A Contribution to the Theory of Soft Sets via Generalized Relaxed Operations

Abstract

Soft set theory has evolved to provide a set of valuable tools for dealing with ambiguity and uncertainty in a variety of data structures related to real-world challenges. A soft set is characterized via a multivalued function of a set of parameters with certain conditions. In this study, we relax some conditions on the set of parameters and generalize some basic concepts in soft set theory. Specifically, we introduce generalized finite relaxed soft equality and generalized finite relaxed soft unions and intersections. The new operations offer a great improvement in the theory of soft sets in the sense of proper generalization and applicability.

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 $\left({\approx }^s\right)$ and lower soft equality $\left({\approx }_s\right)$. 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 $gf$-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 ($g{f}_r$-soft subset), generalized finite relaxed soft equality ($g{f}_r$-soft equality), and generalized finite relaxed soft unions and intersections ($g{f}_r$-soft union and $g{f}_r$-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, $P\left(U\right)$ represents the family of all subsets of U and $P^{*}\left(U\right)$ represents the collection of nonempty subsets of U. For $M,N\subseteq E,$$M\setminus N$ denotes the set difference of M and $N.$

Molodtsov [2] defined a soft set as follows.

Definition 1

([2]). If $M\subseteq E,$ then $F_M$ is called a soft set over U if

$$F_M=\{\left(c,f_M\left(c\right)\right):c\in M)\}$$

where $f_M:M\to P\left(U\right)$ is a set-valued mapping. A soft set $F_M$ can be viewed as a parametrized family of subsets of $U.$ That is, for every $c\in M,$ the set $f_M\left(c\right)\subseteq U$ is a c-approximate element of the soft set $F_M.$ We use the notation $S_E\left(U\right)$ to denote the collection of all soft sets over a common universe $U.$

Maji et al. [15] defined some operations on soft sets as follows.

Definition 2

([15]). A soft set $F_M\in S_E\left(U\right)$ is a null soft set if for all $c\in M,$$f_M\left(c\right)=\varnothing$ (empty set).

Definition 3

([15]). If $F_M,G_N\in S_E\left(U\right),$ then $F_M$ is a soft subset of $G_N$ denoted as $F_M\tilde{\subset }{G}_N$ if

$$M\subseteq N\mathit{and}\mathit{for}\mathit{all}c\in M,f_M\left(c\right)=g_N\left(c\right).$$

Definition 4

([15]). For $F_M,G_N\in S_E\left(U\right),$$F_M$ is soft-equal to $G_N$ if and only if $F_M\tilde{\subset }{G}_N$ and $G_N\tilde{\subset }{F}_M.$

In the following, Zhu and Wen [28] modified Definition 1 by replacing $P\left(U\right)$ with $P^{*}\left(U\right)$. Furthermore, they defined the null soft set as well.

Definition 5

([28]). A soft set $F_M\in S_E\left(U\right)$ is a null soft set if $M=\varnothing$.

Qin and Hong [23] defined soft equalities ${\approx }_s$ and ${\approx }^s$ named as lower soft equality and upper soft equality relations, respectively.

Definition 6

([23]). Let $F_M$ and $G_N$ be soft sets over $U.$ Then $F_M$ is a lower soft equal to $G_N$ denoted as $F_M{\approx }_s{G}_N$ if and only if

$$\begin{array}{ccc}\hfill f_M\left(c\right)& =& g_N\left(c\right),\mathit{if}c\in M\cap N,\hfill \\ \hfill f_M\left(c\right)& =& \varnothing ,\mathit{if}c\in M\setminus N,\mathit{and}\hfill \\ \hfill g_N\left(c\right)& =& \varnothing ,\mathit{if}c\in N\setminus M.\hfill \end{array}$$

Definition 7

([23]). Let $F_M$ and $G_N$ be soft sets over $U.$ Then $F_M$ is an upper soft equal to $G_N$, denoted as $F_M{\approx }^s{G}_N$ if and only if

$$\begin{array}{ccc}\hfill f_M\left(c\right)& =& g_N\left(c\right),\mathit{if}c\in M\cap N,\hfill \\ \hfill f_M\left(c\right)& =& U,\mathit{if}c\in M\setminus N,\mathit{and}\hfill \\ \hfill g_N\left(c\right)& =& U,\mathit{if}c\in N\setminus M.\hfill \end{array}$$

It 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 $F_M\in S_E\left(U\right)$ is a generalized null soft set (g-null soft set) if

$$\mathit{either}M=\varnothing \mathit{or}{f}_M\left(c\right)=\varnothing \mathit{for}\mathit{every}c\in M$$

and it is denoted as ${\varnothing }_M.$

Definition 9

([24]). A soft set $F_M\in S_E\left(U\right)$ is said to be a generalized universal soft set (g-universal soft set) if

(i) 

$M=E\ne \varnothing$ and

(ii) 

$f_M\left(c\right)=U$ for each $c\in E.$

A g-universal soft set is represented by $U_E.$

Definition 10

([24]). If $F_M,G_N\in S_E\left(U\right),$ then $F_M$ is a generalized soft subset (g-soft subset) of $G_N$, denoted as $F_M{\subseteq }_g{G}_N$ if

$$M=\varnothing \mathit{or\; for\; every}c\in M,\mathit{there\; is\; some}{c}^{\stackrel{´}{}}\in N\mathit{such\; as}{f}_M\left(c\right)\subseteq g_N\left(c^{\stackrel{´}{}}\right)$$

Definition 11

([24]). Let $F_M,G_N\in S_E\left(U\right).$ Then the soft sets $F_M$ and $G_N$ are g-soft-equal, that is, $F_M{\approxeq }_g{G}_N$ if and only if $F_M{\subseteq }_g{G}_N$ and $G_N{\subseteq }_g{F}_M.$

Proposition 1

([24]). If $F_M,G_N\in S_E\left(U\right),$ then $F_M{\approx }_s{G}_N$ implies $F_M{\approxeq }_g{G}_N.$

Abbas et al. [25] further generalized the concept of soft subsets, given as follows.

Definition 12

([25]). If $F_M,G_N\in S_E\left(U\right),$ then $F_M$ is a generalized finite soft subset of $G_N$ ($gf$-soft subset of $G_N$) denoted as $F_M{\subseteq }_{gf}{G}_N$ if and only if for every $c\in M,$ there exists a finite subset $N^{\prime }\subseteq N$ such as

$$f_M\left(c\right)\subseteq {\cup }_{c^{\stackrel{´}{}}\in N^{\prime }}{g}_N\left(c^{\prime }\right).$$

If $N^{\prime }$ is a singleton set then $F_M{\subseteq }_{gf}{G}_N$ reduces to $F_M{\subseteq }_g{G}_N$.

Using the above definition, they introduced the $gf$-soft equality of two soft sets.

Definition 13

([25]). If $F_M,G_N\in S_E\left(U\right),$ then $F_M$ and $G_N$ are called generalized finite soft equal ($gf$-soft equal), denoted as $F_M{\approxeq }_{gf}{G}_N$ if and only if $F_M{\subseteq }_{gf}{G}_N$ and $G_N{\subseteq }_{gf}{F}_M.$

Furthermore, they obtained the following proposition.

Proposition 2

([25]). If $F_M,G_N\in S_E\left(U\right),$ then

(1) 

$F_M{\subseteq }_g{G}_N$ implies $F_M{\subseteq }_{gf}{G}_N.$

(2) 

$F_M{\approxeq }_g{G}_N$ implies $F_M{\approxeq }_{gf}{G}_N.$

Definition 14

([15]). If $F_M,G_N\in S_E\left(U\right),$ then for $C=M\cup N,$ the union of $F_M$ and $G_N$, denoted as $H_C=F_M\stackrel{\sim }{\cup }{G}_N,$ is defined as

$$h_C\left(c\right)=\left\{\begin{array}{c}{f}_M\left(c\right),\mathit{if}c\in M\setminus N,\hfill \\ g_N\left(c\right),\mathit{if}c\in N\setminus M,\hfill \\ f_M\left(c\right)\cup g_N\left(c\right),\mathit{if}c\in M\cap N.\hfill \end{array}\right.$$

Definition 15

([15]). If $F_M,G_N\in S_E\left(U\right),$ then for $C=M\cap N,$ the intersection of $F_M$ and $G_N$, denoted as $H_C=F_M\stackrel{\sim }{\cap }{G}_N,$ is defined as

$$h_C\left(c\right)=f_M\left(c\right)\mathrm{or}{g}_N\left(c\right)(\mathit{as}\mathit{both}\mathit{sets}\mathit{are}\mathit{same})$$

for all $c\in C.$

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 $F_M,G_N\in S_E\left(U\right),$ then the restricted union of $F_M$ and $G_N$, denoted as $F_M{\cup }_{\mathcal{R}}{G}_N,$ is a soft set $H_C$ defined as

$$h_C\left(c\right)=f_M\left(c\right)\cup g_N\left(c\right),$$

for all $c\in C=M\cap N$.

Definition 17

([21]). If $F_M,G_N\in S_E\left(U\right),$ then the extended intersection denoted as $F_M{\cap }_{\epsilon }{G}_N$ is a soft set $H_C$, where $C=M\cup N$ and $H_C$ is defined as

$$h_C\left(c\right)=\left\{\begin{array}{c}{f}_M\left(c\right),\mathit{if}c\in M\setminus N,\hfill \\ g_N\left(c\right),\mathit{if}c\in N\setminus M,\hfill \\ f_M\left(c\right)\cap g_N\left(c\right),\mathit{if}c\in M\cap N.\hfill \end{array}\right.$$

Definition 18

([21]). If $F_M,G_N\in S_E\left(U\right),$ then the restricted intersection of $F_M$ and $G_N$ denoted as $F_M⋒G_N$ is a soft set $H_C$, defined as

$$h_C\left(c\right)=f_M\left(c\right)\cap g_N\left(c\right)$$

for all $c\in C=M\cap N.$

Definition 19

([21]). If $F_M\in S_E\left(U\right)$ then the relative complement of $F_M$ is a soft set denoted as ${\left(F_M\right)}^r$, defined as ${\left(F_M\right)}^r=F_M,$ where

$$f_M^r\left(c\right)=U\setminus f_M\left(c\right)$$

for all $c\in M.$

Al-Shami [7] introduced the following classes of soft sets.

Definition 20

([7]). A soft set $F_M\in S_E\left(U\right)$ is called:

(1) 

A huge soft set if there exists $c\in M$ such as $f_M\left(c\right)=U.$

(2) 

A large soft set if there exists a finite set $M^{\stackrel{´}{}}\subseteq M$ such as ${\cup }_{c\in M^{\stackrel{´}{}}}{f}_M\left(c\right)=U.$

2. Generalized Finite Relaxed Soft Equality $\left\{{\mathbf{\approxeq }}_{{\mathbf{gf}}_{\mathbf{r}}}\right\}$

In this section, we present the generalized notion of the equality of two soft subsets. We start with the following definitions.

Definition 21.

Let $F_M$ and $G_N$ be two soft sets over a common universe $U.$ We say that $F_M$ is the generalized finite relaxed soft subset of $G_N$ if either $M=\varnothing$ or, for some $c\in M,$ there exists a finite subset $N^{\stackrel{´}{}}$ of N such as

$$f_M\left(c\right)\subseteq {\cup }_{c^{\stackrel{´}{}}\in N^{\stackrel{´}{}}}{g}_N\left(c\right)$$

and we denote this as $F_M{\subseteq }_{g{f}_r}{G}_N.$

Definition 22.

Let $F_M$ and $G_N$ belong to $S_E\left(U\right).$ Soft sets $F_M$ and $G_N$ are referred to as generalized finite relaxed soft-equal if

$$F_M{\subseteq }_{g{f}_r}{G}_N\mathit{and}{G}_N{\subseteq }_{g{f}_r}{F}_M$$

and we denote this as $F_M{\approxeq }_{g{f}_r}{G}_N.$

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 $F_M,G_N\in S_E\left(U\right),$ then

(1) 

$F_M{\subseteq }_g{G}_N$ implies $F_M{\subseteq }_{gf}{G}_N$ implies $F_M{\subseteq }_{g{f}_r}{G}_N,$

(2) 

$F_M{\approxeq }_g{G}_N$ implies $F_M{\approxeq }_{gf}{G}_N$ implies $F_M{\approxeq }_{g{f}_r}{G}_N.$

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 $U=\left\{u_1,u_2,u_3,u_4,u_5\right\}$ be the given universal set and the sets of parameters $E=\left\{c_1,c_2,c_3\right\},$$M=\left\{c_1,c_2\right\}$ and $N=\left\{c_1,c_2,c_3\right\}$. Define

$$\begin{array}{ccc}\hfill F_M& =& \left\{\left(c_1,\left\{u_3\right\}),(c_2,\left\{u_1,u_2,u_5\right\}\right)\right\},\hfill \\ \hfill G_N& =& \left\{\left(c_1,\left\{u_3,u_5\right\}\right),\left(c_2,\left\{u_4\right\}\right),\left(c_3,\left\{u_5\right\}\right)\right\}.\hfill \end{array}$$

Note that

$$f_M\left(c_2\right)=\left\{u_1,u_2,u_5\right\}⊈{\cup }_{c^{\stackrel{´}{}}\in N^{\stackrel{´}{}}}{g}_N\left(c^{\stackrel{´}{}}\right)$$

for any subset $N^{\stackrel{´}{}}\subseteq N,$ That is $F_M{⊈}_{gf}{G}_N.$ On the other hand, there exists $c_1\in M$, such as

$$f_M\left(c_1\right)=\left\{u_3\right\}\subseteq g_N\left(c_1\right)=\left\{u_3,u_5\right\}\subseteq {\cup }_{c^{\stackrel{´}{}}\in N^{\stackrel{´}{}}}{g}_N\left(c^{\stackrel{´}{}}\right)$$

where $N^{\stackrel{´}{}}\subseteq N$ is any finite subset containing $c_1.$ Hence $F_M{\subseteq }_{g{f}_r}{G}_N.$ Similarly, $G_N{\subseteq }_{g{f}_r}{F}_M.$ This implies that $F_M{\approxeq }_{g{f}_r}{G}_N.$ Note that $F_M{\ncong }_{gf}{G}_N.$

Consider another example to show that there exists soft subsets $F_M,G_N\in S_L\left(U\right)$, such as $F_M{\ncong }_{g{f}_r}{G}_N.$

Example 2.

Let $U=\left\{u_1,u_2,u_3\right\}$ be the given universal set and the sets of parameters $E=\left\{c_1,c_2,c_3\right\},$$M=\left\{c_1,c_2\right\}$ and $N=\left\{c_1\right\}$. Define

$$\begin{array}{ccc}\hfill F_M& =& \left\{\left(c_1,\left\{u_1\right\}\right),\left(c_2,\left\{u_3\right\}\right)\right\}\hfill \\ \hfill G_N& =& \left\{\left(c_1,\left\{u_2\right\}\right)\right\}.\hfill \end{array}$$

As observed, $F_M{⊈}_{{}_{g{f}_r}}{G}_N$ and also $G_N{⊈}_{{}_{g{f}_r}}{F}_N$ for any parameter c. Hence, $F_M{\ncong }_{g{f}_r}{G}_N.$

In [7] the author presented the following proposition.

Proposition 4

([7]).

(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 $gf$-soft-equal.

(3) 

The large and huge soft sets are $gf$-soft-equal.

The following example shows that the converse of statement (1) in the above proposition does not hold true.

Example 3.

Consider $U=\left\{u_1,u_2,u_3\right\}$ and the sets of parameters $M=E=\{c_1,c_2,c_3\}.$ If we define

$$F_M=\left\{\left(c_1,\left\{u_2\right\}\right),\left(c_2,\left\{u_1,u_3\right\}\right),\left(c_3,\varnothing \right)\right\}$$

then $F_M$ 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 $g{f}_r$-soft-equal.

(2) 

The large and huge soft sets are $g{f}_r$-soft-equal.

Proof. 

This proof follows from Propositions 3 and 4. □

Abbas et al. [25] obtained the following result that shows that the operations $\stackrel{\sim }{\cup }$ and ${\cap }_{\epsilon }$ are idempotent, commutative, and associative with respect to the generalized finite soft equality relation ${\approxeq }_{gf}.$

Theorem 1

([25]). If $F_M,G_N,H_C\in S_E\left(U\right),$ then

(a) 

$F_M\stackrel{\sim }{\cup }{F}_M{\approxeq }_{gf}{F}_M,$

(b) 

$F_M\stackrel{\sim }{\cup }{G}_N{\approxeq }_{gf}{G}_N\stackrel{\sim }{\cup }{F}_M,$

(c) 

$\left(F_M\stackrel{\sim }{\cup }{G}_N\right)\stackrel{\sim }{\cup }{H}_C{\approxeq }_{gf}{F}_M\stackrel{\sim }{\cup }\left(G_N\stackrel{\sim }{\cup }{H}_C\right),$

(d) 

$F_M{\cap }_{\epsilon }{F}_M{\approxeq }_{gf}{F}_M,$

(e) 

$F_M{\cap }_{\epsilon }{G}_N{\approxeq }_{gf}{G}_N{\cap }_{\epsilon }{F}_M,$

(f) 

$\left(F_M{\cap }_{\epsilon }{G}_N\right){\cap }_{\epsilon }{H}_C{\approxeq }_{gf}{F}_M{\cap }_{\epsilon }\left(G_N{\cap }_{\epsilon }{H}_C\right).$

The above theorem holds true for the soft equality relation ${\approxeq }_{g{f}_r}.$

Theorem 2.

If $F_M,G_N,H_C\in S_E\left(U\right),$ then

(a)

$F_M\stackrel{\sim }{\cup }{F}_M{\approxeq }_{g{f}_r}{F}_M,$

(b)

$F_M\stackrel{\sim }{\cup }{G}_N{\approxeq }_{g{f}_r}{G}_N\stackrel{\sim }{\cup }{F}_M,$

(c)

$\left(F_M\stackrel{\sim }{\cup }{G}_N\right)\stackrel{\sim }{\cup }{H}_C{\approxeq }_{g{f}_r}{F}_M\stackrel{\sim }{\cup }\left(G_N\stackrel{\sim }{\cup }{H}_C\right),$

(d)

$F_M{\cap }_{\epsilon }{F}_M{\approxeq }_{g{f}_r}{F}_M,$

(e)

$F_M{\cap }_{\epsilon }{G}_N{\approxeq }_{g{f}_r}{G}_N{\cap }_{\epsilon }{F}_M,$

(f)

$\left(F_M{\cap }_{\epsilon }{G}_N\right){\cap }_{\epsilon }{H}_C{\approxeq }_{g{f}_r}{F}_M{\cap }_{\epsilon }\left(G_N{\cap }_{\epsilon }{H}_C\right).$

Proof. 

This proof follows from Proposition 3 and Theorem 1. □

In [24], the authors proved that for $F_M,G_N\in S_E\left(U\right),$$F_M{\approx }_s{G}_N$ (the lower soft equality of $F_M$ and $G_N$) and $F_M{\approx }^s{G}_N$ (the upper soft equality of $F_M$ and $G_N$) imply that $F_M{\approxeq }_g{G}_N$ (the g-soft equality of $F_M$ and $G_N$). Recently Al-shami [7] presented a counter-example (Example 3.1 in [7]) to show that $F_M{\approx }^s{G}_N$ does not imply $F_M{\approxeq }_g{G}_N.$ In the following, we present results that show that the lower and upper soft equalities of $F_M$ and $G_N$ do imply the generalized finite relaxed soft equality of $F_M$ and $G_N.$

Proposition 6.

If $F_M{\approx }_s{G}_N$, then $F_M{\approxeq }_{g{f}_r}{G}_N.$

Proof. 

Suppose that $F_M{\approx }_s{G}_N.$ We will first show that $F_M{\subseteq }_{g{f}_r}{G}_N.$ Since for $M=\varnothing ,$ the result holds trivially. Therefore, we assume that $M\ne \varnothing .$ For any $c\in M,$ if $c\in M\cap N$ then $f_M\left(c\right)=g_N\left(c\right);$ so $f_M\left(c\right)\subseteq g_N\left(c^{\stackrel{´}{}}\right)$ for all $c^{\stackrel{´}{}}=c\in M\cap N\subseteq N.$ If $c\in M$∖$N,$ then $f_M\left(c\right)=\varnothing$ and so $f_M\left(c\right)\subseteq g_N\left(c^{*}\right)$ for all $c^{*}\in N.$ As for some parameter $c\in M,$ there exists a finite subset $N^{\stackrel{´}{}}$ of N, such as

$$f_M\left(c\right)\subseteq {\cup }_{c^{\stackrel{´}{}}\in N^{\stackrel{´}{}}}{g}_N\left(c^{\stackrel{´}{}}\right)$$

implies $F_M{\subseteq }_{g{f}_r}{G}_N.$ On similar lines, it can be shown that $G_N{\subseteq }_{g{f}_r}{F}_M.$ Consequently, we deduce $F_M{\approxeq }_{g{f}_r}{G}_N.$ □

Proposition 7.

If $F_M{\approx }^s{G}_N$ then $F_M{\approxeq }_{g{f}_r}{G}_N$.

Proof. 

For $F_M,G_N\in S_E\left(U\right),$ consider the following cases:

(c-1)

If $M\cap N=\varnothing$ and M and N are nonempty, this implies $f_M\left(c\right)=U$ and $g_N\left(c^{\stackrel{´}{}}\right)=U$ where $c\in M$ and $c^{\stackrel{´}{}}\in N.$ This implies that $F_M{\approxeq }_{g{f}_r}{G}_N.$

(c-2)

If $M\cap N\ne \varnothing ,$ then $f_M\left(c\right)=g_N\left(c\right)$ for each $c\in M\cap N.$ This is enough to conclude that $F_M{\approxeq }_{g{f}_r}{G}_N.$

(c-3)

If $M\cap N=\varnothing$ and either M or N is empty, then trivially the assertion holds true as well.

□

Remark 1.

Consider the soft sets $F_M,$ and $G_N$ as given in Example 1; then, neither $F_M{\not\approx }^s{G}_N$ nor $F_M{\not\approx }_s{G}_N,$ but, on the other hand, $F_M{\approxeq }_{g{f}_r}{G}_N$.

In the following result, lower soft equality holds between the extended and restricted unions and intersections of two soft sets.

Theorem 3

([23]). If $F_M,G_N,H_C\in S_E\left(U\right),$ and $F_M{\approx }_s{G}_N$ then

(a) 

$F_M\stackrel{\sim }{\cup }{G}_N{\approx }_s{F}_M⋒G_N.$

(b) 

$F_M\stackrel{\sim }{\cup }{G}_N{\approx }_s{F}_M{\cap }_{\epsilon }{G}_N.$

(c) 

$F_M{\cup }_{\mathcal{R}}{G}_N{\approx }_s{F}_M⋒G_N.$

(d) 

$F_M{\cup }_{\mathcal{R}}{G}_N{\approx }_s{F}_M{\cap }_{\epsilon }{G}_N.$

The following example shows that the above result can not be extended for the $gfr$-soft equality ${\approxeq }_{g{f}_r}$.

Example 4.

If $U=\left\{u_1,u_2,u_3,u_4,u_5\right\}$, $M=\left\{c_1,c_2\right\}$, the soft sets are given as

$$\begin{array}{ccc}\hfill F_M& =& \left\{\left(c_1,\left\{u_1,u_2,u_3\right\}\right),\left(c_2,\left\{u_4,u_5\right\}\right)\right\},\hfill \\ \hfill G_M& =& \left\{\left(c_1,\left\{u_4,u_5\right\}\right),\left(c_2,\left\{u_1,u_2,u_3\right\}\right)\right\},\hfill \end{array}$$

then $F_M{\approxeq }_{g{f}_r}{G}_M$ and

$$\begin{array}{c}{F}_M\stackrel{\sim }{\cup }{G}_M=\left\{\left(c_1,U\right),\left(c_2,U\right)\right\},F_M⋒G_M=\left\{\left(c_1,\varnothing \right),\left(c_2,\varnothing \right)\right\},\hfill \\ F_M{\stackrel{\sim }{\cup }}_{\mathcal{R}}{G}_M=\left\{\left(c_1,U\right),\left(c_2,U\right)\right\},F_M{\cap }_{\epsilon }{G}_M=\left\{\left(c_1,\varnothing \right),\left(c_2,\varnothing \right)\right\}.\hfill \end{array}$$

On the other hand,

$$\begin{array}{c}{F}_M\stackrel{\sim }{\cup }{G}_M{\ncong }_{g{f}_r}{F}_M⋒G_M,F_M\stackrel{\sim }{\cup }{G}_M{\ncong }_{g{f}_r}{F}_M{\cap }_{\epsilon }{G}_M,\hfill \\ F_M{\cup }_{\mathcal{R}}{G}_M{\ncong }_{g{f}_r}{F}_M⋒G_M,F_M{\cup }_{\mathcal{R}}{G}_M{\ncong }_{g{f}_r}{F}_M{\cap }_{\epsilon }{G}_M,\hfill \end{array}$$

as

$$\begin{array}{c}{F}_M\stackrel{\sim }{\cup }{G}_M{⊈}_{g{f}_r}{F}_M⋒G_M,F_M\stackrel{\sim }{\cup }{G}_M{⊈}_{g{f}_r}{F}_M{\cap }_{\epsilon }{G}_M,\hfill \\ F_M{\cup }_{\mathcal{R}}{G}_M{⊈}_{g{f}_r}{F}_M⋒G_M,F_M{\cup }_{\mathcal{R}}{G}_M{⊈}_{g{f}_r}{F}_M{\cap }_{\epsilon }{G}_M.\hfill \end{array}$$

Theorem 4.

If $F_M,G_N,H_C\in S_E\left(U\right),$ then the following is true.

(1) 

$\left[F_M\stackrel{\sim }{\cup }{G}_N\right]⋒F_M{\approxeq }_{g{f}_r}{F}_M.$

(2) 

$\left[F_M⋒G_N\right]\stackrel{\sim }{\cup }{F}_M{\approxeq }_{g{f}_r}{F}_M.$

Proof. 

This proof follows from Proposition 3 and Theorem 4.4 in [24]. □

Remark 2.

Let $F_M,G_N\in S_E\left(U\right).$ If $F_M{\approxeq }_{g{f}_r}{G}_N$ then $F_M⋒G_N{\approxeq }_{g{f}_r}{F}_M$ and $F_M\stackrel{\sim }{\cup }{G}_N{\approxeq }_{g{f}_r}{G}_N$ 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 $F_M,G_N\in S_E\left(U\right)$, such as $M\cap N\ne \varnothing ,$ then

(a) 

${\left(F_M{\stackrel{\sim }{\cup }}_{\mathcal{R}}{G}_N\right)}^r{\approxeq }_{g{f}_r}{\left(F_M\right)}^r⋒{\left(G_N\right)}^r.$

(b) 

${\left(F_M⋒G_N\right)}^r{\approxeq }_{g{f}_r}{\left(F_M\right)}^r{\cup }_{\mathcal{R}}{\left(G_N\right)}^r.$

(c) 

${\left(F_M\stackrel{\sim }{\cup }{G}_N\right)}^r{\approxeq }_{g{f}_r}{\left(F_M\right)}^r{\cap }_{\epsilon }{\left(G_N\right)}^r.$

(d) 

${\left(F_M{\cap }_{\epsilon }{G}_N\right)}^r{\approxeq }_{g{f}_r}{\left(F_M\right)}^r\stackrel{\sim }{\cup }{\left(G_N\right)}^r.$

Proof. 

This proof follows from Proposition 3 and Theorem 4.14 and Theorem 4.15 in [24]. □

Remark 3.

The lower soft equality ${\approx }_s$ is a congruence relation [23], that is, $F_M{\approx }_s{G}_M$ and $H_M{\approx }_s{I}_M$ imply that

$$F_M⋒H_M{\approx }_s{G}_M⋒I_M\mathit{and}{F}_M\stackrel{\sim }{\cup }{H}_M{\approx }_s{G}_M\stackrel{\sim }{\cup }{I}_M.$$

whereas ${\approxeq }_{g{f}_r}$ is not a congruence relation as shown in the following example.

Example 5.

Consider $U=\left\{u_1,u_2,u_3,u_4\right\}$, $M=\left\{c_1,c_2\right\}$ and the soft sets $F_M,G_M,H_M,I_M\in S_E\left(U\right)$ defined as

$$\begin{array}{c}{F}_M=\left\{\left(c_1,\left\{u_1,u_2\right\}\right),\left(c_2,\varnothing \right)\right\},\hfill \\ G_M=\left\{\left(c_1,\left\{u_3\right\}\right),\left(c_2,\left\{u_{1,}{u}_2\right\}\right)\right\},\hfill \\ H_M=\left\{\left(c_1,\left\{u_3\right\}\right),\left(c_2,\left\{u_4\right\}\right)\right\},\hfill \\ I_M=\left\{\left(c_1,\left\{u_3\right\}\right),\left(c_2,\left\{u_2,u_4\right\}\right)\right\}.\hfill \end{array}$$

Then

$$\begin{array}{c}{F}_M⋒H_M=\left\{\left(c_1,\varnothing \right),\left(c_2,\varnothing \right)\right\},\hfill \\ G_M⋒I_M=\left\{\left(c_1,\left\{u_3\right\}\right),\left(c_2,\left\{u_2\right\}\right)\right\},\hfill \\ F_M\stackrel{\sim }{\cup }{H}_M=\left\{\left(c_1,\left\{u_1,u_2,u_3\right\}\right),\left(c_2,\left\{u_4\right\}\right)\right\},\hfill \\ G_M\stackrel{\sim }{\cup }{I}_M=\left\{\left(c_1,\left\{u_3\right\}\right),\left(c_2,\left\{u_1,u_2,u_4\right\}\right)\right\}.\hfill \end{array}$$

Note that $F_M{\approxeq }_{g{f}_r}{G}_M$ and $H_M{\approxeq }_{g{f}_r}{I}_M$ but

$$F_M⋒H_M{\ncong }_{g{f}_r}{G}_M⋒I_M\mathit{and}{F}_M\stackrel{\sim }{\cup }{H}_M{\ncong }_{g{f}_r}{G}_M\stackrel{\sim }{\cup }{I}_M.$$

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 $F_M,G_N\in S_E\left(U\right)$, denoted by $F_M{\cup }_{gf}{G}_N,$ is the set consisting of all $H_C\in S_E\left(U\right)$ satisfying:

(c-i) 

$F_M{\subseteq }_{gf}{H}_C$ and $G_N{\subseteq }_{gf}{H}_C,$ where $C\subseteq E.$

(c-ii) 

If there exists $J_D\in S_E\left(U\right)$, such as $F_M{\subseteq }_{gf}{J}_D$ and $G_N{\subseteq }_{gf}{J}_D$, then $H_C{\subseteq }_{gf}{J}_D.$

That is, $H_C$ is a minimal $gf$-soft super set of $F_M$ and $G_N$ in the sense that if there exists another soft set $J_D$ satisfying (c-i), then $H_C$ is a $gf$-soft subset of $J_D.$

Furthermore, they introduced the following version of the generalized finite intersection of soft sets.

Definition 24

([25]). The generalized finite intersection of $F_M,G_N\in S_E\left(U\right)$, denoted as $F_M{\cap }_{gf}{G}_N,$ is the set of all $H_C\in S_E\left(U\right)$ satisfying:

(c-iii) 

$H_C{\subseteq }_{gf}{F}_M$ and $H_C{\subseteq }_{gf}{G}_N$, where $C\subseteq E,$

(c-iv) 

If there exists $J_D$$\in S_E\left(U\right)$ such as $J_D{\subseteq }_{gf}{F}_M$ and $J_D{\subseteq }_{gf}{G}_N$ then $J_D{\subseteq }_{gf}{H}_C.$

Here $H_C$ is a maximal $gf$-soft subset of $F_M$ and $G_N$ in the sense that if there exists another soft set $J_D$ satisfying (c-iii), then $J_D$ is the $gf$-soft subset of $H_C.$

We present the definition of generalized finite relaxed unions of soft sets as follows.

Definition 25.

Consider $F_M,G_N\in S_E\left(U\right).$ Then the generalized finite relaxed soft union denoted by $F_M{\cup }_{g{f}_r}{G}_N$ is defined as the collection of all soft sets $H_C\in S_E\left(U\right)$, which satisfies the following two conditions:

(a) 

$F_M{\subseteq }_{g{f}_r}{H}_C$ and $G_N{\subseteq }_{g{f}_r}{H}_C,$ where $C\subseteq E;$

(b) 

If there exists $J_D\in S_E\left(U\right)$, such as $F_M{\subseteq }_{g{f}_r}{J}_D$ and $G_N{\subseteq }_{g{f}_r}{J}_D$, then $H_C{\subseteq }_{g{f}_r}{J}_D.$

That is, $H_C$ is a minimal $gfr$-soft super set of $F_M$ and $G_N$ in the sense that if there exists another soft set $J_D$ satisfying (a), then $H_C$ is a $gfr$-soft subset of $J_D.$

Consider the following example where $F_M{\cup }_{g{f}_r}{G}_N$ is nonempty.

Example 6

(compare Exmple 2.15 in [25]). Suppose that $U=\left\{u_1,u_2,u_3\right\}$ is the universe and $M=\left\{c_1,c_2\right\}$ and $N=\left\{c_1\right\}$ are subsets of the set of parameters $E=\left\{c_1,c_2\right\}.$ Let

$$\begin{array}{ccc}\hfill F_M& =& \left\{\left(c_1,\left\{u_1\right\}\right),\left(c_2,\left\{u_2\right\}\right)\right\},\hfill \\ \hfill G_N& =& \left\{\left(c_1,\left\{u_3\right\}\right)\right\}.\hfill \end{array}$$

Let $H_C\in S_E\left(U\right),$ such as $F_M{\subseteq }_{g{f}_r}{H}_C$ and $G_N{\subseteq }_{g{f}_r}{H}_C,$ where $C\subseteq E.$

If $\left|C\right|=1,$ i.e., $C=\left\{c_i\right\},$ for $i=1$ or $i=2,$ we deduce that if $H_C=\left\{c_i,\left\{u_1,u_3\right\}\right\}$ or $H_C=\left\{c_i,\left\{u_2,u_3\right\}\right\},$ then $H_C$ will satisfy the condition $\left(N\right)$ for some $J_D=\left(c_i,U\right).$

We deduce that both conditions $\left(M\right)$ and $\left(N\right)$ for $gfr$-union hold true and so the set $F_M{\cup }_{g{f}_r}{G}_N$ is nonempty.

Definition 26.

Let $F_M$ and $G_N$ belong to $S_E\left(U\right).$ Then the generalized finite relaxed soft intersection denoted by $F_M{\cap }_{g{f}_r}{G}_N,$ is defined as the collection of all soft sets $H_C\in S_E\left(U\right)$ which satisfies the following two conditions:

(c) 

$H_C{\subseteq }_{g{f}_r}{F}_M$ and $H_C{\subseteq }_{g{f}_r}{G}_N,$ where $C\subseteq E;$

(d) 

If there exists $J_D\in S_E\left(U\right)$ such as $J_D{\subseteq }_{g{f}_r}{F}_M$ and $J_D{\subseteq }_{g{f}_r}{G}_N$, then $J_D{\subseteq }_{g{f}_r}{H}_C.$

Here, $H_C$ is a maximal $gfr$-soft subset of $F_M$ and $G_N$ in the sense that if there exists another soft set $J_D$ satisfying (c), then $J_D$ is the $gfr$-soft subset of $H_C.$

Abbas et al. [25] presented the following result.

Theorem 6

([25]). If $F_M,G_N\in S_E\left(U\right),$ then

$$F_M\stackrel{\sim }{\cup }{G}_N\in F_M{\cup }_{gf}{G}_N.$$

Remark 4.

If $F_M,G_N\in S_E\left(U\right),$ then

$$F_M\stackrel{\sim }{\cup }{G}_N\in F_M{\cup }_{gfr}{G}_N$$

does not hold true, as shown in the following example.

Example 7.

If $U=\left\{u_1,u_2,u_3,u_4\right\}$ and $M=N=E=\left\{c_1,c_2,c_3\right\}$ and the soft sets defined as

$$\begin{array}{ccc}\hfill F_M& =& \left\{\left(c_1,\left\{u_1,u_4\right\}\right),\left(c_2,\left\{u_3,u_4\right\}\right),\left(c_3,\left\{u_1,u_2\right\}\right)\right\},\hfill \\ \hfill G_N& =& \left\{\left(c_1,\left\{u_1,u_2\right\}\right),\left(c_2,\left\{u_3,u_4\right\}\right),\left(c_3,\left\{u_3\right\}\right)\right\},\hfill \end{array}$$

then

$$F_M\stackrel{\sim }{\cup }{G}_N=\left\{\left(c_1,\left\{u_1,u_2,u_4\right\}\right),\left(c_2,\left\{u_3,u_4\right\}\right),\left(c_3,\left\{u_1,u_2,u_3\right\}\right)\right\}.$$

For $J_D=\left\{\left(c_1,\left\{u_1,u_2\right\}\right)\right\},$ clearly,

$$\begin{array}{ccc}\hfill F_M& \subseteq & {}_{gfr}{F}_M\stackrel{\sim }{\cup }{G}_N,G_N{\subseteq }_{g{f}_r}{F}_M\stackrel{\sim }{\cup }{G}_N\mathit{and}\hfill \\ \hfill F_M& \subseteq & {}_{g{f}_r}{J}_D,G_N{\subseteq }_{g{f}_r}{J}_D.\hfill \end{array}$$

On the other hand,

$$F_M\stackrel{\sim }{\cup }{G}_N{\ncong }_{g{f}_r}{J}_D\mathit{implies}{F}_M\stackrel{\sim }{\cup }{G}_N\notin F_M{\cup }_{g{f}_r}{G}_N.$$

Remark 5.

Note that the above Example 7 also shows that

$$F_M{\cap }_{\epsilon }{G}_N\notin F_M{\cap }_{g{f}_r}{G}_N$$

particularly when $M=N=E.$ As observed,

$$H_C=F_M{\cap }_{\epsilon }{G}_N=\left\{\left(c_1,\left\{u_1\right\}\right),\left(c_2,\left\{u_3,u_4\right\}\right),\left(c_3,\varnothing \right)\right\}.$$

For $J_D=\left\{\left(c_1,\left\{u_1,u_2\right\}\right)\right\},$ clearly,

$$J_D{\subseteq }_{g{f}_r}{F}_M\mathit{and}{J}_D{\subseteq }_{g{f}_r}{G}_N$$

but

$$J_D{⊈}_{g{f}_r}{F}_M{\cap }_{\epsilon }{G}_N.$$

Hence,

$$H_C\notin F_M{\cap }_{g{f}_r}{G}_N.$$

Abbas et al. [25] presented the following result.

Proposition 8.

Let $F_M$ and $G_N$ be two soft sets over a common universe $U,$ such as $M\cap N=\varnothing .$ Then $F_M{\cap }_{\epsilon }{G}_N\in F_M{\cap }_{gf}{G}_N.$

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 $U=\left\{u_1,u_2,u_3,u_4\right\}$ be a universal set and $E=\left\{c_1,c_2,c_3\right\},$$M=\{c_1,c_2\},$$N=\left\{c_3\right\}$ are the sets of parameters. Define soft sets as follows: $M\cap N=\varnothing$ and

$$\begin{array}{ccc}\hfill F_M& =& \left\{\left(c_1,\left\{u_1,u_4\right\}\right),\left(c_2,\{u_2,u_3\}\right)\right\}\mathit{and}\hfill \\ \hfill G_N& =& \left\{\left(c_3,\left\{u_1,u_4\right\}\right)\right\}.\hfill \end{array}$$

Note that

$$F_M{\cap }_{\epsilon }{G}_N=\left\{\left(c_1,\left\{u_1,u_4\right\}\right),\left(c_2,\{u_2,u_3\}\right),\left(c_3,\left\{u_1,u_4\right\}\right)\right\}\notin F_M{\cap }_{gf}{G}_N.$$

Abbas et al. [25] presented the following lemma.

Lemma 1

([25]). If $F_M,G_N,H_C\in S_E\left(U\right),$ then

$$F_M{\subseteq }_{gf}{G}_N\mathit{and}{G}_N{\subseteq }_{gf}{H}_C\mathit{implies}\mathit{that}{F}_M{\subseteq }_{gf}{H}_C.$$

Note that Lemma 1 does not hold true if we replace ${\subseteq }_{gf}$ with ${\subseteq }_{gfr}.$ In this context we present the following example.

Example 9.

Let $U=\left\{u_1,u_2,u_3,u_4\right\}$ be a universal set; $E=\left\{c_1,c_2,c_3\right\}$ the set of parameters; and $M=\left\{c_1\right\},$$N=\{c_2,c_3\}$ and $C=\left\{c_1\right\}$. Consider the following soft sets

$$\begin{array}{ccc}\hfill F_M& =& \left\{\left(c_1,\left\{u_1,u_2\right\}\right)\right\},\hfill \\ \hfill G_N& =& \left\{\left(c_2,\left\{u_1,u_2\right\}\right),\left(c_3,\left\{u_3\right\}\right)\right\}\mathit{and}\hfill \\ \hfill H_C& =& \left\{\left(c_1,\left\{u_3\right\}\right)\right\}.\hfill \end{array}$$

As can be observed, although $F_M{\subseteq }_{gfr}{G}_N$ and $G_N{\subseteq }_{gfr}{H}_C$ but $F_M{⊈}_{gfr}{H}_C.$

Using Lemma 1, they [25] obtained the following results.

Theorem 7

([25]). Let $F_M$ and $G_N$ be two soft sets over a common universe U, such as $M\cap N=\varnothing .$ Then

$$F_M{\cup }_{gf}{G}_N=\left\{H_C\in S_E\left(U\right):H_C{\approxeq }_{gf}{F}_M\stackrel{\sim }{\cup }{G}_N\right\}.$$

Theorem 8

([25]). Let $F_M$ and $G_N$ be two soft sets over a common universe U, such as $M\cap N=\varnothing .$ Then

$$F_M{\cap }_{gf}{G}_N=\left\{H_C\in S_E\left(U\right):H_C{\approxeq }_{gf}{F}_M{\cap }_{\epsilon }{G}_N\right\}.$$

Now we present the following result.

Proposition 9.

Let $F_M$ and $G_N$ be two soft sets over a common universe $U,$ such as $M\cap N=\varnothing .$ Then

$$F_M{\cap }_{\epsilon }{G}_N\in F_M{\cap }_{g{f}_r}{G}_N.$$

Proof. 

Let $F_M{\cap }_{\epsilon }{G}_N=H_C$, where $C=M\cup N.$ Then $h_C\left(c\right)=f_M\left(c\right)$ for all $c\in M\setminus N.$ Since $M\cap N=\varnothing ;$ therefore, $c\in M=M\setminus N.$ We know

$$h_C\left(c\right)=f_M\left(c\right)\subseteq {\cup }_{M\in M^{\prime }}{f}_M\left(M\right)$$

where $M\in M^{\prime }\subseteq M.$ Thus, $F_M{\cap }_{\epsilon }{G}_N{\subseteq }_{gfr}{F}_M,$ and similarly $F_M{\cap }_{\epsilon }{G}_N{\subseteq }_{gfr}{G}_N.$ Next consider $D\subseteq E$ and $J_D\in S_E\left(U\right),$ such as $J_D{\subseteq }_{gfr}{F}_M$ and $J_D{\subseteq }_{gfr}{G}_N.$ Then

$$j_D\left(d\right)\subseteq {\cup }_{M\in M^{\prime }}f\left(M\right)\mathit{and}{j}_D\left(d\right)\subseteq {\cup }_{N\in N^{\prime }}g\left(N\right)$$

for some $d\in D,$ where $M^{\prime }\subseteq M,$$N^{\prime }\subseteq N$ are finite. As $M\cap N=\varnothing ,$ so

$$j_D\left(d\right)\subseteq \left({\cup }_{M\in M^{\prime }}f\left(M\right)\right)\cup \left({\cup }_{N\in N^{\prime }}g\left(N\right)\right)={\cup }_{c\in C^{\prime }}{h}_C\left(c\right)$$

for some $d\in D,$ where $C^{\prime }=M^{\prime }\cup N^{\prime }.$ Therefore, $J_D{\subseteq }_{gfr}{F}_M{\cap }_{\epsilon }{G}_N$. This shows that

$$F_M{\cap }_{\epsilon }{G}_N\in F_M{\cap }_{g{f}_r}{G}_N.$$

□

Example 10.

Let $U=\left\{u_1,u_2,u_3,u_4\right\}$ be a universal set, $E=\left\{c_1,c_2,c_3\right\}$ the set of parameters and $M=\{c_1,c_2\},$$N=\left\{c_3\right\}$. Define,

$$\begin{array}{ccc}\hfill F_M& =& \left\{\left(c_1,\left\{u_1,u_4\right\}\right),\left(c_2,\left\{u_3\right\}\right)\right\}\hfill \\ \hfill G_N& =& \left\{\left(c_3,\left\{u_1\right\}\right)\right\}.\hfill \end{array}$$

Then

$$F_M{\cap }_{\epsilon }{G}_N=\left\{\left(c_1,\left\{u_1,u_4\right\}\right),\left(c_2,\left\{u_3\right\}\right),\left(c_3,u_1\right)\right\}.$$

If

$$J_D=\left\{\left(c_1,\left\{u_1\right\}\right),\left(c_2,\left\{u_3\right\}\right)\right\}$$

then $J_D{\subseteq }_{gfr}{F}_M$ and $J_D{\subseteq }_{gfr}{G}_N,$ and, as observed, $J_D{\subseteq }_{gfr}{F}_M{\cap }_{\epsilon }{G}_N.$

Now we present a characterization of the soft sets in $F_M{\cap }_{g{f}_r}{G}_N$ via generalized finite relaxed soft equality as follows.

Theorem 9.

Let $F_M$ and $G_N$ be two soft sets over a common universe U, such as $M\cap N=\varnothing .$ If $H_C\in F_M{\cap }_{g{f}_r}{G}_N,$ then

$$H_C{\approxeq }_{g{f}_r}{F}_M{\cap }_{\epsilon }{G}_N.$$

Proof. 

Let

$$H_C\in F_M{\cap }_{g{f}_r}{G}_N.$$

(1)

Then, according to the definition of $F_M{\cap }_{g{f}_r}{G}_N$, $H_C{\subseteq }_{gfr}{F}_M$ and $H_C{\subseteq }_{gfr}{G}_N,$ where $C\subseteq E.$ Since $M\cap N=\varnothing .$ Furthermore, from Proposition 9 we have

$$F_M{\cap }_{\epsilon }{G}_N\in F_M{\cap }_{g{f}_r}{G}_N.$$

(2)

The inclusions (1) and (2) imply that

$$H_C{\subseteq }_{gfr}{F}_M{\cap }_{\epsilon }{G}_N\mathit{and}{F}_M{\cap }_{\epsilon }{G}_N{\subseteq }_{gfr}{H}_C.$$

Consequently, we can obtain

$$H_C{\approxeq }_{g{f}_r}{F}_M{\cap }_{\epsilon }{G}_N.$$

□

The converse of Theorem 9 does not hold true, as shown in the following example.

Example 11.

Let $U=\left\{u_1,u_2,u_3\right\}$ be a universal set, $E=\left\{c_1,c_2,c_3\right\}$ the set of parameters and $M=\{c_1,c_2\} and$$N=\left\{c_3\right\}$. Consider the soft sets

$$\begin{array}{ccc}\hfill F_M& =& \left\{\left(c_1,\left\{u_1\right\}\right),\left(c_2,\left\{u_2\right\}\right)\right\}\mathit{and}\hfill \\ \hfill G_N& =& \left\{\left(c_3,U\right)\right\}.\hfill \end{array}$$

Note that $M\cap N=\varnothing$ and

$$F_M{\cap }_{\epsilon }{G}_N=\left\{\left(c_1,\left\{u_1\right\}\right),\left(c_2,\left\{u_2\right\}\right),\left(c_3,U\right)\right\}.$$

Consider

$$H_C=\left\{\left(c_3,U\right)\right\},$$

and, as observed, $H_C{\approxeq }_{g{f}_r}{F}_M{\cap }_{\epsilon }{G}_N$ but $H_C{⊈}_{gfr}{F}_M$. Hence,

$$H_C\notin F_M{\cap }_{g{f}_r}{G}_N.$$

4. Conclusions

In an attempt to develop further the soft set theory, we generalized the notions of $gf$-soft subsets and the $gf$-soft equality of two soft subsets by introducing the notions of $g{f}_r$-soft subsets and the $g{f}_r$-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 $gf$-soft intersection does not hold true. However, with the generalized notions, we were able to prove Proposition 8 in regard to $g{f}_r$-soft intersection. Furthermore, Al-Shami [7] pointed out that the upper soft equality ${\approx }^s$ of two soft sets does not imply the $gf$-soft equality but the upper soft equality implies the $gfr$-soft equality of two soft sets. Regarding $g{f}_r$-soft unions, Example 7 shows that Theorem 4 cannot be extended for $g{f}_r$-soft unions. Moreover, with $g{f}_r$-soft intersections, the converse of Theorem 9 does not hold true, as shown in Example 11. We can see that generalized finite relaxed ($g{f}_r$) operations are better when it comes to the $g{f}_r$-soft intersection of soft sets as compared to the $g{f}_r$-soft union of soft sets.

As a future direction, one can investigate $g{f}_r$-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]).