TWO NEW OPERATOR DEFINED OVER INTERVAL VALUED INTUITIONISTIC FUZZY SETS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

TWO NEW OPERATOR DEFINED OVER INTERVAL VALUED INTUITIONISTIC FUZZY SETS S. Sudharsan1, 2 and D. Ezhilmaran3 1

2

Research Scholar, Bharathiar University, Coimbatore -641046, India.

Department of Mathematics, C. Abdul Hakeem College of Engineering & Technology, Melvisharam,Vellore – 632 509, Tamilnadu, India.

3 School of Advanced Sciences, VIT University, Vellore – 632 014, Tamilnadu, India.

ABSTRACT IVIFS with and Multiplication of an IVIFS with the natural number ”are proved.

In this paper, two new Operator defined over IVIFSs were introduced, which will be “Multiplication of an

Key Words: Intuitionistic fuzzy set, Interval valued Intuitionistic fuzzy sets, Operations over interval valued intuitionistic fuzzy sets.

AMS CLASSIFICATION: 03E72

1. INTRODUCTION In 1965, Fuzzy sets theory was proposed by L. A. Zadeh[18]. In 1986, the concept of intuitionistic fuzzy sets (IFSs), as a generalization of fuzzy set were introduced by K. Atanassov[1]. After the introduction of IFS, many researchers have shown interest in the IFS theory and applied in numerous fields, such as pattern recognition, machine learning, image processing, decision making and etc... In 1994, new operations defined over the intuitionistic fuzzy sets was proposed by K. Atanassov[3]. In 2000, Some operations on intuitionistic fuzzy sets were proposed by Supriya Kumar De, Ranjit Biswas and Akhil Ranjan Roy[12]. In 2001, an application of intuitionistic fuzzy sets in medical diagnosis were proposed by Supriya Kumar De, Ranjit Biswas and Akhil Ranjan Roy [13]. In 2006, n-extraction operation over intuitionistic fuzzy sets were proposed by B. Riecan and K. Atanassov[9]. In 2010, Operation division by n over intuitionistic fuzzy sets were proposed by B. Riecan and K. Atanassov [10]. In 2010, Remarks on equalities between intuitionistic fuzzy sets was K. Atanassov[4]. In 2008, properties of some IFS operators and operations were proposed by Liu Q, Ma C and Zhou X [7]. In 2008, Four equalities connected with intuitionistic fuzzy sets was proposed by T. Vasilev [14]. In 2011, Intuitionistic fuzzy sets: Some new results were proposed by R. K. Verma and B. D. Sharma[15].

In 1989, the notion of Interval-Valued Intuitionistic Fuzzy Sets which is a generalization of both DOI : 10.5121/ijfls.2014.4401

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

Intuitionistic Fuzzy Sets and Interval-Valued Fuzzy Sets were proposed by K. Atanassov and G. Gargov [5]. After the introduction of IVIFS, many researchers have shown interest in the IVIFS theory and applied it to the various field. In 1994, Operators over interval-valued intuitionistic fuzzy sets was proposed by K. Atanassov[6]. In 2007, methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making was proposed by Z.S. Xu [17]. In 2007, Some geometric aggregation operators based on interval-valued intuitionistic fuzzy sets and their application to group decision making were proposed by G.W .Wei and X.R.Wang [16]. In 2012, Some Results on Generalized Interval-Valued Intuitionistic Fuzzy Sets were proposed by Monoranjan Bhowmik and Madhumangal Pal[8]. In 2013, Interval-Valued Intuitionistic Hesitant Fuzzy Aggregation Operators and Their Application in Group DecisionMaking were proposed by Zhiming Zhang [19]. In 2014, new Operations over Interval Valued Intuitionistic Hesitant Fuzzy Set were proposed by Broumi and Florentin Smarandache [11]. This paper proceeds as follows: In section 2 some basic definitions related to intuitionistic fuzzy sets, interval valued intuitionistic fuzzy sets and set operations are introduced over the IVIFSs are presented. In section 3 two new operators and defined over IVIFSs are introduced and proved. In section 4 and 5, Conclusion and Acknowledgments are given.

2. PRELIMINARIES Definition 2.1: Intuitionistic Fuzzy Set [1,2]: An intuitionistic fuzzy set A in the finite universe X is defined as ⌊ , , âŒŞ | ∈ , where : → 0,1 and : → 0,1 with the condition 0 sup! " # sup! " 1, for any ∈ . The intervals and denote the degree of membership function and the degree of non-membership of the element x to the set A. Definition 2.2:

Interval valued Intuitionistic Fuzzy Set [5]: An Interval valued intuitionistic fuzzy set A in the finite universe X is defined as A ⌊ , , âŒŞ, | ∈ . The intervals and denote the degree of membership function and the degree of non-membership of the element x to the set A. For every ∈ , and are closed intervals and their Left and Right end points are denoted by % , & , % ,and & .Let us denote % , & ) % , & )âŒŞ & & % '⌊ , (

, (

| ∈ *Where 0 # 1 , + % % & % & 0, + 0. Especially if and then the given IVIFS A is reduced to an ordinary IFS. Let us define the empty IVIFS, the totally uncertain IVIFS and the unit IVIFS by: ,∗ ⌊ , 0,0 , 1,1 âŒŞ | ∈ , . ∗ ⌊ , 0,0 , 1,1 âŒŞ | ∈ / 0 1 ∗ ⌊ , 1,1 , 0,0 âŒŞ | ∈ .

Definition 2.2. Set operations on IVIFSs [5]: Let A and B be two IVIFSs on the universe X, where

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

'2 , ( % , & ) , ( % , & )3 | ∈ * 4

' 2 , ( 5% , 5& ) , ( 5% , 5& )3 | ∈ *

Here, we define some set operations for IVIFSs: âˆŞ 4 7 8 ∊ 4 78

, (9/ ! % , 5% " , 9/ ! & , 5& ") , !9: ! % , 5% " , 9: ! & , 5& " "

, (9: ! % , 5% " , 9: ! & , 5& ") ,

; | â&#x2C6;&#x2C6; <

; | â&#x2C6;&#x2C6; < !9/ ! % , 5% " , 9/ ! & , 5& " " , ! % # 5% @ % 5% , & 5& @ & 5& ", A | â&#x2C6;&#x2C6; B # 4 > ? % , 5% , & 5& , ! % 5% , & , 5& ", â&#x2C6;&#x2122; 4 > ? % A | â&#x2C6;&#x2C6; B # 5% @ % 5% , & # 5& @ & 5& D E F , ! % , & ", ! % , & "G | â&#x2C6;&#x2C6; H I EF , ! % , & ", !1 @ % , 1 @ & "G| â&#x2C6;&#x2C6; H â&#x;Ą EF , !1 @ % , 1 @ & " , ! % , & "G| â&#x2C6;&#x2C6; H

EF , !1 @ !1 @ % " , 1 @ !1 @ & " ", !! % " , ! & " "G| â&#x2C6;&#x2C6; H

EF , !! % " , ! & " ", !1 @ !1 @ % " , 1 @ !1 @ & " "G| â&#x2C6;&#x2C6; H

â&#x2C6;&#x161; >? , L M! % ", M! & "N , L1 @ M!1 @ % ", 1 @ M!1 @ & "NA | â&#x2C6;&#x2C6; B

>? , L1 @ M!1 @ % ", 1 @ M!1 @ & "N , L M! % ", M! & "NA | â&#x2C6;&#x2C6; B,

Where + 1 is natural number.

3. TWO NEW OPERATOR OPO AND PO DEFINED OVER IVIFS ARE O INTRODUCED AND PROVED Q

Q

Operations â&#x20AC;&#x153;extractionâ&#x20AC;? as well as of operation â&#x20AC;&#x153;Multiplication of an IVIFS with and Multiplication of an IVIFS with â&#x20AC;?. It has the form for every IVIFS and for every natural number + 1

Two new Operators, defined over IVIFS are introduced, which will be an analogous as of

^ UX a

, Y1 @ Z1 @ M! % ", 1 @ Z1 @ M! & "[ ,] W S S S S 1 W ] W | â&#x2C6;&#x2C6;

]

T

W Z ] % & Y 1 @ M!1 @ " , Z1 @ M!1 @ " [ S SĚ&#x20AC; SW S ] RV \ _

% " & " , (1 @ !1 @ ! " , 1 @ !1 @ ! " ), bc d | â&#x2C6;&#x2C6; e

(!1 @ !1 @ % " " , !1 @ !1 @ & " " )

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

Theorem 3.1. For any two IVIFSs A and B and for every natural number + 1: / . â&#x2C6;© 4 â&#x2C6;© 4 , f . â&#x2C6;ª 4 â&#x2C6;ª 4

1 1 1 1 1 1 g . h â&#x2C6;© 4 i â&#x2C6;© 4 , 0 . h â&#x2C6;ª 4 i â&#x2C6;ª 4 .

Proof: / . â&#x2C6;© 4

EF , ! ! % " , ! & " ", !1 @ !1 @ % " , 1 @ !1 @ & " "G| â&#x2C6;&#x2C6; H â&#x2C6;© [ Y % " & " % " & " EF , ! ! 5 , ! 5 ", !1 @ !1 @ 5 , 1 @ !1 @ 5 "G| â&#x2C6;&#x2C6; H % % & & UX , (9: !! " , ! 5 " ", 9: !! " , ! 5 " ") ,^ a l q

W ] % % k | â&#x2C6;&#x2C6; p 9/ !1 @ !1 @ " , 1 @ !1 @ 5 " ", TW ` ] m

n 9/ !1 @ !1 @ & " , 1 @ !1 @ 5& " " RV \ _ j o

% " % " , ! ") , 1 @ (1 @ 9: !! ^ UX a

5 , Y [, W ] S S

S S 1 @ (1 @ 9: !! & " , ! 5& " ") W ] | â&#x2C6;&#x2C6;

W

] T (9/ !1 @ !1 @ % " , 1 @ !1 @ 5% " ") , W ] S SÌ&#x20AC; [] SW Y S

& " & " (9/ !1 @ !1 @

, 1 @ !1 @

") 5 5 RV \ _

1 @ (9/ !1 @ ! % " , 1 @ ! 5% " ") , ^ UX a , Y [, ] W S S

S S 1 @ (9/ !1 @ ! & " , 1 @ ! 5& " ") W ] W | â&#x2C6;&#x2C6;

] TW 9/ (!1 @ !1 @ % " " , !1 @ !1 @ 5% " " ) , ] S SÌ&#x20AC; SW Y

[] S & " & " " , !1 @ !1 @

" ) 9/ (!1 @ !1 @

5 RV \ _

% " % " 1 @ 9/ (!1 @ ! " , !1 @ ! 5 " ), ^ UX a , Y

[,] W

S S S S 1 @ 9/ (!1 @ ! & " " , !1 @ ! 5& " " ) ] W W | â&#x2C6;&#x2C6;

] T W 9/ (!1 @ !1 @ % " " , !1 @ !1 @ 5% " " ) , ] S SÌ&#x20AC;

[ ] S WY S & " & " 9/ (!1 @ !1 @

" , !1 @ !1 @

" )

5 \ RV _

% " % " 9: (1 @ !1 @ ! " , 1 @ !1 @ ! 5 " ), ^ UX a , Y

[ ,] W

S S S S 9: (1 @ !1 @ ! & " " , 1 @ !1 @ ! 5& " " ) ] W | â&#x2C6;&#x2C6;

W

] TW 9/ (!1 @ !1 @ % " " , !1 @ !1 @ 5% " " ) , ] S SÌ&#x20AC; SW Y S

[ ] 9/ (!1 @ !1 @ & " " , !1 @ !1 @ 5& " " ) \ RV _

% " & " , (1 @ !1 @ ! " , 1 @ !1 @ ! " ), bc d | â&#x2C6;&#x2C6; eq

l (!1 @ !1 @ % " " , !1 @ !1 @ & " " ) k p p k â&#x2C6;©

k p % " & " , (1 @ !1 @ ! " , 1 @ !1 @ ! " ) , k p 5 5 bc d | â&#x2C6;&#x2C6; e

(!1 @ !1 @ 5% " " , !1 @ !1 @ 5& " " ) j o

4

â&#x2C6;© 4 Hence / is proved and similarly (b) is proved by analogy.

Proof: g .

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

1 h â&#x2C6;© 4 i

% & % & 78 , m L M N , L M Nn , L1 @ M!1 @ ", 1 @ M!1 @ "N; | â&#x2C6;&#x2C6; < l q 1k

p k â&#x2C6;© p k p

% & % & 78 , m L M 5 N , L M 5 Nn , L1 @ M!1 @ 5 ", 1 @ M!1 @ 5 "N; | â&#x2C6;&#x2C6; < j o

UX , m 9: L M % , M % N , 9: L M & , M & Nn ,^ a

5

5 lSW ] Sq SW Sp ] k 1

] | â&#x2C6;&#x2C6; p k W 9/ L1 @ M!1 @ % ", 1 @ M!1 @ 5% "N , kTW l q ] p k p ] kSW SÌ&#x20AC;p

9/ L1 @ M!1 @ & ", 1 @ M!1 @ 5& "N SW S ] j o \ jRV _o

^ UX a % % l1 @ Z m1 @ 9: L M , M 5 Nn ,q ] SW S SW , k S p, ] SW S ] k p

SW S ] 1 @ Z m1 @ 9: L M & , M 5& Nn SW S ] j o | â&#x2C6;&#x2C6;

W ] TW

] % % SW l Z9/ L1 @ M!1 @ ", 1 @ M!1 @ 5 "N ,q] SÌ&#x20AC; SW k S p] S S k p W ]

S

S & & SW Z9/ L1 @ M!1 @ ", 1 @ M!1 @ 5 "N ] S o\ RV j _

^ UX a Z9/ L1 @ M % , 1 @ M 5% N ,q 1 @

l ] SW S SW ] S p, , k SW S ] k p

SW S ] 1 @ Z9/ L1 @ M & , 1 @ M 5& N SW S ] j o ] | â&#x2C6;&#x2C6; W

] TW

% % SWl9/ Y Z1 @ M!1 @ " , Z1 @ M!1 @ 5 "[ ,q ] SÌ&#x20AC; SW k S p] SW k S p ]

p SW k ] S

& & SW 9/ Y Z1 @ M!1 @ " , Z1 @ M!1 @ 5 "[ ] S RV j _ o\

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

^ UX a

% Z1 @ M 5% [ ,q ] ZL1 M 1 @ 9/ Y @ N ,

SW l S p ] SW k S , p, ] SW k S

SW k 1 @ 9/ Y ZL1 @ M & N , Z1 @ M & [ p ] S

5 SW ] S j o ] W | â&#x2C6;&#x2C6; TW ]

SWl9/ Y Z1 @ M!1 @ % " , Z1 @ M!1 @ 5% "[ ,q ] SÌ&#x20AC; SW k ] S p SW k S p ]

SW k S p ]

& & SW 9/ Y Z1 @ M!1 @ " , Z1 @ M!1 @ 5 "[ ] S RV j o\ _

^ UX a

% % Z S W l9: Y1 @ ZL1 @ M N , 1 @ 1 @ M 5 [ ,q ] S p] SW k S , p ,] SW k S

S W k 9: Y1 @ ZL1 @ M & N , 1 @ Z1 @ M & [ p ] S

5 SW ] S j o] W | â&#x2C6;&#x2C6; TW ]

S W l9/ Y Z1 @ M!1 @ % " , Z1 @ M!1 @ 5% "[ ,q ] SÌ&#x20AC; SW k ] S p SW k S p ]

SW k S p ]

& & S W 9/ Y Z1 @ M!1 @ " , Z1 @ M!1 @ 5 "[ ] S RV j o \ _

UX , L1 @ M(1 @ r % ) , 1 @ M(1 @ r & )N ,^ a Sq ] l SW ] | â&#x2C6;&#x2C6; p k W

kTW Y Z1 @ M!1 @ % " , Z1 @ M!1 @ & " [ ] p

W ] S SÌ&#x20AC;p k \ _p k RV k â&#x2C6;© p

k UX % & ap , L1 @ M(1 @ r 5 ) , 1 @ M(1 @ r 5 )N ,^ k SW Sp ] k W ] | â&#x2C6;&#x2C6; p

k TW p

] % & SW Y Z1 @ M!1 @ 5 " , Z1 @ M!1 @ 5 " [ ] SÌ&#x20AC; j RV \ _o 1 1 â&#x2C6;© 4

Hence g is proved and similarly (d) is proved by analogy.

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

Theorem 3.2. For every IVIFS A and for every natural number + 1:

/ . I I ; f . â&#x;¡ â&#x;¡ ; g . I I ; 0 .â&#x;¡ â&#x;¡ .

1 I

1

I 78 , m L M % N , L M & Nn , L1 @ M!1 @ % ", 1 @ M!1 @ & "N ; | â&#x2C6;&#x2C6; <

Proof: (a).

I

U T R

c , Y1 @ ZL1 @ M % N , 1 @ ZL1 @ M & N[ , Y Z1 @ M!1 @ % " , Z1 @ M!1 @ & " [d

| â&#x2C6;&#x2C6;

^ UX a , L1 @ M(1 @ r % ) , 1 @ M(1 @ r & )N , SW S ] | â&#x2C6;&#x2C6;

W ]

TW

% & S m1 @ L1 @ M(1 @ r )N , 1 @ L1 @ M(1 @ r )Nn] SÌ&#x20AC; RV \ _

a ` _

>? , L1 @ M(1 @ r % ) , 1 @ M(1 @ r & )N , L M(1 @ r % ) , M(1 @ r & )NA | â&#x2C6;&#x2C6; B

(3.1)

1 I

1 % & % & I bc8 , m L M N , L M Nn , L1 @ M!1 @ ", 1 @ M!1 @ "N;d | â&#x2C6;&#x2C6; e

1

% & % >? , L M , M N , L 1 @ M , 1 @ M & NA | â&#x2C6;&#x2C6; B

% & % & >? , L1 @ M(1 @ r ) , 1 @ M(1 @ r )N , L M(1 @ r ) , M(1 @ r )NA | â&#x2C6;&#x2C6; B

(3.2)

From (3.1) and (3.2), we get

1 1 I I

Hence (a) is proved and similarly (b) is proved by analogy.

I

I EF , !! % " , ! & " ",!1 @ !1 @ % " , 1 @ !1 @ & " "G| â&#x2C6;&#x2C6; H

Proof: g .

% & % & I tu , (1 @ !1 @ ! " " , 1 @ !1 @ ! " " ) , (!1 @ !1 @ " " , !1 @ !1 @ " " )v | â&#x2C6;&#x2C6; w

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

bc

, (1 @ !1 @ ! % " " , 1 @ !1 @ ! & " " ) ,

h1 @ (1 @ !1 @ ! % " " ) , 1 @ (1 @ !1 @ ! & " " )i

d | â&#x2C6;&#x2C6; e

'2 , (1 @ !1 @ ! % " " , 1 @ !1 @ ! & " " ) , (!1 @ ! % " " , !1 @ ! & " " )3 | â&#x2C6;&#x2C6; * (3.3) I

IEF , !! % " , ! & " ", !1 @ !1 @ % " , 1 @ !1 @ & " " G| â&#x2C6;&#x2C6; H

EF , !! % " , ! & " " , !1 @ ! % " , 1 @ ! & " " G| â&#x2C6;&#x2C6; H

'2 , (1 @ !1 @ ! % " " , 1 @ !1 @ ! & " " ) , (!1 @ ! % " " , !1 @ ! & " " ) 3 | â&#x2C6;&#x2C6; * (3.4)

From (3.3) and (3.4), we get I I Hence g is proved and similarly (d) is proved by analogy.

Theorem 3.3. For any two IVIFSs A and B and for every natural number + 1:

1 1 1 1 1 1 / . h # 4 i # 4 , f . h â&#x2C6;&#x2122; 4 i â&#x2C6;&#x2122; 4 ,

g . # 4 # 4 , 0 . â&#x2C6;&#x2122; 4 â&#x2C6;&#x2122; 4 .

Proof: /

1 h # 4 i

1

L>? , r , 1 @ M!1 @ " A| â&#x2C6;&#x2C6; B # >? , r 5 , 1 @ M!1 @ 5 "A | â&#x2C6;&#x2C6; BN

1

>? , ( r # r 5 @ r r 5 ) , L1 @ M!1 @ "N L1 @ M!1 @ 5 "N A | â&#x2C6;&#x2C6; B

^ UX a 1 @ M(1 @ r ) # q ] SW l S

SW k

] S p M(1 @ r 5 ) SW k 1@ p S ]

p , ZL1 @ M!1 @ "N ZL1 @ M!1 @ "N] | â&#x2C6;&#x2C6; W , k 5 ] T W k@ L1 @ M(1 @ r )Np

p ] SW k SÌ&#x20AC; p ] SW k S

SW S ] L1 @ M(1 @ r 5 )N o \ RV j _

bc , 1 @ M(1 @ r ) , ZL1 @ M!1 @ "N d| â&#x2C6;&#x2C6; e

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014 # bc , 1 @ M(1 @ r 5 ) , ZL1 @ M!1 @ 5 "Nd| â&#x2C6;&#x2C6; e

1 1

>? , r , 1 @ M!1 @ "A | â&#x2C6;&#x2C6; B # >? , r 5 , 1 @ M!1 @ 5 "A | â&#x2C6;&#x2C6; B

1 1 # 4

Hence (a) is proved and similarly (b) is proved by analogy.

Proof: g

# 4

!EF , ! " , 1 @ !1 @ " G| â&#x2C6;&#x2C6; H # EF , ! 5 " , 1 @ !1 @ 5 " G| â&#x2C6;&#x2C6; H"

EF , ! " # ! 5 " @ ! " ! 5 " , !1 @ !1 @ " "!1 @ !1 @ 5 " "G| â&#x2C6;&#x2C6; H

1 @ !1 @ ! " " ^ UX a ] S W l #1 @ !1 @ ! " " q S 5

p ] " " W , k , !1 @ !1 @

" !1 @ !1 @

" | â&#x2C6;&#x2C6;

5 k p T W k@ (1 @ !1 @ ! " " )p ] SW SÌ&#x20AC;

]

R V j (1 @ !1 @ ! 5 " " ) o \ _

F " " , !1 @ !1 @ " " G > , 1 @ !1 @ !

B | â&#x2C6;&#x2C6;

F " " , !1 @ !1 @ 5 " " G # > , 1 @ !1 @ ! 5 B | â&#x2C6;&#x2C6;

EF , ! " , 1 @ !1 @ " G| â&#x2C6;&#x2C6; H # EF , ! 5 " , 1 @ !1 @ 5 " G| â&#x2C6;&#x2C6; H

# 4

Hence (c) is proved and similarly (d) is proved by analogy.

Theorem 3.4. For every IVIFS A and for every natural number + 1:

1 1 / . h i f .

Proof. / .

1 h i

1

L >? , L M % , M & N , L1 @ M!1 @ % ", 1 @ M!1 @ & "NA | â&#x2C6;&#x2C6; BN

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

^ UX a

% Z Z1 @ M & [ , ] M , Y 1 @ 1 @ , 1 @

W S S S S W ] W ] | â&#x2C6;&#x2C6; TW

]

SWY Z1 @ M!1 @ % " , Z1 @ M!1 @ & " [] SÌ&#x20AC; S S \ RV _

^ UX a

SW l x1 @ x1 @ Y1 @ Z1 @ M % [y y ,q Y Z1 @ M!1 @ % " [ , ] S l q

SW k S ] p k p] p W , k , | â&#x2C6;&#x2C6;

p k p k TW k p]

SW k x1 @ x1 @ Y1 @ Z1 @ M & [y y p Y Z1 @ M!1 @ & " [ ] SÌ&#x20AC;

SW S ] j o \ RV j o _

U a

Sz , lY1 @ Y Z1 @ M % [ [ , Y1 @ Y Z1 @ M & [ [q , L1 @ M!1 @ % " , 1 @ M!1 @ & "N{S

T j o S SÌ&#x20AC; R _ | â&#x2C6;&#x2C6; 78 , m1 @ L1 @ M % N , 1 @ L1 @ M & Nn , L1 @ M!1 @ % ", 1 @ M!1 @ & "N; | â&#x2C6;&#x2C6; <

>? , L M % , M & N , L1 @ M!1 @ % ", 1 @ M!1 @ & "N A | â&#x2C6;&#x2C6; B

Hence (a) is proved.

Proof. f .

1

1

! EF , !! % " , ! & " ", !1 @ !1 @ % " , 1 @ !1 @ & " " G| â&#x2C6;&#x2C6; H"

% & 1 , ( 1 @ !1 @ ! " " , 1 @ !1 @ ! " " ) d | â&#x2C6;&#x2C6; e bc

, (!1 @ !1 @ % " " , !1 @ !1 @ & " " )

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

^ UX a L1 @ Mh1 @ (1 @ !1 @ ! % " " )iN , l q ] SW S , k SW S p, ]

SW S ] L1 @ Mh1 @ (1 @ !1 @ ! & " " )iN W ]| â&#x2C6;&#x2C6; j o TW ]

]

S W M SÌ&#x20AC; % " & " N , L M!1 @ !1 @ " " N[] S WYL !1 @ !1 @ S SV S \

R _

% " UX ^ a M " N , SW l1 @ L !1 @ !

S q % " & " ] W , k , ! 1 @ !1 @ , 1 @ !1 @ "] | â&#x2C6;&#x2C6; p

TW

] 1 @ L M!1 @ ! & " " N S SÌ&#x20AC; \ RV j _ o

% & % & tu , (1 @ !1 @ ! " ", 1 @ !1 @ ! " ") , !1 @ !1 @ " , 1 @ !1 @ " "v | â&#x2C6;&#x2C6; w

EF , ! ! % " , ! & " ", !1 @ !1 @ % " , 1 @ !1 @ & " "G| â&#x2C6;&#x2C6; H

Hence (b) is proved.

Theorem 3.5: For every IVIFS A and for every natural number n +1:

(a). | h | i , (b). |! | "

Proof. (a).

1 | L | N

1 | h EF , ! % , & ", ! % , & "G | } Hi

% & % & | L> ? , L1 @ M1 @ ! ", 1 @ M1 @ ! "N , L M! ", M! "N A| â&#x2C6;&#x2C6; BN

^ UX a

% " Z Z M1 , Y 1 @ @ ! , 1 @ M1 @ ! & "[ , ]

lS SW Sq S ] k W p | k W ] | â&#x2C6;&#x2C6; p kT W

p ]

S WY1 @ ZL1 @ M! % "N , 1 @ ZL1 @ M! & "N[] SÌ&#x20AC; S S jR V \ _o

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014

^ UX a

, Y1 @ ZL1 @ M! % "N , 1 @ ZL1 @ M! & "N[ ,] W S S S S W ] W | â&#x2C6;&#x2C6;

] TW

]

% & S SĚ&#x20AC; Y Z1 @ M1 @ ! ", Z1 @ M1 @ ! "[ SW S ] RV \ _

1

Hence (a) is proved. |! | "

Proof. (b).

|! EF , ! % , & ", ! % , & "G | } H"

|!E F , !1 @ !1 @ % " , 1 @ !1 @ & " ", !! % " , ! & " "G| â&#x2C6;&#x2C6; H"

% & % & | t2 , (!1 @ !1 @ " " , !1 @ !1 @ " " ) , (1 @ !1 @ ! " " , 1 @ !1 @ ! " " )3 | â&#x2C6;&#x2C6; w

' 2 , (1 @ !1 @ ! % " " , 1 @ !1 @ ! & " " ) , (!1 @ !1 @ % " " , !1 @ !1 @ & " " )3 | â&#x2C6;&#x2C6; *

Hence (b) is proved.

4. CONCLUSION In this paper, two new operators based IVIFS were introduced and few theorems were proved. In future, the application of this operator will be proposed and another two operators based on IVIFS are to be introduced.

5. ACKNOWLEDGMENTS The authors are highly grateful to the Editor-in-Chief and Reviewer Professor Ayad Ghany Ismaeel and the referees for their valuable comments and suggestions.

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