A Study of the Concept of Soft Set TheoryAnd Survey Of Its Literature

VOL. 3, NO. 1, January 2013 ISSN 2225-7217ARPN Journal of Science and Technology©2011-2013. All rights reserved.by;Then, the α – level set of F best and F poor are givenF best (0.2) = {h 1 , h 2 , h 5 , h 6 } ; F poor (0.2) = { h 1 ,h 2 , h 3 h 4 , h 5 }F best (0.7) = {h 2 , h 5 , h 6 } ; F poor (0.3) = { h 1 ,h 3 h 4 }F best (0.9) = {h 5 , h 6 } ; F poor (0.9) = { h 1 ,h 3 h 4 }F best (1.0) = {h 6 } ; F poor (1.0) = {h 3h 4 }Where here A = {0.2, 0.7, 0.9, 1.0}⊂ [0, 1]which can be regarded as the parameter set such thatF best: A⟶ P( ) gives the approximate value set F best (∝),for ∝ ∈ A. Thus, the soft set for the fuzzy set F best can bewritten as;(F best , A) = {(0.2, {h 1 , h 2 , h 5 , h 6 }), (0.7, {h 2 , h 5 ,h 6 }), (0.9, {h 5 , h 6 }), (1.0, {h 6 })}.Similarly the soft set for the fuzzy set F poor , isgiven by;http://www.ejournalofscience.org(a) The lower approximation of X, denoted by R ∗ Xand defined as;R ∗ X = {x ∈ : [x] R ⊆ X} which consist of theset of all objects that certainly belong to X.(b) The upper approximation of X denoted by R*Xdefined as;R*X = {x ∈ : [x] R ∩ X ≠ } consisting of theset of all objects which possibly belong toX. Then the set X is called a rough set if;R*X − R ∗X ≠ and X is crisp if R*X − R ∗X =.(v) A vague set V is a set characterized by twomembership functions viz, a true membership functiont v : ⟶ [0, 1] and a false membership function f v : ⟶[0, 1], with0≤t v (x)+f v (x) ≤ 1 and V is represented as;V = {(x, [t v (x), f v (x)]: x ∈ }, where the interval [t v (x),1−f v (x)] is called the vague value of x in V.(F poor , B) = { (0.2{ h 1 , h 2 , h 3 h 4 , h 5 }), (0.3{ h 1 , h 3h 4 }), (0.9{ h 1 , h 3 h 4 }), (1.0{h 3 h 4 })}:Where B = {0.2, 0.3, 0.9, 1.0}⊂ [0, 1].In order to have a comparative view of soft setwith other existing sets, we briefly define the following:Let be an initial universe, E a set of parameters, P( )the power set of and A ⊆ E.(i) A classical (or crisp) set C ⊆ is a setcharacterized by the function χ C , called the characteristicfunction of C, where χ C : ⟶ {0, 1} and C is defined byC = {x∈ : χ(x) =1,if x C0,if x C(ii) A multi set M ⊆ is a set characterized by anumeric– valued function α M : ⟶{IN ∪ 0} where M isdefined byM = {x ∈ : α M (x) =}IN,if x M0,if x M(iii) A fuzzy set F over is a set characterized by amembership function μ F of Fwhere μ F : ⟶ [0, 1] and F is represented by;F = {(x, μ F (x)): x ∈ , μ F (x) ∈ [0, 1]}.(iv) Let R be an equivalence relation on , and for x∈ , let [x] R denote the equivalence class of Rdetermined by x. Let X ⊆ , X is characterized withrespect to R by a pair of crisp sets called;}.(vi) A soft set (F, A), F A or (f A , E) over ischaracterized by an approximate set valued function; f A :E⟶ P ( ) and (F, A) is represented as a set of orderedpairs defined as;(F, A) = {(x, f A (x): f A (x) ∈ P( ) and f A (x) = ,if x ∉ A}.3. LITERATURE REVIEW OF SOFT SETTHEORYMolodtsov [34] initiated the concept of soft settheory as a general mathematical tool for solvingcomplicated problems dealing with vagueness anduncertainties which classical methods and some modernmathematical methods, such as probability theory [12],fuzzy set theory [49], rough set theory [37], intervalmathematics theory[19], vague set theory [16] etc.,cannot successfully solve due to inadequacy of theirparameterization tools.Molodtsov [34] pointed out that soft set theoryprovides enough parameter and as a resultaccommodates initial approximate descriptions of anobject. This, he said, makes soft set theory free from theabove difficulty and becomes very convenient and easilyapplicable in practice. Molodtsov[34] therefore defines asoft set as a parameterized family of subsets of auniverse set, where each element is considered as the setof approximate elements of the soft set. He alsosuccessfully applied soft set theory in areas such assmoothness of function (where he compared smoothnessof function as being similar to continuity of functions inthe classical case), game theory, operation research,Riemann and Perron integrations, probability theory andmeasurement theory, and introduced the basic notions ofthe theory of soft sets.63

VOL. 3, NO. 1, January 2013 ISSN 2225-7217ARPN Journal of Science and Technology©2011-2013. All rights reserved.Based on the work of Molodtsov [34],Maji etal.,[30] initiated the theoretical study of the soft settheory. This includes, the definition of soft subset, softsuperset, equality of soft sets, and complement of a softset among others, with some illustrative examples. Softbinary operations such as AND OR operations, unionand intersection operations are also defined. Theyverified De Morgan’s law and presented a number ofresults on soft set theory. For the purpose of storing asoft set in a computer, they represented a soft set in theform of a table.Yang [48] was the first to point out error in thework of Maji et al., [30] by giving a counter example.Ali et al.,[3] also pointed out several assertions in Maji etal. ,[30]that are not true in general by counter examplesSome new operations such as restricted union, restrictedintersection, restricted difference and extendedintersection were further introduced. Moreover theyimproved on the notion of complement of a soft set andproved that certain De Morgan’s laws hold in soft settheory with respect to these new operations. They alsoremarked that the incorrectness of the assertionsmentioned above may be as a result of the way andmanner some of the related notions were defined.http://www.ejournalofscience.orgapplication of injective soft set in decision makingproblem.Xiao et al. [46], reviewed the notions of softsets and injective soft sets and proposed the notion ofexclusive disjunctive soft sets. Some studies onoperations of exclusive disjunctive soft sets such asrestricted/ relaxed AND- operations and the dependencybetween exclusive disjunctive soft sets and injective softsets were also established. They finally gave anapplication of exclusive disjunctive soft set to attributereduction of incomplete information system and pointedout that potential studies could be focusing on how toextend it to fuzzy soft set as well as applying it insolving decision- making problems.Babitha and Sunil [6] introduced the concept ofsoft set relation as a sub soft set of the Cartesian productof soft sets. Besides, many related concepts such asequivalence soft set relation, partition of soft set, soft setfunction, composition of soft set functions with relatedresults were proposed. In their concluding remarks, theyproposed a future research on the theoretical aspect ofthese generalized concepts in relation to topologygenerated by soft set relation.Ge and Yang [17], further investigated theoperational rules given by Maji et al. [30] and Ali et al.,[3] and obtained some necessary and sufficientconditions for the rules to hold. They specifically notedthat several assertions in relation to null soft set andabsolute soft set are incorrect, due to the notation of therelated definitions in Maji et al. ,[30].Fu Li [15] also pointed out some errors in [30]and corrected them. He also discussed the distributivelaws with respect to certain operations of union andintersection introduced in [3], using De Morgan laws.Sezgin and Atagun [40], proved that certain DeMorgan’s laws with respect to different operations onsoft set defined by (Maji et al. ,[30], (Pei and Miao [38]and Ali et al. ,[3] hold. They further discussed variousbasic properties of operations on soft set such as union,restricted union, extended and restricted intersections,restricted difference and their interconnection betweenthem. Finally they defined the notion of restrictedsymmetric difference of soft sets and investigated itsproperties with examples.Singh and Onyeozili [42, 43, 44] establishedand investigated (with illustrative examples) someresults on distributive and absorption properties withrespect to various operations on soft sets.Gong et al. [18], proposed the concept ofinjective soft set and some of its operations such as therestricted AND the relaxed AND operations. They alsodefined the dependency between two injective soft setsand the injective soft decision system and gave anIn continuation of their work, Babitha et al. ,[7],introduced the concept of anti- symmetric relation andtransitive closure of a soft set relation and proposed ananalogue of Warshall’s algorithm for calculating thetransitive closure of a soft set relation. They also definedordering on a soft set and established the relationshipbetween different orderings supported with examples.In an attempt to broaden the theoretical aspectof soft set relation, Yang et al. [47], introduced thenotions of anti- reflexive kernel, symmetric kernel,reflexive closure and symmetric closure of a soft setrelation and obtained with proofs some results involvingthem. The notions of soft set relation mapping and itsinverse were also proposed and some of their relatedproperties discussed.Qin and Hong [39] made a theoretical study ofthe algebraic structures of soft sets such as latticestructures and introduced the concept of soft equalityrelation and also discussed its related properties. It wasproved that soft equality relation is a congruence relationwith respect to some operations. They finally establishedthe quotient algebra with respect to restricted union (⋃ R )and extended intersection (⋂ E ).Won Keon Min [33] introduced and studied theconcept of similarity between soft sets which is anextension of the equality for soft sets and investigatedsome properties. He also introduced the concept ofconjunctive ( ∝∧β) and disjunctive ( ∝ ∨ β) of orderedpair parameter ( ∝, β) for soft set theory; modified andinvestigated some operations of soft set theoryintroduced by ( Maji et al. ,2003).64

VOL. 3, NO. 1, January 2013 ISSN 2225-7217ARPN Journal of Science and Technology©2011-2013. All rights reserved.Kharal and Ahmad [27] introduced the notion ofmapping on soft classes and study several properties ofimages and inverse images of soft sets supported withexamples and counter examples. An application of thesenotions to the problem of medical diagnosis in medicalexpect systems where a patient’s complaints orsymptoms may easily be encoded into a soft set, wasfinally given.In line with the work of Kharal and Ahmad[27], Majumdar and Samanta [32] introduced the idea ofsoft mapping and studied some of its properties with theobservation that fuzzy mapping is a special case of softmapping. Majumdar et al. [32] also defined the imageand the respective inverse image of a soft set under a softmapping and studied some of their properties. They alsoapplied the soft mapping in medical diagnosis problemand remarked that the model used in the diagnosis wasvery preliminary and may be improved by incorporatingdetailed disease- symptom information and clinicalresults. For future research, a work on the theory offuzzy soft mapping and the intuitionist fuzzy softmapping was also proposed.Chen et al..[11] pointed out that the result ofsoft set reductions proposed by( Maji et al. ,[31] wasincorrect and explained with example that the algorithmsused to first compute the redact -soft set and then thechoice value for selecting the optimal objects in thedecision problem were unreasonable. As a result, theyproposed a new definition of parameter reduction for theimprovement of Maji et al. ,[31] and concluded that theidea of redact under rough set theory generally cannot beapplied directly in redact under soft set theory, with theobservation that the parameter reduction may well playan important role in some knowledge discoveryproblems.Kong et al.,[28] analyzed that the problems ofsub- optimal choice was not addressed by Chen et al.,[11] and so introduced the definition of normalparameter reduction in soft set theory to overcome theproblems in Chen’s model. Two new definitions viz,-theparameter important degree and the soft decisionpartition were described and use to analyze thealgorithm of normal parameter reduction. They notedthat with this approach, the optimal choices are stillpreserved.http://www.ejournalofscience.orgbased on the notion of multi-soft sets, constructed from amulti-valued information system and obtained a resultwhich turned out to be equivalent to Pawlak’s rough setreduction.Pei and Miao[38] and Xiao et al.[46] discussedthe relationship between soft sets and informationsystems and showed that soft sets are a class of specialinformation systems.Various applications of soft set theory weremade by several authors. Maji et al. ,[31], gave apractical application of soft sets in decision makingproblem using the notion of knowledge reduction inrough set theory of Pawlak [37].Later, Cagman and Enginolu [8] redefined someoperations of soft set theory in order to make them morefunctional for improving several new results. Fourproducts of soft sets viz-AND ( ∧) - product, OR ( ∨)-product AND-Not ( ⊼)- product and OR-NOT ( ⊻)-product were also defined and their properties discussed.Using AND ( ∧)-product, they constructed an uni-int(union- intersection) soft decision making method whichselect a set of optimum elements from the alternativeswithout using the rough sets and fuzzy soft sets. Theyfinally applied the method to a decision making problem,remarking that apart from the uni-int soft decisionmethod, similar types of decision methods using int-uni(intersection-union), uni-uni (union -union) and int-int(intersection-intersection can be constructed for ∨-product, ⊼-product and ⊻-product respectively in asimilar way though with some modifications.Similar to their work in Cagman andEnginoglu,[9], they introduced soft matrices which arerepresentation of soft sets and defined AND (∧), OR (∨),AND-NOT (⊼) and OR-NOT ( ⊻) products of softmatrices. Also using the AND-product of soft matrices,they constructed a soft max-min (maximum -minimum)decision making method and applied it to a real estateproblem, noting that similarly, other products ∨, ⊼ and ⊻of soft matrices could be used for other convenient orsimilar problems.Other works on applications of soft set theoryinclude( Chen et al. ,[11]), (Gong et al. ,[18]), (Kharal etal. ,[27]), (Majundar et al. ,[32]), among others.Zou and Xiao[50], proposed a new techniquefor decision making of soft set theory under incompleteinformation systems which is based on the calculation ofweighted –average of all possible choice values ofobject.Herewan et al.[20] also analyzed the existingworks on attributes and parameterization reduction ofsoft sets proposed by Maji et al. ,[31], Zou et al. ,[50]and remarked that all the techniques were based onBoolean- valued information system. They proposed analternative approach for attribute reduction which wasThe algebraic structures of soft set theory havealso been studied extensively. Aktas and Cagman [2],introduced the basic concepts of soft groups, softsubgroups, normal soft subgroups and softhomomorphism and discussed their basic properties.Feng et al. [13] considered the algebraicstructure of smearing and introduced the notion of softsmearing. They studied and investigated some basicalgebraic properties of soft smearing and some relatednotions such as soft ideals, idealistic soft semirings andsoft smearing homomorphism with illustrative examples65

VOL. 3, NO. 1, January 2013 ISSN 2225-7217ARPN Journal of Science and Technology©2011-2013. All rights reserved.Based on the work of Feng et al. [13]), Acar etal. [1] introduced the basic notions of soft rings as aparameterized family of subrings of a ring over a ringwith some illustrative examples. They also introducedthe notions of soft subrings, soft ideal of a soft ring,idealistic soft rings and soft ring homomorphism withsome corresponding examples.In relation to the work of Acar et al. [1], Celiket al.,[10] introduced some new operations on soft ringsuch as extended sum, restricted sum, extended product,restricted product and established some of their basicproperties.Ali et al.,[4]defined some algebraic structuressuch as semi groups, semirings and lattices associatedwith soft sets, and completely described the distributiveand absorption laws on operations of soft. MV- algebrasand BCK-algebras associated with soft set, with a fixedset of parameters were also studied.Atagun and Sezgin[5], introduced and studiedsome sub structures such as soft subrings and soft idealsof a ring, soft subfield of a field and soft sub module of amodule with several illustrative examples. Some relatedproperties on operations of restricted intersection,product and sum for these soft sub- structures wereestablished and investigated with examples.In another work of Sezgin and Atagun [40],they first corrected some assertions in (Soft set and softgroup by (Aktas and Cagman, [2]) and further extendedthe theoretical aspects of soft group defined in (Aktasand Cagman,[2]) by introducing the concept ofnormalistic soft group, normalistic soft grouphomomorphism, and establishing that the normalisticsoft group isomorphism is an equivalence relation onnormalistic soft groups.Fu Li [14] defined the notion of soft lattice, andthe operations on soft lattice with several examples.(Nagaranjan and Meenambigar,2011)further defined softlattices, soft distributive lattices, soft modular lattices,soft lattice ideals and soft lattice homomorphism andinvestigated several related properties and somecharacterization theorems, while Karaaslan et al. ,[26]also defined soft lattices and their substructures differentfrom the above authors (Fu Li ,[14] and Nagaranjan et al.,[35]).Jun[21] applied the notion of soft sets to thetheory of BCK/BCI- algebras and introduced the notionof soft BCK/ BCI- algebras and soft sub algebras andthen derived their basic properties with some illustrativeexamples.Lee et al.[29] and Jun et al.[23] introduced thenotions of implicative, positive implicative andcommutative soft ideals. They also introduced the notionof idealistic soft BCK- algebra with some illustrativeexamples and derived their basic properties.http://www.ejournalofscience.orgJun et al.[24] introduced the concept of softHilbert algebra, soft Hilbert abysmal algebra and softHilbert deductive algebra and investigated theirproperties. Jun et al. ,[24] also in another paper,introduced the notion of soft p- ideals and p- idealisticsoft BCI- algebras and discussed their basic properties.Some other authors who have studied on the algebraicstructures of soft set theory include (Jun and Park [22]),(Sun et al.[45]), (Qin and Hong,[39]), (Ozturk et al.,[36], among others.4. CONCLUSION AND FUTURE WORKSoft set theory, introduced by Molodtsov in1999 is an effective tool with adequate parameterizationin dealing with problems of uncertainties. Recently,various researches had been done on this theory both intheory and in practice. 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