Neutrosophic Sets: An Overview

In this study, we give some concepts concerning the neutrosophic sets, single valued neutrosophic sets, interval-valued neutrosophic sets, bipolar neutrosophic sets, neutrosophic hesitant fuzzy sets, inter-valued neutrosophic hesitant fuzzy sets, refined neutrosophic sets, bipolar neutrosophic refined sets, multi-valued neutrosophic sets, simplified neutrosophic linguistic sets, neutrosophic over/off/under sets, rough neutrosophic sets, rough bipolar neutrosophic sets, rough neutrosophic hyper-complex set, and their basic operations. Then we introduce triangular neutrosophic numbers, trapezoidal neutrosophic fuzzy number and their basic operations. Also some comparative studies between the existing neutrosophic sets and neutrosophic number are provided.


INTRODUCTION
The concept of fuzzy sets was introduced by L. Zadeh (1965).Since then the fuzzy sets and fuzzy logic are used widely in many applications involving uncertainty.But it is observed that there still remain some situations which cannot be covered by fuzzy sets and so the concept of interval valued fuzzy sets (Zadeh, 1975) came into force to capture those situations, Although Fuzzy set theory is very successful in handling uncertainties arising from vagueness or partial belongingness of an element in a set, it cannot model all New Trends in Neutrosophic Theory and Applications.Volume II sorts of uncertainties prevailing in different real physical problems such as problems involving incomplete information.Further generalization of the fuzzy set was made by Atanassov (1986), which is known as intuitionistic fuzzy sets (IFs).In IFS, instead of one membership grade, there is also a non-membership grade attached with each element.Further there is a restriction that the sum of these two grades is less or equal to unity.The conception of IFS can be viewed as an appropriate/ alternative approach in case where available information is not sufficient to define the impreciseness by the conventional fuzzy sets.Later on intuitionistic fuzzy sets were extended to interval valued intuitionistic fuzzy sets (Atanassov & Gargov, 1989).Neutrosophic sets (NSs) proposed by (Smarandache, 1998(Smarandache, , 1999(Smarandache, , 2002(Smarandache, , 2005(Smarandache, , 2006(Smarandache, , 2010) ) which is a generalization of fuzzy sets and intuitionistic fuzzy set, is a powerful tool to deal with incomplete, indeterminate and inconsistent information which exist in the real world.Neutrosophic sets are characterized by truth membership function (T), indeterminacy membership function (I) and falsity membership function (F).This theory is very important in many application areas since indeterminacy is quantified explicitly and the truth membership function, indeterminacy membership function and falsity membership functions are independent.Wang, Smarandache, Zhang, & Sunderraman (2010) introduced the concept of single valued neutrosophic set.The single-valued neutrosophic set can independently express truth-membership degree, indeterminacy-membership degree and falsity-membership degree and deals with incomplete, indeterminate and inconsistent information.All the factors described by the single-valued neutrosophic set are very suitable for human thinking due to the imperfection of knowledge that human receives or observes from the external world.

BASIC AND FUNDAMENTAL CONCEPTS
2.1.Neutrosophic sets (Smarandache, 1998) Let  be the universe.A neutrosophic set (NS) A in  is characterized by a truth membership function A , , , 0,1 } There is no restriction on the sum of A T (x), A I (x)and A F (x) and so 0 2.2 Single valued neutrosophic sets (Wang et al., 2010) Let X be a space of points (objects) with generic elements in  denoted by x.A single valued neutrosophic set A (SVNS ) is characterized by truth-membership function ( ) x , and a falsity-membership function ( ) A SVNS A can be written as 3 Interval valued neutrosophic sets (Wang et al., 2005) Let  be a space of points (objects) with generic elements in X denoted by x.An interval valued neutrosophic set A (IVNS A) is characterized by an interval truth-membership function ( ) , x is capability, 2 x trustworthiness, 3 x price.The values of 1 x , 2 x and 3 x are in [0,1].They are obtained from questionnaire of some domain experts and the result can be obtained as the degree of good, degree of indeterminacy and the degree of poor.Then an interval neutrosophic set can be obtained as (Deli et al., 2015) A bipolar neutrosophic set A in  is defined as an object of the form A={<x, Florentin Smarandache, Surapati Pramanik (Editors) An empty bipolar neutrosophic set 1 Numerical Example: , 0.5, 0.3, 0.1, 0.6, 0.4, 0.01 , ,0.3, 0.2, 0.4, 0.03, 0.004, 0.05 , ,0.6, 0.5, 0.4, 0.1, 0.5, 0.004 is a bipolar neutrosophic number.
New Trends in Neutrosophic Theory and Applications.Volume II 2.6 Multi-valued neutrosophic sets (Wang & Li, 2015;Peng & Wang, 2015) Let X be a space of points (objects) with generic elements in X denoted by x, then multi-valued neutrosophic sets A in X is characterized by a truth-membership function ( ) .Multi-valued neutrosophic sets can be defined as the following form: { , ( ), ( ), ( ) }, , are sets of finite discrete values, and satisfies the condition 0 , , 1 For the sake of simplicity, , , has only one value, the multi-valued neutrosophic sets is single valued , the multi-valued neutrosophic sets is double hesitant fuzzy sets.If ( ) ( ) , the multi-valued neutrosophic sets is hesitant fuzzy sets.
Numerical example: Investment company have four options (to invest): the car company, the food company, the computer company, and the arms company, and it considers three criteria: the risk control capability, the growth potential, and the environmental impact.Then the decision matrix based on the multivalued neutrosophic numbers is R. the neutrosophic set A ⊂  .Let T (x) , I (x) , F (x) be the functions that describe the degree of membership, indterminate membership and non-membership respectively of a generic element x ∈  with respect to the neutrosophic set A. A neutrosophic overset (NOVs) A on the universe of discourse  is defined as: A= x, T (x), I (x), F (x) , x ∈  and T(x), I(x), F(x) ∈ [0, Ω ] , where T(x), I(x), F(x):  → [0, Ω ], 0 < 1< Ω and Ω is called over limit.Then there exist at least one element in A such that it has at least one neutrosophic component > 1, and no element has neutrosophic component < 0.
Florentin Smarandache, Surapati Pramanik (Editors) 2.7.2 Definition of neutrosophic underset: Let  be a universe of discourse and the neutrosophic set A ⊂  .Let T (x) , I (x) , F (x) be the functions that describe the degree of membership, indterminate membership and non-membership respectively of a generic element x ∈  with respect to the neutrosophic set A. A neutrosophic under set (NUs) A on the universe of discourse  is defined as: Where and Ψ is called lowerlimit.Then there exist at least one element in A such that it has at least one neutrosophic component < 0, and no element has neutrosophic component > 1.
2.7.3 Definition of neutrosophic offset: Let  be a universe of discourse and the neutrosophic set A ⊂  .Let T (x) , I (x) , F (x) be the functions that describe the degree of membership, indterminate membership and non-membership respectively of a generic element x ∈  with respect to the neutrosophic set A. A neutrosophic offset (NOFFs) A on the universe of discourse  is defined as: where Then there existe some elments in A such that at least one neutrosophic component > 1, and at least another neutrosophic component < 0.

Some operations of neutrosophic over/off/under sets
Definition 1: The complement of a neutrosophic overset/ underset/offset A is denoted by C(A) and is defined by Let  be a universe of discourse and A the neutrosophic set A⊂U.Let T (x), I (x),F (x) be the functions that describe the degree of membership, indterminate membership and non-membership respectively of a generic element x ∈ with respect to the neutrosophic set A. A neutrosophic overset (NOV) A on the universe of discourse U is defined as: A= x, T (x), I (x), F (x) , x ∈  and T(x), I(x), F(x) ∈ [0, Ω ] , where Then there exist at least one element in A such that it has at least one neutrosophic component >1, and no element has neutrosophic component <0.

OPERATIONS ON SOME NEUTROSOPHIC NUMBERS AND NEUTROSOPHIC SETS 2.1 Single valued neutrosophic number
be two single valued neutrosophic number.Then, the operations for SVNNs are defined as below; i.

SCORE FUNCTION, ACCURACY FUNCTION AND CERTAINTY FUNCTION OF NEUTROSOPHIC NUMBERS
A convenient method for comparing of single valued neutrosophic number is described as follows: be a single valued neutrosophic number.Then, the score function 1 ( )

RANKING OF NEUTROSOPHIC NUMBERS
Suppose that 1 are two single valued neutrosophic numbers.Then, the ranking method is defiend as follows: New Trends in Neutrosophic Theory and Applications.Volume II A single valued triangular neutrosophic number (SVTrN-number) is a special neutrosophic set on the real number set R, whose truth membership, indeterminacy-membership, and a falsity-membership are given as follows:  be a single valued triangular neutrosophic number, then the truth membership, indeterminacy membership and falsity membership are expressed as follows be two single valued triangular neutrosophic numbers.Then, the operations for SVTrN-numbers are defined as below; ( a , a , a ); m in ( , ), m ax(I , I ), m ax(F , F )

Score function and accuracy function of single valued triangular neutrosophic numbers
The convenient method for comparing of two single valued triangular neutrosophic numbers is described as follows: be a single valued triangular neutrosophic number.Then, the score function 1 ( ) s A  and accuracy function 1 ( ) a A  of a SVTrN-numbers are defined as follows:

Ranking of single valued triangular neutrosophic numbers
Let 1 A  and 2 A  be two SVTrN-numbers.The ranking of 1 A  and 2 A  by score function and accuracy function is defined as follows: New Trends in Neutrosophic Theory and Applications.Volume II A single valued trapezoidal neutrosophic number (SVTN-number) is a special neutrosophic set on the real number set R, whose truth membership, indeterminacy-membership, and a falsity-membership are given as follows

6.2.2Score function and accuracy function of single valued trapezoidal neutrosophic numbers
The convenient method for comparing of two single valued trapezoidal neutrosophic numbers is described as follows:

Ranking of single valued trapezoidal neutrosophic numbers
Let 1 A  and 2 A  be two SVTN-numbers.The ranking of 1 A  and 2 A  by score function is defined as follows: Later on, Liang et al. (2017) redefined the score function, accuracy function and certainty function as follows: Let a = < [a ,a ,a ,a ], (T , I , F ) > be a SVTNN.Then, the score function, accuracy function, and certainty function of SVTNN a are defined, respectively, as: where (COG) denotes the center of gravity of K and can be defined as follows: New Trends in Neutrosophic Theory and Applications.Volume II 6.3 Interval valued neutrosophic number 6.3.1Operations on interval valued neutrosophic number be two interval valued neutrosophic numbers.Then, the operations for IVNNs are defined as below; ) , (I ) ,(I ) , ( ) ,( ) is said to be empty if and only if and is denoted by

Score function and accuracy functions of interval valued neutrosophic number
The convenient method for comparing of interval valued neutrosophic numbers is described as follows: of an IVNN are defined as follows: (i) are two interval valued neutrosophic numbers.Then, the ranking method for comparing two IVNS is defiend as follows: 1 ( 1) ,(I ) ,( ) , ( ) , ( ) , ( 1(1 ( F )) ) where 0  .

Score function, accuracy function and certainty function of bipolar neutrosophic number
In order to make comparison between two BNNs.Deli et al. (2015) introduced a concept of score function.The score function is applied to compare the grades of BNS.This function shows that greater is the value, the greater is the bipolar neutrosophic sets and by using this concept paths can be ranked.Let (i) New Trends in Neutrosophic Theory and Applications.Volume II ix.
7. TRAPEZOIDAL NEUTROSOPHIC SETS (Ye, 2015b;Biswas et al., 2014) Assume that X be the finite universe of discourse and F [0, 1] be the set of all trapezoidal fuzzy numbers on [ 0, 1].A trapezoidal fuzzy neutrosophic set (TrFNS) A  in X is represented as: The trapezoidal fuzzy numbers ( ) ( ) , respectively , denote the truth-membership, indeterminacy-membership and a falsity-membership degree of x in A  and for every xX, 0 ≤ 4 T ( ) For notational convenience, the trapezoidal fuzzy neutrosophic value (TrFNV) A  is denoted by , and ( 1 ( ) The parameters satisfy the following relations 1 The truth membership function is defined as follows x a a x a a a otherwise The indeterminacy membership function is defined as follows: Florentin Smarandache, Surapati Pramanik (Editors) otherwise and the falsity membership function is defined as follows:  (e ,e ,e ,e ),(f ,f ,f ,f ),(g ,g ,g ,g ) A   be two TrFNVs in the set of real numbers, and 0  .Then, the operational rules are defined as follows;
Ye (2015b) presented the following definitions of score function and accuracy function.The score function S and the accuracy function H are applied to compare the grades of TrFNSs.These functions show that greater is the value, the greater is the TrFNS.(i)

Score function and accuracy function of trapezoidal fuzzy neutrosophic value
In order to make a comparison between two TrFNV, Ye (2015b) presented the order relations between two TrFNVs.(e ,e ,e ,e ),(f ,f ,f ,f ),(g ,g ,g ,g ) A   be two TrFNVs in the set of real numbers.Then, we define a ranking method as follows:

Ranking of trapezoidal fuzzy neutrosophic value
8. TRIANGULAR FUZZY NEUTROSOPHIC SETS (Biswas et al., 2014) Assume that X be the finite universe of discourse and F [0, 1] be the set of all triangular fuzzy numbers on [ 0, 1].A triangular fuzzy neutrosophic set (TFNS) A  in X is represented ( ) , respectively, denote the truth-membership, indeterminacy- membership and a falsity-membership degree of x in A  and for every xX For notational convenience, the triangular fuzzy neutrosophic value (TFNV) A  is denoted by ( , , ),( , , ),( , , ) A a b c e f g r s t   where,( 1 T ( ) , f, g), and ( 1 ( )

Operation on triangular fuzzy neutrosophic value
c e f g r s t   be two TFNVs in the set of real numbers, and 0  .Then, the operational rules are defined as follows; (i) a a a a b b b c c c c A A e e f f g g r r s s t t

b b c c
A A e e e e f f f f g g g g r r r r s s s s t t t t ) ,1 (1 ) ,1 (1 ) ) , (1 (1 ) ,1 (1 ) ,1 (1 ) ) Ye (2015b) introduced the concept of score function and accuracy function TFNS.The score function S and the accuracy function H are applied to compare the grades of TFNS.These functions show that greater is the value, the greater is the TFNS.
New Trends in Neutrosophic Theory and Applications.Volume II

Score function and accuracy function of triangular fuzzy neutrosophic value
Let 1 1 1 1  1 1 1  1 1 1 ( , , ),( , , ),( , , ) A a b c e f g r s t   be a TFNV.Then, the score function 1 ( ) S A  and an accuracy function 1 ( ) H A  of TFNV are defined as follows: In order to make a comparison between two TFNVs, Ye (2015b) presented the order relations between two TFNVs.

Ranking of triangular fuzzy neutrosophic values
( , , ),( , , ),( , , ) A a b c e f g r s t   be two TFNVs in the set of real numbers.Then, the ranking method is defined as follows:  is a convex intuitionistic fuzzy set on the set  of real numbers, whose membership and non-membership functions are follows x a w a a a x a w a x a x a x w a a a a a x a x a a a a a a x u x a a a a x a u a x a x x a u a x a a a x a x a x a where 0 , a w  and a u  respectively represent the maximum membership degree and the minimum membership degree of a , ( ) 1 ( ) ( ) is called as the intuitionistic fuzzy index of an element xin a  . 1 a and 2 a respectively represent the minimum and maximum values of the most probable value of the fuzzy number a , a represents the minimum value of the a , and a represents the maximum value of the a .

Trapezoidal neutrosophic fuzzy number
Definition 2. Let X be a universe of discourse, then a trapezoidal fuzzy neutrosophic set N  in X is defined as the following form: { , ( ), ( ), ( ) }, , and .

Difference and comparison between trapezoidal intuitionistic fuzzy number and trapezoidal neutrosophic fuzzy number
The difference and comparison between the trapezoidal intuitionistic fuzzy number and trapezoidal neutrosophic fuzzy number are represented in the following way: It can be observed from the Fig. 1, there are some differences between trapezoidal intuitionistic fuzzy number and trapezoidal neutrosophic fuzzy number.On one hand, the membership degree, non-membership degree and hesitancy of trapezoidal intuitionistic fuzzy number are mutually constrained, and the maximum value of the sum of them is not more than 1.However, the truth membership, indeterminacy membership and falsity membership functions of trapezoidal neutrosophic fuzzy number are independent, and their values are between 0 and 3.And the maximum value of their sum is not more than 3. On the other hand, trapezoidal neutrosophic fuzzy number is a generalized representation of trapezoidal fuzzy number and trapezoidal intuitionistic fuzzy number, and trapezoidal intuitionistic fuzzy number is a special case of trapezoidal neutrosophic fuzzy number.

DIFFERENCE BETWEEN TRIANGULAR FUZZY NUMBERS, INTUITIONISTIC TRIANGULAR FUZZY NUMBER AND SINGLED VALUED NEUTROSOPHIC SET
Fuzzy sets have been introduced by Zadeh (1965) in order to deal with imprecise numerical quantities in a practical way.A fuzzy number (Kaufmann& Gupta, 1988) is a generalization of a regular, real number in the sense that it does not refer to one single value but rather to a connected set of possible values, where each possible value has its own weight between 0 and 1.This weight is called the membership function.A fuzzy number is thus a special case of a convex, normalized fuzzy set of the real line .
10.1 Triangular fuzzy number (Lee, 2005) A triangular fuzzy number A TIFN (See Fig. 2) A is a subset of IFS in R with the following membership functions and non-membership function as follows Florentin Smarandache, Surapati Pramanik (Editors) where 1 It can be observed from the membership functions that in case of triangular intuitionistic fuzzy number, membership and non-membership degrees are triangular fuzzy numbers.Further it can be noted that the neutrosophic components are best suited in the presentation of indeterminacy and inconsistent information whereas intuitionistic fuzzy sets cannot handle indeterminacy and inconsistent information.The difference between the fuzzy numbers and singled valued neutrosophic set can be understood clearly with the help of an example.Suppose it is raining continuously for few days in a locality.Then one can guess whether there would be a flood like situation in that area.Observing the rainfall of this year and recalling the incidents of previous years one can only give his judgment on the basis of guess in terms of yes or no but still there remains an indeterminate situation and that indeterminate situation is expressed nicely by the single valued neutrosophic set.
Triangular fuzzy numbers (TFNs) and single valued neutrosophic numbers (SVNNs) are both generalizations of fuzzy numbers that are each characterized by three components.TFNs and SVNNs have been widely used to represent uncertain and vague information in various areas such as engineering, medicine, communication science and decision science.However, SVNNs are far more accurate and convenient to be used to represent the uncertainty and hesitancy that exists in information, as compared to TFNs.SVNNs are characterized by three components, each of which clearly represents the degree of truth membership, indeterminacy membership and falsity membership of a the SVNNs with respect to a an attribute.Therefore, we are able to tell the belongingness of a SVNN to the set of attributes that are being studied, by just looking at the structure of the SVNN.This provides a clear, concise and comprehensive method of representation of the different components of the membership of the number.This is in contrast to the structure of the TFN which only provides us with the maximum, minimum and initial values of the TFN, all of which can only tell us the path of the TFN, but does not tell us anything about the degree of non-belongingness of the TFN with respect to the set of attributes that are being studied.Furthermore, the 11.REFINED NEUTROSOPHIC SETS (Smarandache, 2013;Deli et al., 2015b) Refined neutrosophic sets can be expressed as follows: Let E be a universe.A neutrosophic refined set (NRS) A on E can be defined as follows , ( (x), (x),..., (x)), ( (x), (x),..., (x)), ( (x), (x),..., (x)) Where 1 2 (x ), (x ), ..., : E  [ 0, 1] and 1 2 (x), (x), ..., (x)

., I (x))
is respectively the truth membership sequence, indeterminacy membership sequence and falsity membership sequence of the element x.Also, P is called the dimension of BNRS.
The set of all bipolar neutrosophic refined sets on E is denoted by BNRS(E).

Let ( ), ( ), ( )
are any two MVNNs, and 0  .The operations for MVNNs are defined as follows. (1) 13.2 Score function, accuracy function and certainty function of multi-valued neutrosophic number 13.3 Comparison of multi-valued neutrosophic numbers Let ( ), ( ), ( ) are two multi-valued neutrosophic numbers.Then the comparision method can be defined as follows: i.
If ( ) ( ) 14. Simplified neutrosophic linguistic sets (SNLSs) (Tian et al., 2016) 14.1 SNLSs Definition 1.Let X be a space of points (objects) with a generic element in X, denoted by x and be a finite and totally ordered discrete term set, where t is a nonnegative real number.

COMPARISON ANALYSIS
Refined neutrosophic set is a generalization of fuzzy set, intuitionistic fuzzy set, neutrosophic set, intervalvalued neutrosophic set, neutrosophic hesitant fuzzy set and interval-valued neutrosophic hesitant fuzzy set.Also differences and similarities between these sets are given in Table 1.Bosc and Pivert (2013) said that "Bipolarity refers to the propensity of the human mind to reason and make decisions on the basis of positive and negative effects.Positive information states what is possible, satisfactory, permitted, desired, or considered as being acceptable.On the other hand, negative statements express what is impossible, rejected, or forbidden.Negative preferences correspond to constraints, since they specify which values or objects have to be rejected (i.e., those that do not satisfy the constraints), while positive preferences correspond to wishes, as they specify which objects are more desirable than others (i.e., satisfy user wishes) without rejecting those that do not meet the wishes."Therefore, Lee (2000Lee ( , 2009) ) introduced the concept of bipolar fuzzy sets which is a generalization of the fuzzy sets.Bipolar neutrosophic refined sets which is an extension of the fuzzy sets, bipolar fuzzy sets, intuitionistic fuzzy sets and bipolar neutrosophic sets.Also differences and similarities between these sets are given in Table 2.

CONCLUSIONS
NSs are characterized by truth, indeterminacy, and falsity membership functions which are independent in nature.NSs can handle incomplete, indeterminate, and inconsistent information quite well, whereas IFSs and FSs can only handle incomplete or partial information.However, SVNS, subclass of NSs gain much popularity to apply in concrete areas such as real engineering and scientific problems.Many extensions of NSs have been appeared in the literature.Some of them are discussed in the paper.New hybrid sets derived from neutrosophic sets gain popularity as new research topics.Extensions of neutrosophic sets have been developed by many researchers.This paper presents some of their basic operations.Then, we investigate their properties and the relation between defined numbers and function on neutrosophic sets.We present comparison between bipolar fuzzy sets and its various extensions.We also present the comparison between different types of neutrosophic sets and numbers.The paper can be extended to review different types of neutrosophic hybrid sets and their theoretical development and applications in real world problems.
the truth membership, indeterminate membership and false membership of an element  corresponding to a bipolar neutrosophic set A and the negative membership degree ( ) false membership of an element  to some implicit counter-property corresponding to a bipolar neutrosophic set A.
6. DIFFERENT TYPES OF NEUTROSOPHIC NUMBERS AND RELATED TERMS ASSOCIATED WITH THEM6.1 Single valued-triangular neutrosophic numbers(Ye 2015b) FlorentinSmarandache, Surapati Pramanik (Editors) of a SVTN-numbers are defined as follows: be a single valued neutrosophic number.Then, the score function 1

,
neutrosophic number.Then, the score function ( ) s A  , accuracy function ( ) a A  and certainty function ( ) c A  of an BNN are defined as follows: The trapezoidal fuzzy neutrosophic number is a particular case of trapezoidal neutrosophic number when all the three vector are equal: 1 ,a ,a ),(b ,b ,b ,b ),(c ,c ,c ,c ) ,a ,a ),(b ,b ,b ,b ),(c ,c ,c ,c) of TrFNV are defined as follows: ,a ,a ),(b ,b ,b ,b ),(c ,c ,c ,c )

Fig. 2 .
Fig.2.Graphical representation of triangular intuitionistic fuzzy number Neutrosophic Theory and Applications.Volume II structure of the TFN is not able to capture the hesitancy that naturally exists within the user in the process of assigning membership values.These reasons clearly show the advantages of SVNNs compared to TFNs.
New Trends in Neutrosophic Theory and Applications.Volume II membership and falsity-membership of the element x in X to the linguistic term ( ) A t x , ( )A i x and ( ) A f x represent, respectively, the degree of truth-membership, indeterminacy-

Table 1 .
Comparison of fuzzy set anditsextensive set theory

Table 2 .
Comparison of bipolar fuzzy set and its various extensions

Table 3 .
Comparison of different types of neutrosophic setsNew Trends in Neutrosophic Theory and Applications.Volume II