Interval-Valued Fuzzy mα-Continuous Mappings on Interval-Valued Fuzzy Minimal Spaces
Won Keun Min and Young Ho Yoo
Department of Mathematics, Kangwon National University, Chuncheon, 200-701, Korea Department of Statics, Kangwon National University, Chuncheon, 200-701, Korea
Abstract
We introduce the concepts of interval-valued fuzzy mα-open sets and interval-valued fuzzy mα-continuous mappings.
And we study some characterizations and properties of such concepts.
Key Words : interval-valued fuzzy minimal spaces, interval-valued fuzzy mα -open sets, interval-valued fuzzy m-continuous mappings, interval-valued fuzzy mα-continuous mappings
1. Introduction and Preliminaries
Zadeh [8] introduced the concept of fuzzy set and several researchers were concerned about the generaliza- tions of the concept of fuzzy sets, intuitionistic fuzzy sets [1] and interval-valued fuzzy sets [3]. In [2], Alimo- hammady and Roohi introduced fuzzy minimal structures and fuzzy minimal spaces and gave some results. In [6], Min introduced the concept of interval valued fuzzy min- imal structure (simply, IVF minimal structure) as a gen- eralization of interval-valued fuzzy topology introduced by Mondal and Samanta [7]. Also he introduced the concept of interval-valued fuzzy m-continuous mappings which are generalizations of interval-valued fuzzy contin- uous mappings. In this paper, we introduce the concepts of interval-valued fuzzy mα-open sets and interval-valued fuzzy mα-continuous mappings defined on interval-valued fuzzy minimal spaces. These notions are a generaliza- tion of interval-valued fuzzy m-open sets and interval- valued fuzzy m-continuity in interval-valued fuzzy mini- mal spaces. We study characterizations and properties of such concepts.
Let D[0, 1] be the set of all closed subintervals of the interval [0, 1]. The elements of D[0, 1] are gener- ally denoted by capital letters M, N, · · · and note that M = [ML, MU], where ML and MU are the lower and the upper end points of M , respectively. Especially, we denote 0 = [0, 0], 1 = [1, 1], and a = [a, a] for a ∈ (0, 1).
We also note that
(1) (∀M, N ∈ D[0, 1])(M = N ⇔ ML = NL, MU = NU).
(2) (∀M, N ∈ D[0, 1])(M ≤ N ⇔ ML ≤ NL, MU ≤ NU).
For every M ∈ D[0, 1], the complement of M , denoted by Mc, is defined by Mc= 1 − M = [1 − MU, 1 − ML].
Let X be a nonempty set. A mapping A : X → D[0, 1]
is called an interval-valued fuzzy set(simply, IVF set) in X. For each x ∈ X, A(x) is a closed interval whose lower and upper end points are denoted by A(x)L and A(x)U, respectively. For a point p ∈ X and for [a, b] ∈ D[0, 1]
with b > 0, the IVF set which takes the value [a, b] at p and 0 elsewhere in X is called an interval-valued fuzzy point (simply, IVF point) and is denoted by [a, b]p. In particu- lar, if b = a, then it is also denoted by ap. Denoted by IVF(X) the set of all IVF sets in X. An IVF point Mx, where M ∈ D[0, 1], is said to belong to an IVF set A in X, denoted by Mxe∈A, if A(x)L ≥ MLand A(x)U ≥ MU. In [7], it has been shown that A = ∪{Mx: Mx∈A}.e
For every A, B ∈ IV F (X), we define A = B ⇔ (∀x ∈ X)([A(x)]L= [B(x)]Land
[A(x)]U = [B(x)]U),
A ⊆ B ⇔ (∀x ∈ X)([A(x)]L⊆ [B(x)]Land [A(x)]U ⊆ [B(x)]U).
The complement Acof A is defined by, for all x ∈ X, [Ac(x)]L = 1 − [A(x)]Uand [Ac(x)]U = 1 − [A(x)]L.
For a family of IVF sets {Ai : i ∈ J} where J is an index set, the union G = ∪i∈JAi and F = ∩i∈JAi are defined by
(∀x ∈ X) ([G(x)]L = supi∈J[Ai(x)]L,
Manuscript received may. 12, 2009; revised Sep. 30, 2009.
Corresponding Author : [email protected] (Won Keun Min)
[G(x)]U = supi∈J[Ai(x)]U), (∀x ∈ X) ([F (x)]L= infi∈J[Ai(x)]L,
[F (x)]U = infi∈J[Ai(x)]U), respectively.
Let f : X → Y be a mapping and let A be an IVF set in X. Then the image of A under f , denoted by f (A), is defined as follows
[f (A)(y)]L=
½ supf (x)=y[A(x)]L, if f−1(y) 6= ∅,
0, otherwise ,
[f (A)(y)]U =
½ supf (x)=y[A(x)]U, if f−1(y) 6= ∅,
0, otherwise ,
for all y ∈ Y .
Let B be an IVF set in Y . Then the inverse image of B under f , denoted by f−1(B), is defined by
([f−1(B)(x)]L = [B(f (x))]L, [f−1(B)(x)]U = [B(f (x))]U) for all x ∈ X.
Definition 1.1 ([7]). A family τ of IVF sets in X is called an interval-valued fuzzy topology on X if it satisfies:
(1) 0, 1 ∈ τ .
(2) A, B ∈ τ ⇒ A ∩ B ∈ τ .
(3) For i ∈ J , Ai∈ τ ⇒ ∪i∈JAi∈ τ .
Every member of τ is called an IVF open set. An IVF set A is called an IVF closed set if the complement of A is an IVF open set. And (X, τ ) is called an interval-valued fuzzy topological space.
Definition 1.2 ([4, 5]). An IVF set A in an IVF topo- logical space (X, τ ) is called an IVF α-open set in X if A ⊆ Int(Cl(Int(A)))).
And an IVF set A is called an IVF α-closed set if the complement of A is an IVF α-open set. We denote the set of all IVF α-open (resp., IVF α-closed) sets by IVFα(X) (resp., IVFαC(X)).
Let (X, τ1) and (Y, τ2) be two IVFTS’s. Then a map- ping f : X → Y is said to be IVF α-continuous [5] if for every IVF open set B in Y , f−1(B) is IVF α-open in X.
Definition 1.3 ([6]). A family M of interval-valued fuzzy sets in X is called an interval-valued fuzzy minimal struc- ture on X if
0, 1 ∈ M.
In this case, (X, M) is called an interval-valued fuzzy min- imal space (simply, IVF minimal space). Every member of M is called an interval-valued fuzzy m-open set (simply IVF m-open set). An IVF set A is called an IVF m-closed set if the complement of A (simply, Ac) is an IVF m-open set.
Let (X, M) be an IVF minimal space and A in IVF(X).
The IVF minimal-closure and the IVF minimal-interior of
A [6], denoted by mC(A) and mI(A), respectively, are defined as
mC(A) = ∩{B ∈ IV F (X) : Bc∈ Mand A ⊆ B}, mI(A) = ∪{B ∈ IV F (X) : B ∈ Mand B ⊆ A}.
Theorem 1.4 ([6]). Let (X, M) be an IVF minimal space and A, B in IVF(X).
(1) mI(A) ⊆ A and if A is an IVF m-open set, then mI(A) = A.
(2) A ⊆ mC(A) and if A is an IVF m-closed set, then mC(A) = A.
(3) If A ⊆ B, then mI(A) ⊆ mI(B) and mC(A) ⊆ mC(B).
(4) mI(A) ∩ mI(B) ⊇ mI(A ∩ B) and mC(A) ∪ mC(B) ⊆ mC(A ∪ B).
(5) mI(mI(A)) = mI(A) and mC(mC(A)) = mC(A).
(6) 1 − mC(A) = mI(1 − A) and 1 − mI(A) = mC(1 − A).
Definition 1.5 ([6]). Let (X, MX) and (Y, MY) be two IVF minimal spaces. Then f : X → Y is said to be interval-valued fuzzy minimal continuous (simply, IVF m- continuous) mapping if for every A ∈ MY, f−1(A) is in MX.
2. Interval-Valued Fuzzy mα-Open Sets and Interval-Valued Fuzzy mα-Continuous
Mappings
Definition 2.1. Let (X, M) be an IVF minimal space and A in IVF(X). Then an IVF set A is called an interval- valued fuzzy mα-open (simply IVF mα-open) set in X if
A ⊆ mI(mC(mI(A))).
And an IVF set A is called an interval-valued fuzzy mα- closed (simply IVF mα-closed) set if the complement of A is an IVF mα-open set.
Remark 2.2. Let (X, M) be an IVF minimal space and A in IVF(X). If the IVF minimal structure M is an IVF topology, clearly an IVF mα-open set is IVF α-open by Definition 1.2.
Obviously the following statements are obtained:
Lemma 2.3. Let (X, M) be an IVF minimal space. Then the statements are hold.
(1) Every IVF m-open set is IVF mα-open.
(2) A is an IVF mα-closed set if and only if mC(mI(mC(A))) ⊆ A.
Theorem 2.4. Let (X, M) be an IVF minimal space. Any union of IVF mα-open sets is IVF mα-open.
Proof. Let Ai be an IVF mα-open set for i ∈ J . From Theorem 1.4 (3), it follows
Ai⊆ mI(mC(mI((Ai)))) ⊆ mI(mC(mI(∪Ai))).
This implies ∪Ai ⊆ mI(mC(mI(∪Ai))). Hence ∪Ai is an IVF mα-open set.
Remark 2.5. Let (X, M) be an IVF minimal space. The intersection of any two IVF mα-open sets may not be IVF mα-open set as shown in the next example.
Example 2.6. Let X = {a, b}, let A and B be IVF sets defined as follows:
A(a) = [0.1, 0.6], A(b) = [0.3, 0.7]
and
B(a) = [0.2, 0.5], B(b) = [0.7, 0.3].
Consider M = {0, A, B, 1} as an IVF minimal structure on X. Then A, B are IVF mα-open but C = A ∩ B, that is, C(a) = [0.1, 0.5] and C(b) = [0.3, 0.3] is not IVF mα- open.
Definition 2.7. Let (X, M) be an IVF minimal space. For A ∈ IV F (X), the α-closure and the α-interior, denoted by αmC(A) and αmI(A) in (X, M), respectively, are de- fined as the following:
αmC(A) = ∩{F ∈ IV F (X) : A ⊆ F, F is IVF mα-closed in X},
αmI(A) = ∪{U ∈ IV F (X) : U ⊆ A, U is IVF mα-open in X}.
Theorem 2.8. Let (X, M) be an IVF minimal space and A, B ∈ IV F (X). Then
(1) αmI(A) ⊆ A.
(2) If A ⊆ B, then αmI(A) ⊆ αmI(B).
(3) A is IVF mα-open iff αmI(A) = A.
(4) αmI(αmI(A)) = αmI(A).
(5) αmC(X −A) = X −αmI(A) and αmI(X −A) = X − αmC(A).
Proof. (1), (2) Obvious.
(3) It follows from Theorem 2.4.
(4) It follows from (3).
(5) For A ∈ IV F (X), X − αmI(A)
= X − ∪{U : U ⊆ A, U is IVF mα-open in X}
= ∩{X − U : U ⊆ A, U is IVF mα-open}
= ∩{X − U : X − A ⊆ X − U, U is IVF mα-open}
= αmC(X − A).
Similarly, we have αmI(X −A) = X −αmC(A).
Theorem 2.9. Let (X, M) be an IVF minimal space and A, B, F ∈ IV F (X). Then
(1) A ⊆ αmC(A).
(2) If A ⊆ B, then αmC(A) ⊆ αmC(B).
(3) F is IVF mα-closed iff αmC(F ) = F . (4) αmC(αmC(A)) = αmC(A).
Proof. It is similar to the proof of Theorem 2.8.
Theorem 2.10. Let (X, M) be an IVF minimal space and A, U, V ∈ IV F (X). Then
(1) Mx∈αmC(A) if and only if A ∩ V 6= 0 for everye IVF mα-open set V containing Mx.
(2) Mx∈αmI(A) if and only if there exists an IVF mα-e open set U such that U ⊆ A.
Proof. (1) Suppose there is an IVF mα-open set V con- taining Mxsuch that A ∩ V = 0. Then X − V is an IVF mα-closed set such that A ⊆ X −V and Mxe/∈X −V . This implies Mxe/∈αmC(A).
The reverse relation is obvious.
(2) Obvious.
Definition 2.11. Let (X, MX) and (Y, MY) be two IVF minimal spaces. Then f : X → Y is said to be interval- valued fuzzy mα-continuous (simply, IVF mα-continuous) if for each IVF point Mxand each IVF m-open set V con- taining f (Mx), there exists an IVF mα-open set U con- taining Mxsuch that f (U ) ⊆ V .
Every IVF m-continuous mapping is IVF mα- continuous but the converse may not be true as shown in the next example.
IVF m-continuity ⇒ IVF mα-continuity
Example 2.12. Let X = {a, b}, let A and B be IVF sets defined as in Example 2.6. Consider M = {0, A, B, 1}
and N = {0, A, B, A ∪ B, 1} as IVF minimal structures on X. Consider the identity mapping f : (X, M) → (X, N ).
Note that C is IVF mα-open but it is not IVF m-open in (X, M). Thus f is IVF mα-continuous but it is not IVF m-continuous.
Remark 2.13. Let f : X → Y be an IVF mα-continuous mapping between IVF minimal spaces (X, MX) and (Y, MY). If the IVF minimal structures MX and MY
are IVF topologies on X and Y , respectively, then f is IVF α-continuous [5].
Theorem 2.14. Let f : X → Y be a mapping on IVF min- imal spaces (X, MX) and (Y, MY). Then the following statements are equivalent:
(1) f is IVF mα-continuous.
(2) f−1(V ) is an IVF mα-open set for each IVF m- open set V in Y .
(3) f−1(B) is an IVF mα-closed set for each IVF m- closed set B in Y .
(4) f (αmC(A)) ⊆ mC(f (A)) for A ∈ IV F (X).
(5) αmC(f−1(B)) ⊆ f−1(mC(B)) for B ∈ IV F (Y ).
(6) f−1(mI(B)) ⊆ αmI(f−1(B)) for B ∈ IV F (Y ).
Proof. (1) ⇒ (2) Let V be an IVF m-open set in Y and Mx∈fe −1(V ). By hypothesis, there exists an IVF mα-open set UMx containing Mxsuch that f (UMx) ⊆ V . This im- plies Mx∈Ue Mx⊆ f−1(V ) for all Mx∈fe −1(V ). Hence by Theorem 2.4, f−1(V ) is IVF mα-open.
(2) ⇒ (3) Obvious.
(3) ⇒ (4) For A ∈ IV F (X), f−1(mC(f (A)))
= f−1(∩{F ∈ IV F (Y ) : f (A) ⊆ F and F is IVF m-closed})
= ∩{f−1(F ) ∈ IV F (X) : A ⊆ f−1(F )and F is IVF mα-closed}
⊇ ∩{K ∈ IV F (X) : A ⊆ Kand Kis IVF mα-closed}
= αmC(A).
Hence f (αmC(A)) ⊆ mC(f (A)).
(4) ⇒ (5) For B ∈ IV F (Y ), from (4), it follows f (αmC(f−1(B))) ⊆ mC(f (f−1(B))) ⊆ mC(B).
Hence we get (5).
(5) ⇒ (6) For B ∈ IV F (Y ), from mI(B) = Y − mC(Y − B), it follows:
f−1(mI(B)) = f−1(Y − mC(Y − B))
= X − (f−1(mC(Y − B)))
⊆ X − αmC(f−1(Y − B))
= αmI(f−1(B)).
Hence (6) is obtained.
(6) ⇒ (1) Let Mxbe an IVF point in X and V an IVF m-open set containing f (Mx). Then from (6), it follows Mx∈fe −1(V ) = f−1(mI(V )) ⊆ αmI(f−1(V )). So from Theorem 2.10, there exists an IVF mα-open set U con- taining Mxsuch that Mx∈U ⊆ fe −1(V ). Hence f is IVF mα-continuous.
Theorem 2.15. Let (X, MX) be an IVF minimal space and A ∈ IV F (X). Then
(1) mC(mI(mC(A))) ⊆ mC(mI(mC(αmC(A))))
⊆ αmC(A).
(2) αmI(A) ⊆ mI(mC(mI(αmI(A)))) ⊆ mI(mC(mI(A))).
Proof. (1) For A ∈ IV F (X), by Theorem 2.9, αmC(A) is an IVF mα-closed set. Hence from Lemma 2.3, we have mC(mI(mC(A))) ⊆ mC(mI(mC(αmC(A)))) ⊆ αmC(A).
(2) It is similar to the proof of (1).
Theorem 2.16. Let f : X → Y be a function on IVF min- imal spaces (X, MX) and (Y, MY). Then the following statements are equivalent:
(1) f is IVF mα-continuous.
(2) f−1(V ) ⊆ mI(mC(mI(f−1(V )))) for each IVF m-open set V in Y .
(3) mC(mI(mC(f−1(F )))) ⊆ f−1(F ) for each IVF m-closed set F in Y .
(4) f (mC(mI(mC(A)))) ⊆ mC(f (A)) for A ∈ IV F (X).
(5) mC(mI(mC(f−1(B)))) ⊆ f−1(mC(B)) for B ∈ IV F (Y ).
(6) f−1(mI(B)) ⊆ mI(mC(mI(f−1(B)))) for B ∈ IV F (Y ).
Proof. (1) ⇔ (2) It follows from Theorem 2.14 and defi- nition of IVF mα-open sets.
(1) ⇔ (3) It follows from Theorem 2.14 and Lemma 2.3.
(3) ⇒ (4) Let A ∈ IV F (X). Then since f is IVF mα-continuous, from Theorem 2.14 (4) and Theo- rem 2.15, it follows mC(mI(mC(A))) ⊆ αmC(A) ⊆ f−1(f (αmC(A))) ⊆ f−1(mC(f (A))).
Hence f (mC(mI(mC(A)))) ⊆ mC(f (A)).
(4) ⇒ (5) Obvious.
(5) ⇒ (6) From (5) and Theorem 1.4, it follows f−1(mI(B)) = f−1(Y − mC(Y − B))
= X − (f−1(mC(Y − B)))
⊆ X − mC(mI(mC(f−1(Y − B))))
= mI(mC(mI(f−1(B)))).
Hence, (6) is obtained.
(6) ⇒ (1) Let V an IVF m-open set in Y . Then by (6) and Theorem 1.4, we have f−1(V ) = f−1(mI(V )) ⊆ mI(mC(mI(f−1(V )))). This implies f−1(V ) is an IVF mα-open set. Hence by Theorem 2.14 (2), f is IVF mα- continuous.
Won Keun Min
He received his Ph. D. degree in the Department of Mathematics from Korea University, Seoul, Korea, in 1987. He is currently a professor in the Department of Mathematics, Kangwon National University. His research interests include general to pologyand fuzzy to pology.
E-mail:[email protected]
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Won Keun Min
He received his Ph. D. degree in the Department of Mathematics from Korea University, Seoul, Korea, in 1987. He is currently a professor in the Department of Mathematics, Kangwon National University.
His research interests include general to pologyand fuzzy to pology.
E-mail:[email protected]
Young Ho Yoo
He received his Ph. D. degree in the Department of Statistics from Okayama University, Japan, in 1992. Heiscur-rently a professor in the Department of Statistics, Kangwon National University. His research interests include statistics and fuzzy theory.
E-mail:[email protected]
