1. Introduction and Preliminaries
Along with various mathematical models, an essential tool for the handling of imprecise and vague data was introduced by Zadeh [
1], known as “fuzzy set theory”. It was observed that fuzzy set theory does not provide enough parameters to tackle the uncertainty in various data structures, so Molodtsov [
2] in 1999 initiated the study of “soft sets” to overcome the deficiencies and shortcomings affiliated with the parameters in connection with fuzzy set theory. Since that time, soft set theory has been used to solve problems in various fields, for example, in decision making [
3], game theory [
4], and medical diagnosis [
5].
Soft set theory has further been developed in many directions, for instance, soft algebraic structures and extensions, hybrid models such as fuzzy soft sets, rough soft sets, soft topology and the semantics of soft sets and decision making (compare [
2,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19]). Maji et al. [
15] presented an elaborated study related to some algebraic operations on soft sets. Furthermore, Maji et al. [
3] used soft set theory to solve problems in decision making. Yang [
20], Ali et al. [
21], and Li [
22] pointed out some shortcomings in the existing literature related to basic operations in soft set theory and introduced new and improved operations.
In 2010, Qin and Hong [
23] discussed the generalized soft equality relations known as upper soft equality
and lower soft equality
. In 2014, Abbas et al. [
24] presented definitions which preserve assertions from the classical theory of crisp sets (that an empty set is a subset of every set). Furthermore, they presented a version of generalized soft equality named
g-soft equality by generalizing the upper and lower soft equality. In 2017, by relaxing the conditions on the parameter set, another definition of generalized finite soft equality, known as
-soft equality, was introduced by Abbas et al. [
25]. Moreover, they introduced generalized finite soft unions and intersections as well. Very recently, Al-shami [
7] pointed out and corrected some shortcomings in [
25].
The reduction of parameters is an important operation which is used to improve the performance of decision-making processes linked with various uncertainty theories; for details, see [
26,
27]. As in soft set theory, enough parameters can be used to handle fuzziness in the data. In this paper, by further relaxing the conditions on parameters, we introduce new notions, which we refer to as the generalized finite relaxed soft subset (
-soft subset), generalized finite relaxed soft equality (
-soft equality), and generalized finite relaxed soft unions and intersections (
-soft union and
-soft intersection) of two soft sets and via these notions, we extend various results given in [
7,
23,
24,
25]. Although they have obtained new results, we have noted the shortcomings pointed out in [
7] regarding the previous notions of soft equality.
Some required elementary definitions, results, and concepts associated with soft sets will be presented in this section before we prove the main results.
Let U be an initial universe and E a set of parameters. Throughout this paper, represents the family of all subsets of U and represents the collection of nonempty subsets of U. For denotes the set difference of M and
Molodtsov [
2] defined a soft set as follows.
Definition 1 ([
2])
. If then is called a soft set over U ifwhere is a set-valued mapping. A soft set can be viewed as a parametrized family of subsets of That is, for every the set is a c-approximate element of the soft set We use the notation to denote the collection of all soft sets over a common universe Maji et al. [
15] defined some operations on soft sets as follows.
Definition 2 ([
15])
. A soft set is a null soft set if for all (empty set). Definition 3 ([
15])
. If then is a soft subset of denoted as if Definition 4 ([
15])
. For is soft-equal to if and only if and In the following, Zhu and Wen [
28] modified Definition 1 by replacing
with
. Furthermore, they defined the null soft set as well.
Definition 5 ([
28])
. A soft set is a null soft set if . Qin and Hong [
23] defined soft equalities
and
named as lower soft equality and upper soft equality relations, respectively.
Definition 6 ([
23])
. Let and be soft sets over Then is a lower soft equal to denoted as if and only if Definition 7 ([
23])
. Let and be soft sets over Then is an upper soft equal to , denoted as if and only ifIt has been observed that some basic properties discussed in [15,28] do not hold true in general. These were pointed out and improved by Ali et al. [21], Li [22], Yang [20], and Zhu et al. [28] and Abbas et al. [24]. Some new operations on soft sets were considered as follows to remove the existing deficiencies in soft set theory and to develop the theory further. Definition 8 ([
24])
. A soft set is a generalized null soft set (g-null soft set) ifand it is denoted as Definition 9 ([
24])
. A soft set is said to be a generalized universal soft set (g-universal soft set) if- (i)
and
- (ii)
for each
A g-universal soft set is represented by
Definition 10 ([
24])
. If then is a generalized soft subset (g-soft subset) of , denoted as if Definition 11 ([
24])
. Let Then the soft sets and are g-soft-equal, that is, if and only if and Proposition 1 ([
24])
. If then implies Abbas et al. [
25] further generalized the concept of soft subsets, given as follows.
Definition 12 ([
25])
. If then is a generalized finite soft subset of (-soft subset of ) denoted as if and only if for every there exists a finite subset such asIf is a singleton set then reduces to . Using the above definition, they introduced the -soft equality of two soft sets.
Definition 13 ([
25])
. If then and are called generalized finite soft equal (-soft equal), denoted as if and only if and Furthermore, they obtained the following proposition.
Proposition 2 ([
25])
. If then- (1)
implies
- (2)
implies
Definition 14 ([
15])
. If then for the union of and , denoted as is defined as Definition 15 ([
15])
. If then for the intersection of and , denoted as is defined asfor all Ali et al. (see Example 2.2 in [
21]) pointed out that some results in Proposition 2.4 in [
15] are incorrect. They further introduced the following versions of the union and intersection of soft sets.
Definition 16 ([
21])
. If then the restricted union of and , denoted as is a soft set defined asfor all . Definition 17 ([
21])
. If then the extended intersection denoted as is a soft set , where and is defined as Definition 18 ([
21])
. If then the restricted intersection of and denoted as is a soft set , defined asfor all Definition 19 ([
21])
. If then the relative complement of is a soft set denoted as , defined as wherefor all Al-Shami [
7] introduced the following classes of soft sets.
Definition 20 ([
7])
. A soft set is called:- (1)
A huge soft set if there exists such as
- (2)
A large soft set if there exists a finite set such as
2. Generalized Finite Relaxed Soft Equality
In this section, we present the generalized notion of the equality of two soft subsets. We start with the following definitions.
Definition 21. Let and be two soft sets over a common universe We say that is the generalized finite relaxed soft subset of if either or, for some there exists a finite subset of N such asand we denote this as Definition 22. Let and belong to Soft sets and are referred to as generalized finite relaxed soft-equal ifand we denote this as Now we present the following proposition that shows that the generalized finite relaxed soft equality of soft sets is more general than the existing concepts of soft equalities.
Proposition 3. If then
- (1)
implies implies
- (2)
implies implies
Proof. This proof follows from Definitions 12, 13, 21, 22 and Proposition 2. □
The following example shows that Definitions 21 and 22 are proper generalizations of Definitions 12 and 13.
Example 1. Let be the given universal set and the sets of parameters and . Define Note thatfor any subset That is On the other hand, there exists , such aswhere is any finite subset containing Hence Similarly, This implies that Note that Consider another example to show that there exists soft subsets , such as
Example 2. Let be the given universal set and the sets of parameters and . Define As observed, and also for any parameter c. Hence,
In [
7] the author presented the following proposition.
Proposition 4 - (1)
Every huge soft set is a large soft set.
- (2)
Any two huge soft sets are g-soft-equal, which implies that they are -soft-equal.
- (3)
The large and huge soft sets are -soft-equal.
The following example shows that the converse of statement (1) in the above proposition does not hold true.
Example 3. Consider and the sets of parameters If we definethen is large but it is not a huge soft set. We present the following proposition regarding huge and large soft sets.
Proposition 5. - (1)
Any two huge soft sets are -soft-equal.
- (2)
The large and huge soft sets are -soft-equal.
Proof. This proof follows from Propositions 3 and 4. □
Abbas et al. [
25] obtained the following result that shows that the operations
and
are idempotent, commutative, and associative with respect to the generalized finite soft equality relation
Theorem 1 ([
25])
. If then- (a)
- (b)
- (c)
- (d)
- (e)
- (f)
The above theorem holds true for the soft equality relation
Theorem 2. If then
- (a)
- (b)
- (c)
- (d)
- (e)
- (f)
Proof. This proof follows from Proposition 3 and Theorem 1. □
In [
24], the authors proved that for
(the lower soft equality of
and
) and
(the upper soft equality of
and
) imply that
(the
g-soft equality of
and
). Recently Al-shami [
7] presented a counter-example (Example 3.1 in [
7]) to show that
does not imply
In the following, we present results that show that the lower and upper soft equalities of
and
do imply the generalized finite relaxed soft equality of
and
Proposition 6. If , then
Proof. Suppose that
We will first show that
Since for
the result holds trivially. Therefore, we assume that
For any
if
then
so
for all
If
∖
then
and so
for all
As for some parameter
there exists a finite subset
of
N, such as
implies
On similar lines, it can be shown that
Consequently, we deduce
□
Proposition 7. If then .
Proof. For consider the following cases:
- (c-1)
If and M and N are nonempty, this implies and where and This implies that
- (c-2)
If then for each This is enough to conclude that
- (c-3)
If and either M or N is empty, then trivially the assertion holds true as well.
□
Remark 1. Consider the soft sets and as given in Example 1; then, neither nor but, on the other hand, .
In the following result, lower soft equality holds between the extended and restricted unions and intersections of two soft sets.
Theorem 3 ([
23])
. If and then- (a)
- (b)
- (c)
- (d)
The following example shows that the above result can not be extended for the -soft equality .
Example 4. If , , the soft sets are given asthen and Theorem 4. If then the following is true.
- (1)
- (2)
Proof. This proof follows from Proposition 3 and Theorem 4.4 in [
24]. □
Remark 2. Let If then and do not necessarily hold true, as shown in the examples (see Remark 4.6 and Example 4.7 in [24]). The following result is related to De Morgan’s laws for soft sets, involving generalized finite relaxed soft equality.
Theorem 5. If , such as then
- (a)
- (b)
- (c)
- (d)
Proof. This proof follows from Proposition 3 and Theorem 4.14 and Theorem 4.15 in [
24]. □
Remark 3. The lower soft equality is a congruence relation [23], that is, and imply thatwhereas is not a congruence relation as shown in the following example. Example 5. Consider , and the soft sets defined as Note that and but 3. Generalized Finite Relaxed Soft Union and Intersection
In this section, we present the notions of generalized finite relaxed soft unions and intersections of soft sets.
Recently, Abbas et al. [
25] introduced generalized finite unions of soft sets as follows.
Definition 23 ([
25])
. The generalized finite soft union of , denoted by is the set consisting of all satisfying:- (c-i)
and where
- (c-ii)
If there exists , such as and , then
That is, is a minimal -soft super set of and in the sense that if there exists another soft set satisfying (c-i), then is a -soft subset of
Furthermore, they introduced the following version of the generalized finite intersection of soft sets.
Definition 24 ([
25])
. The generalized finite intersection of , denoted as is the set of all satisfying:- (c-iii)
and , where
- (c-iv)
If there exists such as and then
Here is a maximal -soft subset of and in the sense that if there exists another soft set satisfying (c-iii), then is the -soft subset of
We present the definition of generalized finite relaxed unions of soft sets as follows.
Definition 25. Consider Then the generalized finite relaxed soft union denoted by is defined as the collection of all soft sets , which satisfies the following two conditions:
- (a)
and where
- (b)
If there exists , such as and , then
That is, is a minimal -soft super set of and in the sense that if there exists another soft set satisfying (a), then is a -soft subset of
Consider the following example where is nonempty.
Example 6 (compare Exmple 2.15 in [
25]).
Suppose that is the universe and and are subsets of the set of parameters LetLet such as and where If i.e., for or we deduce that if or then will satisfy the condition for some
We deduce that both conditions and for -union hold true and so the set is nonempty.
Definition 26. Let and belong to Then the generalized finite relaxed soft intersection denoted by is defined as the collection of all soft sets which satisfies the following two conditions:
- (c)
and where
- (d)
If there exists such as and , then
Here, is a maximal -soft subset of and in the sense that if there exists another soft set satisfying (c), then is the -soft subset of
Abbas et al. [
25] presented the following result.
Theorem 6 ([
25])
. If then Remark 4. If thendoes not hold true, as shown in the following example. Example 7. If and and the soft sets defined asthen For clearly, Remark 5. Note that the above Example 7 also shows thatparticularly when As observed, For clearly,but Hence,
Abbas et al. [
25] presented the following result.
Proposition 8. Let and be two soft sets over a common universe such as Then
Al-Shami [
7] presented the following example to show that Proposition 8 does not hold true.
Example 8 (compare Example 3.11 in [
7]).
Let be a universal set and are the sets of parameters. Define soft sets as follows: andNote that Abbas et al. [
25] presented the following lemma.
Lemma 1 ([
25])
. If then Note that Lemma 1 does not hold true if we replace with In this context we present the following example.
Example 9. Let be a universal set; the set of parameters; and and . Consider the following soft setsAs can be observed, although and but Using Lemma 1, they [
25] obtained the following results.
Theorem 7 ([
25])
. Let and be two soft sets over a common universe U, such as Then Theorem 8 ([
25])
. Let and be two soft sets over a common universe U, such as Then Now we present the following result.
Proposition 9. Let and be two soft sets over a common universe such as Then Proof. Let
, where
Then
for all
Since
therefore,
We know
where
Thus,
and similarly
Next consider
and
such as
and
Then
for some
where
are finite. As
so
for some
where
Therefore,
. This shows that
□
Example 10. Let be a universal set, the set of parameters and . Define,ThenIfthen and and, as observed, Now we present a characterization of the soft sets in via generalized finite relaxed soft equality as follows.
Theorem 9. Let and be two soft sets over a common universe U, such as If then Proof. Let
Then, according to the definition of
,
and
where
Since
Furthermore, from Proposition 9 we have
The inclusions (
1) and (
2) imply that
Consequently, we can obtain
□
The converse of Theorem 9 does not hold true, as shown in the following example.
Example 11. Let be a universal set, the set of parameters and . Consider the soft setsNote that andConsiderand, as observed, but . Hence, 4. Conclusions
In an attempt to develop further the soft set theory, we generalized the notions of
-soft subsets and the
-soft equality of two soft subsets by introducing the notions of
-soft subsets and the
-soft equality of two soft sets via a further relaxation of constraints on the parameter sets. Example 8, developed by Al-Shami [
7], shows that Proposition 8 about
-soft intersection does not hold true. However, with the generalized notions, we were able to prove Proposition 8 in regard to
-soft intersection. Furthermore, Al-Shami [
7] pointed out that the upper soft equality
of two soft sets does not imply the
-soft equality but the upper soft equality implies the
-soft equality of two soft sets. Regarding
-soft unions, Example 7 shows that Theorem 4 cannot be extended for
-soft unions. Moreover, with
-soft intersections, the converse of Theorem 9 does not hold true, as shown in Example 11. We can see that generalized finite relaxed (
) operations are better when it comes to the
-soft intersection of soft sets as compared to the
-soft union of soft sets.
As a future direction, one can investigate
-soft operations to develop further refinements so that the above-mentioned shortcomings can be removed. Moreover, based on these new operations, one can consider the extension of existing soft topologies and various hybrid structures involving soft sets (compare [
8,
17,
29]).