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 [18], rough sets [16] and intuitionistic fuzzy sets [1]. Fuzzy sets and intuitionistic fuzzy sets are characterized by membership functions, membership and nonmembership 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 [13] 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 [9] to solve complicated problems and various types of uncertainties. In [10], 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.[12] and Yang [17]. In 2010, Çağman and Enginoğlu [5] made some modifications the operations of soft sets and filled in these gap. In 2014, Çağman [4] redefined soft sets using the single parameter set and compared definitions with those defined before.
Maji [11] 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 [11] 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 [4] 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 [15] and soft sets [9]. 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
[15] A neutrosophic set on the universe of discourse is defined as
where and . From philosophical point of view, the neutrosophic set takes the value from real standard or nonstandard 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 nonstandard subset of . Hence we consider the neutrosophic set which takes the value from the subset of .
Definition 2.2
[9] 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
From now on, we will use definitions and operations of soft sets which are more suitable for pure mathematics based on study of Çağman [4].
Definition 2.4
[4] A soft set over is a set valued function from to
. It can be written a set of ordered pairs
Note that if , then the element is not appeared in . Set of all soft sets over is denoted by .
Definition 2.5
[4] Let . Then,

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

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

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

, if and .

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

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

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
Definition 2.7
[11] 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,
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:
Definition 2.9
[11] 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
[11] 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
[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
[11] 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
[11] 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 truthmembership, indeterminacymembership and falsitymembership of are as follow.
Definition 2.14
[11] Let and be two over the common universe . Then, intersection of and is defined by , where and the truthmembership, indeterminacymembership and falsitymembership of are as follow.
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
[11]
For any two , and over the same universe , we have the following propositions.
Proposition 2.16
[11]
Definition 2.17
[11] Let and be two over the common universe . Then ’AND’ operation on them is denoted by ’’ and is defined by , where the truthmembership, indeterminacymembership and falsitymembership of are as follow.
Definition 2.18
[11] Let and be two over the common universe . Then ’OR’ operation on them is denoted by ’’ and is defined by , where the truthmembership, indeterminacymembership and falsitymembership of are as follow.
Notes on neutrosophic soft sets [11]
In this section, we verify that some propositions in the study of Maji [11] are incorrect by counterexamples.
We verify these notes by counterexamples.
Example 2.19
Example 2.20
3 Neutrosophic soft sets
In this section, we will redefine the neutrosophic soft set based on paper of Çağman [4].
Definition 3.1
A neutrosophic soft set (or namely nsset) over is a neutrosophic set valued function from to . It can be written as
where, denotes all neutrosophic sets over . Note that if , the element is not appeared in the neutrosophic soft set .Set of all nssets 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 nsset and denoted by .
Definition 3.4
Let .If and for all and for all , then is called universal nsset and denoted by .
Definition 3.5
Let . Then union and intersection of nssets and denoted by and respectively, are defined by as follow
and nsintersection of and is defined as
Definition 3.6
Let . Then complement of nsset , denoted by , is defined as follow
Proposition 3.7
Let . Then,




and
Proposition 3.8
Let . Then



.
Theorem 3.9
Let . Then,

and

and

and

and

and

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.

From Definition 3.6, we have

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
here, , and are defined by
Definition 3.12
Let . Then ’OR’ product of nssets and denoted by , is defined as follow
Definition 3.13
Let . Then ’AND’ product of nssets and denoted by , is defined as follow
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