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WEAKLY θ-CLOSED FUNCTIONS BETWEEN FUZZY TOPOLOGICAL SPACES

Miguel Caldas, Govindappa Navalagi and Ratnesh Saraf

Abstract. In this paper we introduce and characterize fuzzy weakly θ-closed functions between fuzzy topological spaces and also study these functions in relation to some other types of already known functions.

1. Introduction and preliminaries

The concept of fuzzy sets was introduced by L.A. Zadeh in his classical paper [22]. After the discovery of fuzzy subsets, much attention has been paid to generalize the basic concepts of classical topology in fuzzy setting and thus a modern theory of fuzzy topology is developed. The notion of fuzzy subsets naturally plays a significant role in the study of fuzzy topology which was introduced by C.L. Chang [5]. Ming and Ming [14] introduced the concepts of quasi-coincidence and q-neighbourhoods by which the extensions of functions in fuzzy setting can very interestingly and effectively be carried out. In [18] D.A. Rose defined weakly closed functions in topological spaces. J.H. Park, Y.B. Park and J.S. Park introduced in [17] the notion of weakly closed functions between fuzzy topological spaces. In this paper we introduce and discuss the concept of fuzzy θ-closed function and fuzzy weakly θ-closed function. Also, we obtain several characterizations of these functions comparing with other already existing known functions. Here it is seen that fuzzy θ-closedness implies fuzzy weakly θ-closedness but not conversely. But under a certain condition the converse is also true.

Throughout this paper by (X, τ) or simply by X we mean a fuzzy topological space (fts, shortly) due to Chang [5]. A fuzzy point in X with support x ∈ X and value p (0 < p ⩽ 1) is denoted by x_{p}. Two fuzzy sets λ and β are said to be quasi-coincident (q-coincident, shortly), denoted by λqβ, if there exists x ∈ X such that λ(x)+β(x) > 1 [14] and by q̄ we denote “is not q-coincident”. It is known [14] that λ ⩽ β if and only if λq̄(1 − β). A fuzzy set λ is said to be q-neighbourhood (q-nbd) of x_{p} if there is a fuzzy open set μ such that x_{p}qμ ⩽ λ. The interior, closure and the complement of a fuzzy set λ ∈ X are denoted by Int(λ), Cl(λ) and 1 − λ = λ^{c}, respectively.

AMS Subject Classification: 54 A 40, 54 C 10, 54 D 10

Keywords and phrases: Fuzzy θ-open sets, fuzzy weakly open, fuzzy θ-continuous, fuzzy weakly θ-continuous, fuzzy extremally disconnected spaces, fuzzy almost compact spaces.

For definitions and results not explained in this paper, the reader is referred to [1,5,8,9,11,15,17,18] assuming them to be well known.

Definiton 1.1. A fuzzy set λ in an fts X is called: fuzzy preopen [4] if λ ⩽ Int(Cl(λ)); fuzzy preclosed [4] if Cl(Int(λ)) ⩽ λ; fuzzy regular open [2] if λ = Int(Cl(λ)); fuzzy regular closed [2] if λ = Cl(Int(λ)); fuzzy α-open [4] if λ ⩽ Int(Cl(Int(λ))); fuzzy α-closed [4] if Cl(Int(Cl(λ))) ⩽ λ.

Definition 1.2. Let f: (X, τ_{1}) → (Y, τ_{2}) be a function from an fts (X, τ_{1}) into an fts (Y, τ_{2}). The function f is called:

(i) fuzzy closed [5] if f(λ) is a fuzzy closed set of Y, for each fuzzy closed set λ ∈ X;

(ii) fuzzy open [5] if f(λ) is a fuzzy open set in Y, for each fuzzy open set λ ∈ X;

(iii) fuzzy preclosed [19] if f(λ) is a fuzzy preclosed set of Y, for each fuzzy closed set λ ∈ X.

Recall that a fuzzy point x_{p} is said to be a fuzzy θ-cluster point of a fuzzy set λ [15], if for every fuzzy open q-nbd μ of x_{p}, Cl(μ) is q-coincident with λ. The set of all fuzzy θ-cluster points of λ is called the fuzzy θ-closure of λ and will be denoted by Cl_{θ}(λ). A fuzzy set λ will be called θ-closed if λ = Cl_{θ}(λ). The complement of a fuzzy θ-closed set is called fuzzy θ-open and the θ-interior of λ denoted by Int_{θ}(λ) is defined as Int_{θ}(λ) = { x_{p} : for some fuzzy open q-nbd, β of x_{p}, Cl(β) ⩽ λ }.

Lemma 1.3. [3] Let λ be a fuzzy set in an fts X. Then:

1) λ is fuzzy θ-open if and only if λ = Int_{θ}(λ);

2) 1 − Int_{θ}(λ) = Cl_{θ}(1 − λ) and Int_{θ}(1 − λ) = 1 − Cl_{θ}(λ);

3) Cl_{θ}(λ) (resp. Int_{θ}(λ)) is a fuzzy closed set (resp. fuzzy open set) but not necessarily is a fuzzy θ-closed set (resp. fuzzy θ-open set).

Result 1.4. (i) It is easy to see that Cl(λ) ⩽ Cl_{θ}(λ) and Int_{θ}(λ) ⩽ Int(λ) for any fuzzy set λ in an fts X.

(ii) For a fuzzy open (resp. fuzzy closed) set λ in an fts X, Cl(λ) = Cl_{θ}(λ) (resp. Int_{θ}(λ) = Int(λ)).

Definition 1.5. [8] An fts (X, τ) is fuzzy T_{2} (FT_{2})-space in the sense of Ganguly and Saha if for two distinct fuzzy points x_{λ}, y_{μ} in X:

(i) x ≠ y implies that x_{λ}, y_{μ} have fuzzy open nbds which are not q-coincident; (ii) x = y and λ < μ (say) imply that x_{λ} has a fuzzy open nbd and y_{μ} has a

fuzzy open q-nbd which are not q-coincident.

Weakly θ-closed functions between fuzzy topological spaces 15

Definition 1.6. [9] An fts (X, τ) is said to be fuzzy normal if:

(i) for two fuzzy closed sets λ and μ in X with different supports, there exist fuzzy open sets η and γ such that λ ⩽ η, μ ⩽ γ and ηq̄γ;

(ii) if x ∈ Supp(λ) and x_{μ}qλ then there exist fuzzy open sets η and γ such that x_{μ}qη, λ ⩽ γ and ηq̄γ.

2. Fuzzy weakly θ-closed functions

Now we define a generalized form of weakly closed and closed functions in fuzzy setting.

Definition 2.1. A function f: (X, τ_{1}) → (Y, τ_{2}) is said to be fuzzy weakly θ-closed if Cl_{θ}(f(Int(β))) ⩽ f(β) for each fuzzy closed set β in X.

Definition 2.2. A function f: (X, τ_{1}) → (Y, τ_{2}) is said to be fuzzy θ-closed if f(β) is a fuzzy θ-closed set in Y for each fuzzy closed set β in X.

Clearly, every fuzzy θ-closed function is a fuzzy weakly θ-closed function since Cl_{θ}(f(Int(β))) ⩽ Cl_{θ}(f(β)) = f(β) for every fuzzy closed subset β of X. Also, every fuzzy θ-closed function is a fuzzy closed function since every fuzzy θ-closed set is a fuzzy closed set, but the converse is not generally true.

Example 2.3. Let X = {x, y, z} and Y = {a, b, c}. Fuzzy sets A and B are defined as: A(x) = 0, A(y) = 0.7, A(z) = 0.4; B(a) = 0, B(b) = 0.2, B(c) = 0.3. Let τ_{1} = {0, A, 1_{X}} and τ_{2} = {0, B, 1_{Y} }. Then the function f: (X, τ_{1}) → (Y, τ_{2}) defined by f(x) = a, f(y) = b and f(z) = c is fuzzy weakly θ-closed but not fuzzy θ-closed.

Example 2.4. Let X = {a, b, c} and Y = {x, y, z}. Fuzzy sets A and H are defined as: A(a) = 0, A(b) = 0.2, A(c) = 0.7; H(x) = 0, H(y) = 0.2, H(z) = 0.2. Let τ = {0, A, 1_{X}} and σ = {0, H, 1_{Y} }. Then the function f: (X, τ) → (Y, σ) defined by f(a) = x, f(b) = y and f(c) = y is fuzzy closed but not fuzzy θ-closed.

Theorem 2.5. For a function f: (X, τ_{1}) → (Y, τ_{2}) the following conditions are equivalent:

(i) f is fuzzy weakly θ-closed; (ii) Cl_{θ}(f(λ)) ⩽ f(Cl(λ)) for every fuzzy open set λ in X.

Proof. (i) → (ii). Let λ be any fuzzy open subset of X. Then Cl_{θ}(f(λ)) = Cl_{θ}(f(Int(λ))) ⩽ Cl_{θ}(f(Int(Cl(λ)))) ⩽ f(Cl(λ)).

(ii) → (i). Let β be any fuzzy closed subset of X. Then, Cl_{θ}(f(Int(β))) ⩽ f(Cl(Int(β))) ⩽ f(Cl(β)) = f(β).

Since the proof of the following theorem is mostly straightforward, it is omitted.

Theorem 2.6. For a function f: (X, τ_{1}) → (Y, τ_{2}) the following conditions are equivalent:

(i) f is fuzzy weakly θ-closed; (ii) Cl_{θ}(f(λ)) ⩽ f(Cl(λ)) for each fuzzy open set λ in X; (iii) Cl_{θ}(f(Int(β))) ⩽ f(β) for each fuzzy closed subset β in X; (iv) Cl_{θ}(f(Int(β))) ⩽ f(β) for each fuzzy preclosed subset β in X;

(v) Cl_{θ}(f(Int(β))) ⩽ f(β) for every fuzzy α-closed subset β in X.

Theorem 2.7. For a function f: (X, τ_{1}) → (Y, τ_{2}) the following conditions are equivalent:

(i) f is fuzzy weakly θ-closed; (ii) Cl_{θ}(f(Int(Cl(λ)))) ⩽ f(Cl(λ)) for each fuzzy set λ in X; (iii) Cl_{θ}(f(Int(Cl_{θ}(λ)))) ⩽ f(Cl_{θ}(λ)) for each fuzzy set λ in X; (iv) Cl_{θ}(f(λ)) ⩽ f(Cl(λ)) for each fuzzy preopen set λ in X.

Proof. It is clear that (i) → (ii) → (iv) → (i) and (i) → (iii). For (iii) → (iv) note that Cl_{θ}(λ) = Cl(λ) for each fuzzy preopen subset λ ∈ X [11].

Theorem 2.8. If Y is a fuzzy regular space, then for a function f: (X, τ_{1}) → (Y, τ_{2}) the following conditions are equivalent:

(i) f is fuzzy weakly θ-closed; (ii) Cl_{θ}(f(λ)) ⩽ f(Cl(λ)) for each fuzzy regular open subset U of X; (iii) for each fuzzy subset β in Y and each fuzzy open set μ in X with f^{−1}(β) ⩽ μ,

there exists a fuzzy θ-open set δ in Y with β ⩽ δ and f^{−1}(δ) ⩽ Cl(μ); (iv) for each fuzzy point y_{p} in Y and each fuzzy open set μ in X with f^{−1}(y_{p}) ⩽ μ,

there exists a fuzzy θ-open set δ in Y containing y_{p}, and f^{−1}(δ) ⩽ Cl(μ). Proof. It is clear that (i) → (ii) and (iii) → (iv). To show that (ii) → (iii),

let β be a fuzzy subset in Y and let μ be fuzzy open in X with f^{−1}(β) ⩽ μ. Then f^{−1}(β)q̄ Cl(1_{X} −Cl(μ)) and consequently βq̄f(Cl(1_{X} −Cl(μ))). Since 1_{X} −Cl(μ) is fuzzy regular open, βq̄ Cl_{θ}(f(1_{X} −Cl(μ))) by (ii). Let δ = 1_{Y} −Cl_{θ}(f(1_{X} −Cl(μ))). Then by Corollary 3.6 of [15] δ is fuzzy θ-open with β ⩽ δ and f^{−1}(δ) ⩽ 1_{X} − f^{−1}(Cl_{θ}(f(1_{X} − Cl(μ)))) ⩽ 1_{X} − f^{−1}f(1_{X} − Cl(μ)) ⩽ Cl(μ).

(iv) → (i). Let β be fuzzy closed in X and let y_{p} ∈ 1_{Y} −f(β). Since f^{−1}(y_{p}) ⩽ 1_{X} − β, there exists a fuzzy θ-open δ in Y with y_{p} ∈ δ and f^{−1}(δ) ⩽ Cl(1_{X} − β) = 1_{X} − Int(β) by (iv). Therefore δq̄f(Int(β)), so that y_{p} ∈ 1_{Y} − Cl_{θ}(f(Int(β))). Thus (iv) → (i).

Note that the fact of Y being fuzzy regular space in Theorem 2.8 is only used in (ii) → (iii).

Theorem 2.9. If f: (X, τ_{1}) → (Y, τ_{2}) is fuzzy weakly θ-closed, then for each fuzzy point y_{p} in Y and each fuzzy open q-nbd μ of f^{−1}(y_{p}) in X, there exists a fuzzy θ-open q-nbd δ of y_{p} in Y such that f^{−1}(δ) ⩽ Cl(μ).

Proof. Let μ be any fuzzy open q-nbd of f^{−1}(y_{p}) in X. Then μ(x) + p > 1 and hence there exists a positive real number α such that μ(x) > α > 1 − p. Then μ is a fuzzy open q-nbd of f^{−1}(y_{α}). By Theorem 2.8(iv) there exists a fuzzy θ-open set δ containing y_{α} in Y such that f^{−1}(δ) ⩽ Cl(μ). Now, δ(y) > α and hence δ(y) > 1 − p. Thus δ is a fuzzy θ-open q-nbd of y_{p}.

Next we investigate conditions under which fuzzy weakly θ-closed functions are fuzzy θ-closed.

Theorem 2.10. If f: (X, τ_{1}) → (Y, τ_{2}) is fuzzy weakly θ-closed and if for each fuzzy closed subset β of X and each fiber f^{−1}(y_{p}) ⩽ 1_{X} −β there exists a fuzzy open q-nbd μ in X such that f^{−1}(y_{p}) ⩽ μ ⩽ Cl(μ) ⩽ 1_{X} − β. Then f is fuzzy θ-closed.

Proof. Let β be any fuzzy closed subset of X and let y_{p} ∈ 1_{Y} − f(β). Then f^{−1}(y_{p})q̄β and hence f^{−1}(y_{p}) ⩽ 1_{X} − β. By hypothesis, there exists a fuzzy open q-nbd μ in X such that f^{−1}(y_{p}) ⩽ μ ⩽ Cl(μ) ⩽ 1_{X} − β. Since f is fuzzy weakly θ-closed, by Theorem 2.9, there exists a fuzzy θ-open q-nbd ν in Y with y_{p} ∈ ν and f^{−1}(ν) ⩽ Cl(μ). Therefore, we obtain f^{−1}(ν)q̄β and hence νq̄f(β), this shows that y_{p} ∉ Cl_{θ}(f(β)). Therefore, f(β) is fuzzy θ-closed in Y and f is fuzzy θ-closed.

Now we give the following

Definition 2.11. Let f: (X, τ_{1}) → (Y, τ_{2}) be a function from an fts (X, τ_{1}) into an fts (Y, τ_{2}). The function f is called:

(i) fuzzy contra-closed if f(λ) is a fuzzy open subset of Y for each fuzzy closed set λ ∈ X.

(ii) fuzzy contra-θ-open if f(λ) is a fuzzy θ-closed subset of Y for each fuzzy open set λ ∈ X.

Theorem 2.12. (i) If f: (X, τ_{1}) → (Y, τ_{2}) is fuzzy preclosed and fuzzy contraclosed, then f is fuzzy weakly θ-closed.

(ii) If f: (X, τ_{1}) → (Y, τ_{2}) is fuzzy contra-θ-open, then f is fuzzy weakly θclosed.

Proof. (i) Let β be a fuzzy closed subset of X. Since f is fuzzy preclosed, Cl_{θ}(Int(f(β))) = Cl(Int(f(β))) ⩽ f(β) and since f is fuzzy contra-closed f(β) is fuzzy open. Therefore Cl_{θ}(f(Int(β))) ⩽ Cl_{θ}(f(β)) = Cl_{θ}(Int(f(β))) ⩽ f(β).

(ii) Let β be a fuzzy closed subset of X. Then, Cl_{θ}(f(Int(β))) = f(Int(β)) ⩽ f(β).

Theorem 2.13. If Y is a fuzzy regular space and if f: (X, τ_{1}) → (Y, τ_{2}) is a fuzzy weakly θ-closed function, then for every fuzzy subset β in Y and every fuzzy open set λ in X with f^{−1}(β) ⩽ λ, there exists a fuzzy θ-closed set δ in Y such that β ⩽ δ and f^{−1}(δ) ⩽ Cl(λ).

Proof. Let β be a fuzzy subset of Y and let λ be a fuzzy open subset of X with f^{−1}(β) ⩽ λ. Put δ = Cl_{θ}(f(Int(Cl(λ)))). By Corollary 3.6 of [15] δ is a fuzzy θ-closed set in Y such that β ⩽ δ since β ⩽ f(λ) ⩽ f(Int(Cl(λ))) ⩽ Cl_{θ}(f(Int(Cl(λ)))) = δ. And since f is fuzzy weakly θ-closed, f^{−1}(δ) ⩽ Cl(λ).

Corollary 2.14. Let Y be a fuzzy regular space. If f: (X, τ_{1}) → (Y, τ_{2}) is fuzzy weakly θ-closed, then for every fuzzy point y_{p} in Y and every fuzzy open set λ in X with f^{−1}(y_{p}) ⩽ λ, there exists a fuzzy θ-closed set δ in Y containing y_{p} such that f^{−1}(δ) ⩽ Cl(λ).

A fuzzy set β in an fts X is fuzzy θ-compact if for each cover Ω of β by fuzzy open q-nbd μ in X, there is a finite family {μ_{1},…, μ_{n}} in Ω such that β ⩽ Int(⋃{Cl(μ_{i}) : i = 1, 2,…, n}).

Theorem 2.15. If f: (X, τ_{1}) → (Y, τ_{2}) is fuzzy weakly θ-closed with all fibers fuzzy θ-closed, then f(β) is fuzzy θ-closed for each fuzzy θ-compact β in X.

Proof. Let β be fuzzy θ-compact and let y_{p} ∈ 1_{Y} − f(β). Then f^{−1}(y_{p})q̄β. Since f^{−1}(y_{p}) is fuzzy θ-closed, then for each x_{p} ∈ β there is a fuzzy open q-nbd μ_{x_{p}} containing x_{p} in X and Cl(μ_{x_{p}})q̄f^{−1}(y_{p}). Clearly Ω = {μ_{x_{p}} : x_{p} ∈ β} is a fuzzy open q-nbd cover of β and since β is fuzzy θ-compact, there is a finite family {μ_{x_{1}},…, μ_{x_{n}}} in Ω such that β ⩽ Int(λ), where λ = ⋃{Cl(μ_{x_{i}}) : i = 1,…, n}. Since f is fuzzy weakly θ-closed, by Theorem 2.9 there exists a fuzzy θ-open q-nbd δ in Y containing y_{p} and f^{−1}(δ) ⩽ Cl(1_{X} − λ) = 1_{X} − Int(λ) ⩽ 1_{X} − β. Therefore δq̄f(β). Thus y_{p} ∈ 1_{Y} − Cl_{θ}(f(β)). This shows that f(β) is fuzzy θ-closed.

Two non-empty fuzzy subsets λ and β in X are strongly fuzzy separated [1] if there exist fuzzy open sets μ and ν in X with λ ⩽ μ and β ⩽ ν and Cl(μ)q̄ Cl(ν). If λ and β are fuzzy singleton sets we may speak of fuzzy points being strongly fuzzy separated. We will use the fact that in a fuzzy normal space [9], disjoint fuzzy closed sets are strongly fuzzy separated.

Theorem 2.16. If f: (X, τ_{1}) → (Y, τ_{2}) is fuzzy weakly θ-closed surjection and all pairs of disjoint fibers are strongly fuzzy separated, then Y is fuzzy T_{2}.

Proof. Let y_{p} and y_{q} be two fuzzy points in Y. Let μ and ν be fuzzy open sets in X such that f^{−1}(y_{p}) ⩽ μ and f^{−1}(y_{q}) ⩽ ν with Cl(μ)q̄ Cl(ν). By fuzzy weak θ-closedness (Theorem 2.8(iv)) there are fuzzy θ-open sets λ and β in Y such that y_{p} ⩽ λ and y_{q} ⩽ β, f^{−1}(λ) ⩽ Cl(μ) and f^{−1}(β) ⩽ Cl(ν). Therefore λq̄β, because Cl(μ)q̄ Cl(ν) and f is surjective. Then Y is fuzzy T_{2}.

Corollary 2.17. If f: (X, τ_{1}) → (Y, τ_{2}) is weakly fuzzy θ-closed surjection with all fuzzy closed fibers and X is fuzzy normal, then Y is fuzzy T_{2}.

Definition 2.18. A family {λ_{α} : α ∈ Ω} of fuzzy open subsets of a fuzzy topological space (X, τ) is an open cover if ⋃{λ_{α} : α ∈ Ω} = X. An fts X is said to be fuzzy almost compact [12,6] (resp. fuzzy c-compact) if every fuzzy open cover (resp. fuzzy closed cover) contains a finite subfamily {λ_{α_{i}} : i = 1, 2,…, n} such that X = ⋃_{i=1}^{n} Cl(λ_{α_{i}}). A fuzzy subset λ of X is fuzzy almost compact relative to X (resp. fuzzy c-compact relative to X) if every cover of λ by fuzzy open (resp. fuzzy closed) sets of X has a finite subfamily whose fuzzy closures cover X.

Recall that an fts (X, τ) is said to be fuzzy extremally disconnected if the closure of every fuzzy open set of X is fuzzy open in X [11].

Lemma 2.19. [21] A function f: (X, τ_{1}) → (Y, τ_{2}) is fuzzy open if and only if for each fuzzy subset β in Y, f^{−1}(Cl(β)) ⩽ Cl(f^{−1}(β)).

Theorem 2.20. Let f: (X, τ_{1}) → (Y, τ_{2}) be a fuzzy θ-open and fuzzy weakly θ-closed function from a fuzzy extremally disconnected space X into a fuzzy regular space Y such that f^{−1}(y_{p}) is fuzzy almost compact relative to X for each fuzzy point y_{p} in Y. If λ is fuzzy c-compact relative to Y, then f^{−1}(λ) is fuzzy almost compact.

Proof. Let {ν_{β} : β ∈ I} be a fuzzy open cover of f^{−1}(λ). Then for each y_{p} ∈ λ ∩ f(X), f^{−1}(y_{p}) ⩽ ⋃{Cl(ν_{β}) : β ∈ I(y_{p})} = δ_{y_{p}} for some finite subfamily I(y_{p}) of I. Since X is fuzzy extremally disconnected, each Cl(ν_{β}) is fuzzy open, hence δ_{y_{p}} is fuzzy open in X. So by Corollary 2.14, there exists a fuzzy θ-closed set μ_{y_{p}} containing y_{p} such that f^{−1}(μ_{y_{p}}) ⩽ Cl(δ_{y_{p}}). Then, {μ_{y_{p}} : y_{p} ∈ λ∩f(X)}∪{1_{Y} − f(X)} is a fuzzy θ-closed cover of λ, λ ⩽ ⋃{Cl(μ_{y_{p}}) : y_{p} ∈ K} ∪ {Cl(1_{Y} − f(X))} for some finite fuzzy subset K of λ ∩ f(X). Hence by Lemma 2.19, f^{−1}(λ) ⩽ ⋃{f^{−1}(Cl(μ_{y_{p}})) : y_{p} ∈ K} ∪ {f^{−1}(Cl(1_{Y} − f(X)))} ⩽ ⋃{Cl(f^{−1}(μ_{y_{p}})) : y_{p} ∈ K} ∪ {Cl(f^{−1}(1_{Y} − f(X)))} ⩽ {Cl(f^{−1}(μ_{y_{p}})) : y_{p} ∈ K}, so f^{−1}(λ) ⩽ ⋃{Cl(ν_{β}) : β ∈ I(y_{p}), y_{p} ∈ K}. Therefore f^{−1}(λ) is fuzzy almost compact.

Corollary 2.21. Let f: (X, τ_{1}) → (Y, τ_{2}) be as in Theorem 2.20. If Y is c-compact, then X is almost compact.

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(received 20.10.2001)

M. Caldas, Departamento de Matematica Aplicada, Universidade Federal Fluminense, Rua Mario Santos Braga, s/n, CEP: 24020-140, Niteroi, RJ Brasil E-mail: gmamccs@vm.uff.br G. B. Navalagi, Department of Mathematics, KLE Societys, G. H. College, Haveri–581110, Karnataka, India E-mail: gnavalagi@hotmail.com R. K. Saraf, Department of Mathematics, Govt. K.N.G. College, Damoh–470661, M.P., India