A short history of fuzzy, intuitionistic fuzzy, neutrosophic and plithogenic sets

Recently, research on uncertainty modeling is progressing rapidly and many essential and breakthrough studies have already been done. There are various ways such as fuzzy, intuitionistic and neutrosophic sets to handle these uncertainties. Although these concepts can handle incomplete information in various real-world issues, they cannot address all types of uncertainty such as indeterminate and inconsistent information. Also, plithogenic sets as a generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets, which is a set whose elements are characterized by many attributes’ values. In this paper, our aim is to demonstrate and review the history of fuzzy, intuitionistic and neutrosophic sets. For this purpose, we divided the paper as: section 1. History of Fuzzy Sets, section 2. History of Intuitionistic Fuzzy Sets and section 3. History of Neutrosophic Theories and Applications, section 4.

that the content of perception is never identical with the perceived entity.However, others assert that this content is identical with the perceived object, and provided that perception is veridical.Identity theories were strongly criticized by B. Russell, 113 who clearly distinguished the properties of words from the properties of things.In this famous paper, he writes that vagueness and precision are characteristics which can only belong to a language and that "apart from representation there can be no such thing as vagueness or precision".He means that vagueness is a characteristic of words, not of things. 58The aim of fuzzy set theory is to built a formal framework for unfinished and gradual knowledge in natural language.The concepts of ambiguity and imprecision of information can be variously understood by researchers as vagueness, uncertainty, etc.When these concepts are regarded as related for representational systems, everyone agree that language is a typical example.
In the twentieth century, Zadeh introduced the degree of membership/truth (T) in 1965 and defined the fuzzy set. 148In contrast to classical binary logic, this aims to express thoughts and notions which do not mathematically explain in natural language such as uncertainty and randomness.Thus, Zadeh preferred classes of objects consisting of relative notions in natural language terms such as age, size, height, temperature.However, classical binary logic is not appropriate to analyze such classes where it appears that membership is a gradual notion rather than an all-or-nothing matter.The fuzzy theory summarizes significantly perceptive phenomenas that expound the complexity of the world.Analytical representations of physical phenomena can model natural language but they are sometimes difficult to understand since they have nebulous explanations and is not be obvious for the non-specialist people.Mental representations has uncertainty and vagueness, the lack of specificity of linguistic terms and the lack of well-defined boundaries of the class of preferred objects.In natural language, objects are represented by proper names, properties are expressed by adjectives and nouns, whereas prepositions and verbs tend to express relations between two or more things. 114In artificial languages, for example, in a language based on first order logic, objects are represented by singular names, whereas properties and relations are represented by predicates. 58or years, producing of the devices simulated human thoughts and behaviors is many scientists' dream.Since we have entered the era of information management as soon as the emergence of computers, this dream have started to realize.The developments of science and technology needing detailed knowledge to produce of the devices human thoughts and behaviors and easy attainableness to computers helps to realize the dream.An important problem is to stock and make use of the information in various areas.Because the fuzzy theory attend to this tendency, 56 it has close connection with artificial intelligence.
When Zadeh propounded fuzzy sets, his concerns concentrated on their potential contribution in the fields of model classification, 38 processing and transportation of knowledge, abstraction and summarization. 149Despite the claims that fuzzy sets were related in these fields emerged in the early sixties, the developments of sciences, technologies and engineering showed that these intuitions were true.
The terms "depressed" and "old" are fuzzy concepts since they cannot be sharply defined.However, people use it to decide.The fuzzy concepts is absolutely contrast to the terms as married, under 25 years old or over 2.5 feet tall.Pure mathematics is interested in classes of objects such as the subset of rational numbers in the set of real numbers.However, when the subset of young people in a given set of people, it may be impossible to decide whether a person is in that subset or not.It is possible to answer it but there can be losing of information when it does.Even though this case has existed from time immemorial, the dominant context is the situation in which statements are exact.
In this day and age, quickly developing technology and the dream of producing devices that mimic human reasoning human thoughts and behaviors which is based on uncertain and imprecise knowledge has drawn many scientists' attention.The theory and applications of fuzzy concepts is largely in the fieds of engineering and applied sciences.The widely usage of fuzzy concepts in technology and science are witnessed by means of the developments of automatic control and expert systems.Although the mathematical elements that form the basis of fuzzy notions have existed for a long time, the rising of its applications has satisfied a motivation for mathematics.Until the rising of fuzzy set theory as a crucial tool in practical applications, there did not exist a force majeure to analyse its mathematics.By virtue of the practical significance of such improvements, studying the mathematical basis of this theory has become important.
In the literature of fuzzy sets, the word "fuzzy" means the word "vague".According to the Oxford English Dictionary, the word "fuzzy" has some various meanings.Unlike any other meaning, "fuzzy" can be used to form a predicate of the form "something is fuzzy".For instance, it can sound normal to say that "a fox is fuzzy" but sound weirdo to say that "old is fuzzy".In this case, the adjective "old" is vague (but not fuzzy in the material sense) since its sense is not set by certain boundaries.However, the word "fuzzy" is applied to predicates and needs to state the gradual nature of some of these words that appear as vague.Additionally, "vagueness" specifies mostly a much larger kind of tabulation for words including ambiguity.Vagueness is interested the meaning of signs in a language and is treated as a certain type of ambiguity.If any notion Doi: https://doi.org/10.54216/IJNS.180109Received Aug. 30, 2021 Accepted: Jan 16, 2022 is not sharply defined, then it is called vague.Since there exists a graceful degradation between situations to which a vague concept fully applies and situations to which it does not never apply, there exist a whole range of situations to which a vague concept fractionally apply. 58This is called "membership gradience" by Lakoff. 90It is keynoted that the fuzziness is an main property of the notion of vagueness which is different from uncertainty.Many researchers think that full applicability of a vague notion to a particular situation may be dubious. 112If a vague concept applies or not to a particular situation, then such doubt can only emerge in the case which is forced to decide.If a proposition involves gradual predicates, then it is called vague.A characteristic property of these propositions is that they can be neither true nor false when they applied to describe a given situation. 49In a word, they can be stated as a degree.Symbolic forms of these degrees of truth exist in natural languages, for example, "very", "rather", "almost no", "less", etc.If a person of the class of "depressive people" characterizes "less depressive", then it is said that the truth level of the statement "this person is depressive" is "less" and this person has a low degree of membership to the class of "depressive people".These linguistic restrictions apply only to gradual predicates.For example, "married" is not a gradual predicate since the term "less married" does not exist.
There exist two reasons that there are fuzziness and ambiguity in all languages.A first reason is that any language is discrete and the external world seems to be continuous.According to Aristotle, this represents the extensity of ambiguity in languages.Specially, a certain number of words import ostensible continuous numerical scales.For example, let the word "old" apply to humans.In order to fix an age threshold below which old fully applies and it does not at all is difficult.There exists a conflict between the linguistic representation of the age scale (for example, {old, teen, young, mature}) and its numerical representation (for instance, the real interval [0, 120] years for humans).The range of functions which Zadeh 148 calls membership functions, can involve as many elements as in the age scale and continuous itself.Terms such as old, young, teen, etc. is called fuzzy predicates.A second reason is that natural language overlooks the existence of exceptions.For instance, some of snakes are venomous but other is not since some don't have venom.Thus, the collection of snakes is not just a set, and also it is a set partially ordered by a relation of compared typicality.Then the membership function is only a partially arbitrary encoding of the partial ordering relation.
Imprecision is also characteristic of language and relates to measurable concepts.These concepts are represented by numbers.This imprecision originates from the fact that any measurement results bounded truth.Then it is possible to mention many correspondences between the results of measurement and real numbers.Chwistek suggested the usage of intervals instead of single numbers 47 and Mellor improved in various direction by Mellor ( 97 98 ).Imprecision seems in the form of disjunctions in logic.For instance, a proposition "p or q is true" contains imprecision since it is not known whether p is true, q is true, or both.Hence, imprecision is represented by sets as a disjunction (not conjunction) of elements.R. Young attempts to improve a formal theory allowing manipulations with indefinite quantities. 147This leading work is called interval mathematics.R. Young substituted a variable which supposes severally considered values by a notion of many-valuedness.This notion was stated as a set of values but it considered collectively.S. Lesniewski studied widely on these sets ( 91 92 ).
Despite imprecision and vagueness mention the contents of a part of knowledge in a language, uncertainty states the ability of an agent to claim whether a proposition holds or not.Uncertainty of Boolean propositions can be gradual as in probability theory where partial doctrine ranges on the unit interval while uncertainty is three-valued in propositional logic, i.e., a proposition is known, its negation is known or both is not known.Even though stating and reasoning these three cases explicitly requires the features of modal logic, 78 the modal possibility has the lack of certainty and certainty reflects logical consistency.However, uncertainty modeled by probability is different from uncertainty modeled by propositional logic.Probability theory often models uncertainty irritating from conflicting, exactly observed, part of knowledge.This situation seems in statistics where a random experiment runs several times an it does not produce the same outcomes.Probability logic is a tool for making inferences from data.Also, Probability logic is used in cases where propositions are either true or false, but the knowledge is imprecise.If the knowledge is fuzzy, then a model of gradual uncertainty is produced since this uncertainty stems from a lack of knowledge.Hence, fuzzy sets causes gradual uncertainty which is different from the one of frequentest probability.This is possibility theory ( 58 59 ).
In complex real world cases, many types of uncertainty exist.For example, "morality" or "beauty" of each population of humans chosen at random can be researched and so on.These examples are fuzzy concepts that is accurately formulated by fuzzy sets.Each type of uncertainty has its mathematical model.Different mathematical theories are like tools which are advantageous to use than another in a given situation.Some of these theories may be used individually or in conjunction.However, the right combination of mathematical theories should be chosen.
This discussion points out that uncertainty differs from imprecision and vagueness and only result from them.In order to see better the differences between the three notions discussed here, let us consider the Doi: https://doi.org/10.54216/IJNS.180109Received Aug. 30, 2021 Accepted: Jan 16, 2022 following assertions about some car: • This car is between 10 and 15 years old (pure imprecision).
• This car is very big (imprecision and vagueness).
• This car was probably made in Germany (uncertainty).In the first case there is a lack of knowledge, due to a lack of ability to measure or to evaluate numerical features.In the second case there is a lack of precise definition of the notion "big" and the modifier "very" indicates a rough degree of "bigness", and the third case expresses uncertainty about a well-defined proposition, perhaps based on statistics. 58he originality of fuzzy sets is to catch the idea of partial membership.The membership function of a fuzzy set is a function whose range is an ordered membership set containing more that two values.Thus, a fuzzy set can be thought as a function.Also, the fuzzy set theory are to handle functions as if they were subsets of their domains because such functions are used to state gradual classes.Therefore, the concepts such as intersection, union, complement, inclusion are enlarged in order to combine functions ranging on an ordered membership set.In fuzzy set theory, the union, intersection, inclusion and complementation of functions are committed by taking their point wise maximum, minimum, inequality between functions, and by means of order-reversing automorphism of the membership scale, respectively.Because degrees of membership may be thought as degrees of truth, intersection as conjunction, union as disjunction, complementation as negation and set-inclusion as implication, it seems easily that the fuzzy set theory is closely related to many-valued logics.Moreover, many mathematical notions are extended to fuzzy sets.It is defined special types of fuzzy sets such as convex fuzzy sets, fuzzy numbers, fuzzy intervals and fuzzy relations, and some types of non-classical sets that are different from fuzzy sets.
Gottwald 71 and Ostasiewicz ( 106 107 ) defined the early developments of fuzzy sets in detailed.Also, Ostasiewicz gave the basic philosophical parts of fuzzy set theory. 105The first published book on fuzzy sets wrote by Kaufmann in French (1973) and translated into English in 1975. 83Up to now.Then a mathematical dissertation is followed by Negoita and Ralescu. 100Dubois and Prade ( 60 61 ), Kandel, 80 Novak, 102 Yager et al. 146 and Klir and Bo Yuan ( 84 85 ) are fundamental books on fuzzy sets.Kruse et al. (1994) focused on probabilistic foundations. 89Gottwald, 72 Lowen 96 and Nguyen and Walker 101 studied on a mathematical structure of fuzzy sets.Zimmerman, 150 Terano, Asai and Sugeno, 141 Klir and Folger, 86 Klir et al 87 and Pedrycz and Gomide 108 investigate methodological problems and applications.Also, the anthology edited by Dubois, Prade and Yager is seen for some basic fuzzy papers of the first twenty five years and other books. 57

History of Intuitionistic Fuzzy Sets
The set theory based on the concept of element membership to sets has proved itself to be one of the most powerful tools of Modern Mathematics and it has allowed to model and to improve other sciences.However, the concept of element membership to a set is a bivalent concept, useded by the values 0 (there is no membership) and 1 (there is membership) and it does not approve other set possibilities in logic.
In today's word, science, technology, biology, social sciences, linguistic, psychology and economics consists of complicated operations and situations for which complete information does not always exist.For these situations, mathematical models are improved to treat various types of systems containing elements of uncertainty.Some of these models are based fuzzy sets.The fuzzy set (FS, briefly) theory introduced by Zadeh regards membership degree and the non-membership degree is the complement of the membership degree.However, this linguistic complement does not provide the logical negation.Since a membership function can be Gaussian, triangular, exponential or any other membership function, and so, there exist a hesitation when a membership function is defined, non-membership degree is less than or equal to the complement of the membership degree.Because different results are obtained with different membership functions.
In 1985, Atanassov introduced the degree of non-membership (T) and falsehood (F) (or non-membership) and defined the intuitionistic fuzzy set (IFS, for short).An intuitionistic fuzzy set may be thought as an alternative approach to describe a fuzzy sets on is not sufficient for the definition of an incomplete notion by means of an ordinary fuzzy set.In the intuitionistic fuzzy set theory, the non-membership degree is not equal to the complement of the membership degree owing to the fact that a lack of information exists when the membership function is defined.In contrast to fuzzy sets, a intuitionistic fuzzy set has two uncertainties called membership and non-membership degrees.Actually, the idea of intuitionistic fuzziness was a coincidence as a mathematical game.Atanassov obtained extension of the ordinary fuzzy set by adding a second degree (degree of non-membership).Since he did not immediately notice properties of intuitionistic fuzzy sets that are different from properties of fuzzy sets, his first studies in this area followed step by step the existing results in fuzzy sets.Some notions of fuzzy sets are not very difficult to extend formally notions of intuitionistic fuzzy sets.However, it is interesting to show that the relevant extension has certain properties which do not exist in the basic concept.Since the method of fuzzification has the idea of intuitionism, the name "intuitionistic fuzzy set" (IFS) is suggested by George Gargov who is Atanassov's former lecturer in the Mathematical Faculty of Sofia University ( 6 77 ).The intuitionistic fuzzy set is a successful generalization of the fuzzy set.The fuzzy set has already achieved a great success in theoretical researches and practical applications. 94Therefore, it is expected that the intuitionistic fuzzy set could be used to simulate human decision-making processes and any activities requiring human expertise, experience, and knowledge, which are inevitably imprecise or not totally reliable.Thus, it is completely believed that the intuitionistic fuzzy set has a wide prospect of applications to the fields such as management, economics, business and environment as well as military. 95he first example for intuitionistic fuzzy sets is given as follow: 6 This is a story with Johnny and Mary, characters in many Bulgarian anecdotes.They bought a box of chocolates with 10 pieces inside.Being more nimble, Johnny ate seven of them, while Mary-only two.
One of the candies fell into the floor.
In this moment, a girl friend of Mary came and Mary said, "We can't treat you with chocolates, because Johnny ate them all".
Let us estimate the truth value of this statement at the moment of speaking, i.e., before we have any knowledge of subsequent events.
From classical logic point of view, which uses for estimations the members of the set {0, 1}, the statement has truth value of 0, since Mary has also taken part in eating the candies and Johnny was not the only one who has eaten them.On the other hand, we are intuitively convinced that the statement is more true than false.Mary greets her guest -statement's truth value is obviously 0.7.However, at the next moment Johnny can take the fallen candy and place it back into the box of treat the guest, preserving the truth value of Mary's statement at 0.7, and falsity 0.3.But he can always take advantage of the distraction and eat the last candy which would result in truth value of 0.8 and falsity of 0.2.In this sense the statement depends to a great extent on Johnny's actions.Therefore, the apparatus of IFSs gives us the most accurate answer to the question: [0.7, 0.2].The degree of uncertainty now is 0.1 and it corresponds to our ignorance of the boundaries of Johnny's gluttony.
In the beginning of the last century, L. Brouwer introduced the notion of the intuitionism. 41He proposed to the mathematicians to remove Aristoteles' law of excluded middle.He said An immediate consequence was that for a mathematical assertion the two cases of truth and falsehood, formerly exclusively admitted, were replaced by the following three: (1) has been proved to be true; (2) has been proved to be absurd; (3) has neither been proved to be true nor to be absurd, nor do we know a finite algorithm leading to the statement either that is true or that is absurd". 143hus, if a proposition A exists, then it can be said that either A is true, or A is false, or that it is not known whether A is true or false.Also, the proposition A ∨ ¬A is always valid in the first order l ogic.It has truth value "true" (or 1) in Boolean algebras bu this proposition can take value smaller than 1 in the ordinary fuzzy logic of Zadeh and in many-valued logics starting with Lukasiewicz.The same is true for intuitionistic fuzzy sets but the situation happen on semantical and prediction' level.the prediction in Brouwer's sense is fuzzified by clarifying the three possibilities.Thus, Gargov proposed the name "intuitionistic fuzzy set".
By now, the relations between the intuitionistic fuzzy set theory and Brouwer's intuitionism don't have been researched in detail.
The intuitionistic fuzzy sets allow the definition o f o perators w hich a re, i n a w ay, s imilar t o t he modal operators.Since these operators reduce to identity, they are meaningless in ordinary fuzzy sets. 7G. Takeuti and S. Titani just attribute a very different meaning in the notion of an "intuitionistic fuzzy sets". 140Atanassov's first two communications in B ulgarian 7 and English. 8tanassaov started a discussion on the accuracy of the name of the intuitionistic fuzzy sets. 62Besides, T. Trifonov and Atanassov built a similar of Takeuti and Titani's research (, 910 ).H. Bustince and P. Burillo showed to coincide vague sets with intuitionistic fuzzy sets. 42he notion of "interval-valued fuzzy set" (IVFS, shortly) are defined in. 70Atanassov and Gargov discussed that this notion equals 88 to the intuitionistic fuzzy sets ( 11 33 ).They showed that each intuitionistic fuzzy set can be stated by an interval-valued fuzzy set and vice versa.Since the intuitionistic fuzzy sets are extensions of the ordinary fuzzy sets, every studies on fuzzy sets may be described in terms of intuitionistic fuzzy sets.
The relations between ordinary fuzzy sets and intuitionistic fuzzy sets are examined from geometrical and probabilistical aspects.Also, some scientists discuss the statement µ ∨ ¬µ ≤ 1 in ordinary fuzzy sets as a declaration of the idea of intuitionism.This inequality does not provide the Law of Excluded Middle.However, Doi: https://doi.org/10.54216/IJNS.180109Received Aug. 30, 2021 Accepted: Jan 16, 2022 this is not the situation in a geometrical interpretation.Having in mind that in fuzzy set theory ¬µ = 1 − µ, it is obtained that the geometrical interpretation as follows: The situation in the intuitionistic fuzzy sets case is different and as below: Now, the geometrical sums of both degrees can really be smaller than 1, i.e., the Law of Excluded Middle is not valid.From probabilistic point of view, for case of the ordinary fuzzy sets, if µ&¬µ = 0, then the probability p(µ ∨ ¬µ) = p(µ) + p(¬µ) = 1 as in the geometrical case, while in the case of intuitionistic fuzzy sets, the inequality p(µ ∨ ¬µ) ≤ 1 follows that for proper elements of intuitionistic fuzzy sets will be strong.All of these constructions are only on the level of the definition of the set (fuzzy set or intuitionistic fuzzy sets), i.e., not related to the possible operations that can be defined over these sets. 6he notion of vague sets is defined. 64It is proved that the notion of vague sets is equivalent to the notion of intuitionistic fuzzy sets. 42 non-probabilistic type of entropy measure is offered for intuitionistic fuzzy sets. 134Li and Cheng examined similarity measures of intuitionistic fuzzy sets and their application to pattern recognitions. 54e et al. studied on Sanchez's approach for medical diagnosis, generalized this concept with the notion of intuitionistic fuzzy set theory and described some operations of intuitionistic fuzzy sets ( 50 51 ).
Some types of fuzzy connectedness in intuitionistic fuzzy topological spaces are defined. 142Szmidt and Kacprzyk contemplated the usage of intuitionistic fuzzy sets for constructing soft decision-making models with incomplete information and they offered two solutions about the intuitionistic fuzzy core and the concurrence winner for group decision-making by using intuitionistic fuzzy sets ( 135 136 137 138 ).Besides, they analyzed distances between intuitionistic fuzzy sets are analyzed. 139A novel and effective approach to deal with decision-making in medical diagnosis using the composition of intuitionistic fuzzy relations was suggested. 55ifferent theorems for building intuitionistic fuzzy relations on a set with preestablished features are given. 46ifferent forms of fuzzy propositional representations and their relationships are investigated. 48owever, there has been barely existed researches on multicriteria or multiattribute in discrete decision situations and/or group decision-making using intuitionistic fuzzy sets, which is in fact one of the most crucial fields in decision analysis as in real world decision problems contain multiple criteria and a group of decision makers (, 5294 ).Multiattribute decision-making using intuitionistic fuzzy sets is researched, a lot of suitable linear programming models are built to produce optimal weights of attributes and the suitable decision-making methods are suggested. 93oreover, various types of intuitionistic fuzzy sets are introduced ( 30 31 32 ).Two versions of intuitionistic fuzzy propositional calculus, intuitionistic fuzzy predicate calculus, two versions of intuitionistic fuzzy modal logic and temporal intuitionistic fuzzy logic (TIFL, briefly) were defined and investigated in a series of communications ( 33 34 35 65 ).

History of Neutrosophic Theories and Applications
The neutrosophic theory that was founded by F. Smarandache in 1998 constitutes a further generalization of fuzzy set, intuitionistic fuzzy set, picture fuzzy set, Pythagorean fuzzy set, spherical fuzzy set, etc.Since then, this logic has been applied in various domains of science and engineering.F. Smarandache introduced the degree of indeterminacy/neutrality (I) as independent component in 1995 (published in 1998) and he defined the neutrosophic set on three components: (T, I, F) = (Truth, Indeterminacy, Falsehood), where in general T, I, In 2018, F. Smarandache introduced the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), which is a set whose elements are characterized by many attributes' values.An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria.In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set, and for a more exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined between each attribute value and the dominant (most important) attribute value.The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement, inclusion (inequality), equality are influenced by the attribute values contradiction (dissimilarity) degrees. 123lithogeny is the genesis or origination, creation, formation, development, and evolution of new entities from dynamics and organic fusions of contradictory and/or neutrals and/or non-contradictory multiple old entities.While plithogenic means what is pertaining to plithogeny.
A plithogenic set P is a set whose elements are characterized by one or more attributes, and each attribute may have many values.Each attribute's value v has a corresponding degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria.
In order to obtain a better accuracy for the plithogenic aggregation operators, a contradiction (dissimilarity) degree is defined between each attribute value and the dominant (most important) attribute value.
{However, there are cases when such dominant attribute value may not be taking into consideration or may Doi: https://doi.org/10.54216/IJNS.180109Received Aug. 30, 2021 Accepted: Jan 16, 2022 not exist [therefore it is considered zero by default], or there may be many dominant attribute values.In such cases, either the contradiction degree function is suppressed, or another relationship function between attribute values should be established.}The plithogenic aggregation operators (intersection, union, complement, inclusion, equality) are based on contradiction degrees between attributes' values, and the first two are linear combinations of the fuzzy operators' tnorm and tconorm.
Plithogenic set is a generalization of the crisp set, fuzzy set, intuitionistic fuzzy set, and neutrosophic set, since these four types of sets are characterized by a single attribute value (appurtenance): which has one value (membership) -for the crisp set and fuzzy set, two values (membership, and nonmembership) -for intuitionistic fuzzy set, or three values (membership, nonmembership, and indeterminacy) -for neutrosophic set (see 123 ).http://fs.unm.edu/NSS/PlithogenicSetAnExtensionOfCrisp.pdf