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Generalized fuzzy b-closed and generalized * -fuzzy b-closed sets in double fuzzy topological spaces

Fatimah M. Mohammed a>*>1i M.S.M. Noorania, A. Ghareeb b

a School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia b Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt

ARTICLE INFO ABSTRACT

Article history: Received 28 July 2015 Received in revised form 2 September 2015 Accepted 2 September 2015 Available online

Keywords:

Double fuzzy topology (r, s)-generalized fuzzy b-closed sets (r, s)-generalized *-fuzzy b-closed sets

The purpose of this paper is to introduce and study a new class of fuzzy sets called (r, s)-generalized fuzzy b-closed sets and (r, s)-generalized *-fuzzy b-closed sets in double fuzzy topological spaces. Furthermore, the relationships between the new concepts are introduced and established with some interesting examples.

© 2015 Mansoura University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-

nc-nd/4.0/).

Mathematics subject classification:

1. Introduction

A progressive development of fuzzy sets [1] has been made to discover the fuzzy analogues of the crisp sets theory. On the other hand, the idea of intuitionistic fuzzy sets was first introduced by Atanassov [2]. Later on, Qoker [3] presented the

notion of intuitionistic fuzzy topology. Samanta and Mondal [4], introduced and characterized the intuitionistic gradation of openness of fuzzy sets which is a generalization of smooth topology and the topology of intuitionistic fuzzy sets. The name "intuitionistic" is discontinued in mathematics and applications. Garcia and Rodabaugh [5] concluded that they work under the name "double".

* Corresponding author. Tel.: +20 1014725551.

E-mail address: nafea_y2011@yahoo.com (F.M. Mohammed). 1 Permanent Address: College of Education, Tikrit University, Iraq. http://dx.doi.org/10.1016/j.ejbas.2015.09.001

2314-808X/© 2015 Mansoura University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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In 2009, Omari and Noorani [6] introduced generalized b-closed sets (briefly, gb-closed) in general topology. As a generalization of the results in References 6 and 7, we introduce and study (r, s)-generalized fuzzy b-closed sets in double fuzzy topological spaces, then a new class of fuzzy sets between an (r, s)-fuzzy b-closed sets and an (r, s)-generalized fuzzy b-closed sets namely (r, s)-generalized *-fuzzy b-closed sets is introduced and investigated. Finally, the relationships between (r, s)-generalized fuzzy b-closed and (r, s)-generalized *-fuzzy b-closed sets are introduced and established with some interesting counter examples.

2. Preliminaries

Throughout this paper, X will be a non-empty set, I = [0,1], I0 = (0, 1] and I1 = [0,1). A fuzzy set X is quasi-coincident with a fuzzy set j (denoted by, Xqj) iff there exists x e X such that X(x) + )(x) > 1 and they are not quasi-coincident otherwise (denoted by, Xq)).The family of all fuzzy sets on X is denoted by IX. By 0 and 1, we denote the smallest and the greatest fuzzy sets on X. For a fuzzy set X e IX, 1 -X denotes its complement. All other notations are standard notations of fuzzy set theory.

Now, we recall the following definitions which are useful in the sequel.

Definition 2.1. (see [4]) A double fuzzy topology (t, t*) on X is a pair of maps t, t* : IX ^ I, which satisfies the following properties:

The triplet (X, t, t*) is called a double fuzzy topological space (briefly, dfts). A fuzzy set X is called an (r, s)-fuzzy open (briefly, (r, s)-fo) if t(X) > r and t*(X) < s. A fuzzy set X is called an (r, s)-fuzzy closed (briefly, (r, s)-fc) set iff 1 -X is an (r, s)-fo set.

Theorem 2.1. (see [8]) Let (X, t, t*) be a dfts. Then double fuzzy closure operator and double fuzzy interior operator of X e IX are defined by

CTT* (X, r, s) = a{) e IX | X < ), t(1 - )) > r, t*(1 - )) < s},

IT,T.(X, r, s) = v{) e IX|) < X, t()) > r, t*()) < s}.

Where r e I0 and s e I1 such that r + s < 1.

Definition 2.2. Let (X, t, t*) be a dfts. For each X e IX, r e I0 and s e Ii. A fuzzy set X is called:

1. An (r, s)-fuzzy semiopen (see [9]) (briefly, (r, s)-fso) if X< CT,T.(IT,T.(X, r,s), r, s). X is called an (r, s)-fuzzy semi closed (briefly, (r, s)-fsc) iff 1 - X is an (r, s)-fso set.

2. An (r, s)-generalized fuzzy closed (see [10]) (briefly, (r, s)-gfc) if CT,T.(X, r, s) < fi, X < /, t(j) > r and t*(/) < s. X is called an

(r, s)-generalized fuzzy open (briefly, (r, s)-gfo) iff 1 -X is (r, s)-gfc set.

Definition 2.3. (see [11,12]) Let (X, t, t*) be a dfts. For each X, j e IX and r e I0, s e I1. Then, a fuzzy set X is said to be (r, s)-fuzzy generalized yp-closed (briefly, (r, s)-fgyp-closed) if yCv-(X, r, s) < ) such that X < j and j is (r, s)-fuzzy p-open set. X is called (r, s)-fuzzy generalized yp-open (briefly, (r, s)-fgyp-open) iff 1 - X is (r, s)-fgyp-closed set.

3. (r, s)-generalized fuzzy b-closed sets

In this section, we introduce and study some basic properties of a new class of fuzzy sets called an (r, s)-fuzzy b-closed sets and an (r, s)-generalized fuzzy b-closed.

Definition 3.1. Let (X, t, t*) be a dfts. For each X e IX, r e I0 and s e I1. A fuzzy set X is called:

1. An (r, s)-fuzzy b-closed (briefly, (r, s)-fbc) if

X > (V(CT,T.(X, r, s), r, s)) A (CZZ*(IZZ*(X, r, s), r, s)).

X is called an (r, s)-fuzzy b-open (briefly, (r, s)-fbo) iff 1 -X is (r, s)-fbc set.

2. An (r, s)-generalized fuzzy b-closed (briefly, (r, s)-gfbc) if bCT,T(X, r, s) < ), X < j, t(/) > r and t*(/) < s. X is called an (r, s)-generalized fuzzy b-open (briefly, (r, s)-gfbo) iff 1 -X is (r, s)-gfbc set.

Definition 3.2. Let (X, t, t*) be a dfts. Then double fuzzy b-closure operator and double fuzzy b-interior operator of X e IX are defined by

bCT,T.(X, r, s) = a{) e IX| X<) and ) is (r, s)-fbc},

bITT*(X, r, s) = v{) e IX|) < X and ) is (r, s)-fbo}.

Where r e I0 and s e I1 such that r + s < 1.

Remark 3.1. Every (r, s)-fbc set is an (r, s)-gfbc set.

The converse of the above remark may be not true as shown by the following example.

Example 3.1. Let X = {a, b}. Defined /, a and p by: )(a) = 0.3, )(b) = 0.4,

a(a) = 0.4, a(b) = 0.5,

P(a) = 0.3, P(b) = 0.7,

i1, if X e {0,1}, i0, if Xe {0,1},

t(X) = j if X = ), t*(X) = jif X = ),

[0, otherwise. [1, otherwise.

(01) t(X) < 1 - t*(X) for each X e IX.

(02) t(X1 aX2) >t(X1) at(X2) and t'(X1 aX2) <t'(X1) v t*(X) for each X1, X2 e IX.

(03) T(vierXj) > AierT(Xj) and T*(vierX) < vierT*(Xj) for each Xi e IX, i e r.

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Then /} is an (!,-J)-gfbc set but not an (1, 2)-fbc set.

Definition 3.3. Let (X, t, t*) be a dfts, X e IX, r e I0 and s e I1. X is called an (r, s)-fuzzy b-Q-neighborhood of xt e Pt(X) if there exists an (r, s)-fbo set p e IX such that xtqp and p < X.

The family of all (r, s)-fuzzy b-Q-neighborhood of xt denoted by b-Q(xt, r, s).

Theorem 3.1. Let (X, t, t*) be a dfts. Then for each X, p e IX, r e I0 and s e I1, the operator bCT,T* satisfies the following statements:

(C1) bC„.(0, r, s) = 0, bC„.(1, r, s) = 1,

(C2) X < bC,T-(X, r, s),

(C3) If X < p, then bCTT.(X, r, s) < bCTT.(p, r, s),

(C4) If X is an (r, s)-fbc, then X = bCT T.(X, r, s),

(C5) If p is an (r, s)-fbo, then pqX iff pqbCT T.(X, r, s),

(C6) bC„.(bC„.(X, r, s), r, s) = bC„.(X, r, s),

(C7) bCTT.(X, r, s) v bCT T.(p, r, s) < bCT T.(X v p, r, s),

(C8) bCTT.(X, r, s) л bCT T.(p, r, s) > bCT T.(X л p, r, s),

1. bIT,T.(1 -X, r,s) = 1 -bCTT*(X, r,s), bCT,T.(1 -X, r,s) = 1 -bITT* (X, r, s),

2. bIT,T.(0, r, s) = 0, bIT,T.(1, r, s) = 1,

3. bIT,T.(X,r,s)<X,

4. If X is an (r, s)-fbo, then X = bIT,T*(X, r, s),

5. If X < p, then bIT,T.(X, r, s) < bIT,T.(p, r, s),

6. bIT,T.(bIT,T.(X, r, s), r, s) = bIzz* (X, r, s),

7. bITT*(X v p, r, s) > bITT*(X, r, s) v bITT*(p, r, s),

8. bIT,T.(X a p, r, s) < bIT,T.(X, r, s) a bIT,T.(p, r, s).

Proof. It is similar to Theorem 3.1.

Theorem 3.3. Let (X, t, t*) be a dfts. X e IX is (r, s)-gfbo set, r e I0 and s e I1 if and only if p< bIT,T*(X, r, s) whenever p < X, t(1 -p) > r and t*(1 -p) < s.

Proof. Suppose that X is an (r, s)-gfbo set in IX, and let t(1 - p) > r and t*(1 - p) < s such that p < X. By the definition, 1 - X is an (r, s)-gfbc set in IX. So,

Proof. (1), (2), (3), and (4) are proved easily.

(5) Let pqX and p is an (r, s)-fbo set, then X < 1 - p. But we have, pqX iff pqbCv-(X, r, s) and

bCT,T. (X, r, s) < bCT,T. (1 - p, r, s) = 1 - p,

so pqbCTT-(X, r, s), which is contradiction. Then nqX iff

pqbCT T-(X, r, s).

(6) Let xt be a fuzzy point such that xt ^ bCT,T.(X, r, s). Then there is an (r, s)-fuzzy b-Q neighborhood n of xt such that pqX. But by (5), we have an (r, s)-fuzzy b-Q-neighborhood p of xt such that

pqbCT,T*(X, r, s) Also,

xt ^ bCT,T.(bCT,T.(X, r, s), r, s). Then

bCT,T.(bCT,T.(X, r, s), r, s) < bCT,T.(X, r, s).

bCT,T.(1 -X, r, s) < 1 -p Also,

1 - bîT,T.(X, r, s) < 1 -p.

And then, p< bIT T.(X, r, s).

Conversely, let p < X, t(1 - p) > r and т*(1 - p) < s, r e I0 and s e Ii such that p < bITy(X, r, s). Now

1 - bIT,T.(X, r, s) < 1 -p,

bCT,T.(1 -X, r, s) < 1 -p.

That is, 1 -X is an (r, s)-gfbc set, then X is an (r, s)-gfbo set.

Theorem 3.4. Let (X, т, т*) be a dfts, X e IX, r e Io and s e I1. If X is an (r, s)-gfbc set, then

But we have, bCT.T.(bCT.T.(A, r, s), r, s) > bCT,T.(A, r, s).

Therefore bCT,T.(bCT,T.(A, r, s), r, s) = bCT,T.(A, r, s). (7) and (8) are obvious.

1. bCT,T.(X, r, s)-X does not contain any non-zero (r, s)-fc sets.

2. X is an (r, s)-fbc iff bCT,T.(X, r, s)- X is (r, s)-fc.

3. p is (r, s)-gfbc set for each set p e IX such that X<p< bCT,T(X, r, s).

4. For each (r, s)-fo set p e IX such that p < X, p is an (r, s)-gfbc relative to X if and only if p is an (r, s)-gfbc in IX.

5. For each an (r, s)-fbo set p e IX such that bCT,T.(X, r, s)qp iff Xqp.

Theorem 3.2. Let (X, t, t*) be a dfts. Then for each X, p e IX, r e I0 and s e I1, the operator bIT,T* satisfies the following statements:

Proof. (1) Suppose that t(1 -p) > r and t*(1 -p) < s, r e I0 and s e I1 such that p < bCT,T.(X, r, s) - X whenever X e IX is an (r, s)-gfbc set. Since 1 - p is an (r, s)-fo set,

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X < (1 - )) ^ bCT,T (X, r, s) < (1 - )) (1 - bCT,T.(X, r, s)) ^ ) < (1 - bCzz* (X, r, s)) a (bCT,T. (X, r, s) - X) = 0

and hence ) = 0 which is a contradiction. Then bCT,T. (X, r, s) - X does not contain any non-zero (r, s)-fc sets.

(2) Let X be an (r, s)-gfbc set. So, for each r e I0 and s e I1 if X is an (r, s)-fbc set then,

bCT,T.(X, r, s) -X = 0

which is an (r, s)-fc set.

Conversely, suppose that bCT,T*(X, r, s) -X is an (r, s)-fc set. Then by (1), bCT,T.(X, r, s) - X does not contain any non-zero an (r, s)-fc set. But bCT,T.(X, r, s)-X is an (r, s)-fc set, then

bCTT* (X, r, s) - X = 0 ^ X = bCTT* (X, r, s).

So, X is an (r, s)-fbc set.

(3) Suppose that T(a) > r and T*(a) < s where r e I0 and s e I1 such that j < a and let X be an (r, s)-gfbc set such that X < a. Then

bCT,T.(X, r, s) < a. So,

bCT,T.(X, r, s) = bCT,T.(), r, s),

Therefore bCTT*(), r, s) < a.

So, j is an (r, s)-gfbc set.

(4) Let X be an (r, s)-gfbc and t(X) > r and t*(X) < s, where r e I0 and s e I1. Then bCT,T.(X, r, s) <X. But, j < X so,

bCzy(), r, s) < bCTT*(X, r, s) < X.

Also, since j is an (r, s)-gfbc relative to X, then

X a bCT,T.(X)(), r, s) = bCTT*(), r, s),

bCTT*(), r, s) = bCTT*(X)(), r, s) < X.

Now, if j is an (r, s)-gfbc relative to X and T(a) > r and T*(a) < s where r e I0 and s e I1 such that j < a, then for each an (r, s)-fo set a a X, ) = ) a X < a a X. Hence j is an (r, s)-gfbc relative to X,

bCTT*(), r, s) = bCTT*(X)(), r, s) < (a a X) < a. Therefore, j is an (r, s)-gfbc in IX.

Conversely, let j be an (r, s)-gfbc set in IX and T(a) > r and T*(a) < s whenever a < X such that j < a, r e I0 and s e I1. Then for each an (r, s)-fo set p e IX, a = p a X. But we have, j is an (r, s)-gfbc set in IX such that j < p,

bCTT*(), r, s) < ß ^ bCT,T.(A)(^, r, s) = bCTT*(), r, s) aX< ß aX = a.

That is, ¡i is an (r, s)-gfbc relative to X. (5) Suppose i is an (r, s)-fbo and Xqju, r e I0 and s e Ii. Then X < (1 - p). Since (1 - p) is an (r, s)-fbc set of IX and X is an (r, s)-gfbc set, then

bCT,T.(X, r, s)qp.

Conversely, let i be an (r, s)-fbc set of IX such that X < i, r e I0 and s e I1. Then

Xq(1 -)). But

bCT,T. (X, r, s)q(1 -)) ^ bCT,T. (X, r, s) < ).

Hence X is an (r, s)-gfbc. Proposition 3.1. Let (X, t, t*) be a dfts, X e IX, r e I0 and s e I1.

1. If X is an (r, s)-gfbc and an (r, s)-fbo set, then X is an (r, s)-fbc set.

2. If X is an (r, s)-fo and an (r, s)-gfbc, then X a i is an (r, s)-gfbc set whenever ) < bCTT*(X, r, s).

Proof. (1) Suppose X is an (r, s)-gfbc and an (r, s)-fbo set such that X < X, r e I0 and s e I1. Then

bCT,T(X, r, s) < X.

But we have,

X < bCT,T.(X, r, s).

X = bCT,T.(X, r, s).

Therefore, X is an (r, s)-fbc set.

(2) Suppose that X is an (r, s)-fo and an (r, s)-gfbc set, r e I0 and s e I1. Then

bCTT*(X, r, s) < X ^ X is an (r, s)-fbc set ^ X a ) is an (r, s)-fbc ^ X a ) is an (r, s)-gfbc.

4. (r, s)-generalized * -fuzzy b-closed sets

In this section, we introduce and study some properties of a new class of fuzzy sets called an (r, s)-generalized *-fuzzy closed sets and an (r, s)-generalized *-fuzzy b-closed sets

Definition 4.1. Let (X, t, t*) be a dfts. For each A e IX, r e Io and s e I1. A fuzzy set X is called:

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1. An (r, s)-generalized *-fuzzy closed (briefly, (r, s)-g *fc) if CT,T*(X, r, s) < p whenever X < p and p is an (r, s)-gfo set in IX. X is called an (r, s)-generalized *-fuzzy open (briefly, (r, s)-g *fo) iff 1 -X is (r, s)-g *fc set.

2. An (r, s)-generalized *-fuzzy b-closed (briefly, (r, s)-g *fbc) if bCT,T.(X, r, s) < p whenever X < p and p is an (r, s)-gfo set in IX. X is called an (r, s)-generalized *-fuzzy b-open (briefly, (r, s)-g *fbo) iff 1 -X is (r, s)-g *fbc set.

Theorem 4.1. Let (X, t, t*) be a dfts. X e IX is an (r, s)-g *fbo set if and only if p < bITT*(X, r, s) whenever p is an (r, s)-gfc, r e I0 and s e I1.

Proof. Suppose that X is an (r, s)-g *fbo set in IX, and let p is an (r, s)-gfc set such that p < X, r e I0 and s e I1. So by the definition, we have 1 - X is an (r, s)-gfo set in IX and 1 - X < 1 - p. But 1 -X is an (r, s)-g *fbc set, then bCT,T.(1 -X,r,s)< 1 -p. But

bCT,T- (1 - X, r, s) = 1 - bIT,T* (X, r, s) < 1 - p.

Therefore, p< bIT,T.(X, r, s).

Conversely, suppose that p < bIT,T*(X, r, s) whenever p < X and p is an (r, s)-gfc set, r e I0 and s e I1. Now

1 - bIT,T.(X, r, s) < 1 -p,

bCT,T.(1 -X, r, s) < 1 -p.

Therefore, 1 -X is an (r, s)-gfbc set and X is an (r, s)-gfbo set.

Proposition 4.1. Let (X, t, t*) be dfts's. For each X e IX, r e I0 and

s e I1

1. If a fuzzy set X is an (r, s)-g *fbc, then bCT,T.(X, r, s)-X contains no non-zero (r, s)-gfc set.

2. If a fuzzy set X is an (r, s)-g *fbc, then bCTT.(X, r, s) -X is an (r, s)-g *fbo.

3. An (r, s)-g *fbc set X is an (r, s)-fbc iff bCTT.(X, r, s)-X is an (r, s)-fbc set.

4. If a fuzzy set X is an (r, s)-g *fbc, then p = 1, whenever p is an (r, s)-gfo set and bITT*(X, r, s) v (1 - X) < p.

Proof. (1) Suppose that X is an (r, s)-g *fbc set and p is an (r, s)-gfc set of IX, r e I0 and s e I1 such that

p < bCT,T.(X, r, s)

X < 1 -p.

But X is an (r, s)-g *fbc set and 1 - p is an (r, s)-gfo set, then bCT,T. (X, r, s) < 1 -p^p< bCT,T. (X, r, s) a (1 - bCT,T. (X, r, s)) = 0.

Therefore bCT,T.(X, r, s) - X contains no non-zero (r, s)-gfc set.

(2) Let X be an (r, s)-g *fbc set, r e I0 and s e I1. Then by (1) we have, bCT,T.(X, r, s)-X contains no non-zero (r, s)-gfc set. So, bCT,T.(X, r, s)-X is an (r, s)-g *fbo set.

(3) Let X be an (r, s)-g *fbc set. If X is an (r, s)-fbc, r e I0 and s e I1, then

bCT,T.(X, r, s) -X = 0.

Conversely, let bCTT*(X, r, s) -X is an (r, s)-fbc set in IX and X is an (r, s)-g *fbc, r e I0 and s e I1, then by (1) we have, bCT,T.(X, r, s)-X contains no non-zero (r, s)-gfc set. Then,

bCT,T.(X, r, s) -X = 0,

that is

bCT,T.(X, r, s) = X.

Hence X is an (r, s)-fbc set.

(4) Let pbe an (r, s)-gfcsetand bIT,T.(X, r, s) v(1 -X) <p, r e I0 and s e I1. Hence

1 - p < bCT,T.(1 - X, r, s) a X = bCT,T.(1 - X, r, s) - (1 - X).

But (1 -p) is an (r, s)-gfc and 1 -X is an (r, s)-g *fbc by (1), 1 -p = 0 and hence p = 1.

Proposition 4.2. Let (X, t, t*) be dfts's. For each X and p e IX, r e I0 and s e I1.

If X and p are (r, s)-g *fbc, then Xap is an (r, s)-g *fbc. 1. 2. If X is an (r, s)-g *fbc and T(p) > r, T*(p) < s, then X a p is an (r, s)-g *fbc.

Proof. (1) Suppose that X and p are (r, s)-g *fbc sets in IX such that X a p < v for each an (r, s)-gfo set v e IX, r e I0 and s e I1. Since X is an (r, s)-g *fbc,

bCT,T.(X, r, s) < v

for each an (r, s)-gfo set v e IX and X < v. Also, p is an (r, s)-g *fbc, bCT,T.(p, r, s) < v

for each an (r, s)-gfo set v e IX and p < v. Then we have, bCT,T(X, r, s) a bCTT*(p, r, s) < v,

whenever X a p < v, Therefore, X a p is an (r, s)-g *fbc.

(2) Since every an (r, s)-fc set is an (r, s)-g *fbc and from (1) we get the proof.

Proposition 4.3. Let (X, t, t*) be dfts's. For each X and p e IX, r e I0 and s e I1.

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If X is both an (r, s)-gfo and an (r, s)-g *fbc, then X is an1(r, s)-fbc set.

2. If X is an (r, s)-g *fbc and X<)< bCTT*(X, r, s), then i is an (r, s)-g *fbc.

Proof. (1) Suppose that X is an (r, s)-gfo and an (r, s)-g *fbc in

IX such that bCT T*(X, r, s) <n, r e I0 and s e I1. But

X < bCT,T.(X, r, s).

Therefore X = bCT,T(X, r, s).

Hence X is an (r, s)-fbc set.

(2) Suppose that X is an (r, s)-g *fbc and v is an (r, s)-gfo set in IX such that i < v for each i e IX, r e I0 and s e I1.So X < v. But we have, X is an (r, s)-g *fbc, then

bCT,T.(X, r, s) < v.

(X a bCTT* (), r, s)) v (1 - bCT,T.(), r, s)) < v v (1 - bCT,T.(), r, s)). ^Xv (1 - bCTT. (), r, s)) < v v (1 - bCTT* (), r, s)).

Since X is an (r, s)-g *fbc, then

bCT,T.(X, r, s) < vv (1 -)).

)<X^ bCT,T.(), r, s) < bCT,T.(X, r, s). Thus

bCT,T.(), r, s) < bCT,T.(X, r, s) < v v (1 -bCT,T.(), r, s)).

Therefore bCTT*(), r, s) < v, but bCTT*(), r, s) is not contained in (1 -bCTT*(), r, s)).That is, j is an (r, s)-g *fbc relative to X.

bCT,T.(), r, s) < bCT,T.(bCT,T.(X, r, s), r, s) = bCT,T.(X, r, s) < v.

Therefore i is an (r, s)-g *fbc set.

Theorem 4.2. Let (X, t t*) and (Y, t2, t*) be dfts's. If X < 1y < 1x such that X is an (r, s)-g *fbc in IX, r e I0 and s e I1, then X is an (r, s)-g *fbc relative to Y.

Proof. Suppose that (X, t1, t*) and (Y, t2, t*) are dfts's such that X < 1Y < 1x, r e I0, s e I1 and X is an (r, s)-g *fbc in IX. Now, let X < 1Y a ) such that i is an (r, s)-gfo set in IX. But we have, X is an (r, s)-g *fbc in IX,

X< bCTT(X, r, s) <).

Interrelations

The following implication illustrates the relationships between different fuzzy sets:

So that 1Y a bCz z* (X, r, s) < 1Y a p.

Hence X is an (r, s)-g *fbc relative to Y.

None of these implications is reversible where A ^ B represents A implies B, as shown by the following examples. But at this stage we do not have information regarding the relationship between an (r, s)-gfbc and (r, s)-g *fc sets.

Theorem 4.3. Let (X, t1, t*) be adfts. For each X and i e IX, r e I0 Example 5.1. (1) Let X = (a, b, c} and let i and a are fuzzy sets

and s e I1 with i < X. If i is an (r, s)-g *fbc relative to X such defined by: that X is both an (r, s)-gfo and (r, s)-g *fbc of IX, then i is an (r,

s)-g *fbc relative to X. )(a) =1.0, )(b) = a5, )(c) = 0.0,

Proof. Suppose that n is an (r, s)-g *fbc and Tv) > r and t*(v) < s such that ¡i < v, r e I0, s e Ii. But we have, n<X< 1, therefore ¡i < X and i < v. So

fl<X A v.

Also we have, ¡i is an (r, s)-g *fbc relative to X,

a(a) = 0.0, a(b) = 0.4, a(c) = 1.0.

Define (t, t*) on X as follows:

ii, if X e {0,1}, Î0, if Xe {0,1},

t(X) = j -2, if X = /u, t*(X) = j-2, if X = /u,

[0, otherwise. [1, otherwise.

X a bCT,T.(), r, s) <Xav^Xa bCT,T.(), r, s) < v.

Then a is an (1,1)-gfbc set, but not an (1,-i)-g *fbc set.

ARTICLE IN PRESS

Egyptian journal of basic and applied sciences

(2015)

(2) Take X = (a, b} in (1) and define p, a and ß by: p(a) = 0.6, p(b) = 0.6,

a(a) = 0.3, a(b) = 0.2,

ß(a) = 0.4, ß(b) = 0.5.

Then ß is an -2)-g *fbc set, but not an (§,f)-fbc set.

(3) Let X = (a, b, c}. Define p, v and 7by:

p(a) = 1.0, p(b) = 0.5, p(c) = 0.3, v(a) = 1.0, v(b) = 0.6, v(c) = 0.0,

7(a) = 0.0, Y(b) = 0.6, y(c) = 0.0.

Define (т, т") as in (1). Then v is an (1,2)-g *fbc set but not an (1,2)-fc set and not an (2,1)-gfc. And y is an (-j,i)-g *fbc set, but not an (-j,-^)-fsc set.

(4) Take (3) and defined p and v by:

p(a) = 1.0, p(b) = 1.0, p(c) = 0.6,

v(a) = 0.3, v(b) = 0.5, v(c) = 0.5.

Define (т, т") as in (1).Then v is an (1, |)-g *fbc set, but not an ((¡, 2)-g *fc set.

(5) See Example 3.1. Clearly ß is an (2,1)-gfbc set, but not an (, 1)-gfc set.

(6) Let X = (a, b}. Define p, v and y as follows:

p(a) = 0.7, p(b) = 0.6, v(a) = 0.3, v(b) = 0.2,

Y(a) = 0.4, Y(b) = 0.5.

Define (т, f) as in (1).Then v is an (1,2)-fbc set but not an (i,i) -fsc set, also not an (1,1)-gfc.

(7) Let X = (a, b, c} and let p and a as fuzzy sets defined by:

p(a) = 0.9, p(b) = 0.8, p(c) = 0.3,

a(a) = 0.1, a(b) = 0.8, a(c) = 0.3.

Define (т, т.) on X by:

[1, if X e (0,1}, [0, if X e (0,1},

t(X) = <¡0.6, if X = p, t*(X) = <¡0.3, if X = p,

[0, otherwise. [1, otherwise.

Then a is an (0.6,0.3)-fsc set, but not an (0.6,0.3)-fc set.

(8) Let X = (a, b} and let p and a as fuzzy sets defined by: p(a) = 0.9, p(b) = 0.4,

a(a) = 0.1, a(b) = 0.8.

Define (t, t*) on X by:

i1, if X e (0,1}, i0, if Xe (0,1},

t(X) = <! 2, if X = p, t*(X) = <!2, if X = p,

[0, otherwise. [1, otherwise.

Then p is an (2,-2)-g *fc set, but not an (1,2)-fc set.

Acknowledgments

The authors would like to acknowledge the following: UKM Grant DIP-2014-034 and Ministry of Education, Malaysia grant FRGS/1/2014/ST06/UKM/01/1 for financial support.

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