Download On Intuitionistic Fuzzy Special Light Mapping By Khalid Shea.Al

On Intuitionistic Fuzzy Special Light Mapping
By
Khalid Shea.Al-jabri
Department of Mathematics, Colloge of Education , Al- Qadisiyh
University
[email protected]
Abstract:
In this work, we introduce the concept of intuitionistic fuzzy special light
mapping and investigate the several properties of this concept.
1- Introduction:
Fuzzy topological spaces were introduced by Chang.C.L. in [2], Who studied
a number of basic concepts including fuzzy continuous maps and compactness.
Fuzzy topological spaces are a natural generalization of topological spaces. In
1983, Mashhour.A.S in [4] introduced the supra topological spaces and studied
S-continuous functions and S*-continuous functions. In 1987,Abd ElMonsef.M.E. in [1] introduced the fuzzy -supra topological spaces and studied
fuzzy supra-continuous functions and obtained a number of characterizations.
Also fuzzy –supra topological spaces are generalization of supra topological
spaces. In 1996, Won Keun.M. in [3] introduced fuzzy S-continuous, fuzzy Sopen and fuzzy S-closed Maps and established a number of characterizations. In
this work we introduce the intuitionistic fuzzy special light mapping and
establish a number of characterizations.
2-Basic definitions and facts:
2-1Definition[5]: Let X be a non empty set and AX , A1, A2 are two subset of X
such that A1A2= and A1consiste of elements of A and represent of these
elements by the element x .Then we say that A is intuitionistic fuzzy special set
(IFSS'S) and denoted by A=x,A1,A2.
2-2Definition[5]: Let X be a non empty set and A,B are intuitionistic fuzzy
special sets in X such that A=x,A1,A2 and B=x,B1,B2 then
1- AB iff A1B1 and B2A2.
2- A=B iff AB and BA.
_
_
2-3Remark: 1- The complement of A denoted by A and define by A =x,A2,A1.
c
c
2- A1 A 2 , A2 A1 .
3- A=x,A,Ac
4- X=x,X, and =x,,X
5- we define two operation are A,A as follows :c
c
A=x,A1, A1  and A=x, A 2 ,A2.
Now the following definition is given in [5]
1
2-4Definition: Let X be a non empty set and  the family of IFSS in X , then  is
called intuitionistic fuzzy special topology (IFSS) on X if satisfy the following
conditions :1- and X 
2- if A.B then AB  .
3- if Ai  iJ then  A i   .
iJ
The elements of  are called intuitionistic fuzzy special open sets (IFSOS) and
(X,) is called intuitionistic fuzzy special topological space(IFSTS).
2-5Example:Let X={a,b,c} and ={,X,A}, such that A=x,{a},{b} is
intuitionistic fuzzy special set in X then  is intuitionistic fuzzy special topological
on X such that A=x,{a},{b} is intuitionistic fuzzy special open set in X and
_
A =x,{b},{a} is intuitionistic fuzzy special closed set in X.
2-6 Definition:- Let (X,) and (Y, σ) are intuitionistic fuzzy special topological
space(IFSTS) and f:X
Y mapping , then f is called
1- intuitionistic fuzzy special continuous mapping if f -1(A)  τ for all A  σ.
2- intuitionistic fuzzy special open mapping if f(A)  σ for all A  τ.
3- intuitionistic fuzzy special closed mapping if f(B) is intuitionistic fuzzy special
closed set in Y for all intuitionistic fuzzy special closed set B in X.
4- intuitionistic fuzzy special homeomorphism if f is bijective intuitionistic fuzzy
special continuous mapping and f is intuitionistic fuzzy special open (closed)
mapping.
3-Connectedness in IFSTS.
Now we give the following definition:3-1Definition: Let X be a intuitionistic fuzzy special topological space.Then we
say that X is intuitionistic fuzzy disconnected space if there exists A , B ,
A,B  such that AB=X and AB= .otherwise X is called intuitionistic fuzzy
connected space.
3-2 Example: Let X ={a,b,c,d} and ={ , X,A,B,C,D,E} such that
A=x,{b,c},{d} , B=x,{a},{c}, C=x,{a,d},{c}, D=x,{a,b,c},{d} and E=x,,{c,d}
we note that X is intuitionistic fuzzy connected space but *={,
X,A,B,C,D,E}is intuitionistic fuzzy disconnected space note that A,C
 * such that A,C and AC= X and AC=
3-3Example: Let X={1,2,3} and ={, X,A ,B,C}such that A=x,{1},{2,3}
,B=x,{2},{1} and C=x,{1,2}, then we note that X is intuitionistic fuzzy
connected space but *={, X,A,B,C} is intuitionistic fuzzy disconnected
space.
2
Now we give the following definitions:3-4 Definition:- Let X be a intuitionistic fuzzy special topological space and let
A,B are non empty intuitionistic fuzzy special set in X then A/B is said to be a
Intuitionistic fuzzy special disconnection (IFSD) to X iff AB=X and AB=.
3-5 Definition: Let X be a intuitionistic fuzzy special topological space and G be a
non empty intuitionistic fuzzy special subset of X , and A,B are non empty
intuitionistic fuzzy special set in X then A/B is said to be a intuitionistic fuzzy
special disconnection to G in X iff
1- AG and BG.
2- (AG)(BG)=G.
3- (AG)(BG)=.
3-6 Definition: Let X be a intuitionistic fuzzy special topological space then we
say that X is totally intuitionistic fuzzy special disconnected space if every pair of
point a,b X there exist a intuitionistic fuzzy special disconnection A/B to X
such that aA and bB.
Now the following proposition shows that the property of totally intuitionistic
fuzzy special disconnected space define on intuitionistic fuzzy special set is a
topological property.
3-7proposition:let X and Y be two intuitionistic fuzzy special topological space
spaces, and let f:X→Y be a intuitionistic fuzzy special homeomorphism .If X or Y
is a totally intuitionistic fuzzy special disconnected so is the other.
Proof: Suppose that X is totally intuitionistic fuzzy special disconnected and let
y1 , y 2  Y such that y1≠ y2 .since f is bijective mapping then there exists two
points x1, x2  X such that x1≠ x2 and f(x1)=y1 , and f(x2)=y2 since X is totally
intuitionistic fuzzy special disconnected space , then there is a Intuitionistic fuzzy
special disconnection U/V such that x1  U , x2  V since f is Intuitionistic fuzzy
special homeomorphism then each of f(U) and f(V) are Intuitionistic fuzzy special
open sets in Y but f (U )  f (V )  f (U  V )  f ( X )  Y and since f is one-to-one
then f (U )  f (V )  f (U  V )  f ( )   , X and y1  f (U ), y2  f (V ) and hence Y is
totally intuitionistic fuzzy special disconnected space .
4- Intuitionistic Fuzzy special Light mapping.
Now we introduce the following definition:
4-1Definition: a mapping f: X→Y is called Intuitionistic fuzzy special Light
mapping if f -1(y) ,is totally intuitionistic fuzzy special disconnected set for all
y  Y.
4-2Proposition: let f:X→Y be a Intuitionistic fuzzy special Light mapping and let
G  X , then the restriction mapping f│G :G→f(G) , is also Intuitionistic fuzzy
special Light mapping.
3
Proof: to show that for all y  f (G ) , f 1 ( y)  G is totally intuitionistic fuzzy
special disconnected set in G. let a,b
 f 1 ( y)  G then a,b  f 1 ( y) and since f
1
is Intuitionistic fuzzy special Light mapping then for all y  Y , f ( y) is totally
disconnected set in X , that is there exists a Intuitionistic fuzzy special
( A  f 1 ( y))  ( B  f 1 ( y))  f 1 ( y)and
disconnection A/B such that
( A  f 1 ( y))  ( B  f 1 ( y))  
Such that A,B are disjoint Intuitionistic fuzzy special open subsets of X, and
a  A, b  B now to show that A/B is Intuitionistic fuzzy special disconnection to
f 1 ( y)  G also .since
((G  f 1 ( y ))  A)  ((G  f 1 ( y ))  B)  (G  (( f 1 ( y )  A))  (G  ( f 1 ( y )  B))
 G  [( f 1 ( y )  A)  ( f 1 ( y )  B)]  G  f 1 ( y )and
((G  f 1 ( y ))  A)  ((G  f 1 ( y ))  B)  (G  ( f 1 ( y )  A))  (G  ( f 1 ( y )  B))
 G  [( f 1 ( y )  A)  ( f 1 ( y )  B)]  G    
Such that (G  f 1 ( y))  A, (G  f 1 ( y))  B are two disjoint Intuitionistic fuzzy
special open sets , hence f 1 ( y)  G is totall intuitionistic fuzzy special y
disconnected set , f│G is Intuitionistic fuzzy special Light mapping.
Now we introduce the following definitions:
4-3Definition: A mapping f:X→Y is said to be totally intuitionistic fuzzy special
disconnected mapping if each totally intuitionistic fuzzy special disconnected set
U in X, f(U) is totally intuitionistic fuzzy special disconnected set in Y.
4-4Definition: A mapping f:X→Y is said to be inversely totally intuitionistic fuzzy
special disconnected mapping if each totally intuitionistic fuzzy special
disconnected set U in Y, f -1(U) is totally intuitionistic fuzzy special disconnected
set in X.
4-5Proposition: let f1:X→Y and f2:Y→Z be two mappings, then f=f2 of1:X→Z is
Intuitionistic fuzzy special Light mapping if f1 is inversely totally intuitionistic
fuzzy special disconnected and f2 is Intuitionistic fuzzy special Light mapping .
Proof: to prove that for z  Z , f 1 ( z ) is totally intuitionistic fuzzy special
1
1
disconnected set in X. let z  Z then f 1 ( z )  ( f 2of1 ) 1 ( z )  f1 ( f 2 ( z )) and since
f2 is Intuitionistic fuzzy special light mapping then f2-1(z) is totally intuitionistic
fuzzy special disconnected set in Y and since f1 is inversely totally intuitionistic
1
1
fuzzy special disconnected mapping then f1 ( f 2 ( z )) is totally intuitionistic
fuzzy special disconnected set in X, but f 1 ( z )  f11 ( f 2 1 ( z )) , then f 1 ( z ) is
totally intuitionistic fuzzy special disconnected set in X.
4-6Proposition: let f:X→Y be the composition f  f 2of1 of two mapping f1:X→Y
' and f :Y '→Y , then
2
4
1- If f2 is an bijective mapping and f1 Intuitionistic fuzzy special Light mapping
then f is Intuitionistic fuzzy special Light mapping .
2- If f is Intuitionistic fuzzy special Light mapping and f2 is an injective mapping
then f1 is Intuitionistic fuzzy special Light mapping.
3- If f is Intuitionistic fuzzy special Light mapping and f1 is a totally intuitionistic
fuzzy special disconnected mapping then f2 is Intuitionistic fuzzy special Light
mapping.
Proof: 1- let y  Y , since f2 is a bijective mapping then there exists one point
y'  Y such that f 2 (y')=y and since
f 1 ( y)  (f 2 of1 ) 1 ( y)  f11 (f 2 1 ( y))  f11 (f 2 1 (f 2 ( y ' ))  f11 ( y' )
and f1 is Intuitionistic fuzzy special light mapping then f11 ( y' ) is totally
1
Intuitionistic fuzzy special disconnected set in X, so f1 ( y ) is totally Intuitionistic
fuzzy special disconnected set in X and hence f is Intuitionistic fuzzy special light
mapping.
2- let y' Y' , then f 2 ( y ' )  Y and since f is Intuitionistic fuzzy special light then
f 1 (f 2 ( y' )) is totally Intuitionistic fuzzy special disconnected set in X but
1
1
1
f 1 (f 2 ( y' )) = f1 (f 2 (f 2 ( y' ))  f1 ( y' ), then f1 is a Intuitionistic fuzzy special
light mapping.
3- let y  Y ,since f is a Intuitionistic fuzzy special light , then f -1(y) is a totally
intuitionistic fuzzy special disconnected set in X and since f1 is totally
intuitionistic fuzzy special disconnected mapping then f1(f -1(y)) is a totally
intuitionistic fuzzy special disconnected set in Y ', but
1
1
1
f1 ( f 1 ( y ))  f1 (( f 2of1 ) 1 ( y ))  f1 ( f1 ( f 2 ( y )))  f 2 ( y ). then f2 -1(y) is a totally
intuitionistic fuzzy special disconnected set in Y ', so f2 is Intuitionistic fuzzy
special light mapping.
Now we will show when the product of two mappings is a Intuitionistic fuzzy
special light mapping.
4-7Proposition: let f1:X1→Y1 , f2:X2→Y2 be two mappings , then the product
mapping f1хf2 :X1хX2→Y1хY2 is a Intuitionistic fuzzy special light mapping if f1
is an bijective mapping and f2 is a Intuitionistic fuzzy special light mapping.
Proof: let (y1,y2)  Y1хY2 , then (f1хf2)-1(y1,y2)=(f1-1хf2-1)(y1,y2)=f1-1(y1)хf2-1(y2) and
since f1 is bijective then there exists x1  X1 such that
1
1
1
f1 ( y1 )  f1 ( f1 ( x1 ))  x1 , then ( f1  f 2 ) 1 ( y1 , y2 )  x1  f 2 ( y2 ) , and since f2 is a
1
Intuitionistic fuzzy special light mapping , then f 2 ( y2 ) is a totally intuitionistic
1
fuzzy special disconnected set in X2 and since x1  f 2 ( y2 ) is homeomorphic to
1
1
f 2 ( y2 ) , then from (3-7) x1  f 2 ( y2 ) is totally intuitionistic fuzzy special
disconnected , so , ( f1  f 2 )1 ( y1, y2 ) is totally intuitionistic fuzzy special
5
disconnected set in
X1хX2 and hence f1хf2 is Intuitionistic fuzzy special light
mapping.
4-8Corollary: let
f1 : X1  Y1 ,
f 2 : X 2  Y2 be two maps such that if
f1  f 2 : X1  X 2  Y1  Y2 is a Intuitionistic fuzzy special light mapping, then if one
of them is a surjective , then the other is a Intuitionistic fuzzy special light
mapping.
Now the following definition is given in [6].
4-9Definition: let f : X  Y and g:Y '→Y be maps and X ' is the subspace of the
product space XхY ' defined as follows:-
X '  ( x, y' )  X  Y ' f ( x)  g ( y' ), X ' is called the fibered product of Y ' and X
over Y . let f ':X '→Y ' be the restriction of the second projection then f ' is called
the pull back of f by g.
4-10Proposition: The pull back of Intuitionistic fuzzy special light mapping is
also Intuitionistic fuzzy special light mapping.
Proof: let f:X→Y be Intuitionistic fuzzy special light mapping and f ':X '→Y ' be a
pull back of f by g:Y '→Y , now let y '  Y ' , g(y ')  Y , since f is Intuitionistic
fuzzy special light mapping then
f 1 (g ( y' )) is
a totally intuitionistic fuzzy special
disconnected set in X. now for fixed y '  Y ' ,
f
' 1


( y' )  ( x, y' )  X  Y ' f ' ( x, y' )  y' for all x  X and f(x)=g(y')


 ( x, y ' )  X  Y ' x  f 1 ( g ( y ' ))  f 1 ( g ( y ' ))  y ' and since
f
1
 
( g ( y ' ))  y ' is homeomrphic to f
-1(g(y')
and f -1(g(y') is totally intuitionistic
fuzzy special disconnected then f 1 ( g ( y ' ))  y ' is totally intuitionistic fuzzy
special disconnected by(3-7) and hence f '1 ( y ' ) , is totally intuitionistic fuzzy
special disconnected , then f ' is Intuitionistic fuzzy special light mapping.
6
References
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[3] Kwon,K.M."On Fuzzy S-Continuous functions ", Kangweon-Kyungki
Math.J.4 (1996).no.1, 77-82.
[4] Mashhour,A.S, Allan, F.S. and Khder,F.H," On supra topological spaces " ,
Indian J.pure and Appl.Math .14(1983),no.4, 502-510.
[5] OZGAG,S. and Coker,D.," On connectedness Intuitionistic fuzzy special
topological spaces" ,J.Math.Sci.vol.21, no.1(1998) 33-40.
[6]
Spainer,E.H,"
Algebraic
topology",
California(1966).
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