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SOMEWHAT FUZZY γ-IRRESOLUTE CONTINUOUS MAPPINGS
SOMEWHAT FUZZY γ-IRRESOLUTE CONTINUOUS MAPPINGS
Journal of Applied Mathematics & Informatics. 2014. Jan, 32(1_2): 203-209
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : March 20, 2013
  • Accepted : August 04, 2013
  • Published : January 28, 2014
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About the Authors
YOUNG BIN IM
JOO SUNG LEE
YUNG DUK CHO

Abstract
We define and characterize a somewhat fuzzy γ -irresolute con-tinuous mapping and a somewhat fuzzy irresolute γ -open mapping on a fuzzy topological space. AMS Mathematics Subject Classification : 54A40.
Keywords
1. Introduction
The concept of fuzzy γ -continuous mappings on a fuzzy topological space was introduced and studied by I. M. Hanafy in [2] . Also, the concept of fuzzy γ -irresolute continuous mappings on a fuzzy topological space were introduced and studied by Y. B. Im et al. in [8] and fuzzy irresolute γ -open mappings on a fuzzy topological space was introduced and studied by Y. B. Im in [3] .
Recently, somewhat fuzzy γ -continuous mappings on a fuzzy topological space were introduced and studied by G. Thangaraj and V. Seenivasan in [9] .
In this paper, we define and characterize a somewhat fuzzy γ -irresolute con-tinuous mapping and a somewhat fuzzy irresolute γ -open mapping which are stronger than a somewhat fuzzy γ -continuous mapping and a somewhat fuzzy γ -open mapping respectively. Besides, some interesting properties of those map-pings are also given.
2. Preliminaries
A fuzzy set μ on a fuzzy topological space X is called fuzzy γ -open if μ ≤ ClInt μ ∨ IntCl μ and μ is called fuzzy γ - closed if μc is a fuzzy γ -open set on X .
A mapping f : X Y is called fuzzy γ - continuous if f −1 ( ν ) is a fuzzy γ -open set on X for any fuzzy open set ν on Y and a mapping f : X Y is called fuzzy γ -open if f ( μ ) is a fuzzy γ -open set on Y for any fuzzy open set μ on X . It is clear that every fuzzy continuous mapping is a fuzzy γ -continuous mapping. And every fuzzy open mapping is a fuzzy γ -open mapping from the above definitions. But the converses are not true in general [2] .
A mapping f : X Y is called fuzzy γ-irresolute continuous if f −1 ( ν ) is a fuzzy γ -open set on X for any fuzzy γ -open set ν on Y and a mapping f : X Y is called fuzzy irresolute γ-open if f ( μ ) is a fuzzy γ -open set on Y for any fuzzy γ -open set μ on X . It is clear that every fuzzy γ -irresolute continuous mapping is a fuzzy γ -continuous mapping. And every fuzzy irresolute γ -open mapping is a fuzzy open mapping from the above definitions. But the converses are not true in general [8] and [3] .
A mapping f : X Y is called somewhat fuzzy γ - continuous if there exists a fuzzy γ -open set μ ≠ 0 X on X such that μ f −1 ( ν )≠ 0 X for any fuzzy open set ν on Y . It is clear that every fuzzy γ -continuous mapping is a somewhat fuzzy γ -continuous mapping. But the converse is not true in general.
A mapping f : X Y is called somewhat fuzzy γ - open if there exists a fuzzy γ -open set ν ≠ 0 Y on Y such that ν f ( μ )≠0 Y for any fuzzy open set μ on X . Every fuzzy open mapping is a somewhat fuzzy γ -open mapping but the converse is not true in general [9] .
3. Somewhat fuzzy γ-irresolute continuous mappings
In this section, we introduce a somewhat fuzzy γ -irresolute continuous map-ping and a somewhat fuzzy irresolute γ -open mapping which are stronger than a somewhat fuzzy γ -continuous mapping and a somewhat fuzzy γ -open mapping respectively. And we characterize a somewhat fuzzy γ -irresolute continuous mapping and a somewhat fuzzy irresolute γ -open mapping.
Definition 3.1. A mapping f : X Y is called somewhat fuzzy γ -irresolute continuous if there exists a fuzzy γ -open set μ ≠ 0 X on X such that μ f −1 ( ν ) for any fuzzy γ -open set ν ≠ 0 Y on Y .
It is clear that every fuzzy γ -irresolute continuous mapping is a somewhat fuzzy γ -irresolute continuous mapping. And every somewhat fuzzy γ -irresolute continuous mapping is a fuzzy γ -continuous mapping from the above definitions. But the converses are not true in general as the following examples show.
Example 3.2. Let μ 1 , μ 2 and μ 3 be fuzzy sets on X = { a , b , c } and let ν 1 , ν 2 and ν 3 be fuzzy sets on Y = { x , y , z } with
  • μ1(a) = 0.1,μ1(b) = 0.1,μ1(c) = 0.1,
  • μ2(a) = 0.2,μ2(b) = 0.2,μ2(c) = 0.2,
  • μ3(a) = 0.5,μ3(b) = 0.5,μ3(c) = 0.5 and
  • ν1(x) = 0.3,ν1(y) = 0.2,ν1(z) = 0.3,
  • ν2(x) = 0.5,ν2(y) = 0.5,ν2(z) = 0.5,
  • ν3(x) = 0.5,ν3(y) = 0.2,ν3(z) = 0.5.
Let
PPT Slide
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be fuzzy topologies on X and let τ = {0 Y , ν 1 , ν 2 , 1 Y } be fuzzy topologies on Y . Consider the mapping f : ( X , τ ) → ( Y , τ ) defined by f ( a ) = y , f ( b ) = y and f ( c ) = y . Then we have μ 1 f −1 ( ν 1 ) = μ 2 , f −1 ( ν 2 ) = μ 3 and μ 1 f −1 ( ν 3 ) = μ 2 . Since μ 1 is a fuzzy γ -open set on ( X , τ ), f is somewhat fuzzy γ -irresolute continuous. But f −1 ( ν 1 ) = μ 2 and f −1 ( ν 3 ) = μ 2 are not fuzzy γ -open sets on ( X , τ ). Hence f is not a fuzzy γ -irresolute continuous mapping.
Example 3.3. Let μ 1 , μ 2 and μ 3 be fuzzy sets on X = { a , b , c } and let ν 1 , ν 2 and ν 3 be fuzzy sets on Y = { x , y , z } with
  • μ1(a) = 0.2,μ1(b) = 0.2,μ1(c) = 0.2,
  • μ2(a) = 0.5,μ2(b) = 0.5,μ2(c) = 0.5, and
  • ν1(x) = 0.3,ν1(y) = 0.2,ν1(z) = 0.3,
  • ν2(x) = 0.5,ν2(y) = 0.5,ν2(z) = 0.5.
Let
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be fuzzy topologies on X and let τ = {0 Y , ν 2 , 1 Y } be fuzzy topologies on Y . Consider the mapping f : ( X , τ ) → ( Y , τ ) defined by f ( a ) = y , f ( b ) = y and f ( c ) = y . Since f −1 ( ν 2 ) = μ 2 is fuzzy γ -open sets on ( X , τ ), f is fuzzy γ -continuous. But the inverse images 0 X f −1 ( ν 1 ) = μ 1 of a fuzzy γ -open set ν 1 on ( Y , τ ) is not fuzzy γ -open on ( X , τ ). Hence f is not a fuzzy somewhat γ -irresolute continuous mapping.
Definition 3.4 ( [9] ). A fuzzy set μ on a fuzzy topological space X is called fuzzy γ -dense if there exists no fuzzy γ -closed set ν such that μ < ν < 1.
Theorem 3.5. Let f : X Y be a mapping. Then the following are equivalent:
  • (1) f is somewhat fuzzy γ-irresolute continuous.
  • (2) If ν is a fuzzy γ-closed set of Y such that f−1(ν)≠ 1X,then there exists a fuzzy γ-closed set μ≠ 1Xof X such that f−1(ν) ≤μ.
  • (3) If μ is a fuzzy γ-dense set on X, then f(μ)is a fuzzy γ-dense set on Y.
Proof . (1) implies (2): Let ν be a fuzzy γ -closed set on Y such that f −1 ( ν )≠ 1 X . Then ν c is a fuzzy γ -open set on Y and f −1 ( ν c ) = ( f −1 ( ν )) c ≠ 0 X . Since f is somewhat fuzzy γ -irresolute continuous, there exists a fuzzy γ -open set λ ≠ 0 X on X such that λ f −1 ( ν c ). Let μ = λ c . Then μ ≠ 1 X is fuzzy γ -closed such that f −1 ( ν ) = 1 − f −1 ( ν c ) ≤ 1 − λ = λ c = μ .
(2) implies (3): Let μ be a fuzzy γ -dense set on X and suppose f ( μ ) is not fuzzy γ -dense on Y . Then there exists a fuzzy γ -closed set ν on Y such that f ( μ ) < ν < 1. Since ν < 1 and f −1 ( ν ) ≠ 1 X , there exists a fuzzy γ -closed set δ ≠ 1 X such that μ f −1 ( f ( μ )) < f −1 ( ν ) ≤ δ . This contradicts to the assumption that μ is a fuzzy γ -dense set on X . Hence f ( μ ) is a fuzzy γ -dense set on Y .
(3) implies (1): Let ν ≠ 0 Y be a fuzzy γ -open set on Y and f −1 ( ν ) ≠ 0 X . Suppose there exists no fuzzy γ -open μ ≠ 0 X on X such that μ f −1 ( ν ). Then ( f −1 ( ν )) c is a fuzzy set on X such that there is no fuzzy γ -closed set δ on X with ( f −1 ( ν )) c < δ < 1. In fact, if there exists a fuzzy γ -open set δ c such that δ c f −1 ( ν ), then it is a contradiction. So ( f −1 ( ν )) c is a fuzzy γ -dense set on X . Then f (( f −1 ( ν )) c ) is a fuzzy γ -dense set on Y . But f (( f −1 ( ν )) c ) = f ( f −1 ( ν c ))≠ ν c < 1. This is a contradiction to the fact that f (( f −1 ( ν )) c ) is fuzzy γ -dense on Y . Hence there exists a γ -open set μ ≠ 0 X on X such that μ f −1 ( ν ). Consequently, f is somewhat fuzzy γ -irresolute continuous.
A fuzzy topological space X is product related to a fuzzy topological space Y if for fuzzy sets μ on X and ν on Y whenever
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(in which case( γ c × 1) ∨ (1 × δ c ) ≥ ( μ × ν )) where γ is a fuzzy open set on X and δ is a fuzzy open set on Y , there exists a fuzzy open set γ 1 on X and a fuzzy open set δ 1 on Y such that
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and
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[1] .
Theorem 3.6. Let X 1 be product related to X 2 and Y 1 be product related to Y 2 . Then the product f 1 × f 2 : X 1 × X 2 Y 1 × Y 2 of somewhat fuzzy γ - irresolute continuous mappings f 1 : X 1 Y 1 and f 2 : X 2 Y 2 is also somewhat fuzzy γ - irresolute continuous .
Proof . Let λ = ∨ i,j ( μ i × ν j ) be a fuzzy γ -open set on Y 1 × Y 2 where
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are fuzzy γ -open sets on Y 1 and Y 2 respectively. Then
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Since f 1 is somewhat fuzzy γ -irresolute continuous, there exists a fuzzy γ -open set
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such that
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And, since f 2 is somewhat fuzzy γ -irresolute continuous, there exists a fuzzy γ -open set
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Lager Image
such that
PPT Slide
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Now
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and
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is a fuzzy γ -open set on X 1 × X 2 . Hence
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is a fuzzy γ -open set on X 1 × X 2 such that
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Therefore, f 1 × f 2 is somewhat fuzzy γ -irresolute continuous.
Theorem 3.7. Let f : X Y be a mapping . If the graph g : X X × Y of f is a somewhat fuzzy γ - irresolute continuous mapping , then f is also somewhat fuzzy γ - irresolute continuous .
Proof . Let ν be a fuzzy γ -open set on Y . Then f −1 ( ν ) = 1∧ f −1 ( ν ) = g −1 (1× ν ). Since g is somewhat fuzzy γ -irresolute continuous and 1× ν is a fuzzy γ -open set on X × Y , there exists a fuzzy γ -open set μ ≠ 0 X on X such that μ g −1 (1× ν ) = f −1 ( ν ) ≠ 0 X . Therefore, f is somewhat fuzzy γ -irresolute continuous.
Definition 3.8. A mapping f : X Y is called somewhat fuzzy irresolute γ -open if there exists a fuzzy γ -open set ν ≠ 0 Y on Y such that ν f ( μ ) for any fuzzy γ -open set μ ≠ 0 X on X .
It is clear that every fuzzy irresolute γ -open mapping is a somewhat fuzzy ir-resolute γ -open mapping. And every somewhat fuzzy irresolute γ -open mapping is a fuzzy γ -open mapping. Also, every fuzzy γ -open mapping is a somewhat fuzzy γ -open mapping from the above definitions. But the converses are not true in general as the following examples show.
Example 3.9. Let μ 1 and μ 2 be fuzzy sets on X = { a , b , c } and let ν 1 and ν 2 be fuzzy sets on Y = { x , y , z } with
  • μ1(a) = 0.1,μ1(b) = 0.1,μ1(c) = 0.1,
  • μ2(a) = 0.2,μ2(b) = 0.2,μ2(c) = 0.2 and
  • ν1(x) = 0.0,ν1(y) = 0.1,ν1(z) = 0.0,
  • ν2(x) = 0.0,ν2(y) = 0.2,ν2(z) = 0.0,
  • ν3(x) = 0.0,ν3(y) = 0.8,ν3(z) = 0.0,
  • ν4(x) = 0.0,ν4(y) = 0.9,ν4(z) = 0.0.
Let
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be fuzzy topologies on X and let
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be fuzzy topologies on Y . Consider the mapping f : ( X , τ ) → ( Y , τ ) defined by f ( a ) = y , f ( b ) = y and f ( c ) = y . Since
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Lager Image
and
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f is somewhat fuzzy irresolute γ -open. But f ( μ 2 ) = ν 2 is not a fuzzy γ -open set on ( Y , τ ). Hence f is not a fuzzy irresolute γ -open mapping.
Example 3.10. Let μ 1 , μ 2 and μ 3 be fuzzy sets on X = { a , b , c } and let ν 1 and ν 2 be fuzzy sets on Y = { x , y , z } with
  • μ1(a) = 0.4,μ1(b) = 0.1,μ1(c) = 0.4,
  • μ2(a) = 0.5,μ2(b) = 0.5,μ2(c) = 0.5,
  • μ3(a) = 0.1,μ3(b) = 0.0,μ3(c) = 0.1 and
  • ν1(x) = 0.0,ν1(y) = 0.1,ν1(z) = 0.0,
  • ν2(x) = 0.0,ν2(y) = 0.5,ν2(z) = 0.0.
Let τ = {0 X , μ 1 , μ 2 , 1 X } be fuzzy topologies on X and let τ = {0 Y , ν 2 , 1 Y } be fuzzy topologies on Y . Consider the mapping f : ( X , τ ) → ( Y , τ ) defined by f ( a ) = y , f ( b ) = y and f ( c ) = y . Since f ( μ 1 ) = ν 1 and f ( μ 2 ) = ν 2 are fuzzy γ -open sets on ( Y , τ ), f is fuzzy γ -open. But μ 3 ≠ 0 X is a fuzzy γ -open set on ( X , τ ) and f ( μ 3 ) = 0 Y . Hence f is not a fuzzy somewhat irresolute γ -open mapping.
Example 3.11. Let μ 1 , μ 2 and μ 3 be fuzzy sets on X = { a , b , c } with
  • μ1(a) = 0.1,μ1(b) = 0.1,μ1(c) = 0.1,
  • μ2(a) = 0.2,μ2(b) = 0.2,μ2(c) = 0.2 and
  • μ3(a) = 0.3,μ3(b) = 0.3,μ3(c) = 0.3.
Let
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and τ = {0 X ; μ 1 , μ 3 , 1 X }be fuzzy topologies on X . Consider the identity mapping iX : ( X , τ ) → ( X , τ ). We have
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Since μ 3 is a fuzzy γ -open set on ( X , τ ), iX is somewhat fuzzy γ -open. But
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is not a fuzzy γ -open set on ( X ; τ ). Hence iX is not a fuzzy γ -open mapping.
Theorem 3.12. Let f : X Y be a bijection . Then the following are equivalent:
(1) f is somewhat fuzzy irresolute γ - open .
(2) If μ is a fuzzy γ - closed set on X such that f ( μ ) ≠ 1 Y , then there exists a fuzzy γ - closed set ν ≠ 1 Y on Y such that f ( μ ) < ν .
Proof . (1) implies (2): Let μ be a fuzzy γ -closed set on X such that f ( μ ) ≠ 1 Y . Since f is bijective and μ c is a fuzzy γ -open set on X , f ( μ c ) = ( f ( μ )) c ≠ 0 Y . And, since f is somewhat fuzzy irresolute γ -open, there exists a γ -open set δ ≠ 0 Y on Y such that δ < f ( μ c ) = ( f ( μ )) c . Consequently, f ( μ ) < δ c = ν ≠ 1 Y and ν is a fuzzy γ -closed set on Y .
(2) implies (1): Let μ be a fuzzy γ -open set on X such that f ( μ ) ≠ 0 Y . Then μ c is a fuzzy γ -closed set on X and f ( μ c ) ≠ 1 Y . Hence there exists a fuzzy γ -closed set ν ≠ 1 Y on Y such that f ( μ c ) < ν . Since f is bijective, f ( μ c ) = ( f ( μ )) c < ν . Hence ν c < f ( μ ) and ν c ≠ 0 X is a fuzzy γ -open set on Y . Therefore, f is somewhat fuzzy irresolute γ -open.
Theorem 3.13. Let f : X Y be a surjection . Then the following are equiva-lent:
(1) f is somewhat fuzzy irresolute γ - open .
(2) If ν is a fuzzy γ - dense set on Y , then f −1 ( ν ) is a fuzzy γ - dense set on X .
Proof . (1) implies (2): Let ν be a fuzzy γ -dense set on Y . Suppose f −1 ( ν ) is not fuzzy γ -dense on X . Then there exists a fuzzy γ -closed set μ on X such that f −1 ( ν ) < μ < 1. Since f is somewhat fuzzy irresolute γ -open and μ c is a fuzzy γ -open set on X , there exists a fuzzy γ -open set δ ≠ 0 Y on Y such that δ f (Int μ c ) ≤ f ( μ c ). Since f is surjective, δ f ( μ c ) < f ( f −1 ( ν c )) = ν c . Thus there exists a γ -closed set δ c on Y such that ν < δ c < 1. This is a contradiction. Hence f −1 ( ν ) is fuzzy γ -dense on X .
(2) implies (1): Let μ be a fuzzy open set on X and f ( μ ) ≠ 0 Y . Suppose there exists no fuzzy γ -open ν ≠ 0 Y on Y such that ν f ( μ ). Then ( f ( μ )) c is a fuzzy set on Y such that there exists no fuzzy γ -closed set δ on Y with ( f ( μ )) c < δ < 1. This means that ( f ( μ )) c is fuzzy γ -dense on Y . Thus f −1 ((f( μ )) c ) is fuzzy γ -dense on X . But f −1 (( f ( μ )) c ) = ( f −1 ( f ( μ ))) c μ c < 1. This is a contradiction to the fact that f −1 (( f ( ν )) c is fuzzy γ -dense on X . Hence there exists a γ -open set ν ≠ 0 Y on Y such that ν f ( μ ). Therefore, f is somewhat fuzzy irresolute γ -open.
BIO
Young Bin Im received his B.S. and Ph.D. at Dongguk University under the direction of Professor K. D. Park. Since 2009 he has been a professor at Dongguk University. His research interests are fuzzy topological space and fuzzy matrix.
Faculty of General Education, Dongguk University, Seoul 100-715, Korea.
e-mail : philpen@dongguk.edu
Joo Sung Lee received his B.S. from Dongguk University and Ph.D. at University of Florida under the direction of Professor B. Brechner. Since 1995 he has been a professor at Dongguk University. His research interests are topological dynamics and fuzzy theory.
Department of Mathematics, Dongguk University, Seoul 100-715, Korea.
e-mail : jsl@dongguk.edu
Yung Duk Cho received his B.S. and Ph.D. at Dongguk University under the direction of Professor J. C. Lee. Since 2008 he has been a professor at Dongguk University. His research interests are fuzzy category and fuzzy algebra.
Faculty of General Education, Dongguk University, Seoul 100-715, Korea.
e-mail : joyd@dongguk.edu
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