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Abstract
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.
1 Introduction
A progressive development of fuzzy sets [Citation1] 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 [Citation2]. Later on, Çoker [Citation3] presented the notion of intuitionistic fuzzy topology. Samanta and Mondal [Citation4], 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 [Citation5] concluded that they work under the name “double”.
In 2009, Omari and Noorani [Citation6] introduced generalized b-closed sets (briefly, gb-closed) in general topology. As a generalization of the results in References Citation6 and Citation7, 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 λ is quasi-coincident with a fuzzy set µ (denoted by, λqµ) iff there exists x ∈ X such that and they are not quasi-coincident otherwise (denoted by,
). The family of all fuzzy sets on X is denoted by IX. By
and
, we denote the smallest and the greatest fuzzy sets on X. For a fuzzy set λ ∈ IX,
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 [Citation4]) A double fuzzy topology (τ, τ*) on X is a pair of maps τ, τ* : IX → I, which satisfies the following properties:
Table
The triplet (X, τ, τ*) is called a double fuzzy topological space (briefly, dfts). A fuzzy set λ is called an (r, s)-fuzzy open (briefly, (r, s)-fo) if τ(λ) ≥ r and τ*(λ) ≤ s. A fuzzy set λ is called an (r, s)-fuzzy closed (briefly, (r, s)-fc) set iff is an (r, s)-fo set.
Theorem 2.1
(see [Citation8]) Let (X, τ, τ*) be a dfts. Then double fuzzy closure operator and double fuzzy interior operator of λ ∈ IX are defined by
Definition 2.2
Let (X, τ, τ*) be a dfts. For each λ ∈ IX, r ∈ I0 and s ∈ I1. A fuzzy set λ is called:
1. | An (r, s)-fuzzy semiopen (see [Citation9]) (briefly, (r, s)-fso) if |
2. | An (r, s)-generalized fuzzy closed (see [Citation10]) (briefly, (r, s)-gfc) if |
Definition 2.3
(see [Citation11,Citation12]) Let (X, τ, τ*) be a dfts. For each λ, µ ∈ IX and r ∈ I0, s ∈ I1. Then, a fuzzy set λ is said to be (r, s)-fuzzy generalized ψρ-closed (briefly, (r, s)-fgψρ-closed) if such that λ ≤ µ and µ is (r, s)-fuzzy ρ-open set. λ is called (r, s)-fuzzy generalized ψρ-open (briefly, (r, s)-fgψρ-open) iff
is (r, s)-fgψρ-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, τ, τ*) be a dfts. For each λ ∈ IX, r ∈ I0 and s ∈ I1. A fuzzy set λ is called:
1. | An (r, s)-fuzzy b-closed (briefly, (r, s)-fbc) if |
2. | An (r, s)-generalized fuzzy b-closed (briefly, (r, s)-gfbc) if |
Definition 3.2
Let (X, τ, τ*) be a dfts. Then double fuzzy b-closure operator and double fuzzy b-interior operator of λ ∈ IX are defined by
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 µ, α and β by:
Then β is an -gfbc set but not an
-fbc set.
Definition 3.3
Let (X, τ, τ*) be a dfts, λ ∈ IX, r ∈ I0 and s ∈ I1. λ is called an (r, s)-fuzzy b-Q-neighborhood of xt ∈ Pt(X) if there exists an (r, s)-fbo set µ ∈ IX such that xtqµ and µ ≤ λ.
The family of all (r, s)-fuzzy b-Q-neighborhood of xt denoted by .
Theorem 3.1
Let (X, τ, τ*) be a dfts. Then for each λ, µ ∈ IX, r ∈ I0 and s ∈ I1, the operator satisfies the following statements:
Table
Proof
(1), (2), (3), and (4) are proved easily.
(5) Let and µ is an (r, s)-fbo set, then
. But we have, µqλ iff
and
so
, which is contradiction. Then µqλ iff
.
(6) Let xt be a fuzzy point such that . Then there is an (r, s)-fuzzy b-Q neighborhood µ of xt such that
. But by (5), we have an (r, s)-fuzzy b-Q-neighborhood µ of xt such that
Also,
Then
But we have,
Therefore
(7) and (8) are obvious.
Theorem 3.2
Let (X, τ, τ*) be a dfts. Then for each λ, µ ∈ IX, r ∈ I0 and s ∈ I1, the operator satisfies the following statements:
1. |
|
2. |
|
3. |
|
4. | If λ is an (r, s)-fbo, then |
5. | If λ ≤ µ, then |
6. |
|
7. |
|
8. |
|
Proof
It is similar to Theorem 3.1.
Theorem 3.3
Let (X, τ, τ*) be a dfts. λ ∈ IX is (r, s)-gfbo set, r ∈ I0 and s ∈ I1 if and only if whenever µ ≤ λ,
and
.
Proof
Suppose that λ is an (r, s)-gfbo set in IX, and let and
such that µ ≤ λ. By the definition,
is an (r, s)-gfbc set in IX. So,
Also,
And then,
Conversely, let µ ≤ λ, and
, r ∈ I0 and s ∈ I1 such that
. Now
Thus
That is, is an (r, s)-gfbc set, then λ is an (r, s)-gfbo set.
Theorem 3.4
Let (X, τ, τ*) be a dfts, λ ∈ IX, r ∈ I0 and s ∈ I1. If λ is an (r, s)-gfbc set, then
1. |
|
2. | λ is an (r, s)-fbc iff |
3. | µ is (r, s)-gfbc set for each set µ ∈ IX such that |
4. | For each (r, s)-fo set µ ∈ IX such that µ ≤ λ, µ is an (r, s)-gfbc relative to λ if and only if µ is an (r, s)-gfbc in IX. |
5. | For each an (r, s)-fbo set µ ∈ IX such that |
Proof
(1) Suppose that and
, r ∈ I0 and s ∈ I1 such that
whenever λ ∈ IX is an (r, s)-gfbc set. Since
is an (r, s)-fo set,
(2) Let λ be an (r, s)-gfbc set. So, for each r ∈ I0 and s ∈ I1 if λ is an (r, s)-fbc set then,which is an (r, s)-fc set.
Conversely, suppose that is an (r, s)-fc set. Then by (1),
does not contain any non-zero an (r, s)-fc set. But
is an (r, s)-fc set, then
So, λ is an (r, s)-fbc set.
(3) Suppose that τ(α) ≥ r and τ*(α) ≤ s where r ∈ I0 and s ∈ I1 such that µ ≤ α and let λ be an (r, s)-gfbc set such that λ ≤ α. Then
So,
Therefore
So, µ is an (r, s)-gfbc set.
(4) Let λ be an (r, s)-gfbc and τ(λ) ≥ r and τ*(λ) ≤ s, where r ∈ I0 and s ∈ I1. Then . But, µ ≤ λ so,
Also, since µ is an (r, s)-gfbc relative to λ, thenso
Now, if µ is an (r, s)-gfbc relative to λ and τ(α) ≥ r and τ*(α) ≤ s where r ∈ I0 and s ∈ I1 such that µ ≤ α, then for each an (r, s)-fo set α ∧ λ, . Hence µ is an (r, s)-gfbc relative to λ,
Therefore, µ is an (r, s)-gfbc in IX.
Conversely, let µ be an (r, s)-gfbc set in IX and τ(α) ≥ r and τ*(α) ≤ s whenever α ≤ λ such that µ ≤ α, r ∈ I0 and s ∈ I1. Then for each an (r, s)-fo set β ∈ IX, α = β ∧ λ. But we have, µ is an (r, s)-gfbc set in IX such that µ ≤ β,
That is, µ is an (r, s)-gfbc relative to λ.
(5) Suppose µ is an (r, s)-fbo and , r ∈ I0 and s ∈ I1. Then
. Since
is an (r, s)-fbc set of IX and λ is an (r, s)-gfbc set, then
Conversely, let µ be an (r, s)-fbc set of IX such that λ ≤ µ, r ∈ I0 and s ∈ I1. Then
But
Hence λ is an (r, s)-gfbc.
Proposition 3.1
Let (X, τ, τ*) be a dfts, λ ∈ IX, r ∈ I0 and s ∈ I1.
1. | If λ is an (r, s)-gfbc and an (r, s)-fbo set, then λ is an (r, s)-fbc set. |
2. | If λ is an (r, s)-fo and an (r, s)-gfbc, then λ ∧ µ is an (r, s)-gfbc set whenever |
Proof
(1) Suppose λ is an (r, s)-gfbc and an (r, s)-fbo set such that λ ≤ λ, r ∈ I0 and s ∈ I1. Then
But we have,
Then,
Therefore, λ is an (r, s)-fbc set.
(2) Suppose that λ is an (r, s)-fo and an (r, s)-gfbc set, r ∈ I0 and s ∈ I1. Then
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, τ, τ*) be a dfts. For each λ ∈ IX, r ∈ I0 and s ∈ I1. A fuzzy set λ is called:
1. | An (r, s)-generalized ⋆-fuzzy closed (briefly, (r, s)-g ⋆fc) if |
2. | An (r, s)-generalized ⋆-fuzzy b-closed (briefly, (r, s)-g ⋆fbc) if |
Theorem 4.1
Let (X, τ, τ*) be a dfts. λ ∈ IX is an (r, s)-g ⋆fbo set if and only if whenever µ is an (r, s)-gfc, r ∈ I0 and s ∈ I1.
Proof
Suppose that λ is an (r, s)-g ⋆fbo set in IX, and let µ is an (r, s)-gfc set such that µ ≤ λ, r ∈ I0 and s ∈ I1. So by the definition, we have is an (r, s)-gfo set in IX and
. But
is an (r, s)-g ⋆fbc set, then
. But
Therefore,
Conversely, suppose that whenever µ ≤ λ and µ is an (r, s)-gfc set, r ∈ I0 and s ∈ I1. Now
Thus
Therefore, is an (r, s)-gfbc set and λ is an (r, s)-gfbo set.
Proposition 4.1
Let (X, τ, τ*) be dfts's. For each λ ∈ IX, r ∈ I0 and s ∈ I1
1. | If a fuzzy set λ is an (r, s)-g ⋆fbc, then |
2. | If a fuzzy set λ is an (r, s)-g ⋆fbc, then |
3. | An (r, s)-g ⋆fbc set λ is an (r, s)-fbc iff |
4. | If a fuzzy set λ is an (r, s)-g ⋆fbc, then |
Proof
(1) Suppose that λ is an (r, s)-g ⋆fbc set and µ is an (r, s)-gfc set of IX, r ∈ I0 and s ∈ I1 such that
And
But λ is an (r, s)-g ⋆fbc set and is an (r, s)-gfo set, then
Therefore contains no non-zero (r, s)-gfc set.
(2) Let λ be an (r, s)-g ⋆fbc set, r ∈ I0 and s ∈ I1. Then by (1) we have, contains no non-zero (r, s)-gfc set. So,
is an (r, s)-g ⋆fbo set.
(3) Let λ be an (r, s)-g ⋆fbc set. If λ is an (r, s)-fbc, r ∈ I0 and s ∈ I1, then
Conversely, let is an (r, s)-fbc set in IX and λ is an (r, s)-g ⋆fbc, r ∈ I0 and s ∈ I1, then by (1) we have,
contains no non-zero (r, s)-gfc set. Then,
that is
Hence λ is an (r, s)-fbc set.
(4) Let µ be an (r, s)-gfc set and , r ∈ I0 and s ∈ I1. Hence
But is an (r, s)-gfc and
is an (r, s)-g ⋆fbc by (1),
and hence
.
Proposition 4.2
Let (X, τ, τ*) be dfts's. For each λ and µ ∈ IX, r ∈ I0 and s ∈ I1.
1. | If λ and µ are (r, s)-g ⋆fbc, then λ∧µ is an (r, s)-g ⋆fbc. |
2. | If λ is an (r, s)-g ⋆fbc and τ(µ) ≥ r, τ*(µ) ≤ s, then λ ∧ µ is an (r, s)-g ⋆fbc. |
Proof
(1) Suppose that λ and µ are (r, s)-g ⋆fbc sets in IX such that λ ∧ µ ≤ ν for each an (r, s)-gfo set ν ∈ IX, r ∈ I0 and s ∈ I1. Since λ 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, τ, τ*) be dfts's. For each λ and µ ∈ IX, r ∈ I0 and s ∈ I1.
1. | If λ is both an (r, s)-gfo and an (r, s)-g ⋆fbc, then λ is an (r, s)-fbc set. |
2. | If λ is an (r, s)-g ⋆fbc and |
Proof
(1) Suppose that λ is an (r, s)-gfo and an (r, s)-g ⋆fbc in IX such that , r ∈ I0 and s ∈ I1. But
Therefore
Hence λ is an (r, s)-fbc set.
(2) Suppose that λ is an (r, s)-g ⋆fbc and ν is an (r, s)-gfo set in IX such that µ ≤ ν for each µ ∈ IX, r ∈ I0 and s ∈ I1. So λ ≤ ν. But we have, λ is an (r, s)-g ⋆fbc, then
Now
Therefore µ is an (r, s)-g ⋆fbc set.
Theorem 4.2
Let and
be dfts's. If
such that λ is an (r, s)-g ⋆fbc in IX, r ∈ I0 and s ∈ I1, then λ is an (r, s)-g ⋆fbc relative to Y.
Proof
Suppose that and
are dfts's such that
, r ∈ I0, s ∈ I1 and λ is an (r, s)-g ⋆fbc in IX. Now, let
such that µ is an (r, s)-gfo set in IX. But we have, λ is an (r, s)-g ⋆fbc in IX,
So that
Hence λ is an (r, s)-g ⋆fbc relative to Y.
Theorem 4.3
Let be adfts. For each λ and µ ∈ IX, r ∈ I0 and s ∈ I1 with µ ≤ λ. If µ is an (r, s)-g ⋆fbc relative to λ such that λ is both an (r, s)-gfo and (r, s)-g ⋆fbc of IX, then µ is an (r, s)-g ⋆fbc relative to X.
Proof
Suppose that µ is an (r, s)-g ⋆fbc and τ(ν) ≥ r and τ*(ν) ≤ s such that µ ≤ ν, r ∈ I0, s ∈ I1. But we have, , therefore µ ≤ λ and µ ≤ ν. So
Also we have, µ is an (r, s)-g ⋆fbc relative to λ,
Thus
Since λ is an (r, s)-g ⋆fbc, then
Also,
Thus
Therefore , but
is not contained in
. That is, µ is an (r, s)-g ⋆fbc relative to X.
5 Interrelations
The following implication illustrates the relationships between different fuzzy sets:
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.
Example 5.1
(1) Let and let µ and α are fuzzy sets defined by:
Define (τ, τ*) on X as follows:
Then α is an -gfbc set, but not an
-g ⋆fbc set.
(2) Take X = {a, b} in (1) and define µ, α and β by:
Then β is an -g ⋆fbc set, but not an
-fbc set.
(3) Let . Define µ, ν and γ by:
Define (τ, τ*) as in (1). Then ν is an -g ⋆fbc set but not an
-fc set and not an
-gfc. And γ is an
-g ⋆fbc set, but not an
-fsc set.
(4) Take (3) and defined µ and ν by:
Define (τ, τ*) as in (1). Then ν is an -g ⋆fbc set, but not an
-g ⋆fc set.
(5) See Example 3.1. Clearly β is an -gfbc set, but not an
-gfc set.
(6) Let X = {a, b}. Define µ, ν and γ as follows:
Define (τ, τ*) as in (1). Then ν is an -fbc set but not an
-fsc set, also not an
-gfc.
(7) Let and let µ and α as fuzzy sets defined by:
Define (τ, τ*) on X by:
Then α is an -fsc set, but not an
-fc set.
(8) Let X = {a, b} and let µ and α as fuzzy sets defined by:
Define (τ, τ*) on X by:
Then µ is an -g ⋆fc set, but not an
-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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