 # Neutrosophic soft sets with applications in decision making

We firstly present definitions and properties in study of Maji maji-2013 on neutrosophic soft sets. We then give a few notes on his study. Next, based on Çağman cagman-2014, we redefine the notion of neutrosophic soft set and neutrosophic soft set operations to make more functional. By using these new definitions we construct a decision making method and a group decision making method which selects a set of optimum elements from the alternatives. We finally present examples which shows that the methods can be successfully applied to many problems that contain uncertainties.

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## 1 Introduction

Many problems including uncertainties are a major issue in many fields of real life such as economics, engineering, environment, social sciences, medical sciences and business management. Uncertain data in these fields could be caused by complexities and difficulties in classical mathematical modeling. To avoid difficulties in dealing with uncertainties, many tools have been studied by researchers. Some of these tools are fuzzy sets , rough sets  and intuitionistic fuzzy sets . Fuzzy sets and intuitionistic fuzzy sets are characterized by membership functions, membership and non-membership functions, respectively. In some real life problems for proper description of an object in uncertain and ambiguous environment, we need to handle the indeterminate and incomplete information. But fuzzy sets and intuitionistic fuzzy sets don’t handle the indeterminant and inconsistent information. Samarandache  defined the notion of neutrosophic set which is a mathematical tool for dealing with problems involving imprecise and indeterminant data.

Molodtsov introduced concept of soft sets  to solve complicated problems and various types of uncertainties. In , Maji et al. introduced several operators for soft set theory: equality of two soft sets, subsets and superset of soft sets, complement of soft set, null soft sets and absolute soft sets. But some of these definitions and their properties have few gaps, which have been pointed out by Ali et al. and Yang . In 2010, Çağman and Enginoğlu  made some modifications the operations of soft sets and filled in these gap. In 2014, Çağman  redefined soft sets using the single parameter set and compared definitions with those defined before.

Maji  combined the concept of soft set and neutrosophic set together by introducing a new concept called neutrosophic soft set and gave an application of neutrosophic soft set in decision making problem. Recently, the properties and applications on the neutrosophic sets have been studied increasingly [2, 3, 7, 8].The propose of this paper is to fill the gaps of the Maji’s neutrosophic soft set  definition and operations redefining concept of neutrosophic soft set and operations between neutrosophic soft sets. First, we present Maji’s definitions and operations and we verify that some propositions are incorrect by a counterexample. Then based on Çağman’s  study we redefine neutrosophic soft sets and their operations. Also, we investigate properties of neutrosophic soft sets operations. Finally we present an application of a neutrosophic soft set in decision making.

## 2 Preliminaries

In this section, we will recall the notions of neutrosophic sets  and soft sets . Then, we will give some properties of these notions. Throughout this paper , and denote initial universe, set of parameters and power set of , respectively.

###### Definition 2.1

 A neutrosophic set on the universe of discourse is defined as

 A={⟨x,TA(x),IA(x),FA(x)⟩:x∈X}

where and . From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of . But in real life application in scientific and engineering problems it is difficult to use neutrosophic set with value from real standard or non-standard subset of . Hence we consider the neutrosophic set which takes the value from the subset of .

###### Definition 2.2

 Let consider a nonempty set , . A pair is called a soft set over , where is a mapping given by .

###### Example 2.3

Let be the universe which are eight houses and be the set of parameters. Here, stand for the parameters “modern”, “with parking”, “expensive”, “cheap”, “large” and “near to city” respectively. Then, following soft sets are described respectively Mr. A and Mr. B who are going to buy

 F = {(e1,{x1,x3,x4}),(e2,{x1,x4,x7,x8}),(e3,{x1,x2,x3,x8})} G = {(e2{x1,x3,x6}),(e3,X),(e5,{x2,x4,x4,x6})}.

From now on, we will use definitions and operations of soft sets which are more suitable for pure mathematics based on study of Çağman .

###### Definition 2.4

 A soft set over is a set valued function from to

. It can be written a set of ordered pairs

 F={(e,F(e)):e∈E}.

Note that if , then the element is not appeared in . Set of all soft sets over is denoted by .

###### Definition 2.5

 Let . Then,

1. If for all , is said to be a null soft set, denoted by .

2. If for all , is said to be absolute soft set, denoted by .

3. is soft subset of , denoted by , if for all .

4. , if and .

5. Soft union of and , denoted by , is a soft set over and defined by such that for all .

6. Soft intersection of and , denoted by , is a soft set over and defined by such that for all .

7. Soft complement of is denoted by and defined by such that for all .

###### Example 2.6

Let us consider soft sets , in the Example 2.3. Then, we have

 F~∪G = {(e1,{x1,x3,x4}),(e2,{x1,x3,x4,x6,x7,x8}), (e3,X),(e5,{x2,x4,x4,x6})} F~∩G = {(e2{x1}),(e3,{x1,x2,x3,x8})} F~c = {(e1,{x2,x5,x6,x7,x8}),(e2,{x2,x3,x5,x6}), (e3,{x4,x5,x6,x7}),(e4,X),(e5,X),(e6,X)}.
###### Definition 2.7

 Let be an initial universe set and be a set of parameters. Consider . Let denotes the set of all neutrosophic sets of . The collection is termed to be the soft neutrosophic set over , where is a mapping given by

For illustration we consider an example.

###### Example 2.8

Let be the set of houses under consideration and is the set of parameters. Each parameter is a neutrosophic word or sentence involving neutrosophic words. Consider . In this case, to define a neutrosophic soft set means to point out beautiful houses, wooden houses, houses in the green surroundings and so on. Suppose that, there are five houses in the universe given by, and the set of parameters , where stands for the parameter ’beautiful’, stands for the parameter ’wooden’, stands for the parameter ’costly’ and the parameter stands for ’moderate’. Suppose that,

 F(beautiful) = {⟨h1,0.5,0.6,0.3⟩,⟨h2,0.4,0.7,0.6⟩,⟨h3,0.6,0.2,0.3⟩, ⟨h4,0.7,0.3,0.2⟩,⟨h5,0.8,0.2,0.3⟩}, F(wooden) = {⟨h1,0.6,0.3,0.5⟩,⟨h2,0.7,0.4,0.3⟩,⟨h3,0.8,0.1,0.2⟩, ⟨h4,0.7,0.1,0.3⟩,⟨h5,0.8,0.3,0.6⟩}, F(costly) = {⟨h1,0.7,0.4,0.3⟩,⟨h2,0.6,0.7,0.2⟩,⟨h3,0.7,0.2,0.5⟩, ⟨h4,0.5,0.2,0.6⟩,⟨h5,0.7,0.3,0.4⟩}, F(moderate) = {⟨h1,0.8,0.6,0.4⟩,⟨h2,0.7,0.9,0.6⟩,⟨h3,0.7,0.6,0.4⟩, ⟨h4,0.7,0.8,0.6⟩,⟨h5,0.9,0.5,0.7⟩}.

The neutrosophic soft set is a parameterized family of all neutrosophic sets of and describes a collection of approximation of an object.

Thus we can view the neutrosophic soft set as a collection of approximation as below:

 (F,A) = {beautifulhouses={⟨h1,0.5,0.6,0.3⟩,⟨h2,0.4,0.7,0.6⟩, ⟨h3,0.6,0.2,0.3⟩,⟨h4,0.7,0.3,0.2⟩,⟨h5,0.8,0.2,0.3⟩}, woodenhouses={⟨h1,0.6,0.3,0.5⟩,⟨h2,0.7,0.4,0.3⟩, ⟨h3,0.8,0.1,0.2⟩,⟨h4,0.7,0.1,0.3⟩,⟨h5,0.8,0.3,0.6⟩}, costlyhouses=f⟨h1,0.7,0.4,0.3⟩,⟨h2,0.6,0.7,0.2⟩, ⟨h3,0.7,0.2,0.5⟩,⟨h4,0.5,0.2,0.6⟩,⟨h5,0.7,0.3,0.4⟩}, moderatehouses=⟨h1,0.8,0.6,0.4⟩,⟨h2,0.7,0.9,0.6⟩, ⟨h3,0.7,0.6,0.4⟩,⟨h4,0.7,0.8,0.6⟩,⟨h5,0.9,0.5,0.7⟩}}.
###### Definition 2.9

 Let and be two neutrosophic sets over the common universe . is said to be neutrosophic soft subset of is , and , , , . We denote it by . is said to be neutrosophic soft super set of if is a neutrosophic soft subset of . We denote it by .

if is neutrosophic soft subset of and is neutrosophic soft subset of . We denote it

###### Definition 2.10

 NOT set of a parameters. Let be a set of parameters. The NOT set of , denoted by is defined by , where not (it may be noted that and are different operators).

###### Definition 2.11

 Complement of a neutrosophic soft set denoted by and is defined as , where is mapping given by neutrosophic soft complement with , and .

###### Definition 2.12

 Empty or null neutrosophic soft set with respect to a parameter. A neutrosophic soft set over the universe is termed to be empty or null neutrosophic soft set with respect to the parameter if and ,

In this case the null neutrosophic soft set is denoted by

###### Definition 2.13

 Union of two neutrosophic soft sets. Let and be two over the common universe . Then the union of and is defined by , where and the truth-membership, indeterminacy-membership and falsity-membership of are as follow.

 TK(e)(m) = TH(e)(m),ife∈A−B = TG(e)(m),ife∈B−A = max(TH(e)(m),TG(e)(m)),ife∈A∩B IK(e)(m) = IH(e)(m),ife∈A−B = IG(e)(m),ife∈B−A = IH(e)(m)+IG(e)(m)2,ife∈A∩B. FK(e)(m) = FH(e)(m),ife∈A−B = FG(e)(m),ife∈B−A = min(FH(e)(m),FG(e)(m)),ife∈A∩B
###### Definition 2.14

 Let and be two over the common universe . Then, intersection of and is defined by , where and the truth-membership, indeterminacy-membership and falsity-membership of are as follow.

 TK(e)(m) = min(TH(e)(m),TG(e)(m)),ife∈A∩B IK(e)(m) = IH(e)(m)+IG(e)(m)2,ife∈A∩B. FK(e)(m) = max(FH(e)(m),FG(e)(m)),ife∈A∩B

For any two and over the same universe and on the basis of the operations defined above, we have the following propositions.

###### Proposition 2.15



For any two , and over the same universe , we have the following propositions.



###### Definition 2.17

 Let and be two over the common universe . Then ’AND’ operation on them is denoted by ’’ and is defined by , where the truth-membership, indeterminacy-membership and falsity-membership of are as follow.

 TK(α,β)(m) = min(TH(e)(m),TG(e)(m)) IK(α,β))(m) = IH(e)(m)+IG(e)(m)2 FK(α,β))(m) = max(FH(e)(m),FG(e)(m)),∀α∈A,∀b∈B
###### Definition 2.18

 Let and be two over the common universe . Then ’OR’ operation on them is denoted by ’’ and is defined by , where the truth-membership, indeterminacy-membership and falsity-membership of are as follow.

 TO(α,β))(m) = max(TH(e)(m),TG(e)(m)), IO(α,β))(m) = IH(e)(m)+IG(e)(m)2, FO(α,β))(m) = min(FH(e)(m),FG(e)(m)),∀α∈A,∀b∈B

### Notes on neutrosophic soft sets 

In this section, we verify that some propositions in the study of Maji  are incorrect by counterexamples.

1. If Definition (2.9) is true, then Definition (3.3) is incorrect.

2. Proposition (2.15)-(5) and (6), and are incorrect.

We verify these notes by counterexamples.

###### Example 2.19

Let us consider neutrosophic soft set in Example (2.8) and null neutrosophic soft set . If Definition (2.9) is true, it is required that null soft set is neutrosophic soft subset of all neutrosophic soft sets. But, since and but , .

###### Example 2.20

Let us consider neutrosophic soft set in Example (2.8) and null neutrosophic soft set . Then,

 (F,A)∩Φ = {e1={⟨h1,0,0.3,0.3⟩,⟨h2,0,0.35,0.6⟩, ⟨h3,0,0.1,0.3⟩,⟨h4,0,0.15,0.2⟩,⟨h5,0,0.1,0.3⟩}, e2={⟨h1,0,0.15,0.5⟩,⟨h2,0,0.2,0.3⟩,⟨h3,0,0.05,0.2⟩, ⟨h4,0,0.05,0.3⟩,⟨h5,0,0.15,0.6⟩}, e3={⟨h1,0,0.2,0.3⟩,⟨h2,0,0.35,0.2⟩,⟨h3,0,0.1,0.5⟩, ⟨h4,0,0.1,0.6⟩,⟨h5,0,0.15,0.4⟩}, e5={⟨h1,0,0.3,0.4⟩,⟨h2,0,0.45,0.6⟩,⟨h3,0,0.3,0.4⟩, ⟨h4,0,0.4,0.6⟩,⟨h5,0,0.25,0.7⟩}}. ≠ Φ

and

 (F,A)∪Φ = {e1={⟨h1,0.5,0.3,0⟩,⟨h2,0.40.35,0⟩, ⟨h3,0.6,0.1,0⟩,⟨h4,0.7,0.15,0⟩,⟨h5,0.8,0.1,0⟩}, e2={⟨h1,0.6,0.15,0⟩,⟨h2,0.7,0.2,0⟩,⟨h3,0.8,0.05,0⟩, ⟨h4,0.7,0.05,0⟩,⟨h5,0.8,0.15,0⟩}, e3={⟨h1,0.7,0.2,0⟩,⟨h2,0.6,0.35,0⟩,⟨h3,0.7,0.1,0⟩, ⟨h4,0.5,0.1,0⟩,⟨h5,0.7,0.15,0⟩}, e5={⟨h1,0.8,0.3,0⟩,⟨h2,0.7,0.45,0⟩,⟨h3,0.7,0.3,0⟩, ⟨h4,0.7,0.4,0⟩,⟨h5,0.9,0.25,0⟩}}. ≠ (F,A)

## 3 Neutrosophic soft sets

In this section, we will redefine the neutrosophic soft set based on paper of Çağman .

###### Definition 3.1

A neutrosophic soft set (or namely ns-set) over is a neutrosophic set valued function from to . It can be written as

 f={(e,{⟨x,Tf(e)(x),If(e)(x),Ff(e)(x)⟩:x∈X}):e∈E}

where, denotes all neutrosophic sets over . Note that if , the element is not appeared in the neutrosophic soft set .Set of all ns-sets over is denoted by .

###### Definition 3.2

Let . is said to be neutrosophic soft subset of , if , , , . We denote it by . is said to be neutrosophic soft super set of if is a neutrosophic soft subset of . We denote it by .

If is neutrosophic soft subset of and is neutrosophic soft subset of . We denote it

###### Definition 3.3

Let . If and for all and for all , then is called null ns-set and denoted by .

###### Definition 3.4

Let .If and for all and for all , then is called universal ns-set and denoted by .

###### Definition 3.5

Let . Then union and intersection of ns-sets and denoted by and respectively, are defined by as follow

 f⊔g = {(e,{⟨x,Tf(e)(x)∨Tg(e)(x),If(e)(x)∧Ig(e)(x),

and ns-intersection of and is defined as

 f⊓g = {(e,{⟨x,Tf(e)(x)∧Tg(e)(x),If(e)(x)∨Ig(e)(x), Ff(e)(x)∨Fg(e)(x)⟩:x∈X}):e∈E}.
###### Definition 3.6

Let . Then complement of ns-set , denoted by , is defined as follow

 f~c={(e,{⟨x,Ff(e)(x),1−If(e)(x),Tf(e)(x)⟩:x∈X}):e∈E}.
###### Proposition 3.7

Let . Then,

1. and

Proof. The proof is obvious from Definition (3.2), (3.3) and Definition (3.4).

###### Proposition 3.8

Let . Then

1. .

Proof. The proof is clear from Definition (3.3), (3.4) and (3.6).

###### Theorem 3.9

Let . Then,

1. and

2. and

3. and

4. and

5. and

6. and

Proof. The proof is clear from definition and operations of neutrosophic soft sets.

###### Theorem 3.10

Let . Then, De Morgan’s law is valid.

Proof. is given.

1. From Definition 3.6, we have

 (f⊔g)~c = {(e,{⟨x,Tf(e)(x)∨Tg(e)(x),If(e)(x)∧If(e)(x), = {(e,{⟨x,Ff(e)(x)∧Ff(e)(x),1−(If(e)(x)∧If(e)(x)), Tf(e)(x)∨Tg(e)(x)⟩:x∈X}):e∈E} = {(e,{X}):e∈E} ⊓ {(e,{⟨x,Fg(e)(x),1−Ig(e)(x),Tg(e)(x)⟩:x∈X}):e∈E} = f~c⊓g~c.
2. It can be proved similar way (i.)

###### Definition 3.11

Let . Then, difference of and , denoted by is defined by the set of ordered pairs

 f∖g={(e,{⟨x,Tf∖g(e)(x),If∖g(e)(x),Ff∖g(e)(x)⟩:x∈X}):e∈E}

here, , and are defined by

 Tf∖g(e)(x)={Tf(e)(x)−Tg(e)(x),Tf(e)(x)>Tg(e)(x)0,otherwise
 If∖g(e)(x)={Ig(e)(x)−If(e)(x),If(e)(x)
 Ff∖g(e)(x)={Fg(e)(x)−Ff(e)(x),Gf(e)(x)
###### Definition 3.12

Let . Then ’OR’ product of ns-sets and denoted by , is defined as follow

 f⋁g = {((e,e′),{⟨x,Tf(e)(x)∨Tg(e)(x),If(e)(x)∧Ig(e)(x), Ff(e)(x)∧Fg(e)(x)⟩:x∈X}):(e,e′)∈E×E}.
###### Definition 3.13

Let . Then ’AND’ product of ns-sets and denoted by , is defined as follow

 f⋀g = {((e,e′),{⟨x,Tf(e)(x)∧Tg(e)(x),If(e)(x)∨Ig(e)(x), Ff(e)(x)∨Fg(e)(x)⟩:x∈X}):(e,e′)∈E×E}.