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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 AX , A1, A2 are two subset of X such that A1A2= 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- AB iff A1B1 and B2A2. 2- A=B iff AB and BA. _ _ 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 AB . 3- if Ai iJ then A i . iJ 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 AB=X and AB= .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 AC= X and AC= 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 AB=X and AB=. 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- AG and BG. 2- (AG)(BG)=G. 3- (AG)(BG)=. 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 aA and bB. 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 ) f11 ( 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) f11 (f 2 1 ( y)) f11 (f 2 1 (f 2 ( y ' )) f11 ( y' ) and f1 is Intuitionistic fuzzy special light mapping then f11 ( 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 [1] Abd El-Monsef,M.E. and Ramadan,A.E."On Fuzzy supra topological spaces" , Indian J. Pure and Appl. Math. Vol 18(1987), no. 4, 322-329. [2] Chang,C.L."Fuzzy topological spaces", J.Math. Annl. Appl.Vol 24(1968).128190. [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). 7 Magraw Hill, San Francisco,