Equivalence and Correct Operations for Soft Sets

A concept of equivalence of soft sets is introduced in the article. Concepts of correct operations and correct relationships for soft sets are introduced on the basis of equivalence. Examples of correct and incorrect operations and relations are presented


The Specificity of the Notion of Soft Sets and Equivalence of Soft Sets
The formal definition of soft sets is following.Let A Bea set of parameters that can have an arbitrary nature (numbers, functions, sets of words, etc.).Let X Bea universal set.

Definition
A pair ( ) , S A will be called a soft set over X if S is a mapping from set A to the set of all subsets of X, i.e. : 2 X S A → .In fact a soft set is a parameterized family of subsets.For better understanding of the specifics of this formal definition, we should discuss the meaning of a notion "soft set".Although mathematics is usually described by science with precisely defined concepts and objects, but the practical application of mathematics is almost always associated with some blurring of the concepts or objects.This is the so-called approximate solutions.For objective reasons it is not possible to find the exact solution in many problems.For example, if the differential equation has no solution in the form of quadrature, it is necessary to use grid solution methods, which basically cannot give exact solution.Even if there is an analytical form of the solution, for example 2 y x = we still can't accurately calculate the solution at any point due to the fact that we use real numbers.
A similar situation exists for many other areas of mathematics.For practical work with objects and concepts, we are forced to introduce a collection of sets that define an approximate understanding of these objects and concepts.A concept of soft sets is a mathematical tool for dealing with such objects and concepts.The family

S S S F a
From our point of view, soft sets ( )

Introduction
The rapid development of the soft sets theory began with the appearance of [1,2].Later many authors have introduced new operations and relations for soft sets and used these

Definition
Two soft sets ( ) , S A and ( ) ′ ′ , defined over a universal set X are called equivalent, and written ( ) ( ) . Each soft set is a representative of its equivalence class.The difference between equivalent soft sets consists only in the selection of the names for the subsets (including the use of multiple names for a single subset).Therefore, the construction of the theory of soft sets should be produced considering the fact that the replacement of soft set to equivalent does not lead to any changes in results.Let us formulate this notion more formally for operations and relationships with the soft sets.

Correct Operations and Relationships with Soft Sets
We first consider the operations with soft sets.We will consider only unary and binary operations with soft sets.It is easy to transfer all of the proposed constructions and operations to a more complex structure.A unary operation on a soft set ( ) S, A , defined over the universal set X is the mapping Φ ,that for any soft set ( ) S, A corresponds the soft set ( )

Definition
A unary operation Φ is called correct if for any pair of equivalent soft sets ( ) ( ) S , ′ ′ defined over the universal set X , the results of this operation are also equivalent, i.e.

( ) ( )
. The naturalness of this requirement in the theory of soft sets is obvious.A result of the correct operation on the soft set should not depend on the parameterization method (giving names to the subsets) of a family of sets.A binary operation on a pair of soft sets ( )

Definition
Binary operation Θ is called correct if for any four pair wise equivalent soft sets ( ) ( ) ( ) ( ) ′ , defined over the universal set X , the results of this operation are also equivalent, i.e. ( ) ( ) . Only correct operations with soft sets are natural for the soft sets theory.When considering incorrect operations, a detailed explanation of the meaning of these operations and reasons for their introduction appears to need.A relationship Ω for two soft sets ( ) ( ) S, A , , F D , defined over a universal set X , is a mapping

Definition
The relationship Ω is called correct if for any four pair wise equivalent soft sets ( ) ( ) , defined over the universal set X , the equality It seems reasonable to build correctly all relationships for soft sets.

Examples of Operations and Relationships with Soft Sets
Consider first the operations with soft sets proposed in [2,3].The unary operation "complement" ( ) ( ) has a following definition.The set of parameters is the same, and the mappingis given by the formula for all a A ∈ .Binary operation "intersection" and "union" for a couple of soft sets ( ) ( ) , , , S A F D , defined on a universal set X is defined as follows.The set of parameters is chosen to be the direct product of a sets of parameters of arguments, that is equal to A D × , and the corresponding mappings are given by .

Soft set ( )
, S A is an external approximation for soft set ( ) On the basis ofinternal and external approximations for softsets we can introduce relevant concepts of equivalence.

Definition
Soft set ( ) , S A is internally equivalent to a soft set ( )

Soft set ( )
, S A is externally equivalent to a soft set ( )

Soft set ( )
, S A is weakly equivalent to a soft set ( ) and ( ) ( ) . Here is the simples properties of these relations.
The caseof a finite family of sets ( ) , S A ℑ is most interesting for the practical use.Therefore,we will examine which kind of softsets canbe internally and externally equivalent in this case.We introduce notation forthe minimum and maximum for the inclusion for the sets in ( )

Definition
For two soft sets ( ) ⊂ , and We write ) ( ) .»It is obvious that the condition of (i) leads to a lackof correct ness of the relation ( ) ( ) , and thus the relation ( ) ( ) introduced the operation of complement for softsets,based on the mysterious operation with parameters, which hasthe following definition.e not e i ¬ = ∀ .When defininga softset there are no restrictions on the set of parameters.Different objects may play role of parameters.It can be numbers, words, sentences, subsets -generally speaking everything that will choose the authorintroducinga softset.Therefore expression

Definition
e not e i ¬ = ∀ looks a completely mystery.Authors [1] introduced two more operations for softsets.

Definition
Union of two soft sets of ( ) . Secondly, even ifin definition 6.2.instead writing ( ) ( ) ) , these two operations are not correct anyway.We make one more comment [1].Itsauthors introduce the concept of absolute and null soft sets.= .These are incorrect operations with soft sets and other incorrect operations and relations are found in many papers.The list of such papers is too large to bring it here.It would certainly be useful for the theory of soft sets to try to restructure all such results on the basis of correct operations and relationships.

Fuzzy Setsand Equivalentsoftsets
A fuzzy set over the universal set X is described by membership function

Conclusion
The notion of equivalence for soft sets and concepts of the correct operations and relationsfor soft sets area fundamental conceptsof the theory of soft sets.It seems necessary todevelop the theoryof soft sets using onlythe correct operations and relationships.Using the incorrect operations and relationsto be justified by weighty practical necessity.

,,
S A , ( ) , F D ,defined over a universal set X ,is the mapping Θ , that for any pair of soft sets ( ) , S A , ( ) , F D corresponds the soft set ( ) H, B , defined over the universal set X  are correct.The proof is obvious.Consider now therelationshipfor softsets introduced in[3].This relationships are defined similarly to topology comparison.Suppose we havea pair of soft sets ( ) ( )

,FA
and ( ) , G B over the common universe U is the soft set ( ) , H C , where C A B = ∪ , and e C ∀ ∈ , (asbotharesameset).Firstly, it is not necessary that ( ) F e and ( ) G e are samesetsin determining 6.1.
NULL SOFT SET.A soft set ( ), F A over U is said to be a NULL soft set denoted by Φ , if A ε ∀ ∈ , ( )F ε = ∅ , (null-set,).Definition ABSOLUTE SOFT SET.A soft set ( ) , F A over U is said to be absolute soft set denoted by A  , if A ε ∀ ∈ , ( ) F U ε = .Each of these definitions does not define a singlesoft set, and determines the class of soft sets and number of soft sets in this class can be as many as different sets of parameters can be imagined.So you can get to the set of all sets that carries a contradiction.It is much more convenient and easier to introduce the concepts required by using the equivalence of soft sets.In fact, the null soft set is the class of equivalencefor relation ≅ , which is determined by the condition ( ) { } , S A ℑ = ∅ .Similarly, for the absolute soft set we have condition ( ) { } , S A X ℑ Figure1.It is easy to see that the relation Not Set of A Set of Parameters.Let