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Transcript
Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014
Some kinds of fuzzy connected and fuzzy
continuous functions
Hanan Ali Hussein
Dept.of Math. College of Education for Girls Kufa University
[email protected]
Abstract
The main aim of this paper to study new classes of fuzzy connected and fuzzy C-continuous
functions , for this aim ,the nation of fuzzy open, fuzzy connected, fuzzy compact set, and fuzzy
connected space introduced , and we shall study the relationship between fuzzy continuous function
and fuzzy connected function .
Keywords : o-connected function ,fuzz m –connected function,fuzzy weakly connected.
‫الخالصة‬
‫لددث الف اياددة‬
‫اساساددن هد لددثا الدرددة لد ف اياددة لد ل افمددف هد الددف ال اله صددسة ردددسابسا الددف ال الهاد هي ردددسابسا ل ددي‬
‫الهدف‬
‫لس للس هفه م الهاه عدة الهف ردة رددسابسا الهاه عدة اله صدسة رددسابسا الهاه عدة الهيص صدة رددسابسا نهدس لس للدس النالادة ادم الدف ال‬
. ‫الها هي ردسابسا الف ال اله صسة ردسابسا‬
.‫الفالة اله صسة ردسابس الفالة الها هي ردسابس‬: ‫الكلمات المفتاحية‬
Introduction
With the emergence of the fundamental paper [L.A.Zadeh, 1965] by Zadeh in
1965 number of papers have appeared in literature featuring the application of fuzzy
sets to pattern recognition ,decision problems ,function approximation, system theory,
fuzzy logic, fuzzy algorithms, fuzzy automata, fuzzy grammars, fuzzy languages,
fuzzy algebras, fuzzy topology, etc. .In this note, our interests are in the study of
certain concepts in fuzzy topology.
The concepts of continuity, compactness in the context of a fuzzy topological
space are well known [C.V.Negoita and D.A.Raleescu,1975].In this paper the
concept of fuzzy connected function is introduced and some kinds of fuzzy
continuous function are studied. Throughout this paper, simply by X and Y we shall
denote fuzzy topological spaces ( X , T1 ) and (Y , T2 ) and f : X  Y will mean that f is a
fuzzy function from ( X , T1 ) to (Y , T2 ) . This paper includes three sections, in the first
section, we recall the concepts of fuzzy sets and some properties , in the second
section, we have dealt with the concepts fuzzy connected ,fuzzy O-connected, fuzzy
M- connected and fuzzy L-connected functions, and studied the relationship between
their functions . Finally ,in the third section we have discussed the concepts fuzzy Ccontinuous, fuzzy weakly continuous and fuzzy almost continuous, and proved that
every fuzzy continuous function is fuzzy connected functions .
1.Prilimeries
1.1 Definition [S.Dang,A.Behra and S.Nanda, 1994]
A fuzzy point x in X is a fuzzy set defined as follows :
ify  x

x ( y)  
yx
0
Where 0    1;  is called its value and x is support of x .
1.2 Definition[M.H.Rashid and D.M.Ali,2008]
A fuzzy point x is said to belong to a fuzzy set A in X ( denoted by: x  A ) if
and only if   A(x ) ,for some x  X .
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1.3 Definition[S.Dang,A.Behra and S.Nanda, 1994]
Let A and B are fuzzy sets in X . Then A  B if and only if x  B for all x  A .
1.4 Definition[D.H.Foster,1979]
Let A and B are fuzzy sets in X ,then :
1) A  B if and only if A( x)  B( x) x  X .
2) A  B if and only if A( x)  B( x) x  X .
3) Z  A  B if and only if Z ( x)  min{ A( x), B( x)} x  X .
4) D  A  B if and only if D( x)  max{ A( x), B( x)} x  X .
5) E  Ac ( The complement of A )if and only if E ( x)  1  A( x) x  X .
1.5 Theorem[A.B.Saeid,2006]
Let X and Y be two fuzzy topological spaces and let f : X  Y be a
function ,let { Ai }iI , {B j } jJ be families of fuzzy sets in X and Y respectively ,then :
1) f (  Ai )   f ( Ai ) .
iI
iI
2) f (  Ai )   f ( Ai ) .
iI
iI
1
3) f (  B j )   f 1 ( B j ) .
jJ
4) f
1
jJ
(  Bj )   f
jJ
jJ
1
(B j ) .
1.6 Theorem[S.M.Al-Khafaji,2010]
Let X and Y be two fuzzy topological spaces and let f : X  Y be a function,
then the following statements are holds :
1) if B1  B2 ,then f 1 ( B1 )  f 1 ( B2 ) , B1 , B2 are fuzzy sets in Y .
2) if A1  A2 ,then f ( A1 )  f ( A2 ) , A1 , A2 are fuzzy sets in X .
3) A  f 1 ( f ( A)) , A is fuzzy set in X .
4) if f : X  Y is an injective function ,then f 1 ( f ( A))  A .
5) f ( f 1 ( B))  B , B is fuzzy set in Y .
6) if f : X  Y is an serjective function, then f ( f 1 ( B))  B .
7) if A is a fuzzy set in X ,and B is fuzzy set in Y ,then f ( A)  B if and only if
A  f 1 ( B) .
1.7 Definition[C.L.Change,1968]
Let A be a fuzzy set in X ,then :
1) The union of all fuzzy open sets contained in A is called the fuzzy interior of A and
denoted by A0 .
i.e; A0  {B : B  A, B  T } .
2) The intersection of all fuzzy closed sets containing A is called the fuzzy closure of
A ,and denoted by A .i.e; A  {B : A  B, B c  T } .
1.8 Remarks[C.L.Change,1968]
1) The interior of a fuzzy set A is the largest fuzzy open set contained in A , and
trivially a fuzzy set is fuzzy open set if and only if A  A0 .
2)The closure of a fuzzy set A is the smallest fuzzy closed set containing A , and
trivially a fuzzy set A is fuzzy closed if and only if A  A .
1.9 Theorem[X.Tang,2004]
Let A and B are two fuzzy sets in X ,then :
1) 0 X  0 X ,1X  1X .
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Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014
2) A0  A  A .
3) if A  B, then A  B .
4) if A  B, then A0  B 0 .
1.10 Definition[S.Carlson,2005]
Let f be a function from X to Y ,and let B be a fuzzy set in Y ,then the inverse
image of B under f is the fuzzy set f 1 ( B) in X with membership function defined
by the rule : f 1 ( B)( x)  B( f ( x)) for x  X .(i.e; f 1 ( B)  B  f )
For a fuzzy set A in X ,the image of A under f is the fuzzy set f ( A) in Y with
membership function f ( A)( y ) , y  Y defined by :

f ( A)( y )  sup A( x )
 xf -1 ( y )
0
if f
1
( y ) is not empty
otherewise
1
where f ( y)  {x : f ( x)  y} .
1.11 Definition[D.H.Foster,1979]
The union (respectively intersection ) of the fuzzy sets Ai is defined by :
iI
( Ai )( x)  sup{ Ai ( x) : i  I }
iI
( Ai )( x)  inf{ Ai ( x) : i  I }
iI
11.2 Definition[K.S,Raja Sethupathy and S.Lakshmivarahan, 1977]
A fuzzy topological space X is said to be connected if X can not be represented
as the union of two non-empty disjoint open fuzzy sets on X ,otherwise X is called
disconnected space .
1.13 Remark[K.S,Raja Sethupathy and S.Lakshmivarahan, 1977]
If 1X  A  B where A  B  0 X , A and B are non-empty fuzzy open sets of
X , then they are complements to each other ,and hence both are fuzzy open and
fuzzy closed set .
1.14 Definition[A.M.Zahran,2000]
A family  of fuzzy sets is called a cover of a fuzzy set A if and only if
A  {Bi : Bi  } ,and it is called a fuzzy open cover if each member Bi is a fuzzy
open sets .A sub cover of  is a sub family of  which is also a cover of A .
1.15 Definition[D.H.Foster,1979]
Let A be a fuzzy set in a fuzzy topological space X .Then A is said to be a fuzzy
compact set if for every fuzzy open cover of A has a finite sub cover of A . Let
A  X , then X is called a fuzzy compact space ,that is Ai  T for every i  I and
 A  1 X ,then there are finitely many indices i1, i2 ,..., in  I such that
iI i
 A  1X .
ijI ij
2. Some kinds of fuzzy connected functions .
2.1 Definition[K.S,Raja Sethupathy and S.Lakshmivarahan, 1977]
A function f : X  Y is said to be fuzzy connected if and only if f (W ) is fuzzy
connected set in Y for each W is fuzzy connected set in X ,otherwise f is called
fuzzy disconnected function .
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2.2 Definition
A function f : X  Y is said to be fuzzy O-connected function if and only if
f (W ) is fuzzy connected set in Y for each W is fuzzy open and fuzzy connected set in
X.
2.3 Proposition
Every fuzzy connected function is fuzzy O-connected function .
Proof:Let f : X  Y be a fuzzy connected function , and let W be a fuzzy open and fuzzy
connected set in X .
Since W is connected set in X , and since f is connected function .
Then f (W ) is connected set in Y .
Hence f is fuzzy O-connected function .□
The converse of the above proposition is not true in general as shown in the
following example .
2.4 Example
Let X  {a, b, c} , Y  {x, y, z} and T1  {0 X ,1X },T2  {0Y ,1Y , B1 , B2 } be two fuzzy
X and Y respectively, such that B1 : Y  [0,1] defined by
topologies on
and
let
defined
B1 ( x)  0.1 , B1 ( y)  0.5
B1 ( z)  0.2
B2 : Y  [0,1]
be B2 ( x)  0.2 , B2 ( y)  0.3
B2 ( z)  0.3 .
And let f : X  Y be a function defined by f (a)  x , f (b)  y , f (c)  z
Then f is fuzzy O-connected, because 0 X ,1X are all fuzzy open and fuzzy connected
sets in X , such that
f (0 X )  0Y is fuzzy connected set in Y .
And f (1X )  1Y is fuzzy connected set in Y .
But f is fuzzy disconnected because a fuzzy set A : X  [0,1] which is defined by
A(a)  0.2
, A(b)  0.5 A(c)  0.3 is
fuzzy
connected
set
,but
f ( A)  {x0.2 , y0.5 , z 0.3 } is fuzzy disconnected ,since f ( A)  B1  B2 and
f ( A) B1 and f ( A)
B2 .
2.5 Definition
A function f : X  Y is said to be fuzzy L-connected function if and only if
f (W ) is fuzzy connected set in Y for each W is fuzzy closed and fuzzy connected set
in X .
2.6 Proposition
Every fuzzy connected function is fuzzy L-connected function .
Proof:Let f : X  Y be a fuzzy connected function , and let W be a fuzzy closed and fuzzy
connected set in X .
Since f is fuzzy connected function and W is fuzzy connected set .
Then f (W ) is fuzzy connected set in Y .
Then f is fuzzy L-connected function . □
The converse of the above proposition is not true in general as the following
example .
2.7 Example
Let X  {a, b} , Y  {x, y} and T1  {0 X ,1X },T2  {0Y ,1Y , C1 , C2 } be two fuzzy
topologies on X and Y respectively, such that C1 : Y  [0,1] defined by
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Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014
and let C2 : Y  [0,1] defined by
C1 ( x)  0.4 , C1 ( y)  0.9
C2 ( x)  0.6 , C2 ( y)  0.7 .
And let f : X  Y be a function defined by f (a)  x , f (b)  y
Then f is fuzzy L-connected, because 0 X ,1X are all fuzzy closed and fuzzy
connected sets in X , such that
f (0 X )  0Y is fuzzy connected set in Y .
And f (1X )  1Y is fuzzy connected set in Y .
But f is fuzzy disconnected because a fuzzy set A : X  [0,1] which is defined by
A(a)  0.6
, A(b)  0.9 is fuzzy connected set ,but f ( A) is fuzzy
disconnected ,since f ( A)  C1  C2 and f ( A)
C1 and f ( A) C2 .
2.8 Remarks
1) not every fuzzy L-connected function is fuzzy O-connected , and the converse is
true .
2) not every fuzzy L-connected and fuzzy O-connected function is fuzzy connected .
2.9 Definition
A function f : X  Y is said to be fuzzy M-connected function if and only if
f (W ) is fuzzy connected set in Y for each W is fuzzy compact and fuzzy connected
set in X .
2.10 Proposition
Every fuzzy connected function is fuzzy M-connected function .
Proof:Let f : X  Y be a fuzzy connected function , and let W be a fuzzy compact and
fuzzy connected set in X .
Since f is fuzzy connected function .
Then f (W ) is fuzzy connected set in Y .
Then f is fuzzy M-connected function .
3. Some kinds of fuzzy continuous functions
3.1 Definition
A fuzzy set A of a fuzzy topological space is said to be a fuzzy continuum if and
only if A is fuzzy connected and fuzzy compact set .
3.2 Example
A singleton set in any finite fuzzy discreet topological space is fuzzy continuum .
3.3 Definition[Güner,Erdal,2007]
A function f : X  Y is said to be fuzzy continuous function if and only if
f 1 (W ) is fuzzy open (closed ) set in X for each W is fuzzy open ( closed ) set
in Y ,otherwise f is called fuzzy discontinuous.
3.4 Definition
A function f : X  Y is said to be fuzzy continuum function if and only if
f (W ) is fuzzy continuum set in Y for each W is fuzzy continuum set in X .
3.5 Proposition
Every fuzzy continuous function is fuzzy continuum function .
Proof:Let f : X  Y be a fuzzy continuous function , and suppose that f is not fuzzy
continuum function .
Then ,there is fuzzy continuum set W in X ,and f (W ) is not be fuzzy connected set .
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Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014
Then there is two non-empty fuzzy open sets A1 , A2 in Y such that
f (W )  A1  A2 and A1  A2  0Y .
Since f is fuzzy continuous function ,then f 1 ( A1 ) , f 1 ( A2 ) are fuzzy open sets in
X.
Then W  f 1 ( A1  A2 )  f 1 ( A1 )  f 1 ( A2 ) , and
f 1 ( A1 )  f 1 ( A2 )  f 1 ( A1  A2 )  f 1 (0Y )  0 X .
Then W is fuzzy disconnected set , and this contradiction .
Then f is fuzzy continuum function . □
3.6 Definition
A function f : X  Y is said to be fuzzy C-continuous function if and only if
1
f (W ) is fuzzy open (closed ) set in X for each W is fuzzy open ( closed ) and
fuzzy compact set in Y .
3.7 Proposition
Every fuzzy continuous function is fuzzy C-continuum function .
Proof:Let f : X  Y be a fuzzy continuous function , and let W is fuzzy closed and fuzzy
compact set in Y .
Since W is fuzzy closed and f is fuzzy continuous function , then f 1 (W ) is fuzzy
closed set in X .
Then f is fuzzy C-continuous function . □
3.8 Example
Let X  {x1 , x2 } , Y  {a1 , a2 } and T1  {0 X ,1X },T2  {0Y ,1Y } be two fuzzy
topologies on X and Y respectively, and let f : X  Y defined by
f ( x1 )  a1 , f ( x2 )  a2 .
Since 0Y ,1Y are fuzzy closed and fuzzy compact sets in Y such that f 1 (0Y )  0 X is
fuzzy closed set in X ,and f 1 (1Y )  1X is fuzzy closed set in X .
Then f is fuzzy C-continuous function .
3.9 Definition
A function f : X  Y is said to be fuzzy weakly continuous if and only if for all
fuzzy point x in X ,and for all fuzzy open set V in Y contains f ( x ) there exist
open set U in X contain x such that f (U )  V .
3.10 proposition
A function f : X  Y is fuzzy weakly continuous if and only if for each fuzzy
open set V in Y ,such that f 1 (V )  ( f 1 (V )) 0 .
Proof:
Let x be a fuzzy point in X and V be a fuzzy open set in Y contains f ( x ) .
Since V be a fuzzy open set in Y ,then f
Since f ( x )  V ,then x  f 1 (V )
1
(V )  ( f
1
(V )) 0
Since ( f 1 (V )) 0 is fuzzy open set in X , then f 1 (V )  ( f 1 (V )) 0
Since x  f 1 (V ) ,then x  f 1 (V ) , and so x  ( f 1 (V )) 0
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Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014
Let U  ( f 1 (V )) 0 ,then U is a fuzzy open set in
1
X contains
x such that
1
f (U )  f (( f (V )) )  f ( f (V ))  V . See ( theorem 1.6(5) )
Then f is fuzzy weakly continuous function .

Let V be a fuzzy open set in Y ,and let x  f 1 (V ) , then f ( x )  V
Since f is fuzzy weakly continuous function, then there exists fuzzy open set U
contains x such that f (U )  V .
0
Then U  f 1 (V ) . See ( theorem 1.6(7) )
Since x  U  f 1 (V ) ,and since U is fuzzy open set
Then x  ( f
1
(V )) 0
Then f 1 (V )  ( f 1 (V )) 0 . □
3.11 Proposition
Every fuzzy weakly continuous onto function is fuzzy connected
. Proof:Let f : X  Y is fuzzy weakly continuous onto function .
Let X is fuzzy connected space, and suppose that Y is fuzzy disconnected space .
then there exist two non-empty fuzzy open sets A1 , A2 such that
1Y  A1  A2 ,and A1  A2  0Y ,then f 1 ( A1  A2 )  f 1 (0Y )  0 X .
since 1X  f 1 (1Y )  f 1 ( A1  A2 )  f 1 ( A1 )  f 1 ( A2 ) .
Since A1 is fuzzy open set in Y .
Then by proposition ( 3.10) f 1 ( A1 )  ( f 1 ( A1 )) 0
Then A1 is both fuzzy open and fuzzy closed, see remark ( 1.13)
Then A1  A1 ,so f 1 ( A1 )  ( f 1 ( A1 )) 0 ,but ( f 1 ( A1 )) 0  f 1 ( A1 ) .
Then ( f 1 ( A1 )) 0  f 1 ( A1 )
Then f 1 ( A1 ) is fuzzy open set in X .
Similarity we prove that f 1 ( A2 ) is fuzzy open set in X .
Then X is fuzzy disconnected space ,this contradicts
Then Y is fuzzy connected space .
Then f is fuzzy connected function . □
3.12 Definition
A function f : X  Y is said to be fuzzy almost continuous if and only if for all
fuzzy point x in X ,and for all fuzzy open set V in Y contains f ( x ) there exist
_______
0
open set U in X contain x such that f (U )  V
.
3.13 Example
Let X  {a, b}, Y  {x, y} and let T1 be discrete fuzzy topological space on X and
T2  {0Y ,1Y } be fuzzy topological space on Y .
If f : X  Y be a function defined by f (a)  x , f (b)  y .
Then f is fuzzy almost continuous function .
3.14 Proposition
i)Every fuzzy almost continuous function is fuzzy weakly continuous .
ii) Every fuzzy continuous functions is fuzzy connected .
Proof:2339
Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014
1) Let f : X  Y is a fuzzy almost continuous function .
Let x  X ,and let V be a fuzzy open set in Y contains f ( x ) .
Since f is a fuzzy almost continuous function ,then there exist fuzzy open set U in
_______
X such that x  U and f (U )  V 0 .............(1)
_______
Since V 0  V ,then from theorem (1.9(3)) V 0  V ……….(2)
Substitute (2) in (1) ,then we have f (U )  V .
Then f is fuzzy weakly continuous function .
ii) since every fuzzy weakly continuous function is fuzzy connected ,see proposition
(3.11) ,and since every fuzzy almost continuous function is fuzzy connected ,see
proposition ( 3.14(i)) .
then every fuzzy continuous function is fuzzy connected .□
and the following graph explain this relation .
Fuzzy almost
continuous
function
Fuzzy
continuous
function
Fuzzy weakly
continuous
function
Fuzzy connected
function
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Journal of Babylon University/Pure and Applied Sciences/ No.(9)/ Vol.(22): 2014
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