Microsoft Word - 171-183 170 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 Soft Strongly Generalized Mapping With Respect to an Ideal in Soft Topological Space Alyasaa J. Bidaiwi Narjis A. Dawood Dept. of Mathematics/ College of Education for Pure Sciences ( Ibn Al- Haitham)/ University of Baghdad. Received in: 19 March 2015, Accepted in: 26 May 2015 Abstract In this work, we introduced and studied a new kind of soft mapping on soft topological spaces with an ideal, which we called soft strongly generalized mapping with respect an ideal I, we studied the concepts like SSIg-continuous, Contra-SSIg-continuous, SSIg-open, SSIg-closed and SSIg-irresolute mapping and the relations between these kinds of mappings and the composition of two mappings of the same type of two different types, with proofs or counter examples. Key words: SSIg-continuous, contra-SSIg-continuous, SSIg-irresolute, SSIg-closed mapping, SSIg-open mapping. .This paper is a part of M.Sc. Thesis of Alyasaa supervised by Narjis 171 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 Introduction The concept of soft sets was introduced, for the first time, by Molodtstov in (1999) , as a generalization of fuzzy sets, soft sets are used as a tool to ideal with uncertain objects. Recently, in (2013), the study of soft topological spaces was introduced by D. N. Georgiou, A. C. Megaritis, they used the concept of soft set to define a topology ,that leads to a new world in general topology, the study soft strongly generalized closed set with respect to an ideal in soft topological space, and is denoted by SSIg-closed set, introduced by Alyasaa J. and Narjis A. in [1]. We gave the definitions of SSIg-continuous mapping, SSIg-open, SSIg- closed, SSIg-irresolute, Contra-SSIg-continuous mapping. The composition of these mappings are also discussed. 1- Preliminary concepts and results   Definition (1.1)[2]: For A ⊆ E, the pair (F,A) is called a soft set over X, where F is a mapping given by F:A → P(X). In other words, the soft set is a parametrized family of subsets of the set X. Every set F(e), e ∈ E, from this family may be considered as the set of e-elements of the soft set (F,E) , or as the set of e-approximate elements of the soft set. Clearly, a soft set is not a set. Note(1.2): In what follows by SS(X,E) we denote the family of all soft sets over X. Definition(1.3)[1]: For two soft sets (F,A) and (G,B) in SS(X,E) , we say that (F, A) is a soft subset of (G, B) if A B and ( ) ( ),F e G e e A   . Definition (1.4) [1]: The union of two soft sets (F, A) and (G,B) over the common universe X is the soft set (H,C), where C = A∪B and for all e ∈ C, ( ) ( ) ( )H e F e G e  . Definition(1.5)[2]: The intersection of two soft sets (F,A) and (G,B) over the common universe X is the soft set (H,C), where C = A∩B and for all e ∈ C, H(e) = F(e)∩G(e). Definition(1.6) [3]: Let (F,E) be a soft set over X and xX. We say that x (F,E) whenever xF(α) for all αE. Note that for xX, x (F,E) if xF(α) for some α ∈ E. Definition(1.7)[3]: A soft set (F,A) over X is said to be a null soft set, denoted by A , if for all e ∈ A, F(e)= (null set), where ( )A e e A    . Definition(1.8) [3]: A soft set (F,A) over X is said to be an absolute soft set, denoted by AX , if for all e ∈ A , F(e)= X. Clearly, we have c AX = A and c A  = AX . Definition(1.9) [3]: Let τ be a collection of soft sets over X with the fixed set E of parameters , then τ SS(X,E). We say that the family τ defines a soft topology on X if the following axioms are true : 1- AX , A τ , 2- If (G,A) , (H,A) τ , then (G,A)  (H,A) τ, 3- If (G i ,A)τ for every i  , then i  (G i ,A)τ. Then τ is called a soft topology on X and the triple (X,τ,E) is called soft topological spaces over X . 172 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 Definition(1.10) [4]: Let E be a set of parameters, A nonempty collections I of soft subsets over X is called a soft ideal on X if the following holds (1) If (F,A)∈I and (G,B)  (F,A) implies (G,B)∈I (heredity), (2) If (F,A) and (G,A) ∈ I, then (F,A)  (G,A) ∈ I (additivity). If I is ideal on X and Y is subset of X, then IY = { EY  Ii : Ii ∈ I , i} is an ideal on Y. Definition(1.11) [4] :Let (X,τ,E) be a soft topological space with an ideal I . A soft set (F,E) ∈SS(X,E) is called soft generalized closed set with respect to an ideal I (soft Ig-closed) if cl(F,E)-(G,E)∈ I whenever (F,E)  (G,E) and (G,E)∈ τ . The relative complement (F,E)c is called soft generalized open set with respect to an ideal I (soft Ig-open). Definition(1.12) [2]: Let (X,τ,E) be a soft topological space with an ideal I. A soft subset (A,E) of (X,τ,E) is said to be soft strongly generalized closed set with respect to an ideal I ,(briefly SSIg- closed), if cl(int(A,E))-(B,E)∈I whenever (A,E)  (B,E) and (B,E) is soft open set. the relative complement (F,E)c is soft strongly generalized open set with respect to an ideal I,(briefly SSIg- closed). Proposition(1.13) [1]: Let (X,τ,E) be a soft topological space with an ideal I. Then every soft closed set is an SSIg-closed set. Corollary(1.14) [1]: Let (X,τ,E) be a soft topological space with an ideal I. Then every soft open set is an SSIg-open set. Proposition(1.15) [1]: Every soft g-closed set is a soft strongly generalized closed set with respect to an ideal I . Corollary(1.16) [1]: Every soft g-open set is a soft strongly generalized open set with respect to an ideal I . Theorem(1.17) [1]: Every soft Ig- closed set is a soft strongly generalized closed set with respect to a soft ideal I . Definition(1.18) [1]:Let (A,E) be a soft set in a soft topological space ( , ,E) with an ideal I, Then the interior of (A,E) is the union of all SSIg-open sets which are contained in (A,E). and denoted it by int*(A,E) Proposition(1.19) [1]:Let (A,E) be a soft open set in a soft topological space ( , ,E) with an ideal I, then int*(A,E)=(A,E). Proposition (1.20) [1]:Let (A,E) be any soft set in a soft topological space ( , ,E) with an ideal I, then int(A,E)  int*(A,E). Definition(1.22) [1]:For any soft subset (A,E) in a soft topological space ( , ,E) with an ideal I, the SSIg-closure of (A,E), denoted by c *(A,E), is defined by the intersection of all SSIg- closed sets containing (A,E). Theorem (1.23) [1]:If (A,E) is SSIg‐closed set in a soft topological space ( , ,E) with an ideal I ,then (A,E)= c *(A,E). Proposition(1.24) [1]: Let (A,E) be any soft set of a soft topological space ( , ,E) with an ideal I, then c *(A,E)  c (A,E). 173 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 2. Soft strongly generalized mapping with respect to an ideal in soft topological space. Definition(2.1)[6]:Let SS(X,E) and SS(Y,B) be families of soft sets over X and Y respectively, : → and : →B be mappings. Then the mapping :SS(X,E) SS(Y,B) is defined as : 1- If (F,E) SS(X,E) , then the image of (F,E) under , written as (F,E) =( (F),p(E)) is a soft set )in SS(Y,B such that 1 1 ( ) 1 ( ( )) , ( ) . ( )( ) , ( ) . a p b pu u F a p b f F b p b             for all b B. 2- If (G,B) SS(Y,B) , then the inverse image of (G,B) under , written as (G,B) =( (G),p-1(B)) is a soft set in SS(X,E) , such that 1 1 ( ( ( ))) , ( ) . ( )( ) , o.w. pu u G p a p a B G af       for all a E. Definition(2.2):Let :SS(X,E) SS(Y ,K ) and SS(Y,K ) SS(Z,H ) be a soft mappings . Then SS(X,E) SS(Z,H ), if (F,E) SS(X,E), then the image of (F,E) under , written as (F,E) = ( (F), (E)) is a soft set. Remark(2.3):In Definition(3.2.3) . Theorem (2.4)[5]: Let : SS( , ) → SS( , ) , : → , and : → be mappings. Then for soft sets ( ,) and ( , ) in the soft class SS( , ) and ( , ) in SS(Y,K) ,we have the following properties : 1- ( ) E Kpu f   . 2- ( ) E Kpu X Yf  . 3- (( , ) ( , )) ( , ) ( , ) pu pu pu F A G B F A G Bf f f    , in general we get ( ( , )) (( , )) . i i i ii ipu pu F A F A if f      4- (( , ) ( , )) ( , ) ( , ) pu pu pu F A G B F A G Bf f f    , in general we get ( ( , )) (( , )) . i i i ii ipu pu F A F A if f      5- If ( , ) ( , )F A G B , then ( , ) ( , ) pu pu F A G Bf f . 6- 1 ( ) pu K E f    . 7- . 8- 1 1 1(( , ) ( , )) ( , ) ( , ) pu pu pu F A G B F A G Bf f f      , in general we get ( ( , ) ) ( ( , ) ) . i i i ii ip u p u F A F A if f       . 9- 1 1 1(( , ) ( , )) ( , ) ( , ) pu pu pu F A G B F A G Bf f f      , in general we get 1 1( ( , )) (( , )) . pu i i pu i ii i F A F A if f           1 pu f  1 pu f    : qs g  ( )( ) :( ) q p s u g f      ( )( ) ( ) q p s u g f    ( )( )( ) q p s ug f   ( )( )( ) q p s ug f   q p ( )( ) ( ) q p s u qs pu g f g f     1 ( ) pu K E Y Xf   174 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 Note(2.5)[5]:If (X,τ,I) is a topological space with an ideal I , (Y,σ) is a topological space and f : (X,τ,I) →(Y,σ) is a function, then f(I) = {f(Ii) : Ii ∈ I , } is an ideal of Y. So in this research we will depend I as an ideal over X and f(I) is an ideal over Y. Definition (2.6)[5]:Let (X,τX,E,I) and (Y,τY,E) be two soft topological spaces, : SS(X,E) SS(Y,E ) be a soft mapping. then is a said to be soft Ig-continuous if the inverse image under of every soft open set in SS(Y,E) is soft Ig-open set in SS(X,E). Definition(2.7) :Let (X,τX,A) and (Y,τY,B) be two soft topological spaces with an ideal I, : (X, τX,A) (Y,τY,B) be a soft mapping. If for each soft open set (G,E) over Y , ((G,B)) is a SSIg-open set over X, then is said to be SSIg-continuous mapping. Proposition (2.8): Every soft continuous mapping is SSIg-continuous . Proof: Let (X,τ,E) soft topological space with an ideal I, (Y, ,K ) be a soft topological space and : (X,τ,E) (Y, ,K ) be a soft continuous mapping. Let (H,K) be a soft open set in (Y, ,K ), since is a soft continuous mapping. then is soft open set, so c is soft closed set. But we have every soft closed set is SSIg-closed from Proposition(1.13) , then c is SSIg-closed, hence is SSIg-open set, thus is a SSIg-continuous mapping. □ Remark (2.9): The converse of Proposition (2.8) need not to be true by the following example. Example: Let X={a,b,c} ,E = {e1, e2} , Y={h1, h2, h3}, K = {k1, k2}, and τ = { , }, = { , , (G,K)} be two soft topologies defined on X and Y respectively , where (G,E)={( ,{ }), ( ,{ })}. Define : → such that , and : → such that . Then , : (X,τ,E) (Y, ,K) is an SSIg-continuous mapping . But it is not soft continuous since (G,K) is soft open set in (Y, ,K ) but (H,E) which is not soft open set in (X,τ,E) . Therefore is not soft continuous . □ Definition(2.10) :Let (X,τX,A) and (Y,τY,B) be two soft topological spaces with an ideal I, : (X,τX,A) (Y,τY,B) be a mapping. If for each soft open set (G,B) over Y , ((G,B)) is a SSIg-closed set over X, then is said to be contra-SSIg-continuous mapping. Proposition(2.11): Every soft contra-continuous mapping is contra-SSIg-continuous . Proof: Let (X,τ,E) be a soft topological space with an ideal I, (Y, ,K ) be a soft topological space and : (X,τ,E) (Y, ,K ) be a soft contra-continuous mapping. Let (H,K) be a soft open set in (Y, ,K ), since is a soft contra-continuous mapping. then is soft closed set. But we have i    1 pu f   1 ( , ) pu f H K 1( ( , )) pu f H K 1( ( , )) pu f H K 1 ( , ) pu f H K { } E I  E E X K KY 1 k 1 h 2 k 3 2 ,h h 1 2 )( kp e  2 1)( kp e  13 2) , ) , )( ( (h h hu a u b u c     1 (( , )) pu G Kf    1 pu f   1 ( , ) pu f H K 175 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 every soft closed set is SSIg-closed from Proposition(1.13) , then is SSIg-closed, thus is a contra-SSIg-continuous mapping. □ Remark(2.12):The converse of Proposition(2.11) need not be true by the following example. Example : Let X={a,b,c}, E = {e1, e2}, Y={h1, h2, h3}, K = {k1, k2}, , and τ = { , }, = { , , (G,K)} be two soft topologies defined on X and Y respectively , where (G,K) ={( ,{ }), ( ,{ })}. define same as Remark(2.9), then is SSIg- continuous. But it is not soft contra-continuous since (G,K) is soft open set in (Y, ,K) but {( ,{ }), ( ,{ })} is not soft closed set in (X,τ,E) .    Remark(2.13):The concepts of contra-SSIg-continuous and SSIg-continuous are independent by the following examples. Example :Let X={a,b,c} ,E = {e1, e2} , Y={h1, h2, h3}, K = {k1, k2}, and τ = { , ,(F,E) }, = { , , (G,K)} be two soft topologies defined on X and Y respectively , where (F,E)= {( ,{ }), ( ,{ })} ,(G,K)={( ,{ }), ( ,{ })}. Define : → such that , and : → such that . Then , :(X,τ,E) (Y, ,K ) is not contra-SSIg-continuous, since ({( ,{ }), ( ,{ })}) ={( ,{ }), ( ,{b,c})}= (F,E) which is not SSIg-closed . On the other hand, since = (F,E) which is soft open set and so it is SSIg-open set by Corollary(1.15). Therefore is an SSIg-continuous. □ Example: Let X={a,b,c} ,E = {e1, e2}, Y={h1, h2, h3}, K = {k1, k2}, and τ = { , ,(F,E) }, = { , , (G,K)} be two soft topologies defined on X and Y respectively , where (F,E)= {( ,{ }), ( ,{ })} ,(G,K)={( ,{ }), ( ,{ })}. Define same as Remark(2.13), then it is an contra-SSIg- continuous. But it is not SSIg-continuous since (G,K) is soft open set in (Y, ,K ) but ({( ,{ }), ( ,{ })}) ={( ,{a}), ( ,{ })}= (F,E), then cl(int(F,E) c)-(F,E)c . Hence , (F,E) c is not SSIg-closed , therefore is not SSIg-open set. Proposition(2.14) :Let (X,τ,E) be a soft topological space with an ideal I, (Y, ,K ) be a soft topological space and :(X,τ,E) (Y, ,K ) is a soft closed mapping. If (G,E) is a soft closed set in (X,τ,E) , then is SS (I)g-closed in (Y, ,K ) . Proof :Suppose that (G,E) is a closed SSIg-closed in (X,τ,E).Let (H,K) be a soft open set in (Y, ,K ) such that ,( ) pu G Ef  (H,K), then is soft 1 ( , ) pu f H K { } E I  E  E X K KY 1 k 1 h 2 k 3 2 ,h h  1 (( , )) pu G Kf   1e c 2e ,a b { } E I  E  E X K  K Y 1 e a 2 e ,b c 1 k 1 h 2 k 3 2 ,h h 1 1 )( kp e  2 2 )( kp e  1 3 2 ) , ) , )( ( (h h hu a u b u c    1 1(( , )) pu pu G Kf f  1 k 1 h 2 k 3 2 ,h h 1 e a 2 e 1 (( , )) pu G Kf  { } E I  E  E X K  K Y 1 e ,b c 2 e a 1 k 1 h 2 k 3 2 ,h h 1 1(( , )) pu pu G Kf f  1k 1h 2k 3 2,h h 1e 2 e ,b c I 1 (( , )) pu G Kf   ( ), pu G Ef pu f ( ), pu G Ef 176 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 closed set in (Y, ,K ), then by Proposition(1.13) we have get that is SS g-closed. □ Corollary(2.15): Let (X,τ,E) be a soft topological space with an ideal I , (Y, ,K ) be a discrete soft topological space and :(X,τ,E) (Y, ,K ) is a soft mapping . If (G,E) is a SSIg-closed in (X,τ,E) , then is SS (I)g-closed in (Y, ,K ) . Proof :It is clear . □ Proposition (2.16): Every soft g-continuous mapping is SSIg-continuous . Proof: Let (X,τ,E) soft topological space with an ideal I, (Y, ,K ) be a soft topological space and : (X,τ,E) (Y, ,K ) be a soft g-continuous mapping. Let (H,K) be a soft open set in (Y, ,K ), since is an soft g-continuous mapping. then is soft g-open set. But we have every soft g-open set is SSIg-open from Corollary(1.16), then is SSIg-open, thus is an SSIg-continuous mapping. □ Remark(2.17):The converse of Proposition (2.16) need not be true by the following example. Example: Let X={a,b,c} ,E = {e1,e2} , Y={h1, h2, h3}, K = {k1, k2}, and τ = { , ,(F,E) }, = { , , (G,K)} be two soft topologies defined on X and Y respectively , where (F,E)={( ,{ }), ( ,{ })} and (G,K)={( ,{ }), ( ,{ })}. Define : → such that , and : → such that . Then , : (X,τ,E) (Y, ,K ) is a soft mapping and it is a SSIg-continuous . But it is not soft g-continuous since (G,K) is soft open set in (Y, ,K ) since (V,E)c  (F,E) but cl(V,E) c = EX  (F,E). Hence is not soft g-open set, thus is not soft g-continuous. □ Proposition(2.18) :Let (X,τ,E) (Y, ,K ) be an SSIg-continuous mapping and (Y, ,K ) (Z, ,H ) is a soft continuous mapping. Then (X,τ,E) (Z, ,H ) is SSIg-continuous mapping . Proof : Let (X,τ,E) (Y, ,K ) is a SSIg-continuous mapping and (Y, ,K ) (Z, ,H ) is a soft continuous mapping. to prove that (X,τ,E) (Z, ,H ) is SSIg-continuous mapping . Let (M,H) be a soft open set in (Z, ,H ) . Since is a soft continuous mapping . Then 1 ( , ) qs g M H is soft open set in (Y, ,K ) and since is SSIg-continuous mapping and is soft open set in (Y, ,K ), So 1 1( ( , )) pu qs f g M H  is SSIg-open set in (X,τ,E) . Then 1 1 1) ( , ) ( ( , ))( qs pu pu qs g f M H f g M H   . Hence, is SSIg-continuous mapping . □ ( ), pu G Ef ( ) pu f I   ( ), pu G Ef pu f   1 ( , ) pu f H K 1 ( , ) pu f H K { } E I  E  E X K  K Y 1 e ,a c 2e ,a b 1k 1 2,h h 2k 3h 1 1 )( kp e  2 2 )( kp e  1 3 2 ) , ) , )( ( (h h hu a u b u c     1 (( , )) pu G Kf  : pu f  : qs g   : qs pu g f   : pu f   : pu g    :( ) pu g f    qs g  pu f 1 ( , ) qs g M H  qs pu g f 177 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 Proposition(2.19) : Let :(X,τ,E) (Y, ,K) is a SSIg-continuous mapping and (Y, ,K ) (Z, ,H ) is a soft contra-continuous mapping . Then (X,τ,E) (Z, ,H ) is contra-SSIg-continuous mapping . Proof : Let :(X,τ,E) (Y, ,K ) is a SSIg-continuous mapping and (Y, ,K ) (Z, ,H ) is a soft contra-continuous mapping . to prove that (X,τ,E) (Z, ,H ) is contra-SSIg-continuous mapping . Let (M,H) be a soft open set in (Z, ,H ) . Since is a soft contra-continuous mapping . Then is soft closed set in (Y, ,K ) and since is SSIg-continuous mapping and is soft closed set in (Y, ,K ), therefore is SSIg-closed set in (X,τ,E). Hence . Thus, is contra -SSIg-continuous mapping . □ Theorem(2.20): Let : (X,τ,E) (Y, ,K ) be a mapping from a soft space (X,τ,E) with an ideal I to soft space (Y, ,K ). If is SSIg-continuous mapping then for each soft singleton (P,E) in X and each soft open set (O,K) in Y and ( , ) ( , )pu P E O Kf  , there exists a SSIg-open set (U,E) in X such that ( , ) ( , )P E U E and ( , ) ( , )pu U E O Kf  . Proof : Suppose that is SSIg-continuous mapping . Let (P,E) be a soft singleton in X and (O,K) be a soft open set in Y such that ( , ) ( , )pu P E O Kf  . Then 1( , ) ( , ) pu P E O Kf  ,but is SSIg-continuous mapping and (O,K) be a soft open set in Y . By definition of SSIg-continuous mapping we get that 1 ( , ) pu O Kf  is SSIg-open set in X . Put (U,E)= 1 ( , )pu O Kf  . Therefore , ( , ) ( , )P E U E and ( , ) ( , )puO K U Ef .□ Remark(2.21): The converse of Proposition (2.20) need not be true by the following example. Example :Let X={a,b}, E = {e1, e2}, Y={d, c}, K = {k1, k2}, and τ = { , ,(F,E)}, = { , , (G,K)} be two soft topologies defined on X and Y respectively , where (F,E)={( ,{b}), ( , )} , (G,K)={( ,{d}), ( ,Y)} define : → such that , and : → such that . Then , (X,τ,E) (Y, ,K ) is a soft mapping . Since (G,K) is SS g-open and ({( ,{d}), ( ,Y)}) = {( ,{a}), ( ,X)} which is not SSIg-open set. So it is not SSIg-continuous mapping. On the other hand a X, ( , )G K and a {( ,{a}), ( ,{a})} where {( ,{a}), ( ,{a})} is SSIg-open set over X and ( , )a E  {( ,{a}), ( ,{a})} , {( ,{a}), ( ,{a})}  . Also b X and KY , while is SSIg-open set and ( , ) E b E X , ( )pu E KX Yf  .□  : qs g   : qs pu g f    : pu g   : qs pu g f    qs g 1 ( , ) qs g M H 1 ( , ) qs g M H  1 1( ( , ))pu qsf g M H   1 1 1) ( , ) ( ( , ))( qs pu pu qs g f M H f g M H   qs pu g f  { } E I  E  E X K  K Y 1 e 2 e  1 k 2 k ) , )( (d cu a u b  : pu f  ( ) pu If 1 1(( , )) pu pu G Kf f  1k 2k 1 e 2 e  ( , ) pu a Ef Y 1e 2 e 1 e 2 e 1 e 2 e 1 e 2 e ( , ) pu G Kf  ( , ) pu b Ef E X 178 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 Proposition(2.22): Let : (X,τ,E) (Y, ,K ) be a mapping from a soft space (X,τ,E) with an ideal I to soft space (Y, ,K ).Then the following statements are equivalent : 1- is SSIg-continuous mapping . 2- the inverse image under for any soft closed set over Y is SSIg-closed set over X . proof : suppose that is SSIg-continuous mapping . To prove that the inverse image under for any SSIg-closed set over Y is SSIg-closed set over X . Let (F,E) be a soft closed set over Y .We have to show that (F,E) is SSIg-closed over X . Since (F,E) is a soft closed set in Y ,then (F,E)c is a soft open set over Y . Because is SSIg-continuous mapping .Then (F,E)c is SSIg-open over X . Hence, (F,E) is SSIg-closed over X . Suppose that the inverse image under any soft closed set over Y is SSIg-closed set over X and to prove that is SSIg-continuous mapping . Let (F,E) be a soft open set over Y .We have to show that (F,E) is SSIg-open set in X . Since (F,E) is a soft open set over Y , then (F,E)c is a SSIg-closed set over Y . Then (F,E)c is SSIg-closed set over X and (F,E)c = ( (F,E))c . Hence, (F,E) is SSIg-open set over X . Therefore, is SSIg-continuous mapping . □ Proposition(2.23) : Let : (X,τ,E) (Y, ,K ) be an SSIg-continuous mapping from a soft topological space (X,τ,E) with an ideal I to soft topological space (Y, ,K ). If (A,E) is any soft set in over X , then *( ( , )) ( ( , )) pu pu cl A E cl A Ef f . Proof : Let (A,E) be any soft set over X . Then is a soft set over Y and is soft closed set over Y. But is a SSIg-continuous mapping . Then is a SSIg-closed set over X . Then by Proposition(1.26). Then 1 1 1( , ) (( ( , )) ( ( ( , ))) ( , ) ( ( ( , ))) pu pu pu pu pu pu A E f A E f cl A E and A E f cl A Ef f f       Therefore * * 1 1( , ) ( ( ( ( , ))) ( ( ( , ))pu pu pu pucl A E cl f cl A E f cl A Ef f    . Thus, *( ( , )) ( ( , )) pu pu f cl A E cl A Ef .□ Remark (2.24): The equality of Proposition(2.23) need not to be true by the following example Example: Let X={a,b,c} ,E = {e1, e2} , Y={h1, h2, h3}, K = {k1, k2}, and τ = { , ,(F,E)}, = { , , (G,K)} be two soft topologies defined on X and Y respectively, (F,E)={( ,{ }), ( ,{ })}, (G,K)={( ,{ }), ( ,{ })} , define : → such that , and : → such that . Then , (X,τ,E) (Y, ,K ) is a soft mapping . Since ({( ,{ }), ( ,{ })}) ={( ,{ }), ( ,{ })} =(F,E) which is soft open set, so it is soft continuous mapping . Now, let (A,E)={( , X ), ( ,{ })} be a soft set in X .     1 2 1 pu f  1 pu f  1 pu f     2 1 1 pu f  1 pu f  1 pu f  1 pu f  1 pu f  1 pu f   ( , ) pu A Ef ( ( , )) pu cl A Ef 1 *( ( ( , ))) pu pu f cl A Ef * 1 1( ( ( ( , ))) ( ( ( , ))) pu pu pu pu cl f cl A E f cl A Ef f  { } E I  E  E X K  K Y 1 e b 2 e ,a c 1k 1h 2k 2 3,h h 1 1 )( kp e  2 2 )( kp e  1 3 2 ) , ) , )( ( (h h hu a u b u c   : pu f  1 1(( , )) pu pu G Kf f  1k 3h 2 k 1 2 ,h h 1 e b 2 e ,a c 1 e 2 e ,a b 179 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 Then (A,E). We have {( ,Y),( ,{ })}. Then , Hence, *( ( , )) ( ( , ) ) pu K pu f cl A E Y cl f A E  , which mean that ( ( , ))pucl f A E *( ( , ))puf cl A E .□ Remark(2.25) : The converse of Proposition(2.23) need not to be true by the following example. Example: Let X={a,b} ,E = {e1, e2} , Y={h1, h2}, K = {k1, k2}, and τ = { , ,(F,E)}, = { , , (G,K)} be two soft topologies defined on X and Y respectively , where (F,E)={( , ), ( ,{b})} , (G,K)={( ,Y), ( ,{ })} , define : → such that , and : → such that . Then , : (X,τ,E) (Y, ,K ) is a soft mapping . Since ({( ,Y), ( ,{ })}) ={( ,X), ( ,{a})} which is not SSIg-open set. So it is not SSIg-continuous mapping . On the other hand for each soft set (A,E) in SS(X,E) ,then *( ( , )) ( ( , ))pu puf cl A E cl f A E .□ Proposition(2.26) : Let : (X,τ,E) (Y, ,K ) be a SSIg-continuous mapping from a soft space (X,τ,E) with an ideal I to soft space (Y, ,K ). If (A,E) is any soft set in X ,then (int(A,E)  int*( (A,E)) . Proof : Let (A,K) is any soft set in (Y, ,K ) .then int(A,K) is a soft open set in (Y, ,K). But is SSIg-continuous, so (int(A,K)) is SSIg-open by Proposition(1.19) we have (int(A,K))= int * (int(A,K)). But int(A,K)  (A,K), then (int(A,K))  (A,K). Hence, (int(A,K))  int * (A,K). □ Remark(2.27): The converse of Proposition (2.26) need not be true by the following example. Example: Let X={a,b} ,E = {e1, e2} , Y={d, c}, K = {k1, k2}, and τ = { , ,(F,E)}, = { , , (G,K)} be two soft topologies defined on X and Y respectively , where (F,E)={( ,{b}), ( , )} , (G,K)={( ,{d}), ( ,Y)} , define : → such that , and : → such that , . Then, (X,τ,E) (Y, ,K ) is a soft mapping . Since ({( ,{d}), ( ,Y)}) ={( ,{a}), ( ,X)} which is not SSIg-open set. So it is not SSIg-continuous mapping . On the other hand for each soft set (A,K) in SS(Y,K) ,then get (int(A,K))  int * (A,K). □ Remark(2.28): The equality in Proposition(2.26) need not be true in general by the following example. * ( , )cl A E  ( , ) pu f A E  1 k 2 k 1 3 ,h h ( ( , ) ) pu K cl f A E Y { } E I  E  E X K  K Y 1 e  2 e 1 k 2 k 1 h 1 2 ) , )( (h hu a u b    1 1(( , )) pu pu G Kf f  1k 2 k 1 h 1 e 2 e   1 pu f  1 pu f    pu f 1 pu f  1puf  1 pu f  1 pu f  1 pu f  1 pu f  1 pu f  { } E I  E  E X  K  K Y 1 e 2 e  1 k 2 k : pu f   1 1(( , )) pu pu G Kf f  1 k 2 k 1 e 2 e 1 pu f  1 pu f  180 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 Example : Let X={a,b} ,E = {e1, e2} ,Y={d, c}, K = {k1, k2}, and τ = { , ,(F,E)}, = { , , (G,K)} be two soft topologies defined on X and Y respectively , where (F,E)={( ,{a}), ( ,X)} , (G,K)={( ,{d}), ( ,Y)} , define : → such that , and : → such that , . Then, :(X,τ,E) (Y, ,K ) is a soft mapping . It is clear that is an SSIg-continuous mapping . On the other hand let (A,K) be soft set in SS(X,E) such that (A,K) ={( ,{d}), ( , )}, then int(A,E) = so = and ={( ,{a}), ( , )}, so ={( ,{a}), ( , )}, therefore int * (A,K)  (int(A,K)) . □ Proposition(2.29): Let : (X,τ,E) (Y, ,K ) be a SSIg-continuous mapping from a soft topological space (X,τ,E) with an ideal I to soft topological space (Y, ,K ). If (A,E) is any soft set in over X , then *int ( ( , )) (int ( , ))pu puf A E f A E . Proof : Let (A,E) be any soft set over X . Then is a soft set over Y and , is soft open set over Y. But is a SSIg-continuous mapping . Therefore , is a SSIg-open set over X. Then ∗ , , by Proposition(1.19). * 1 * 1 *int ( (int( ( , ))) int ( ( ( , ))) int ( , ) pu pu pu pu f A E f A E A Ef f    , therefore . Thus *int( ( , )) (int ( , ))pu puf A E f A E .□ Remark(2.30): The converse of Proposition(2.29) need not be true in general by the following example. Example: Let X={a,b}, E = {e1, e2}, Y={d, c}, K = {k1, k2}, and τ = { , ,(F,E)}, = { , , (G,K)} be two soft topologies defined on X and Y respectively, where (F,E)={( ,{b}), ( , )}, (G,K)={( ,{d}), ( ,Y)} , define : → such that , and : → such that , . Then, :(X,τ,E) (Y, ,K ) is a soft mapping. Since ({( ,{d}), ( ,Y)}) ={( ,{a}), ( ,X)} which is not SSIg-open set. So it is not SSIg-continuous mapping . On the other hand for each soft set (A,E) in SS(X,E) ,then *int( ( , )) (int ( , ))pu puf A E f A E .□ Remark(2.31): The equality in Proposition(2.29) need not be true in general by the following example. Example: Let X={a,b}, E = {e1, e2}, Y={d, c}, K = {k1, k2}, and τ = { , ,(F,E)}, = { , , (G,K)} be two soft topologies defined on X and Y respectively , where (F,E)={( ,{a}), ( ,X)}, (G,K)={( ,{d}), ( ,Y)}, define : → such that , and : → such that { } E I  E  E X K  K Y 1 e 2 e 1 k 2 k 1 1 )( kp e  2 2 )( kp e    1 k 2 k  K  1 ( ( , ))pu in Atf K  E  1 ( , )puf A K  1 e 2 e  * 1int ( , )puf A K   1 ( , ) pu f A K 1 e 2 e  1 pu f  1 pu f    ( , ) pu A Ef 1 (int( ( , ))) pu pu f A Ef *int ( , )A E { } E I  E  E X  K  K Y 1 e 2 e  1 k 2 k  1 1(( , )) pu pu G Kf f  1k 2k 1e 2 e { } E I  E  E X K  K Y 1 e 2 e 1 k 2 k 181 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 , . Then, : (X,τ,E) (Y, ,K) is a soft mapping . It is clear that is an SSIg-continuous mapping . On the other hand let (A,E) be soft set in SS(X,E) such that (A,E) = {( ,{a}), ( ,{b})}, then = {( ,{d}), ( ,{c})} and ={( ,{d}), ( ,{c})}, therefore *(int ( , )) int( ( , ))pu puf A E f A E .□ Definition(2.32) :Let (X,τX,A) and (Y,τY,B) be two soft topological spaces with an ideal I. Let : (X, τX,A) (Y, τY ,B ) be a mapping. If ((G,B)) is a SSIg-open set over X for each SSIg-open set (G,B) over Y, then is said to be SSIg-irresolute mapping. Proposition(2.33) :Every SSIg-irresolute mapping is an SSIg-continuous mapping. Proof: Let (X,τX,A) and (Y,τY,B) be two soft topological spaces with an ideal I. Let : (X, τX,A) (Y, τY ,B ) be an SSIg- irresolute mapping. To show that is SSIg-continuous. Let (G,B) be a soft open set over Y. Then by Corollary(1.15), (G,B) is an SSIg-open set over Y. Since is an SSIg- irresolute. Then ((G,B)) is a SSIg-open set over X. Therefore, is an SSIg-continuous mapping. □ Remark(2.34) : The converse of Proposition(2.33) need not be true in general by the following example. Example: Let X={a,b} ,E = {e1, e2}, Y={d, c}, K = {k1, k2}, and τ = { , ,(F,E)}, = { , } be two soft topologies defined on X and Y respectively , where (F,E)={( ,{b}),( , )}. Define the same as in Remark(2.31). It is clear that is an SSIg-continuous mapping. But is not SSIg- irresolute mapping since (G,K)= {( ,{d}), ( ,Y)} is an SSIg-open set over Y, but ({( ,{d}), ( ,Y)}) = {( ,{a}), ( ,X)} which is not SSIg-open set over X. □ Remark(2.35) :The notions SSIg-irresolute mapping and soft continuous mapping are independent. The observation follows by Example of Remark(2.3٤), such that is SSIg-continuous mapping but is not SSIg- irresolute mapping. Also in the following example shows that is SSIg- irresolute mapping but is not SSIg- continuous mapping. Example: Let X={a,b} ,E = {e1, e2} , Y={d, c}, K = {k1, k2}, and τX = { , }, = { , ,(G,K)} be two soft topologies defined on X and Y respectively , where (G,K)= {( ,{d}), ( ,Y)} Let define in Remark 2.34 . Since ({( ,{d}), ( ,Y)}) = {( ,{a}), ( ,X)} which is not SSIg-open set over X. Therefore is not SSIg-continuous mapping. But is an SSIg- irresolute mapping , since the inverse image under for every SSIg-open set, (SSIg-closed set),over Y is an SSIg-open set, (SSIg-closed set) . □   1 e 2 e *( ( , ))puf At Ein 1k 2k ( , )puf A E 1 k 2 k  1 pu f   1 pu f  { } E I  E  E X K  K Y 1 e 2 e  1 k 2 k 1 1(( , )) pu pu G Kf f  1k 2k 1 e 2 e { } E I  E  E X K  K Y 1 k 2 k 1 1(( , )) pu pu G Kf f  1k 2k 1e 2e 182 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 References 1.Alyasaa, J. and Narjis A.,(2015) Soft strongly generalized closed set with respect to an ideal in soft topological space, International Journal of Advanced Scientific and Technical Research,5, 319- 329. 2.Georgiou, D. N. and Megaritis A. C.,(2014) Soft set theory and topology, Appl. Gen. Topol. 15, 93-109. 3. Hussain, S.; and Ahmad, B., (2011) Some properties of soft topological spaces, Comput. Math. Appl., 62, 4058-4067. 4. Mustafa, H. I. and F. M. Sleim,(2014) Soft Generalized Closed Sets with Respect to an Ideal in Soft Topological Spaces, Applied Mathematics & Information Sciences, 8, 665-671. 5. Majumdar, P. and Samanta, S.K., (2010) On Soft Mappings, Computers and Mathematics with Applications, 60 , 2666–2672. 183 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد مجلة إبن الھيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (2) 2015 بالنسبة لفضاء تبولوجي ناعم مع مثاليSSIg التطبيقات من النمط اليسع جاسم بديوي داود نرجس عبد الجبار قسم الرياضيات /الھيثم) (ابنكلية التربية للعلوم الصرفة /جامعة بغداد ٢٠١٥أيار ٢٦في: البحث قبل ،٢٠١٥أذار ١٩ :في البحثاستلم الخالصة ، قدمت الدراسة نوعا جديدا من التطبيقات الناعمة في الفضاءات التبولوجية الناعمة مع مثالي، والتي في ھذا العمل - ، حيث درست مفاھيم مثل مستمرة، عكسI في فضاء تبولوجي ناعم مع مثاليSSIg التطبيقات من النمط قد ُسميت " بين ھذه األنواع من التطبيقات والعالقات SSIgمستمرة ، المفتوح ، المغلقة و متردد بالنسبة للتطبيقات الناعمة من النمط وتركيب اثنين من ھذه التطبيقات من النوع نفسه او من نوعين مختلفين، مع البراھين أو أمثلة مضادة. ، SSIg، االرتداد من النمط SSIg، عكس االستمرارية من النمط SSIgاالستمرارية من النمط المفتاحية:الكلمات .SSIg، التطبيق المفتوح من النمط SSIgالتطبيق المغلق من النمط