Modal Type Operators over Interval Valued Intuitionistic Fuzzy Sets of Second Type

Vol. 14, No. 2, 2017, 253-260

ISSN: 2320 –3242 (P), 2320 –3250 (online) Published on 11 December 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/ijfma.v14n2a9

253

International Journal of

Modal Type Operators over Interval Valued Intuitionistic

Fuzzy Sets of Second Type

K. Rajesh and R. Srinivasan

Department of Mathematics

Islamiah College (Autonomous), Vaniyambadi, Tamilnadu.

E-mail: [email protected]

Received 12 November 2017; accepted 10 December 2017

Abstract. In this paper, we introduce modal type operators over interval valued

intuitionistic fuzzy sets of second type and establish some of their properties.

Keywords: Intuitionistic fuzzy sets, intuitionistic fuzzy sets of second type, interval

valued fuzzy sets, interval valued intuitionistic fuzzy sets, interval valued intuitionistic fuzzy sets of second type.

AMS Mathematics Subject Classification (2010): 03E72

1. Introduction

An intuitionistic fuzzy set for a given underlying set x where introduced by Atanassov [2] which is the generalization of ordinary fuzzy sets. Atanassov and Gargov

[3] further introduced the concepts of interval valued intuitionistic fuzzy set. The present authors further introduced the new extension of IVIFS namely interval valued intuitionistic fuzzy sets of second type (IVIFSST) and established some of their properties [4]. The rest of the paper is designed as follows: In Section 2, we give some basic definitions. In Section 3, we introduce modal type operators over interval valued intuitionistic fuzzy sets of second type and establish some of their properties. This paper is concluded in section 4.

2. Preliminaries

In this section, we give some basic definitions.

Definition 2.1. [2] Let X be a non-empty set. An Intuitionistic Fuzzy Set (IFS) A in X is

defined as an object of the following form.

A = 〈, μ , ν 〉| ∈ X

where the functions μ: → 0,1 and ν: → 0,1 denote the degree of membership

and the degree of non-membership of the element x ∈X, respectively, and for every x∈ X.

0 ≤ μ + ν ≤ 1

Definition 2.2. [2] Let a set X be fixed. An intuitionistic fuzzy sets of second type

(IFSST) A in X is defined as an object of the following form.

A = 〈, μ , ν 〉| ∈ X

254

and the degree of non-membership of the element x ∈X, respectively, and for every x ∈X.

0 ≤ μ _{+ ν}

≤ 1

Definition 2.3. [3] An interval valued intuitionistic fuzzy sets (IVIFS) A in X is given by

A = {〈x, MA(x), NA(x)〉| x ∈ X}

where MA : X → [0, 1], NA : X → [0, 1]. The intervals MA(x) and NA(x) denote the degree

of membership and the degree of non-membership of the element x in X, where MA(x)

= [MAL(x), MAU(x)] and NA(x) = [NAL(x), NAU(x)] with the condition that

MAU(x) + NAU(x) ≤ 1 ∀x ∈X

Definition 2.4. [4] An interval valued intuitionistic fuzzy sets of second type A in X is

given by

A = {〈x, MA(x), NA(x)〉| x ∈X}

where MA: X → [0, 1], NA: X → [0, 1]. The intervals MA(x) and NA(x) denote the degree

of membership and the degree of non-membership of the element x inX, where MA(x)

= [MAL(x), MAU(x)] and NA(x) = [NAL(x), NAU(x)] with the condition that

AU(x) + AU(x) ≤ 1 ∀ x ∈X.

Definition 2.5. [4] For every two IVIFSST A and B, we have the following relations and

operations

1. ⊂ " iff %_&'() ≤ %_('()&%_&*() ≤ %_(*()& +&'() ≥ +('()&+&*() ≥ +(*()

2. = " iff ⊂ " & " ⊂

3. ̅ = 〈, N_/(), N₀(), M_/(), M₀()〉| ∈ X

4. ∪ " = 3〈, max7M_/(), M_8/()9 , max7M₀(), M₈₀()9:,

min7N/(), N8/()9 , min7N0(), N80()9〉| ∈ X

5. ∩ " = 3〈, min7M_/(), M_8/()9 , min7M₀(), M₈₀()9:,

max7N/(), N8/()9 , max7N0(), N80()9〉| ∈ X

Definition 2.6. [5] For every IVIFSST, we define the following operators

Necessity operator

□ = >〈, M&*(), M&'(), ?N/(), @1 − %&'()B〉| ∈ XC

Possibility operator

◊ = >〈, ?M&*(), @1 − +&'()B , N/(), N0()〉| ∈ XC

Definition 2.7. [5] Given an IVIFSST A and for every α, β ∈ 0, 1, we define the operators D_H and I_J,K

LJ( ) = MN, O%&*(), P%&'() + Q71 − %&'() − +&'()9R ,::

255

IJ,K( ) = M, O%&*(), P%&'() + Q71 − %&'() − +&'()9R ,:

::OUV(W), PUX(W) + Y7Z − UX(W) − UX(W)9RS | W ∈ [T

3. Modal type operators on IVIFSST

In this section, we define the new modal type operators over IVIFSST and also we establish some of their properties.

Definition 3.1. Given an IVIFSST A and for every α, β ∈ 0,1, we define the operators GH,] , ^_J,K and __J,K

(i). G_H,](A) = 〈, α%_&*(), α%_&'(),::β+_&*(), β+_&'()〉| ∈ X (ii). ^_J,K( ) = 〈, α %_&*(), α %_&'(),:

::O+&*(), P+&'() + `71 − %&'() − +&'()9RS | ∈ XT

(iii). __J,K( ) = MN, O%_&*(), P%&'() + α71 − %&'() − +&'()9R ,::

:`+&*(), `+&'()〉| ∈ X

Definition 3.2. Given an IVIFSST A and for every α, β ∈ 0,1, we define the operators H∗_{J,K and J}∗

J,K

(i). H∗_J,K( ) = 〈, α %_&*(), α %_&'(),:

::O+&*(), P+&'() + `71 − α%&'() − +&'()9RS | ∈ XT

(ii). J∗_J,K( ) = MN, O%_&*(), P%&'() + αd1 − %&'() − `+<sub>&'</sub>()eR ,:: :`+&*(), `+&'()〉| ∈ X

Proposition 3.1. Let X be a non-empty set. For every IVIFSST A and for every Q, ` ∈

0,1 we have the following

(i). ^ggggggggggg = __J,K( f) _K,J( ) (ii). _gggggggggg = ^_J,K( f) _K,J( )

(iii). ^gggggggggg = __h,i( f) ggggggggg = G_i,h( f) _h,h(A)

Proof: Let = 〈, %_&*(), %_&'(), +_&*(), +_&'()〉| ∈ X , Then,

̅ = 〈, +&*(), +&'(), %&*(), %&'()〉| ∈ X ,

256 ^J,K( ) = 〈, α %&*(), α %&'(),:

::O+&*(), P+&'() + `71 − %&'() − +&'()9RS | ∈ XT

_J,K( ) = MN, O%&*(), P%&'() + α71 − %&'() − +&'()9R,::

:`+&*(), `+&'()〉| ∈ X

Now

^J,K( ̅) = 〈, α +&*(), α +&'(),:

::O%&*(), P%&'() + `71 − %&'() − +&'()9RS | ∈ XT

^J,K( ̅)

gggggggggg = MN,O%&*(), P%&'() + `71 − %&'() − +&'()9R::,

:α +&*(), α +&'()〉| ∈ X

= _K,J( )

This is (i)

(ii) __J,K( ̅) = MN, O+_&*(), P+&'() + α71 − %&'() − +&'()9R ,::

`%&*(), `%&'()〉| ∈ X

_J,K( ̅)

ggggggggg = 〈,`%&*(), `%&'(),:

::O+&*(), P+&'() + α71 − %&'() − +&'()9RS| ∈ XT

= ^K,J( )

Therefore,

_J,K( ̅)

ggggggggg = ^K,J( )

(iii) Let = 〈, %_&*(), %_&'(), +_&*(), +_&'()〉| ∈ X ,

Then,

̅ = 〈, +&*(), +&'(), %&*(), %&'()〉| ∈ X

^J,K( ̅) = 〈, α +&*(), α +&'(),:

::O%&*(), P%&'() + `71 − %&'() − +&'()9RS | ∈ XT

^J,K( ̅)

257

:α +&*(), α +&'()〉| ∈ X

Put Q = 1 and ` = 0 in ^gggggggggg_J,K( ̅)

^h,i( ̅)

gggggggggg = MN,O%&*(), P%&'() + 071 − %&'() − +&'()9R ,::

:1+&*(), 1+&'()〉| ∈ X

= 〈,:%&*(), %&'(), :+&*(), +&'()〉| ∈ X (1)

_J,K( ̅) = MN, O+&*(), P+&'() + α71 − %&'() − +&'()9R ,::

`%&*(), `%&'()〉| ∈ X

_J,K( ̅)

ggggggggg = 〈,`%&*(), `%&'(),:

::O+&*(), P+&'() + α71 − %&'() − +&'()9RS| ∈ XT

Put Q = 0 and ` = 1 in _ggggggggg_J,K( ̅)

_i,h( ̅)

ggggggggg = 〈,1%&*(), 1%&'(),:

::O+&*(), P+&'() + 071 − %&'() − +&'()9RS| ∈ XT

= 〈, %&*(), %&'(),::+&*(), +&'()〉| ∈ X (2)

GH,](A) = 〈, α%&*(), α%&'(),::β+&*(), β+&'()〉| ∈ X

Put Q = 1 and ` = 1G_H,](A) we have

Gh,h(A) = 〈, %&*(), %&'(),::+&*(), +&'()〉| ∈ X (3)

From (1), (2) and (3) we have ^h,i( f)

gggggggggg = _ggggggggg = Gi,h( f) h,h(A)

Proposition 3.2. Let X be a non-empty set. For every IVIFSST A and for every Q, ` ∈

0,1 we have the following

(i). Hgggggggggggg = J∗J,K( f) ∗_K,J( )

(ii). Jggggggggggg = H∗_J,K( f) ∗_K,J( )

258 Then,

̅ = 〈, +&* , +&' , %&* , %&' 〉| ∈ X },

H∗_{J,K = 〈, α %&* , α %&' ,:}

::O+&* , P+&' + `71 − α%&' − +&' 9RS | ∈ XT,

J∗

J,K = MN, O%&* , P%&' + αd1 − %&' − `+_&' eR ,::

:`+&* , `+&' 〉| ∈ X}, H∗_{J,K ̅ = 〈, α +&* , α +&' ,:}

:O%&* , P%&' + `_{71 − α+&' − %&' 9RS :| ∈ X},}

and

J∗

J,K ̅ = MN, O+&* , P+&' + α71 − `%&' − +&' 9R ,:: `%&* , `%&' 〉| ∈ X}

Now,

(i) H∗J,K ̅gggggggggggg = MN,O%_&* , P%&' + `71 − %&' − α+&' 9R ,::

:α +&* , α +&' 〉| ∈ X}

= J∗

K,J

Therefore,

H∗J,K ̅ gggggggggggg = J∗

K,J

(ii) J∗_J,K ̅ = MN, O+_&* , P+&' + α71 − `%&' − +&' 9R ,::

`%&* , `%&' 〉| ∈ X}

J∗ J,K ̅

gggggggggg = 〈,`%&* , `%&' ,:

::O+&* , P+&' + α_{71 − `%&' − +&' 9RS| ∈ XT}

= H∗_K,J

Hence,

J∗ J,K ̅

gggggggggg = H∗_K,J

Proposition 3.3. Let X be a non-empty set. For every IVIFSST A and for every Q, ` ∈

259

i. H∗_J,K(A) = F_i,KdG_J,h(A)e

ii. J∗_K,J(A) = F_K,idG_h,J(A)e

Proof: Let = 〈, %_&*(), %_&'(), +_&*(), +_&'()〉| ∈ X }, Then,

̅ = 〈, +&*(), +&'(), %&*(), %&'()〉| ∈ X },

IJ,K = MN, O%&* , P%&' + α71 − %&' − +&' 9R ,::

::O+&* , P+&' + β71 − %&' − +&' 9RS | ∈ XT, GH,] A = 〈, α%&*(), α%&'(),::β+&*(), β+&'()〉| ∈ X},

H∗_{J,K = 〈, α %&*(), α %&'(),:}

::O+&*(), P+&'() + `71 − α%&'() − +&'()9RS| ∈ XT

J∗

J,K = MN, O%&* , P%&' + αd1 − %&' − `+&' eR ,::

:`+&* , `+&' 〉| ∈ X}

(i). Put β = 1 in GJ,K we have

GH,h A = 〈, α%&*(), α%&'(),::+&*(), +&'()〉| ∈ X}

Again put Q = 0 in F_J,K we have

Ii,K = 〈, %&*(), %&'(),:

::O+&*(), P+&'() + β71 − %&'() − +&'()9RS| ∈ XT Ii,KdGH,h Ae = 〈, α%&*(), α%&'(),:

::O+&*(), P+&'() + β71 − α%&'() − +&'()9RS| ∈ XT

From H∗_J,K and I_i,KdGH,h Ae we have

H∗_{J,K = Ii,KdGH,h Ae}

(ii). Put α = β and β = α in __J,K∗ we have

J∗

K,J = MN, O%&* , P%&' + `71 − %&' − Q+&' 9R ,::

:Q+&* , Q+&' 〉| ∈ X}

260

Gh,H(A) = 〈, %&*(), %&'(),::α+&*(), α+&'()〉| ∈ X

Put Q = ` and ` = 0 in F_J,K( ) we have

IK,i( ) = MN, O%&*(), P%&'() + `71 − %&'() − +&'()9R ,::

:+&*(), +&'()〉| ∈ X

Now

IK,idGh,H(A)e = MN, O%&*(), P%&'() + β71 − %&'() − α+&'()9R ,::

::Oα+&*(), Pα+&'() + 071 − α%&'() − +&'()9RS| ∈ XT

= MN, O%&*(), P%&'() + β71 − %&'() − α+&'()9R ,::

:α+&*(), α+&'()〉| ∈ X

From J∗_K,J( )and I_K,idG_h,H(A)e we have

IK,idGh,H(A)e = J∗K,J( )

4. Conclusion

We have introduced modal type operatorsl_m,n, o_p,Y, q_p,Y, o∗_p,Y and q∗_p,Y over IVIFSST also we established some of their properties. It is still open to define some more operators over IVIFSST.

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