. Hassan, J. Intell. Fuzzy Systems 27 (1), 419-424 (2014). F. Adam and N. Hassan, Appl. Math. Sci. 8 (175), 8697-8701 (2014). F. Adam and N. Hassan, AIP Conference Proceedings 1614, 834-839 (2014). F. Adam and N. Hassan, AIP Conference Proceedings 1602, 772-778 (2014). F. Adam and N. Hassan, Appl. Math. Sci. 8 (174), 8689-8695 (2014). F. Adam and N. Hassan, Far East J. Math. Sci. 97 (7), 871-881 (2015). F. Adam and N. Hassan, J. Intell. Fuzzy Systems 30 (2), 943-950 (2016). M. J. Varnamkhasti and N. Hassan, J. Appl. Math. Vol.2012, Article ID 253714 M. J. Varnamkhasti and N. Hassan, J. Intell. Fuzzy Systems 25 (3), 793-796 (2013). M. J. Varnamkhasti and N. Hassan, Pakistan Journal of Statistics 31 (5), 643-651 (2015). P. K. Maji, Ann. Fuzzy Math. Inform. 5 (1), 157-168 (2013). S. Alkhazaleh and A. R. Salleh, Adv. Decis. Sci. Vol.2011, Article ID 757868. M. Sahin, S. Alkhazaleh and V. Ulucay, Appl. Math. 6 (1), 116-127 (2015). D. Ramot, R. Milo, M. Friedman and A. Kandel, IEEE. Trans. Fuzzy. Syst. 10 (2), 171-186 (2002). A. Alkouri and A.R. Salleh, AIP Conference Proceedings 1482, 464-470 (2012). M. Ali and F. Smarandache, Neural. Comput. Appl. (2016), doi: 10.1007/s00521-015-2154-y. A. Kharal and B. Ahmad, New Math. Nat. Comput. 7 (3), 471-481 (2011). A. Kharal and B. Ahmad, Adv. Fuzzy Syst. Vol. 2009, Article ID 407890. S. Alkhazaleh and E. Marei, Neutrosophic Sets and Systems 2, 3-8 (2014). S. Broumi, M. Ali and F. Smarandache, J. New Theory 5, 26-42 (2015).

020111-9

[38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

A. Al-Quran and N. Hassan, Int. J. Appl. Dec. Sci. 9 (2), 212-227 (2016). A. Al-Quran and N. Hassan, J. Intell. Fuzzy Systems 30 (6), 3691-3702 (2016). A. Al-Quran and N. Hassan, AIP Conference Proceedings 1830, 070007, doi: 10.1063/1.4980956 (2017). F. Adam and N. Hassan, Malays. J. Math. Sci. 11 (1), 53-70 (2017). K. Alhazaymeh and N. Hassan, Malays. J. Math. Sci. 11 (1), 23-39 (2017) K. Alhazaymeh and N. Hassan, AIP Conference Proceedings 1830, 070001, doi: 10.1063/1.4980950 (2017). Y. Al-Qudah and N. Hassan, Int. J. Appl. Dec. Sci. 10 (2), 175-179 (2017). A. M. Khalil and N. Hassan, J. Intell. Fuzzy Systems 32 (3), 2309-2314 (2017). A. M. Khalil and N. Hassan, Appl. Math. Model. 41, 684-690 (2017). N. Hassan and M. M. Tabar, Australian Journal of Basic and Applied Sciences 5 (11), 1711-1714 (2011). N. Hassan, M. M. Tabar and P. Shabanzade, Australian Journal of Basic and Applied Sciences 4 (10), 5320- 5325 (2010). N.Hassan, International Journal of Mathematics and Computers in Simulation 10, 77-81 (2016). N.Hassan, Pakistan Journal of Statistics 31 (5), 539-546 (2015).

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