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J. Math. Comput. Sci. 7 (2017), No. 3, 522-536 ISSN: 1927-5307

ON PRE CONTINUOUS AND PRE IRRESOLUTE SOFT MULTIFUNCTIONS

ALKAN ¨OZKAN

Department of Mathematics Computer, University I˘gdır, I˘gdır, Turkey

Copyright c2017 A. ¨Ozkan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

Abstract. In this paper, we define the pre upper and lower continuous multi functions between soft topological spaces and study several properties of these multi functions. Then we define the pre upper and lower irresolute multi functions between soft topological spaces and study several properties of these multifunctions.

Keywords: soft sets; soft topology; soft multifunction; soft upper (lower) pre continuous; soft upper (lower) pre irresolute.

2010 AMS Subject Classification:54A10, 54A20, 54C08.

1. Introduction

The concept of soft set theory is fundamentally important in almost every scientific field. Soft set theory is a new mathematical tool for dealing with uncertainties. Also, this theory is a set associated with parameters and has been applied in several directions. Many complex problems in economics, engineering, environmental science, social science, medical science etc. can not be dealt with by classical methods, because classical methods have inherent difficulties. In 1999, Molodtsov [1] introduced the concept of soft set theory as a new approach for coping with uncertainties and these problems. Also he and et al. [2] presented the basic results of the soft set

E-mail address: alkan.ozkan@igdir.edu.tr Received March 8, 2017; Published May 1, 2017

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theory. He successfully applied the soft set theory into several directions such as smoothness of functions, theory of probability, game theory, Riemann Integration, Perron Integration, theory of optimization, theory of measurement etc. Maji and Roy [3] applied soft sets theory in a multicriteria decision making problems. Then, Shabir and Naz [4] introduced the notions of soft topological spaces. Cagman et al [5] defined a soft topological space. After that, Kharal and Ahmad,[6] defined a mapping on soft classes and studied properties of these mappings. Then Akdag and Erol studied on soft multifunctions in [9, 14, 15, 16].

In this paper, firstly we define the pre upper and lower continuous multifunction from a soft topological spaces(X,τ,E) to a soft topological spaces(Y,σ,K) and study several properties

of these continuous multifunctions. Finally we define the pre upper and lower irresolute multi-functions between soft topological spaces and study several properties of these multimulti-functions.

2. Preliminaries

Definition 2.1. [1] Let X be an initial universe and E be a set of parameters. Let P(X) denote the power set ofX and Abe a non-empty subset ofE. A pair(F,A)is called a soft set over X, whereF is a mapping given by F :A→P(X). In other words, a soft set over X is a parameterized family of subsets of the universeX. Fore∈A,F(e)may be considered as the set ofe−approximate elements of the soft set(F,A).

Definition 2.2. [7] A soft set (F,A)over X is called a null soft set, denoted byΦ, if e∈A, F(e) = /0. IfA=E, then the null soft set is called universal null soft set, denoted byΦ.

Definition 2.3. [7] A soft set(F,A)overX is called an absolute soft set, denoted byAe, ife∈A,

F(e) =X. IfA=E, then the absolute soft set is called universal soft set, denoted byXe.

Definition 2.4. [4] LetY be a non-empty subset ofX, thenYedenotes the soft set(Y,E)overX for whichY(e) =Y, for alle∈E.

Definition 2.5. [7] The union of two soft sets of(F,A)and(G,B)over the common universe

X is the soft set(H,C), whereC=A∪Band for alle∈C,

H(e) =     

   

F(e), ife∈A−B,

G(e), ife∈B−A,

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We write(F,A)∪e(G,B) = (H,C).

Definition 2.6. [7] The intersection (H,C)of two soft sets (F,A)and(G,B)over a common universe X, denoted (F,A)∩e(G,B), is defined asC =A∩B, and H(e) =F(e)∩G(e) for all

e∈C.

Definition 2.7. [7] Let(F,A)and(G,B)be two soft sets over a common universeX. (F,A)⊂e(G,B), ifA⊂BandF(e)⊂G(e)for alle∈A.

Definition 2.8. [8] For a soft set (F,A)over X the relative complement of (F,A) is denoted by (F,A)c and is defined by (F,A)c = (Fc,A), where Fc:A→P(X) is a mapping given by

Fc(α) =X−F(α)for allα ∈A.

Proposition 2.1.[10] Let(G,A),(H,A),(S,A),(T,A)be soft sets inX. Then the following are true;

(a) If(G,A)∩e(H,A) =Φ,then(G,A)⊂e(H,A)

c

. (b)(G,A)∪e(G,A)

c

=Xe.

(c) If(G,A)⊂e(H,A)and(H,A)⊂e(S,A),then(G,A)⊂e(S,A).

(d) If(G,A)⊂e(H,A)and(S,A)⊂e(T,A)then(G,A)∩e(S,A)⊂e(H,A)∩e(T,A). (e)(G,A)⊂e(H,A)if and only if(H,A)

c

e

⊂(G,A)c.

Definition 2.9. [13] The soft set(G,E)overX is called a soft point inXe, denoted byEex,if for

e∈E there existx∈X such thatG(e) ={x}andG(e0) = /0 for alle0∈E− {e}.

Definition 2.10. [9] The soft pointEexis said to be in the soft set(H,E), denoted byEex∈e(H,E), ifx∈H(e).

Proposition 2.2. [10] LetEex be a soft point and(H,E)be a soft set in Xe.IfEex∈e(H,E), then

Eex∈e/(H,E)c.

Definition 2.11.[4] Letτbe the collection of soft sets overX, thenτis said to be a soft topology

onX if satisfies the following axioms. (a)Φ,Xebelong toτ,

(b) the union of any number of soft sets inτ belongs toτ,

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The triplet(X,τ,E)is called a soft topological space overX. Let(X,τ,E)be a soft

topologi-cal space overX, then the members ofτ are said to be soft open sets inX. A soft set(F,A)over

X is said to be a soft closed set inX, if its relative complement(F,A)cbelongs toτ.

Definition 2.12.[17] Let(F,A)be any soft set of a soft topological space(X,τ,E).Then(F,A)

is called

(a) soft pre-open set ofX if(F,A)⊂inte (cl((F,A)))and (b) soft pre-closed set ofX ifcl(int(F,A))⊂e(F,A).

Proposition 2.3.[11] (a) Arbitrary union of soft pre-open sets is a soft pre-open sets. (b) Arbitrary intersection of soft pre-closed sets is a soft pre-closed set.

Definition 2.13. [11] Let(X,τ,E)be a soft topological space overX and(F,A)be a soft set

overX. Then:

(a) spint(F,A) =∪e

(O,E):(O,E) is a soft pre-open set and (O,E)⊂e(F,A) is called soft preinterior.

(b)spcl(F,A) =∩e

(H,E):(H,E) is a soft pre-closed set and (H,E)⊂e(F,A) is called soft preclosure.

Clearly that spint(F,A) is the largest soft pre-open set which is contained in (F,A) and

spcl(F,A)is the smallest soft pre-closed set over which contains(F,A).

Proposition 2.4.[11, 12] Let(F,A)be any soft set of a soft topological space(X,τ,E). Then,

(a)spcl(Xe−(F,A)) =Xe−spint(F,A). (b)spint(Xe−(F,A)) =Xe−spcl(F,A). (c)spcl(F,A) = (F,A)∪cle (int(F,A)). (d)spint(F,A) = (F,A)∩inte (cl(F,A)).

Proposition 2.5. [11] Let (F,A) be any soft set of a soft topological space (X,τ,E). Then,

(F,A)is soft pre-closed (soft pre-open) iff(F,A) =spcl(F,A)((F,A) =spint(F,A)).

Proposition 2.6. [11] Let(F,A)and(G,B)be two soft sets in a soft topological space(X,τ,E).

Then the following are hold; (a)spcl(Φ) =Φ.

(b)spcl(F,A)is soft pre-closed in(X,τ,E)for each soft subset(F,A)ofX.

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(d)spcl(F,A)e∪spcl(G,B)⊂e spcl (F,A)e∪(G,B)

. (e)spcl (F,A)∩e(G,B)

e

⊂spcl(F,A)∩spcle (G,B).

Remark 2.1. [11] Similar results hold for soft preinteriors.

Definition 2.14. [9] Let S(X,E) and S(Y,K) be two soft classes. Let u:X →Y be mul-tifunction and p:E →K be mapping. Then a soft multifunction F :S(X,E) →S(Y,K) is defined as follows: For a soft set(G,A)in(X,E),(F(G,A),K)is a soft set inS(Y,K)given by

F(G,A) (k) =   

 

e∈p−1(k)u(G(e)), if p

−1(k)A6=

∅, otherwise

fork∈K.(F(G,A),K)is called a soft image of a soft set(G,A).Moreover,F(G,A) =∪{Fe (E

x

e):Eex∈e(G,A)}for a soft subset(G,A) ofX .

Definition 2.15. [9] LetF:S(X,E)→S(Y,K)be a soft multifunction.

The soft upper inverse image of(H,K)denoted byF+(H,K)and defined as

F+(H,K) =nEex∈eXe:F(Eex)⊂e(H,K),e∈E o

.

The soft lower inverse image of(H,K)denoted byF−(H,K)and defined as

F−(H,K) =nEex∈eXe:F(Eex)∩e(H,K)6=Φ o

.

Definition 2.16. [9] LetF :S(X,E)→S(Y,K)andG:S(X,E)→S(Y,K)be two soft multi-functions. Then,F equal toGifF(Eex) =G(Eex),for eachEex∈eXe.

Definition 2.17. [9] The soft multifunctionF :S(X,E)→S(Y,K)is called surjective if pand

uare surjective.

Theorem 2.1. [9] LetF:S(X,E)→S(Y,K)be a soft multifunction Then, for soft sets(F,E),

(G,E)and for a family of soft sets(Gi,E)iI in the soft classS(X,E)the following are hold;

(a)F(Φ) =Φ.

(b)F(Xe)⊂eYe.

(c)F (G,A)e∪(H,B)=F(G,A)e∪F(H,B)in generalF ∪ei(Gi,E)

=∪eiF(Gi,E).

(d)F (G,A)∩e(H,B)

e

⊂F(G,A)∩Fe (H,B)in generalF ∩ei(Gi,E)

e

⊂∩eiF(Gi,E).

(e) If(G,E)⊂e(H,E),thenF(G,E)e⊂F(H,E).

Theorem 2.2. [9] LetF:S(X,E)→S(Y,K)be a soft multifunction Then, for soft sets(G,K),

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(b)F−(eY) =XeandF+(eY) =Xe.

(c)F− (G,K)e∪(H,K)=F−(G,K)e∪F−(H,K). (d)F+(G,K)∪Fe

+(H,K)

e

⊂F+ (G,K)

e

∪(H,K). (e)F− (G,K)e∩(H,K)⊂Fe −(G,K)e∩F−(H,K). (f)F+(G,K)∩Fe

+(H,K) =F+ (G,K)

e

∩(H,K). (g) If(G,K)⊂e(H,K),thenF−(G,K)⊂Fe −(H,K)andF

+(G,K)

e

⊂F+(H,K).

Proposition 2.7. [9] Let(Gi,K)be soft sets overY for eachi∈I.The follows are true for a soft

multifunctionF :(X,τ,E)→(Y,σ,K):

(a)F−(∪e

i∈I(Gi,K)) =i∪e∈I(F

(G

i,K))

(b)∩e

i∈I(F

+(G

i,K)) =F+(∩e

i∈I(Gi,K))

(c) ∪e

i∈IF

+(G

i,K)⊂Fe

+

e

i∈I(Gi,K)

(d)F−

e

i∈I(Gi,K)

e

⊂∩e

i∈I(F

(G

i,K)).

Proposition 2.8. [9] Let F :S(X,E)→S(Y,K)be a soft multifunction. Then the follows are true:

(a)(G,A)⊂Fe

+(F(G,A))

e

⊂F−(F(G,A))for a soft subset(G,A)inX.IfFis surjectice then (G,A) =F+(F(G,A)) =F−(F(G,A)).

(b)F(F+(H,B))⊂e(H,B)⊂Fe (F

(H,B))for a soft subset(H,B)inY. (c) For two soft subsets(H,B)and(U,C)inY if(H,B)∩e(U,C) =Φ,thenF

+(H,B)

e

∩F−(U,C) =

Φ.

Proposition 2.9. [9] Let F :S(X,E)→S(Y,K) and G:S(Y,K)→ S(Z,L) be two soft multi-function. Then the follows are true:

(a)(F−)−=F.

(b) For a soft subset (T,C) in Z, (GoF)−(T,C) =F−(G−(T,C)) and (GoF)+(T,C) =

F+(G+(T,C)).

Proposition 2.10. [9] Let(G,K) be a soft set overY. Then the followings are true for a soft multifunctionF :S(X,E)→S(Y,K)

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3. Upper and Lower Pre Continuity of Soft Multifunctions

Definition 3.1. Let(X,τ,E)and(Y,σ,K)be two soft topological spaces.Then a soft

multi-functionF :(X,τ,E)→(Y,σ,K)is said to be;

(a) soft upper pre continuous at a soft pointEex∈eXeif for every soft open set(G,K)such that

F(Eex)⊂e(G,K),there exists a soft pre open neighborhood(P,E)ofEexsuch thatF(Eez)⊂e(G,K) for allEez∈e(P,E).

(b) soft lower pre continuous at a soft point Eex∈eXe if for every soft open set (G,K) such that F(Eαx)∩e(G,K)6= Φ, there exists a soft pre open neighborhood (P,E) of E

x

e such that F(Eez)∩e(G,K)=6 Φfor allEez∈e(P,E).

(c) soft upper(lower) pre continuous ifF has this property at every soft point ofX.

Theorem 3.1. For a soft multifunctionF :(X,τ,E)→(Y,σ,K) the following statements are

equivalent;

(a)F is soft upper pre continuous.

(b)F+(V,K)is a soft pre open set inX for each soft open set(V,K)ofY. (c)F−(H,K)is a soft pre closed set inXfor each soft closed set(H,K)ofY. (d)spcl(F−(V,K))⊂Fe

(cl(V,K))for each soft subset(V,K)ofY.

(e)F+(H,K)is a soft pre neighborhood ofEex, for each soft pointEex inX and for each soft neighborhood(H,K)ofF(Eex).

(f)for each soft pointEex inX and for each soft neighborhood(V,K)ofF(Eex),there exists a soft pre neighborhood(U,E)ofEexsuch thatF(U,E)⊂e(V,K).

(g)F+(int(H,K))⊂spinte (F

+(H,K))for each soft subset(H,K)ofY.

(h)F+(G,K)⊂inte (cl(F

+(G,K)))for each soft open set(G,K)ofY.

(i)cl(F+(V,K))is a soft neighborhood ofEex,for each soft pointEex inX and for each soft neighborhood(V,K)ofF(Eex).

Proof.(a)⇒(b)Let(V,K)be any soft open set ofY andEex∈F+(V,K).

ThenF(Eex)⊂Ve and there exists soft preopen set(U,E)containingE

x

esuch thatF(U,E)⊂e(V,K). SinceEex∈e(U,E)⊂inte (cl(U,E))⊂inte (cl(F

+(V,K))),we haveF+(V,K)

e

⊂int(cl(F+(V,K))).

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(b)⇔(c)Let(H,K)be any soft closed set ofY.Then,Ye−(H,K)is a soft open set inY.By (b), F+Ye−(H,K)

=Xe−F−(H,K) is a soft pre open set inX. Then X−F−(H,K) is a soft pre open set inX.Thus,F−(H,K)is a soft pre closed set inX.

(c)⇒(d) Let (V,K) be any soft subset ofY. Sincecl(V,K) is a soft closed set inY,then

F−(cl(V,K))is soft pre closed set inX.Thusspcl(F−(V,K))⊆F−(cl(V,K)).

(d)⇒(c)Let (H,K)be any soft closed set ofY.Then spcl(F−(H,K))⊂Fe −(cl(H,K)) =

F−(H,K)and henceF−(H,K)is soft pre closed set inX.

(b)⇒(e)LetEex∈eXeand(H,K)be any soft neighborhood ofF(Eex).Then there exists a soft open set(G,K)ofY such thatF(Eex)⊂e(G,K)⊂e(H,K).Then we haveE

x e∈Fe

+(G,K)

e

⊂F+(H,K)

and sinceF+(G,K)is soft pre open set inX thenF+(H,K)is a soft pre neighborhood ofEex.

(e)⇒(f) Let Eex∈eXe and (V,K) be any soft neighborhood of F(Eex). If we take (U,E) =

F+(V,K),then(U,E)is soft pre neighborhood ofEexandF(U,E)⊂e(V,K).

(f)⇒(a) Let Eex∈eXe and (V,K) be any soft open set ofY such that F(Eex)⊂e(V,K). Then (V,K) is a soft neighborhood of F(Eex) and there exists (U,E) soft pre neighborhood of Eex

such that F(U,E)⊂e(V,K). Therefore, there exists soft pre open set (H,E) in X such that

Eex∈e(H,E)⊂e(U,E) and hence F(H,E)⊂e(V,K). This implies that F is soft upper pre con-tinuous.

(b)⇒(g)For any subset(H,K)ofY, int(H,K)is soft open set inY.ThenF+(int(H,K))is soft pre open set inX.

HenceF+(int(H,K)) =spint(F+(int(H,K)))⊂spinte (F

+(H,K)).

(g)⇒(b)Let(V,K)be any soft open set ofY.Then

F+(V,K) =F+(int(V,K))⊂spinte (F

+(V,K))and henceF+(V,K)is soft pre open set inX.

(b)⇔(h)Obvious.

(h)⇒(i)LetEex∈eXeand(V,K)be any soft open neighborhood ofF(Eex). ThenEex∈Fe

+(V,K) =

int(F+(V,K))⊂inte (cl(F

+(V,K)))

e

⊂cl(F+(V,K)) and hence cl(F+(V,K)) is a neighborhood ofEex.

(i)⇒(h)Let(G,K)be any soft open set ofYandEex∈Fe

+(G,K).Thencl(F+(G,K))is a soft

neighborhood ofEexand thusEex∈inte (cl(F

+(G,K))).HenceF+(G,K)

e

⊂int(cl(F+(G,K))).

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(a)F is soft lower pre continuous.

(b)F−(V,K)is a soft pre open set inX for each soft open set(V,K)ofY.

(c)F+(H,K)is a soft pre closed set inX for each soft closed set(H,K)ofY.

(d)spcl(F+(V,K))⊂Fe

+(cl(V,K))for each subset(V,K)ofY.

(e)for eachEex∈eXeand for each soft neighborhood(H,K)which intersectsF(Eex),F−(H,K) is a soft pre neighborhood ofEex.

(f)for eachEex∈eXeand for each soft neighborhood(V,K)which intersectsF(Eex),there exists a soft pre neighborhood(U,E)ofEexsuch that aF(Eez)∩e(V,K)6=Φfor everyEez∈e(U,E).

(g)F−(int(G,K))⊂spinte (F

(G,K))for each soft subset(G,K)ofY. (h)F−(H,K)⊂inte (cl(F

(H,K)))for each soft open set(H,K)ofY.

(i)for eachEex∈eXeand for each soft neighborhood(V,K)which intersectsF(Eex),cl(F−(V,K)) is a soft neighborhood ofEex.

Proof.The proof is similar to previous theorem.

Definition 3.2. [9] Let (X,τ,E) and (Y,σ,K) be two soft topological spaces. Then a soft multifunctionF :(X,τ,E)→(Y,σ,K)is said to be

(a) soft upper continuous at a soft pointEex∈eXeif for each soft open(G,K)such thatF(Eex)⊂e(G,K), there exists a soft open neighborhoodP(Eex)ofEexsuch thatF(Eez)⊂e(G,K)for allE

z e∈Pe (E

x e).

(b) soft lower continuous at a pointEex∈eXeif for each soft open(G,K)such thatF(Eex)∩e(G,K)6=

Φ, there exists a soft open neighborhood P(Eex) of Eex such that F(Eez)∩e(G,K) 6=Φ for all

Eez∈Pe (Eex).

(c) soft upper(lower) continuous ifF has this property at every soft point ofX.

Proposition 3.1.[9] A soft multifunctionF:(X,τ,E)→(Y,σ,K) is;

(a)soft upper continuous if and only if F+(G,K)is soft open in X, for every soft open set (G,K).

(b) soft lower continuous if and only if F−(G,K) is soft open set in X, for every soft set (G,K).

Remark 3.1. Every soft upper (lower) continuous multifunctions is soft upper (lower) pre continuous. But the converse is not true as shown by the following examples.

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τ=

n

Φ,Xe,(G,E) o

,where(G,E) ={(e1,{x1}),(e2,{x2})}and

σ =

n

Φ,Ye,(H,K) o

,where(H,K) ={(k1,{y1}),(k2,{y2})}.

Letu:X →Y be multifunction defined as:u(x1) ={y1,y2},u(x2) ={y2} and p:E→K be mapping defined as:p(e1) =k1, p(e2) =k2.

Then the soft multifunctionF :(X,τ,E)→(Y,σ,K)is soft upper pre continuous but is not

soft upper continuous .Because for a soft open set(H,K)inY,F+(H,K) ={(e2,{x2})}is soft pre open set but is not soft open set inX.

Example 3.2. LetX ={x1,x2},Y ={y1,y2},E ={e1,e2},K={k1,k2}.

τ=

n

Φ,Xe,(G,E) o

,where(G,E) ={(e1,{x1}),(e2,{x2})}and

σ =

n

Φ,Ye,(H,K) o

,where(H,K) ={(k1,{y1}),(k2,{y2})}.

Letu:X →Y be multifunction defined as:u(x1) ={y1,y2},u(x2) ={y2}

and p:E→Kbe mapping defined as: p(e1) =k1,p(e2) =k2.

Then the soft multifunctionF :(X,τ,E)→(Y,σ,K) is soft lower pre continuous but is not

soft lower continuous .Because for a soft open set(H,K)inY,F−(H,K) ={(e1,{x2}),(e2,X)} is soft pre open set but is not soft open set inX.

4. Upper and Lower Irresolute Soft Multifunctions

Definition 4.1. Let(X,τ,E) and (Y,σ,K) be two soft topological spaces. Then a soft

multi-functionF :(X,τ,E)→(Y,σ,K)is said to be;

(a) soft upper pre irresolute at a soft pointEex∈eXeif for every soft pre open set(G,K)such that

F(Eex)⊂e(G,K),there exists a soft pre open neighborhood(P,E)ofEexsuch thatF(Eez)⊂e(G,K) for allEez∈e(P,E).

(b) soft lower pre irresolute at a soft pointEex∈eXe if for every soft pre open set(G,K)such that F(Eex)∩e(G,K)6= Φ, there exists a soft pre open neighborhood (P,E) of E

x

e such that F(Eez)∩e(G,K)6=Φfor allE

z

e∈e(P,E).

(c) soft upper (lower) pre irresolute ifF has this property at every soft point ofX.

Example 4.1. LetX ={x1,x2},Y ={y1,y2},

E={e1,e2},K={k1,k2}. τ = n

Φ,Xe,(F1,E),(F2,E),(F3,E) o

, where

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σ =

n

Φ,Ye,(H1,K),(H2,K),(H3,K) o

,where

(H1,K) ={(k1,{y1})},(H2,K) ={(k2,{y2})},(H3,K) ={(k1,{y1}),(k2,{y2})}.

Letu:X →Y be multifunction defined as:u(x1) ={y1,y2},u(x2) ={y2} and p:E →K be mapping defined as: p(e1) =k1, p(e2) =k2. Then the soft multifunction F :(X,τ,E)→

(Y,σ,K)is soft upper pre irresolute. Because for the soft pre open sets{(k1,{y1})},{(k2,{y2})},

{(k1,{y1}),(k2,{y2})},{(k1,{y1}),(k2,Y)},{(k1,Y),(k2,{y2})}inY, F+({(k1,{y1})}) =Φ,

F+({(k2,{y2})}) ={(e2,{x2})}=F+({(k1,{y1}),(k2,{y2})}),

F+({(k1,{y1}),(k2,Y)}) ={(e2,X)},

F+({(k1,Y),(k2,{y2})}) ={(e1,X),(e2,{x2})}is soft pre open set inX. Example 4.2. LetX ={x1,x2},Y ={y1,y2},

E={e1,e2},K={k1,k2}.

τ=

n

Φ,Xe,(F1,E),(F2,E),(F3,E),(F4,E),(F5,E), (F6,E),(F7,E) o

, where (F1,E) ={(e1,{x1})},(F2,E) ={(e1,{x1}),(e2,{x1})},(F3,E) ={(e1,X)},

(F4,E) ={(e1,{x1}),(e2,X)},(F5,E) ={(e1,X),(e2,{x1})},(F6,E) ={(e2,{x1})},

(F7,E) ={(e2,X)}andσ=

n

Φ,Ye,(H1,K),(H2,K),(H3,K) o

,where(H1,K) ={(k1,{y1})}, (H2,K) ={(k2,{y2})},(H3,K) ={(k1,{y1}),(k2,{y2})}.

Letu:X →Y be multifunction defined as:u(x1) ={y1,y2},u(x2) ={y2} and p:E →K be mapping defined as: p(e1) =k1, p(e2) =k2. Then the soft multifunction F :(X,τ,E)→

(Y,σ,K)is soft lower pre irresolute. Because for the soft pre open sets{(k1,{y1})},{(k2,{y2})},

{(k1,{y1}),(k2,{y2})},{(k1,{y1}),(k2,Y)},{(k1,Y),(k2,{y2})}inY, F−({(k1,{y1})}) ={(e1,{x1})},F−({(k2,{y2})}) ={(e2,X)},

F−({(k1,{y1}),(k2,{y2})}) ={(e1,{x1}),(e2,X)}=F−({(k1,{y1}),(k2,Y)}),

F−({(k1,Y),(k2,{y2})}) =Yeis soft pre open set inX.

Theorem 4.1. For a soft multifunction F :(X,τ,E)→ (Y,σ,K) the following properties are

equivalent;

(a)F is soft upper pre irresolute.

(b)For each soft pointEex∈eXeand each soft pre open(G,K)containingF(Eex),Eex∈inte (cl(F

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(c)For each soft pointEex∈eXeand each soft pre open(G,K)containingF(Eex), there exists a soft open set(U,E)such thatEex∈e(U,E)⊂cle (F

+(G,K)).

(d)For each soft pointEex∈eXe and each soft pre neighorhood(G,K)ofF(Eex), F+(G,K)is soft pre neighborhood ofEex.

(e) For each soft point Eex∈eXe and each soft pre neighorhood (G,K) ofF(Eex), there exists (U,E)is soft pre neighborhood ofEexsuch thatF(U,E)⊂e(G,K).

(f)F+(G,K)is soft pre open set, for every soft pre open set(G,K)inY.

(g)F−(G,K)is soft pre closed set, for every soft pre closed set(G,K)inY.

(h)spcl(F−(G,K))⊂e(F

(spcl(G,K))),for every soft set(G,K)inY.

Proof. (a)⇒(b) Let (G,K) be any soft pre open set and F(Eex)⊂e(G,K). By (a),there ex-ists (P,E) soft pre open set containing Eex such that F(Eez)⊂e(G,K) for all Eez∈e(P,E). Then

F(P,E)⊂e(G,K).Since(P,E)soft pre open set, then we have

Eex∈e(P,E)⊂inte (cl(P,E))⊂inte (cl(F

+(G,K))).

(b)⇒(c)If we takeint(cl(F+(G,K))) = (U,E)in(b),then

Eex∈e(U,E)⊂cle (F

+(G,K)).

(c)⇒(d)Let(G,K)be soft pre neighorhood ofF(Eex),thenEex∈Fe +(G,K)and by(c),there

exists soft open setEex∈e(U,E)⊂cle (F

+(G,K)).ThenEx

e∈e(U,E) =int(U,E)⊂inte (cl(F

+(G,K))).

Thus we haveF+(G,K)⊂inte (cl(F

+(G,K)))andF+(G,K)is soft pre neighorhood ofEx e.

(d)⇒(e)Let(G,K)be soft pre neighorhood ofF(Eex).If we take(U,E) =F+(G,K),then by(d),(U,E)is soft pre neighorhood ofEexandF(U,E)⊂e(G,K).

(e)⇒(f) Let(G,K) be soft pre open set and Eex∈Fe

+(G,K). By(e), there exists(U,E)a

soft pre neighborhood ofEexsuch thatF(U,E)⊂e(G,K).ThenEex∈e(U,E)⊂Fe

+(G,K)and thus

Eex∈e(U,E)⊂inte (cl(U,E))⊂inte (cl(F

+(G,K))).ThereforeF+(G,K)

e

⊂int(cl(F+(G,K))).Then

F+(G,K)is soft pre open set.

(f)⇒(g)Let(H,K)be soft pre closed set. ThenYe−(H,K)is soft pre open set and by(f),

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(g)⇒(h)Let(G,K)be any soft subset. Thenspcl(G,K)is soft pre closed set and by(g), F−(spcl(G,K))is soft pre closed set. Therefore spcl(F−(G,K))⊂spcle (F−(spcl(G,K))) =

F−(spcl(G,K)).

(h)⇒(a)Let(G,K)be soft pre open set andEex∈Fe

+(G,K).Then

F(Eex)⊂e(G,K)andF(E

x e)∩e

e

Y−(G,K)=Φ.Thus Eex∈Fe/ −

e

Y−(G,K)=F−Ye−spint(G,K)

=F−spclYe−(G,K)

and by(h),Eex∈spcle/

F−Ye−(G,K)

.Then there exists soft pre open set(U,E) containing

Eex such that (U,E)∩Fe −

e

Y−(G,K)=Φ.Then,F(U,E)⊂e(G,K).ThereforeF is upper pre irresolute. Theorem 4.2. For a soft multifunction F :(X,τ,E)→ (Y,σ,K) the following properties are

equivalent;

(a)F is soft lower pre irresolute. (b)Eex∈inte (cl(F

(G,K)))for each soft pointEx

e∈eXeand each soft pre open(G,K)such that

F(Eex)∩e(G,K)6=Φ.

(c)For each soft pointEex∈eXeand each soft pre open(G,K)containingF(Eex), there exists a soft open set(U,E)such that

Eex∈e(U,E)⊂cle (F

(G,K)).

(d)For each soft pre open(G,K)and each soft pointEex∈Fe −(G,K), there exists a soft open set(U,E)such thatEex∈e(U,E)⊂e(F

(G,K)).

(e)F−(G,K)is soft pre open set, for every soft pre open set(G,K)inY.

(f)F+(G,K)is soft pre closed set, for every soft pre closed set(G,K)inY.

(g)cl(int(F+(G,K)))⊂e(F

+(spcl(G,K))),for every soft subset(G,K)inY.

(h)F(cl(int(H,E)))⊂spcle (F(H,E)),for every soft subset(H,E)inX. (k)F(spcl(H,E))⊂spcle (F(H,E))),for every soft subset(H,E)inX. (l)spcl(F+(G,K))⊂e(F

+(spcl(G,K))),for every soft set(G,K)inY.

Proof.(a)⇒(b),(b)⇒(c)and(c)⇒(d)is similar to previous theorem.

(d)⇒(e)Let (G,K) be soft pre open set andEex∈Fe −(G,K),then by (d),there exists pre soft open set(U,E)such thatEex∈e(U,E)⊂e(F

(G,K)).Then

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F−(G,K)⊂inte (cl(F

(G,K)))andF(G,K)is soft pre open set.

(e)⇒(f)Let(H,K)be soft pre closed set. ThenYe−(H,K)is soft pre open set and by(e),

F−(eY−(H,K)) =Ye−(F+(H,K))is soft pre open set. ThereforeF+(H,K)is soft pre closed set.

(f)⇒(g)Let(G,K)be any soft subset. Thenspcl(G,K)is soft pre closed set and by(f), F+(spcl(G,K))is soft pre closed set. Therefore

cl(int(F+(G,K)))⊂cle (int(F

+(spcl(G,K))))

e

⊂spcl(F+(spcl(G,K))) =F+(spcl(G,K)).

(g)⇒(h)Let(H,E) be any soft subset. We know that(H,E)⊂Fe

+(F(H,E))and by(g),

cl(int(H,E))⊂cle (int(F

+(F(H,E))))

e

⊂F+(spcl(F(H,E))).

ThusF(cl(int(H,E)))⊂spcle (F(H,E)).

(h)⇒(k)Let(H,E)be any soft subset. We know that

spcl(H,E) = (H,E)∪cle (int(H,E)). Then

F(spcl(H,E)) =F (H,E)∪cle (int(H,E))

e

⊂F(H,E)∪Fe (cl(int(H,E)))⊂espcl(F(H,E)).

(k)⇒(l)Let(G,K)be any soft subset. ForF+(G,K)by(k), F(spcl(F+(G,K)))⊂e spcl(F(F

+(G,K)))and thus we have

spcl(F+(G,K))⊂Fe

+(spcl(F(F+(G,K))))

e

⊂F+(spcl(G,K)).

(l)⇒(a)Let(G,K)be any soft subset withEex∈Fe

(G,K).Then F(Ex

e)∩e(G,K)6=Φand

F(Eex)⊂eYe−(G,K).ThenEex∈Fe/ +

e

Y−(G,K)

.Since(G,K)soft pre open set thenYe−(G,K) is soft pre closed set and Eex∈Fe/ +

spcl

e

Y−(G,K). By (l), Eex∈spcle/

F+

e

Y−(G,K)

and there exists soft pre open set(U,E)containingEexsuch thatΦ= (U,E)∩Fe

+

e

Y−(G,K)=

(U,E)∩eYe−(F−(G,K))and(U,E)⊂Fe

(G,K).Therefore,F is soft lower pre irresolute.

5. Conclusion

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Conflict of Interests

The authors declare that there is no conflict of interests. REFERENCES

[1] D. Molodtsov, Soft set theory-First results, Comput. Math. Appl., 37 (4/5) (1999), 19-31.

[2] D.V. Kovkov, V. M. Kolbanov and D.A. Molodtsov, Soft sets theory-based optinization, J. Comput. Syst. Sci. Int. 46 (6) (2007), 872-880.

[3] P. K. Maji and A. R. Roy, An Application of Soft Sets in a Decision Making Problem, Comput. Math. Appl., 44 (2002), 1077-1083.

[4] M. Shabir and M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786-1799. [5] N. Cagman, S. Karatas¸ and S. Engino˘glu, Soft topology, Comput. Math. Appl., 62 (2011), 351-358. [6] A. Kharal and B. Ahmad, Mappings on soft classes. New Math. Nat. Comput., 7(3) (2011), 471-481. [7] P.K. Maji, R. Biswas and A.R. Roy, Soft set theory. Comput. Math. Appl., 45 (2003), 555-562.

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[10] I. Zorlutuna, M. Akda˘g, W. K. Min and S. Atmaca, Remarks on soft topological spaces, Ann. Fuzzy Math. Inform., 3(2) (2012), 171-185.

[11] M. Akda˘g and A. ¨Ozkan, On Soft Preopen Sets and Soft Pre Separation Axioms, Gazi Univ. J. Sci. 27(4) (2014), 1077-1083.

[12] M. Akda˘g and A. ¨Ozkan, Soft b-open sets and soft b-continuous functions, Math Sci 8 (2014), Article ID 124.

[13] S. K. Nazmul and S. K. Samanta. Neighbourhood properties of soft topological spaces, Annal. Fuzzy Math. Info., 6(1) (2013), 1-15.

[14] M. Akda˘g and F. Erol, Upper and Lower Continuity of Soft Multifunctions, Appl. Math. Inf. Sci., 7 (2013), 1-8.

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References

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