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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014 DOI : 10.5121/ijfls.2014.4401 1 TWO NEW OPERATOR DEFINED OVER INTERVAL VALUED INTUITIONISTIC FUZZY SETS S. Sudharsan 1, 2 and D. Ezhilmaran 3 1 Research Scholar, Bharathiar University, Coimbatore -641046, India. 2 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 In this paper, two new Operator defined over IVIFSs were introduced, which will be “Multiplication of an IVIFS with and Multiplication of an IVIFS with the natural number ”are proved. 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

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In this paper, two new Operator defined over IVIFSs were introduced, which will be “Multiplication of an IVIFS  with  and Multiplication of an IVIFS    with the natural number   ”are proved.

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Page 1: Two new operator defined over

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

DOI : 10.5121/ijfls.2014.4401 1

TWO NEW OPERATOR DEFINED OVER

INTERVAL VALUED INTUITIONISTIC FUZZY

SETS

S. Sudharsan1, 2

and D. Ezhilmaran3

1Research Scholar, Bharathiar University, Coimbatore -641046, India.

2Department of Mathematics, C. Abdul Hakeem College of Engineering & Technology,

Melvisharam,Vellore – 632 509, Tamilnadu, India. 3School of Advanced Sciences, VIT University, Vellore – 632 014, Tamilnadu, India.

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

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

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

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

2

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 Decision-

Making 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 � � ⟨�, �����, ������⟩|� ∈ ��, where ��: � → 0,1� and ��: � → 0,1� with the

condition0 � 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� ⟨�, �����, ������⟩, |� ∈ ��. 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 � � '⟨�, (��%���, ��&���) , (��%���, ��&���)⟩ |� ∈ �*Where0 � ��&��� # ��&��� � 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�⟩|� ∈ ��, .∗ � ⟨�, �0,0�, �1,1�⟩|� ∈ ��/�01∗ �⟨�, �1,1�, �0,0�⟩|� ∈ ��.

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

where

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3

� � '2�, (��%���, ��&���) , (��%���, ��&���)3|� ∈ �* 4� '2�, (�5% ���, �5&���) , (�5%���, �5&���)3|� ∈ �*

Here, we define some set operations for IVIFSs:

� ∪ 4 � 78�, (9/�!��%���, �5% ���", 9/�!��&���, �5&���") ,!9:�!��%���, �5%���", 9:�!��&���, �5&���"" ; |� ∈ �< � ∩ 4 � 78�, (9:�!��%���, �5% ���", 9:�!��&���, �5&���") ,!9/�!��%���, �5%���", 9/�!��&���, �5&���"" ; |� ∈ �< � # 4 � >?�, !��%��� # �5% ��� @ ��%����5% ���, ��&����5&��� @ ��&����5&���",���%���, �5%���, ��&����5&���� A |� ∈ �B

� ∙ 4 � >? �, !��%����5% ���, ��&���, �5&���",���%��� # �5%��� @ ��%����5%���, ��&��� # �5&��� @ ��&����5&����A |� ∈ �B �D � EF�, !��%���, ��&���", !��%���, ��&���"G|� ∈ �H I� � EF�, !��%���, ��&���", !1 @ ��%���, 1 @ ��&���"G|� ∈ �H ⟡ � � EF�, !1 @ ��%���, 1 @ ��&���", !��%���, ��&���"G|� ∈ �H �� � EF�, !1 @ !1 @ ��%���"�, 1 @ !1 @ ��&���"�", !!��%���"�, !��&���"�"G|� ∈ �H �� � EF�, !!��%���"� , !��&���"�", !1 @ !1 @ ��%���"�, 1 @ !1 @ ��&���"�"G|� ∈ �H √�� � >?�, LM!��%���"� , M!��&���"� N , L1 @ M!1 @ ��%���"� , 1 @ M!1 @ ��&���"� NA |� ∈ �B

��� � >?�, L1 @ M!1 @ ��%���"� , 1 @ M!1 @ ��&���"� N , LM!��%���"� , M!��&���"� NA |� ∈ �B, Where � + 1 is natural number.

3. TWO NEW OPERATOR OPO AND

QO PQO DEFINED OVER IVIFS ARE

INTRODUCED AND PROVED

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

Operations “extraction” as well as of operation “Multiplication of an IVIFS ��� with �� and

Multiplication of an IVIFS �� with �”. It has the form for every IVIFS and for every natural

number � + 1

1� ��� �RSSTSSU

VWWWWWX�, Y1 @ Z1 @ M!��%���"�� , 1 @ Z1 @ M!��&���"�� [,YZ1 @ M!1 @ ��%���"� � , Z1 @ M!1 @ ��&���"� � [ \]

]]]]̂ |� ∈ �

_SS̀SSa

��� � bc�, (1 @ !1 @ !��%���"�"�, 1 @ !1 @ !��&���"�"�) ,(!1 @ !1 @ ��%���"�"�, !1 @ !1 @ ��&���"�"�) d|� ∈ �e

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Theorem 3.1. For any two IVIFSs A and B and for every natural number � + 1: �/�.���� ∩ 4�� � ��� ∩ �4� , �f�.���� ∪ 4�� � ��� ∪ �4� �g�. 1� h��� ∩ 4��i � 1� ��� ∩ 1� 4��, �0� .1� h��� ∪ 4��i � 1� ��� ∪ 1� 4��. Proof: �/�. ���� ∩ 4�� � � YEF�, !!��%���"�, !��&���"�", !1 @ !1 @ ��%���"�, 1 @ !1 @ ��&���"�"G|� ∈ �H∩EF�, !!�5% ���"�, !�5&���"�", !1 @ !1 @ �5%���"�, 1 @ !1 @ �5&���"�"G|� ∈ �H[

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

m9/�!1 @ !1 @ ��%���"� , 1 @ !1 @ �5%���"�",9/�!1 @ !1 @ ��&���"�, 1 @ !1 @ �5&���"�"n \]]̂|� ∈ �_̀

aopq

�RSSTSSU

VWWWWWX �, Y1 @ (1 @ 9:�!!��%���"�, !�5% ���"�")� ,1 @ (1 @9:�!!��&���"� , !�5&���"�")� [ ,Y(9/�!1 @ !1 @ ��%���"�, 1 @ !1 @ �5%���"�")� ,(9/�!1 @ !1 @ �5&���"� , 1 @ !1 @ �5&���"�")�[\]

]]]]̂|� ∈ �

_SS̀SSa

�RSSTSSU

VWWWWWX �, Y1 @ (9/�!1 @ !��%���"� , 1 @ !�5% ���"�")� ,1 @ (9/�!1 @ !��&���"�, 1 @ !�5&���"�")� [ ,Y9/� (!1 @ !1 @ ��%���"�"�, !1 @ !1 @ �5%���"�"�) ,9/� (!1 @ !1 @ ��&���"�"� , !1 @ !1 @ �5&���"�"�)[\]]

]]]̂ |� ∈ �_SS̀SSa

�RSSTSSUVWWWWWX �, Y1 @ 9/� (!1 @ !��%���"�"�, !1 @ !�5% ���"�"�) ,1 @ 9/� (!1 @ !��&���"�"� , !1 @ !�5&���"�"�)[ ,Y9/� (!1 @ !1 @ ��%���"�"� , !1 @ !1 @ �5%���"�"�) ,9/� (!1 @ !1 @ ��&���"�"�, !1 @ !1 @ �5&���"�"�)[\]]

]]]̂ |� ∈ �_SS̀SSa

�RSSTSSU

VWWWWWX�, Y9:� (1 @ !1 @ !��%���"�"�, 1 @ !1 @ !�5% ���"�"�) ,9:� (1 @ !1 @ !��&���"�"�, 1 @ !1 @ !�5&���"�"�)[ ,Y9/� (!1 @ !1 @ ��%���"�"� , !1 @ !1 @ �5%���"�"�) ,9/� (!1 @ !1 @ ��&���"�"�, !1 @ !1 @ �5&���"�"�)[ \]]

]]]̂ |� ∈ �_SS̀SSa

�jkkkklbc

�, (1 @ !1 @ !��%���"�"� , 1 @ !1 @ !��&���"�"�) ,(!1 @ !1 @ ��%���"�"�, !1 @ !1 @ ��&���"�"�) d |� ∈ �e∩

bc�, (1 @ !1 @ !�5% ���"�"�, 1 @ !1 @ !�5&���"�"�) ,(!1 @ !1 @ �5%���"�"�, !1 @ !1 @ �5&���"�"�) d |� ∈ �eoppppq

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5

� ��� ∩ �4�

Hence �/�is proved and similarly (b) is proved by analogy.

Proof:�g�. 1� h��� ∩ 4��i

� 1�jkkkl78�, mLM��%���� N , LM��&���� Nn , L1 @ M!1 @ ��%���"� , 1 @ M!1 @ ��&���"� N ; |� ∈ �<

∩78�, mLM�5% ���� N , LM�5&���� Nn , L1 @ M!1 @ �5%���"� , 1 @ M!1 @ �5&���"� N; |� ∈ �<opppq

� 1�jkkkkl

RSSTSSU

VWWWWWWX�, m9:� LM��%���� , M�5% ���� N ,9:� LM��&���� , M�5&���� Nn ,

jkl9/� L1 @ M!1 @ ��%���"� , 1 @ M!1 @ �5%���"� N ,9/� L1 @ M!1 @ ��&���"� , 1 @ M!1 @ �5&���"� No

pq \]

]]]]]̂ |� ∈ �

_SS̀SSa

oppppq

RSSSSSTSSSSSU

VWWWWWWWWWWWX �,

jkkl1 @ Zm1 @9:� LM��%���� , M�5% ���� Nn� ,1 @ Zm1 @ 9:� LM��&���� , M�5&���� Nn� o

ppq ,

jkklZ9/� L1 @ M!1 @ ��%���"� , 1 @ M!1 @ �5%���"� N� ,Z9/� L1 @ M!1 @ ��&���"� , 1 @ M!1 @ �5&���"� N� o

ppq\]]]]]]]]]]]̂|� ∈ �

_SSSSS̀SSSSSa

RSSSSSTSSSSSU

VWWWWWWWWWWWWX �,

jkkl1 @ Z9/� L1 @ M��%���� , 1 @ M�5% ���� N� ,1 @ Z9/� L1 @ M��&���� , 1 @ M�5&���� N� o

ppq ,

jkkkl9/� YZ1 @ M!1 @ ��%���"� � , Z1 @ M!1 @ �5%���"�� [ ,9/� YZ1 @ M!1 @ ��&���"� � , Z1 @ M!1 @ �5&���"�� [o

pppq\]]]]]]]]]]]]̂|� ∈ �

_SSSSS̀SSSSSa

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6

RSSSSSTSSSSSU

VWWWWWWWWWWWWX �,

jkkkl1 @ 9/� YZL1 @ M��%���� N� , Z1 @ M�5% ����� [ ,1 @ 9/� YZL1 @ M��&���� N� , Z1 @ M�5&����� [o

pppq ,

jkkkl9/� YZ1 @ M!1 @ ��%���"� � , Z1 @ M!1 @ �5%���"�� [ ,9/� YZ1 @ M!1 @ ��&���"� � , Z1 @ M!1 @ �5&���"�� [o

pppq\]]]]]]]]]]]]̂

|� ∈ �

_SSSSS̀SSSSSa

RSSSSSTSSSSSU

VWWWWWWWWWWWWX�,

jkkkl9:� Y1 @ ZL1 @ M��%���� N� , 1 @ Z1 @ M�5% ����� [ ,9:� Y1 @ ZL1 @ M��&���� N� , 1 @ Z1 @ M�5&����� [o

pppq ,

jkkkl9/� YZ1 @ M!1 @ ��%���"� � , Z1 @ M!1 @ �5%���"�� [ ,9/� YZ1 @ M!1 @ ��&���"� � , Z1 @ M!1 @ �5&���"�� [o

pppq

\]]]]]]]]]]]]̂

|� ∈ �

_SSSSS̀SSSSSa

jkkkkkkkkklRSTSUVWWWWX�, L1 @ M(1 @ r��%���� )� , 1 @ M(1 @ r��&���� )� N ,YZ1 @ M!1 @ ��%���"� � , Z1 @ M!1 @ ��&���"� � [ \]]

]]̂ |� ∈ �_S̀Sa

RSTSUVWWWWX�, L1 @ M(1 @ r�5% ���� )� , 1 @ M(1 @ r�5&���� )� N ,YZ1 @ M!1 @ �5%���"� � , Z1 @ M!1 @ �5&���"� � [ \]]

]]̂ |� ∈ �_S̀Saopppppppppq

� 1� ��� ∩ 1� 4��

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

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7

Theorem 3.2. For every IVIFS A and for every natural number� + 1: �/�.I ����� � ��I���; �f� .⟡ ����� � �� ⟡ ���; �g�.I��� � �I��; �0�.⟡ ��� � � ⟡ ��.

Proof: (a). I1����

� I1� 78�, mLM��%���� N , LM��&���� Nn , L1 @ M!1 @ ��%���"� , 1 @ M!1 @ ��&���"� N; |� ∈ �< � IRT

Uc�, Y1 @ ZL1 @ M��%���� N� , 1 @ ZL1 @ M��&���� N� [ , YZ1 @ M!1 @ ��%���"� � , Z1 @ M!1 @ ��&���"� � [d|� ∈ � _̀

a

�RSTSUVWWWX �, L1 @ M(1 @ r��%���� )� , 1 @ M(1 @ r��&���� )� N,m1 @ L1 @ M(1 @ r��%���� )� N , 1 @ L1 @ M(1 @ r��&���� )� Nn\]]

]̂|� ∈ �_S̀Sa

� >?�, L1 @ M(1 @ r��%���� )� , 1 @ M(1 @ r��&���� )� N, LM(1 @ r��%���� )� , M(1 @ r��&���� )� NA |� ∈ �B

(3.1) 1�I���

� 1�Ibc8�, mLM��%���� N , LM��&���� Nn , L1 @ M!1 @ ��%���"� , 1 @ M!1 @ ��&���"� N ;d |� ∈ �e

� 1� >?�, LM��%���� , M��&���� N , L1 @ M��%���� , 1 @ M��&���� NA |� ∈ �B

� >?�, L1 @ M(1 @ r��%���� )� , 1 @ M(1 @ r��&���� )� N , LM(1 @ r��%���� )� , M(1 @ r��&���� )� N A |� ∈ �B

(3.2)

From (3.1) and (3.2), we get I1���� � 1�I���

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

Proof:�g�. I��� � I�EF�, !!��%���"�, !��&���"�",!1 @ !1 @ ��%���"� , 1 @ !1 @ ��&���"�"G|� ∈ �H � Itu�, (1 @ !1 @ !��%���"�"�, 1 @ !1 @ !��&���"�"�) , (!1 @ !1 @ ��%���"�"�, !1 @ !1 @ ��&���"�"�) v |� ∈ �w

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8

� bc �, (1 @ !1 @ !��%���"�"�, 1 @ !1 @ !��&���"�"�),h1 @ (1 @ !1 @ !��%���"�"�) , 1 @ (1 @ !1 @ !��&���"�"�)id |� ∈ �e

� '2�, (1 @ !1 @ !��%���"�"� , 1 @ !1 @ !��&���"�"�), (!1 @ !��%���"�"� , !1 @ !��&���"�"�)3 |� ∈ �* (3.3) �I�� � �IEF�, !!��%���"� , !��&���"�", !1 @ !1 @ ��%���"� , 1 @ !1 @ ��&���"�"G|� ∈ �H � �EF�, !!��%���"� , !��&���"�", !1 @ !��%���"� , 1 @ !��&���"�"G|� ∈ �H � � '2�, (1 @ !1 @ !��%���"�"� , 1 @ !1 @ !��&���"�"�) , (!1 @ !��%���"�"�, !1 @ !��&���"�"�)3 |� ∈ �* (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� h��� # 4��i � 1� ��� #1�4��, �f�.1� h��� ∙ 4��i � 1� ��� ∙ 1� 4��, �g�.���� # 4�� � ��� # �4�, �0�.���� ∙ 4�� � ��� ∙ �4�.

Proof:�/� 1� h��� # 4��i

� 1� L>?�, r������ , 1 @ M!1 @ �����"� A|� ∈ �B # >?�, r�5���� , 1 @ M!1 @ �5���"� A |� ∈ �BN

� 1� >?�, (r������ # r�5���� @ r������ r�5���� ) , L1 @ M!1 @ �����"� N L1 @ M!1 @ �5���"� N A |� ∈ �B

�RSSSTSSSUVWWWWWWWWX�,jkkkkkkl 1 @ M(1 @ r������ )� #

1 @ M(1 @ r�5���� )� @ L1 @ M(1 @ r������ )� NL1 @ M(1 @ r�5���� )� N o

ppppppq, ZL1 @ M!1 @ �����"� N�

ZL1 @ M!1 @ �5���"� N�

\]]]]]]]]̂|� ∈ �

_SSS̀SSSa

� bc�, 1 @ M(1 @ r������ )� , ZL1 @ M!1 @ �����"� N� d|� ∈ �e

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#bc�, 1 @ M(1 @ r�5���� )� , ZL1 @ M!1 @ �5���"� N� d|� ∈ �e

� 1� >?�, r������ , 1 @ M!1 @ �����"� A |� ∈ �B # 1� >?�, r�5���� , 1 @ M!1 @ �5���"� A |� ∈ �B � 1� �� #1� 4�

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

Proof:�g� ���� # 4�� � �!EF�, !�����"� , 1 @ !1 @ �����"�G|� ∈ �H # EF�, !�5���"�, 1 @ !1 @ �5���"�G|� ∈ �H"

� �EF�, !�����"� # !�5���"� @ !�����"�!�5���"�, !1 @ !1 @ �����"�"!1 @ !1 @ �5���"�"G|� ∈ �H

�RSTSUVWWWWX�,

jkkkl 1 @ !1 @ !�����"�"�#1 @ !1 @ !�5���"�"�@(1 @ !1 @ !�����"�"�)(1 @ !1 @ !�5���"�"�) o

pppq , !1 @ !1 @ �����"�"�!1 @ !1 @ �5���"�"�

\]]]]̂|� ∈ �

_S̀Sa

� >F�, 1 @ !1 @ !�����"�"�, !1 @ !1 @ �����"�"�G|� ∈ � B # >F�, 1 @ !1 @ !�5���"�"�, !1 @ !1 @ �5���"�"�G|� ∈ � B � �EF�, !�����"�, 1 @ !1 @ �����"�G|� ∈ �H #�EF�, !�5���"�, 1 @ !1 @ �5���"�G|� ∈ �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: �/�.� h1� ���i � ����f�.1� ����� � ��

Proof.�/�. � h1� ���i

� � L1� >?�, LM��%���� , M��&���� N , L1 @ M!1 @ ��%���"� , 1 @ M!1 @ ��&���"� NA |� ∈ �BN

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� �RSSTSSU

VWWWWWX�, Y1 @ Z1 @ M��%����� , 1 @ Z1 @ M��&����� [,YZ1 @ M!1 @ ��%���"� � , Z1 @ M!1 @ ��&���"� � [\]]

]]]̂ |� ∈ �_SS̀SSa

�RSSTSSU

VWWWWWWX�,jkkkklx1 @ x1 @ Y1 @ Z1 @ M��%����� [y

�y ,

x1 @ x1 @ Y1 @ Z1 @ M��&����� [y�y oppppq ,

jkkklYZ1 @ M!1 @ ��%���"� � [� ,YZ1 @ M!1 @ ��&���"� � [�

opppq

\]]]]]]̂ |� ∈ �

_SS̀SSa

�RSTSUz�, j

lY1 @ YZ1 @ M��%���� � [�[ , Y1 @ YZ1 @ M��&���� � [�[oq , L1 @ M!1 @ ��%���"� , 1 @ M!1 @ ��&���"� N{

|� ∈ � _S̀Sa

� 78�, m1 @ L1 @ M��%���� N , 1 @ L1 @ M��&���� Nn , L1 @ M!1 @ ��%���"� , 1 @ M!1 @ ��&���"� N; |� ∈ �< � >?�, LM��%���� , M��&���� N , L1 @ M!1 @ ��%���"� , 1 @ M!1 @ ��&���"� NA |� ∈ �B

� ���

Hence (a) is proved.

Proof.�f�. 1� ����� � 1� !�EF�, !!��%���"�, !��&���"�", !1 @ !1 @ ��%���"� , 1 @ !1 @ ��&���"�"G|� ∈ �H"

� 1� bc�, (1 @ !1 @ !��%���"�"�, 1 @ !1 @ !��&���"�"�), (!1 @ !1 @ ��%���"�"�, !1 @ !1 @ ��&���"�"�) d|� ∈ �e

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11

�RSSSTSSSUVWWWWWWWX �,

jklL1 @ Mh1 @ (1 @ !1 @ !��%���"�"�)i� N ,L1 @ Mh1 @ (1 @ !1 @ !��&���"�"�)i� No

pq ,YLM!1 @ !1 @ ��%���"�"�� N , LM!1 @ !1 @ ��&���"�"�� N[\]]

]]]]]̂

|� ∈ �

_SSS̀SSSa

�RSTSUVWWWX�,

jkl1 @ LM!1 @ !��%���"�"�� N ,1 @ LM!1 @ !��&���"�"�� No

pq, !1 @ !1 @ ��%���"�, 1 @ !1 @ ��&���"�"\]]]̂ |� ∈ �

_S̀Sa

� tu�, (1 @ !1 @ !��%���"�", 1 @ !1 @ !��&���"�") , !1 @ !1 @ ��%���"� , 1 @ !1 @ ��&���"�" v |� ∈ �w � EF�, !!��%���"� , !��&���"�", !1 @ !1 @ ��%���"�, 1 @ !1 @ ��&���"�"G|� ∈ �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).

|L1� �|��N��

� |h1� EF�, !��%���, ��&���", !��%���, ��&���"G|�}�Hi��

� |L>?�, L1 @ M1 @ !��%���"� , 1 @ M1 @ !��&���"� N , LM!��%���"� , M!��&���"� NA |� ∈ �BN��

� |jkkklRSSTSSUVWWWWWX �, YZ1 @ M1 @ !��%���"�� , Z1 @ M1 @ !��&���"�� [ ,Y1 @ ZL1 @ M!��%���"� N� , 1 @ ZL1 @ M!��&���"� N� [\]]

]]]̂ |� ∈ �_SS̀SSa

opppq

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12

�RSSTSSUVWWWWWX�, Y1 @ ZL1 @ M!��%���"� N� , 1 @ ZL1 @ M!��&���"� N� [,

YZ1 @ M1 @ !��%���"�� , Z1 @ M1 @ !��&���"�� [ \]]]]]̂ |� ∈ �

_SS̀SSa

� 1� ���

Hence (a) is proved.

Proof. (b). |!��|��"� � |!�EF�, !��%���, ��&���", !��%���, ��&���"G|�}�H"�

� |!EF�, !1 @ !1 @ ��%���"�, 1 @ !1 @ ��&���"�", !!��%���"�, !��&���"�"G|� ∈ �H"�

� | t2�, (!1 @ !1 @ ��%���"�"�, !1 @ !1 @ ��&���"�"�) , (1 @ !1 @ !��%���"�"�, 1 @ !1 @ !��&���"�"�)3 |� ∈ � w � '2�, (1 @ !1 @ !��%���"�"�, 1 @ !1 @ !��&���"�"�) , (!1 @ !1 @ ��%���"�"�, !1 @ !1 @ ��&���"�"�)3|� ∈ �* � ���

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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