한국지능시스템학회 논문지 2009, Vol. 19, No. 6, pp. 848-851
848
접수일자 : 2009년 4월 1일 완료일자 : 2009년 12월 8일
Interval-Valued Fuzzy m-semicontinuous 함수의 특성 연구
Characterizations For Interval-Valued Fuzzy m-semicontinuous Mappings On Interval-Valued Fuzzy Minimal Spaces
민원근
*Won Keun Min
** 강원대학교 수학과
요 약
[5]에서 정의된 IVF m-semiopen 집합과 IVF m-semicontinuous 함수의 기본적 성질과 특성을 조사한다.
Abstract
In [5], we introduced the concepts of IVF m-semiopen sets and IVF m-semicontinuous mappings on interval-valued fuzzy minimal spaces. In this paper, we investigate some properties of IVF m-semiopen sets and characterizations for the IVF m-semicontinuous mapping.
Key Words : interval-valued fuzzy minimal spaces, IVF m-semiopen sets, IVF m-semicontinuous
1. Intorduction and Preliminaries
Zadeh [7] introduced the concept of fuzzy set and several researchers were concerned about the general- izations of the concepts of fuzzy sets, intuitionistic fuz- zy sets [1] and interval-valued fuzzy sets [3].
Alimohammady and Roohi [2] introduced fuzzy minimal structures and fuzzy minimal spaces and some results are given. In [4], Min introduced the concepts of IVF minimal structures and IVF m-continuous mappings which are generalizations of IVF topologies and IVF continuous mappings [6], respectively. In [5], Min et al.
introduced the concepts of IVF
-semiopen sets and IVF-semicontinuous mappings on interval-valued
fuzzy minimal spaces. We investigated basic properties of IVF-semiopen sets and IVF -semicontinuous
mappings. In this paper, we investigate character- izations for the IVF
-semicontinuous mapping and some properties of IVF
-semiopen sets.Let
be the set of all closed subintervals of the interval
. The elements of
are generally denoted by capital letters
,
,⋯
and note that
= [
,
], where
and
are the lower and the upper end points respectively. We also note that(1) (
∀
, ∈
)(
= ⇔
=
,
=
).(2) (
∀
, N∈
)(
≤ ⇔
≤
,
≤
).For each
∈
, the complement of
, denoted by
, is defined by
=[1-
, 1-
].Let
be a nonempty set. A mapping →
is called an interval-valued fuzzy set (simply, IVF set) in
. For each ∈
,
is a closed interval whose lower and upper end points are denoted by
and
, respectively. For any∈
, the IVF set whose value is the interval for all ∈
is denoted by . We denote
and
as follows:
,
. In particular, for any ∈
, the IVF set whose value is
=
for all ∈
is denoted by simply
. For a point ∈
and for∈
with
>0, the IVF set which takes the value at and
elsewhere in
is called an interval-valued fuzzy point (simply, IVF point) and is denoted by
. In particular, if
, then it is also denoted by
. We denote the set of all IVF sets in
by IVF(
). An IVF point
, where
∈
, is said to belong to an IVF set
in
, de- noted by
∈
, if
≥
and
≥
. In [6], it has been shown that
=∪
{
:
∈
}.For every
∊
, we define ⇔
(∀ ∈
)(
=
,
=
).Interval-Valued Fuzzy m-semicontinuous 함수의 특성 연구
849
⊆ ⇔
(∀ ∈
)(
⊆
,
⊆
).The complement
of
is defined by
1-
and
1-
for all ∈
.For a family of IVF sets {
: ∈
} where
is an in- dex set, the union ∪
∈
and ∩
∈
are defined by
=
∈
,
=
∈
and
=
∈
,
=
∈
, respectively, for all ∈
.Let
→
be a mapping and let
be an IVF set in
. Then the image of
under
, denoted by
, defined as follows
∈
i f
≠ ∅
otherwise
∈
i f
≠ ∅
otherwise
for all
∈
.Let
be an IVF set in
. Then the inverse image of
under
, denoted by
, defined as follows
=
,
=
for all ∈
.Definition 1.1 ([6]). A family
of IVF sets in
is called an interval-valued fuzzy topology on
if it satisfies the following properties:(1)
, ∈ .
(2)
, ∈ ⇒ ∩ ∈
.(3) For
∈
,
∈ ⇒ ∪
∈
∈ .
Every member of
is called an IVF open set. An IVF set
is called an IVF closed set if the comple- ment of
is an IVF open set. And the pair
is alled an interval-valued fuzzy topological space.Definition 1.2 ([4]). A family
of interval-valued fuzzy sets in
is called an interval-valued fuzzy mini-mal structure on
if
, ∈
.In this case,
is called an interval-valued fuz-zy minimal space (simply, IVF minimal space). Every
member of
is called an IVF
-open set. An IVF set
is called an IVF
-closed set if the complement of
(simply,
) is an IVF
-open set.Let
be an IVF minimal space and ∈
IVF (
). The IVF minmal-closure and the IVF minimal-in- terior of
[6], denoted by
and
, re- spectively, are defined as
=∩
{ ∈
IVF(
):
∈
and ⊆
},
=∪
{ ∈
IVF(
): ∈
and ⊆
}.Theorem 1.3 ([4]). Let (
,
) be an IVF minimal space and
, ∈
IVF(
). Then the following properties hold:(1)
(
)⊆
and if
is an IVF
-open set, then
(
)=
.(2)
⊆
(
) and if
is an IVF
-closed set, then
(
)=
.(3) If
⊆
, then
(
)⊆
(
) and
(
)⊆
(
).(4)
( ∩
)⊆
(
)∩
(
) and
(
)∪
(
)⊆
( ∪
).(5)
(
(
))=
(
) and
(
(
))=
(
).(6)
-
(
)=
(
-
) and
-
(
)=
(
-
).2. Main Results
Definition 2.1 ([5]). Let
be an IVF minimal space and
in IVF(
). Then an IVF set
is called an IVF -semiopen set in
if ⊆
(mI(A)). An IVF set
is called an IVF -semiclosed set if the
complement of
is an IVF
-semiopen set. Thesemi-closure and the semi-interior of
, denoted by
(
) and
(
), respectively, are defined as the following:
(
)=∩
{ ∈
IVF(
): ⊆
,
is IVF
-semiclosed in
}
(
)=∪
{ ∈
IVF(
): ⊆
,
is IVF-semiopen in
}.Theorem 2.2 ([5]). Let
be an IVF minimal space and ∈
IVF(
). Then(1)
(
)⊆ ⊆
(
).(2) If
⊆
, then
(
)⊆
(
) and
(
)⊆
(
).(3)
is IVF
-semiopen iff
(
)=
and
is IVF
-semiclosed iff
(
)=
.(4)
(
(
))=
(
) and
(
(
))=
(
).(5)
(
-
)=
-
(
) and
(
-
)=
-
(
).한국지능시스템학회 논문지 2009, Vol. 19, No. 6
850
Lemma 2.3 Let
be an IVF minimal space and ∈
IVF(
). Then(1)
(
(
))⊆
(
(
(
)))⊆
(
).(2)
(
)⊆
(
(
(
)))⊆
(
(
)).(3)
(
(
))=
(
(
)).(4)
(
(
))=
(
(A)).Proof. (1) For
∈
IVF(
), since ⊆
(
) and
(
) is an IVF-semiclosed set, we have
(
(
))⊆
(
(
(
)))⊆
(
).(2) It is similar to the proof of (1).
(3) From (1) and Theorem 2.2, it follows
(
(A))⊆
(A)⊆
(
).This implies
(
(
))=
(
(
)).(4) It is similar to the proof of (3).
Definition 2.4 ([5]). Let
and
be two IVF minimal spaces. Then →
is said to be inter-val-valued fuzzy -semicontinuous (simply, IVF -semicontinuous) if for each IVF point
and each IVF
-open set
containing
(
), there exists an IVF
-semiopen set
containing
such that
(
)⊆
.Theorem 2.5 Let
→
be a function on IVF mini- mal spaces
and
. Then
is IVF
-semicontinuous if and only if for each IVF
-open set
in
,
(
)⊆
(
(
(
))).Proof. Let
be an IVF
-open set in
and
∈
(
). By IVF
-semicontinuity of
, there exists an IVF
-semiopen set
containing
such that
(
)⊆
. Thus from IVF
-semiopenness and Theorem 1.3 (3), it follows
∈ ⊆
(
(
))⊆
(
( (
))).Hence we have
(
)⊆
(
( (
))).For the converse, let
be an IVF point of
and
an IVF
-open set in
containing
(
). Then by hypothesis, we have
∈
(
)⊆
(
(
(
))).Put
=
(
). Then
is an IVF
-semiopen set containing
such that(
)⊆
. Thus is IVF
-semicontinuous.Corollary 2.6. Let
→
be a mapping on IVF minimal spaces
and
. Then the follow- ing statements are equivalent:(1)
is IVF
-semicontinuous.(2) For each IVF
-open set
in
,
(
) is IVF
-semiopen.(3) For each IVF
-closed set
in
,
(
) is IVF
-semiclosed.Proof. It follows from Definition 2.1 and Theorem 2.5.
Corollary 2.7. Let
→
be a mapping on IVFminimal spaces
and
. Then
is IVF
-semicontinuous if and only if for each IVF
-open set
in
,
(
)⊆
(
(
(
)))Proof. It follows from Theorem 2.5 and Lemma 2.3.
Theorem 2.8 Let
→
be a function on IVF mini- mal spaces
and
. Then is IVF
-semicontinuous if and only if
(
(
))⊆
(
(
(
))) for ∈
IVF(
).Proof. For each
∈
(
(
)), since
(
) ∈
(
), there exists an IVF
-open set
such that
(
) ∈ ⊆
. From IVF
-semicontinuity of
, there exists an IVF-semiopen set
containing
such that(
)⊆
. This implies
∈ ⊆
(
)⊆ (
).Since
is IVF
-semiopen, it follows
∈ ⊆
(
(
))⊆
(
( (
))).Hence
(
(
))⊆
(
( (
))).For the converse, let
be an IVF point of
and an IVF
-open set
containing(
). Then by hy- pothesis, we have
(
)= (
(
))⊆
(
( (
))).Thus
(V) is an IVF-semiopen set containing
. Put
= (
). Then(
)⊆
and so is IVF
-semicontinuous.Corollary 2.9. Let
→
be a mapping on IVF minimal spaces
and
. Then
is IVF
-semicontinuous if and only if
(
(
))⊆
(
( (
))) for ∈
IVF(
).Proof. It follows from Theorem 2.8 and Lemma 2.3.
Theorem 2.10 Let
→
be a function on IVF minimal spaces
and
. Then the follow- ing statements are equivalent:(1)
is IVF
-semicontinuous.(2)
(
( (
)))⊆ (
(
)) for ∈
IVF(
).(3)
(
(
(
)))⊆
(
(
)) for ∈
IVF(
).Proof. (1)
⇒
(2) Let ∈
IVF(
). Then from Theorem 1.3 and Theorem 2.8, it follows
(
(
))= (
-
(
-
)) =
-( (
(
-
)))⊆
-
(
( (
-
))) =
(
( (
))).(2)
⇒
(3) For ∈
IVF(
), by (2), we have
(
(
))⊆
(
(
(
(
))))⊆
(
(
(
))).This implies
(
(
(
)))⊆
((
)).(3)
⇒
(1) Let
an IVF
-closed set in
. ThenInterval-Valued Fuzzy m-semicontinuous 함수의 특성 연구
851
by (3),
(
(
(
(
))))⊆
(
(
(
)))⊆
(
)=
. This implies
(
) is an IVF
-semiclosed set. Hence by Corollary 2.6,
is IVF
-semicontinuous.Corollary 2.11. Let
→
be a function on IVF minimal spaces
and
. Then the follow- ing statements are equivalent:(1)
is IVF
-semicontinuous.(2)
(
(
(
)))⊆
(
(
)) for ∈
IVF(
).(3)
(
(
(
)))⊆
(
(
)) for ∈
IVF(
).Proof. It follows from Theorem 2.10 and Lemma 2.3.
References
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Chaos, Solutions and Fractals, vol. 27,pp. 599-605,
2006.[3] M. B. Gorzalczany, "A method of inference in ap- proximate reasoning based on interval-valued fuzzy sets", Fuzzy Sets and Systems, vol. 21, pp.
1-17, 1987.
[4] W. K. Min, "Interval-Valued Fuzzy Minimal Structures and Interval-Valued Fuzzy Minimal Spaces", International Journal of Fuzzy Logic and
Intelligent Systems, vol. 8, no. 3, pp. 202-206,
2008.[5] W. K. Min, M. H. KIM and J. I. KIM,
"Interval-Valued Fuzzy
-semiopen sets and Interval-Valued Fuzzy
-preopen sets on Interval-Valued Fuzzy Minimal Spaces", HonamMathetical J., vol. 31, no. 1, pp. 31-43, 2009.
[6] T. K. Mondal and S. K. Samanta, "Topology of interval-valued fuzzy sets", Indian J. Pure Appl.
Math., vol. 30, no. 1, pp. 23-38, 1999.
[7] L. A. Zadeh, "Fuzzy sets", Inform. and Control, vol. 8, pp. 338-353, 1965.
저 자 소 개
민원근(Won Keun Min)
1988년～현재 : 강원대학교 수학과 교수 관심분야 : 퍼지 위상, 퍼지 이론, 일반 위상 Phone : 033-250-8419
Fax : 033-252-7289
E-mail : [email protected]
