Study of Soft Set Theory, Its Algebra, and Application

Study of Soft Set Theory, Its Algebra, and Application

CHAPTER ONE

 Aim and Objectives

The aim of this dissertation is to study the theory of soft sets, its algebraic structures and application in decision making.

In order to achieve this aim, the following objectives are proposed to be accomplished:

1. To conduct a systematic and critical study of the concept and fundamentals of soft sets and obtain some new results:
2. Clarify some conceptual misunderstandings of the operations of Not set of a set and complement of a set.
3. Establish and investigate, with suitable examples, some distributive and absorption properties with respect to various soft set
4. Compare soft set operations with their corresponding soft matrix
5. To study some algebraic structures of soft sets and formulate some new algebraic structures.
6.  Study soft groups, soft rings, soft semirings, soft
7. Formulate certain monoids, semirings, and lattices of soft subsets and partitions of a soft
8. To investigate various techniques used in the application of soft sets to decision making problems and exemplify them and set.

CHAPTER TWO

LITERATURE REVIEW

 In this chapter, an up-to-date literature  survey of  the  concept and  fundamentals  of soft set theory, a systematization of various algebraic structures of soft sets and finally, some applications of soft set theory in decision making are presented.

Concept and Fundamentals of Soft Set Theory

The concept and development of soft set theory whose origin could be traced to the work of (Pawlak, 1993) was initiated by (Molodtsov, 1999). Molodtsov proposed that soft set theory is a general mathematical tool for solving complicated problems dealing with vagueness and uncertainties which classical methods and some modern mathematical methods, such as probability theory (Dempster, 1958), fuzzy set the- ory (Zadeh, 1965), rough set theory (Pawlak, 1982), interval mathematics theory (Gorzalzany, 1987), vague theory (Gau and Buchrer, 1993) etc., cannot successfully solve due to inadequacy of their parameterization tools.

Molodtsov (1999) pointed out that soft set theory provides enough parameter and as a result accommodates initial approximate descriptions of an object. This, makes soft set theory free from the above difficulty and becomes very convenient and easily applicable in practice. Molodtsov (1999) therefore defined a soft set as a parameterized family of subsets of a universe set, where each element is considered as the set of approximate elements of the soft set. He also successfully applied soft set theory in areas such as smoothness of functions (where he compared smoothness of functions as being similar to continuity of functions in the classical case), game theory, operations research, Rieman and Perron integrations, probability theory and measurement theory, and introduced the basic notions of the theory of soft sets.

Based on the work of( Molodtsov,1999), (Maji et al., 2003) initiated the theoretical study of the soft set theory. This includes, the definition of soft subset, soft superset, equality of soft sets, complement of a soft set among others, with some illustrative examples. Soft binary operations such as AND, OR, union and intersection operations were also defined. Verification of De Morgan’s law and a number of re- sults on soft set theory were presented. For the purpose of storing a soft set in a computer, they represented a soft set in the form of a table.

Pei and Miao (2005) discussed the relationship between soft sets and information systems and showed that soft sets are a class of special information systems.

Yang (2008) was the first to point out error in the work of (Maji et al., 2003 ) by giving a counter example.

Ali et al., (2009) also pointed out several assertions in (Maji et al., 2003) that are not true in general by counter examples Some new operations such as  restricted  union, restricted intersection, restricted difference and extended intersection were further introduced.  Moreover the notion of complement of a soft set was improved

and that certain De Morgan’s laws hold in soft set theory with respect to these new operations. Ali et al., (2009) also remarked that the incorrectness of the assertions mentioned above may be as a result of the way and manner some of the related notions were defined.

Gong et al., (2010) proposed the concept of bijective soft set and some of its operations such as the restricted AND and the relaxed AND operations.

Dependency between two bijective soft sets and the bijective soft decision system was also defined. Finally an application of bijective soft set in decision making problem was given.

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CHAPTER THREE

FUNDAMENTALS OF SOFT SET THEORY

In this chapter, a systematic and critical study of the fundamentals of soft set theory, which include operations on soft sets and their properties, soft set relations and functions, and matrix representation of soft sets are presented. Some new results on properties of soft set operations are also presented.

Basic Definitions and Results

In this section, some basic definitions and results on soft sets with suitable examples, much of which were introduced in (Molodtsov, 1999, Maji et al., 2003, Pie and Mio, 2005, Ali et al., 2009) are given.

Molodtsov (1999) in his pioneer work therefore defined soft set as follows:

Definition 3.1.1: Soft set (Molodtsov, 1999)

Let U be an initial universe set and E a set of  parameters with respect to U. Let   P (U ) denote the power set of U and A ⊆ E. A pair (F, A) is called a soft set over U, where F is a mapping given by  F  : A → P(U).  In other words, a soft set over     a universe U is a parameterized family of subsets of the universe U. For e∈A, F(e) may be considered as the set of e- elements or e- approximate elements of the soft set (F, A). Thus,

(F, A) ={ F(e)∈ P (U ): e∈A ⊆ E} .

As an illustration, let us consider the following example:

CHAPTER FOUR

ALGEBRA OF SOFT SETS

In this chapter, soft groups, soft rings, soft semi-rings, and soft lattices are presented. Finally, monoids, semirings and lattices of soft subsets and partitions of a soft set are developed.

 Soft Groups

Soft groups and their substructures, such as soft subgroup, soft normal subgroup, and soft group homomorphism and isomorphism were introduced in (Aktas and Cagman, 2007) and further studied in (Sezgin and Atagun, 2011). The concept of soft set over a group given in the following is the basic tool to define a soft group. Definition 4.1.1

Let G be a group and A be a non empty set.  Let F be a set –valued function from         A to G, defined as: F(x) = {y ∈ G: (x, y) ∈ R, x ∈ A and y ∈ G}, where R⊆A x

Then, (F,A) is called a soft set over a group G.  It may be observed that the function F: A−→ P(G), in fact, defines the relation R:A−→G  given by   R= { (x, y) ∈ AxG, x ∈ A and y ∈ F(x)}.

CHAPTER FIVE

APPLICATIONS OF SOFT SET THEORY IN DECISION MAKING

A number of researchers [Maji et al.,(2002), Chen et al., (2005), Cagman and Enginoglu,(2010), Mitra et al.,(2012), Samsiah and Mohamad,(2012)] etc., have applied the concepts of soft set theory to real life problems involving uncertainties. Such problems involve decision making, medical diagnosis, forecasting, among others. In this chapter, some typical applications using different approaches, especially in decision making, is discussed, and compared. The parameter reduction in soft set theory which was employed by some researchers is first described.

CHAPTER SIX

SUMMARY, CONCLUSION AND RECOMMENDATIONS

  Summary

This dissertation, consisting of six chapters, presents a systematic study of soft set theory, its algebras and applications in decision making.

In chapter one, a general introduction of the dissertation, which includes the background of the study, motivation and justification of the study, the aim and objectives amongst others, is presented.

An up- to- date literature survey of the development of soft set theory, in respect of fundamentals, algebraic structures and applications in decision making, is presented in chapter two.

Chapter three presents a comprehensive and critical study of the fundamentals of soft set theory: soft set operations, soft set relations and functions,  soft matrices  and their properties and some new results, mainly on the distributive and absorption properties, with respect to various soft set operations are presented.  In addition,  soft set operations are shown to be equivalent to their corresponding soft matrix operations. This chapter forms a stepping stone to the study of the underlying algebraic structures and applications in the forthcoming chapters.

A study of various soft set algebras such as soft groups, soft rings, soft semirings, and soft lattices, which have been developed so far in the literature, is presented in chapter four. The main contribution in the part of the dissertation is formulation  and exemplification of monoids, semirings and lattices of soft subsets and partitions of a soft set.

In chapter five, techniques of applications of soft set theory in decision making are presented.

Finally, conclusion and recommendations of the dissertation are presented.

Conclusion

In this dissertation, fundamentals of soft set theory, various algebraic structures of soft sets and their applications in decision making have been critically studied and some new results obtained.

The following new results are the main contributions of this dissertation: Let (F,A) , (G,B) and (H,C) be soft sets over a common universe. Then,

Distributive Properties-Proposition 3.2.2 (Singh and Onyeozili, 2012A, 2012B, 2012C)

(i)  (F, A)∪˜ ((G, B) ∩_ε (H, C)) ƒ= ((F, A)∪˜(G, B)) ∩_ε ((F, A)∪˜(H, C))

(ii)  (F, A) ∩_ε ((G, B)∪˜(H, C)) ƒ= ((F, A) ∩_ε (G, B)) ∪˜ ((F, A) ∩_ε (H, C))

(iii) (F, A)∪_R ((G, B) ×_R (H, C)) ƒ= ((F, A) ∪_R (G, B)) ×_R ((F, A) ∪_R (H, C))

(iv) (F, A)∩_ε ((G, B) ×_R (H, C)) ƒ= ((F, A) ∩_ε (G, B)) ×_R ((F, A) ∩_ε (H, C))

(v)  (F, A)∪˜ ((G, B) ×_R (H, C)) ((F, A)∪˜(G, B)) ×_R ((F, A)∪˜(H, C))

(vi)  (F, A) ∪_R ((G, B)∪˜(H, C)) = ((F, A) ∪_R (G, B)) ∪˜ ((F, A) ∪_R (H, C))

(vii)  (F, A)∩˜ ((G, B) ∩_ε (H, C)) = ((F, A)∩˜(G, B)) ∩_ε ((F, A)∩˜(H, C))

(viii)  (F, A)∪˜ ((G, B) ∪_R (H, C)) ƒ= ((F, A)∪˜(G, B)) ∪_R ((F, A)∪˜(H, C))

(ix)  (F, A) ∩_ε ((G, B)∩˜(H, C)) ƒ= ((F, A) ∩_ε (G, B)) ∩˜ ((F, A) ∩_ε (H, C))

(x)  (F, A)  ∧ ((G, B) ×_R (H, C)) = ((F, A) ∩˜(G, B)) ×_R ((F, A)∩˜(H, C))

(xi)  (F, A) ∨ ((G, B) ×_R (H, C)) ƒ= ((F, A) ∩˜(G, B)) ×_R ((F, A)∩˜(H, C))

[Proofs, see page 44]

Absorption Properties and Absorption Inclusions (Singh and Onyeozili,2012A)

 Absorption Properties-Proposition2.1

(i)  (F, A)∪˜ ((F, A)∩˜(G, B)) = (F, A)

(ii)  (F, A)∩˜ ((F, A)∪˜(G, B)) = (F, A)

(iii) (F, A) ∪_R ((F, A) ∩_ε (G, B)) = (F, A) (iv) (F, A) ∩_ε ((F, A) ∪_R (G, B)) = (F, A)

[Proofs, see page 38 ]

Absorption Inclusions-Proposition 3.2.3

(i)  (F, A) ∪_R ((F, A)∩˜(G, B)) ⊂˜(F, A)

(ii)  (F, A)∩˜ ((F, A) ∪_R (G, B)) ⊂˜(F, A),

(iii)  (F, A)∪˜ ((F, A) ∩_ε (G, B)) ⊃˜(F, A),  and (iv)  (F, A) ∩_ε ((F, A)∪˜(G, B)) ⊃˜(F, A).

[Proofs see page 51 ]

Formulation of Monoids, Semirings and Lattices of Soft subsets and Partitions of a soft set

Commutative, Idempotent Monoids and Semirings of Soft Subsets of a SoftSet

1. 1. Monoids

1. (SS(F,A), ∗ ∈ {∪˜,∩˜})  are  commutative,  idempotent  monoids  with  Φ˜∅  as the identity
2. (SS(F,A),∪_R) is a commutative, idempotent monoid with Φ˜A  as the identity
3. (SS(F,A), ∩_R) is a commutative, idempotent monoid with (F, A) as the identity

Semirings

 (SS(F,A),∪˜,∪_R) is a commutative, idempotent semiring with identity ele- ment Φ˜∅  .

(SS(F,A),∩_R,

element (F,A).

∪˜)  is  a  commutative,  idempotent  semiring  with  identity

(SS(F,A),∪_R,∩˜) is a commutative, idempotent semiring with identity ele- ment Φ˜A  .

(SS(F,A),

element Φ˜∅  .

∩˜,∪_R  )  is  a  commutative,  idempotent  semiring  with  identity

(SS(F,A), ∪_R, ∩_R) is a commutative, idempotent semiring with identity elementΦ˜A  

(SS(F,A), ∩_R,∪_R} ) is a commutative, idempotent semiring with identity element (F,A).

(SS(F,A),

∩˜,  ∩_R  )  is  a  commutative,  idempotent  semiring  with  identity

element Φ˜  ∅  .

(SS(F,A), ∪˜, ∩_R} ) is a commutative, idempotent semiring with identity

element Φ˜  ∅.

[Proofs, see page 96 ]

Bounded, Distributive Lattices of Soft Subsets of a Soft Set

 (SS(F, A), ∩˜,∪_R ) and (SS(F, A), ∪_R,∩˜),  and

(ii)  (SS(F, A), ∩_R, ∪˜) and (SS(F, A), ∪˜, ∩_R)

are bounded, distributive lattices. [Proofs, see page 98 ]

Commmutative, Idempotent Monoids and Semirings of Partitions of a Soft Set

1. Monoids

1 ( ℘(F, A), H) is a commutative, idempotent monoid. 2 (℘(F, A) H) is a commutative, idempotent monoid.

B. Semirings

1. (℘(F, A), H, H) with the partition consisting of the single block as the identity
2. (℘(F, A), H,H) with the partition consisting of the singleton blocks as the identity

[Proofs see page 99 ]

Bounded, Distributive Lattices of Partitions of a Soft Set

(i) (℘(F, A), H_E, H_R),

• (℘(F, A), H _R, H _E),and
• (℘(F, A), H, H) and (iv) (℘(F, A), H , H) are bounded , distributive [Proofs, see page 102 ]

Recommendations

Since soft set theory is a relatively new area in mathematics and also has been recognized to be important in applications to real life decision making problems involving uncertainties, the following recommendations may be taken into consideration:

In view of the techniques of soft set theory being relatively simpler and widely applicable, researches in various hybrid theories like fuzzy soft sets, multi soft sets, soft multi sets, etc., are promising(Maji et al., 2001,

Alkhazaleh et al., 2011, Pinaki and Samanta, 2012, Singh et al., 2014, are som specific references).

Also, as studies in the area of soft topological analysis are appearing recently (Cagman, et al., 2011, Shabir and Naz, 2011, Tariq and Salma, 2014, are some representative references), it may be a promising research area to obtain some deep mathematical results.

In view of the aforesaid facts, soft set theory could be inculcated into the curricula of studies, both at undergraduate and postgraduate levels.

LIST OF PUBLICATIONS

Journal Papers

• Singh, D., Onyeozili, I.A. (2012). Some Conceptual Misunderstandings ofthe Fundamental of Soft Set ARPN Journal of Systems and Software, 2(9):251-254.
• Singh, D., Onyeozili, I.A. (2012). Notes on Soft Matrices Operations. ARPN Journal of Science and Technology,2(9):861-869.
• Singh, D., Onyeozili, I.A. (2012).  Some Results On Distributive and Absorp-  tionProperties of Soft  IOSR Journal of Mathematics, 4(2):18-30.
• Singh, D., Onyeozili, I.A. (2012). On Some New Properties of Soft Set Oper- ations. International Journal of Computer Applications,59(4):39-44.
• Singh, D.,  Onyeozili,  I.A. (2013).  Matrix Representation of Soft Sets and  Its Application to Decision Making Problem. The International Journalof Engineering And Science, 2(2):48-56.
• Singh, D., Onyeozili, I.A., Alkali, A. J. (2014). Notes on Multi Soft Matrices. Journalof Emerging Trends in Computing and Information Sciences, 5(5):421- 427.

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