1. Introduction
Artificial Intelligence (AI) and logical algebra are studied in different non-classical sets like soft sets [
1], fuzzy sets [
2], and others, to solve various problems in our life. For example, in 2021, nano-sets have been used to study COVID-19 [
3]. COVID-19 has also been studied by Arfan and others [
4]. Zhong et al. [
5] showed that the majority of inhabitants with an above-average socioeconomic status, particularly females, expressed optimism about COVID-19. A variety of AI and logic applications have been introduced in diverse domains including the medical field [
6,
7]. The concept of fuzzy set (FS) was introduced by Zadeh [
8] in 1965 and has been then successfully applied in different domains [
9,
10]. The connotation of fuzzy algebra determined by G. Xi [
9] is called fuzzy
-algebra. Several applications of fuzzy
-algebras were discussed by Y. B. Jun [
10].
In [
11], some concepts of fuzzy algebras such as fuzzy
-subalgebra
, fuzzy
-ideal
, and fuzzy
-ideal
were introduced. The mathematical idea of soft sets is a fresh notion studied by D. Molodtsov [
12]. This theory has been applied in various fields, as fuzzy sets theory [
13]. The notion of fuzzy soft algebra was introduced by Jun and others [
14], who called it fuzzy soft
-algebra.
The connotations of interval-valued fuzzy sets (IVFS) were investigated as an extension of FS [
15]. Similar to BCK, IVFS has been applied to various domains and subgroups [
2]. Moreover, the general ideas of algebraic fuzzy systems (AFS) are enriched by introducing the notion of fuzzy subsets. Jun et al. [
16] presented some operations such as P/R-union and P/R-intersection on cubic sets. They described several different ways to find the solutions for intricate problems in engineering, economics, and environment.
While conventional methods have been successfully applied in diverse domains, these methods do not handle uncertainties. Sometimes traditional methods in logical algebra are not sufficient to solve some problems or to obtain good results because different uncertainties models are necessary for those problems. The majority of system algebras are not commutative for any non-fixed pair of their members. Therefore, some algebra structures that are commutative for any non-fixed pair of their members, such as
-algebra [
11] and
-algebra [
17], have been proposed. In this work, we used
-algebra to consider new types of cubic soft algebras, such as (
λ-
CSδ-
SA) and (
CSδ-
SA). These classes in
-algebra are different from any other class, since any pair
in algebra
, they satisfy the condition
. We also proved that P-union is not really a soft cubic subalgebra of two soft cubic
-subalgebras. We revealed that for any R/P-cubic soft subset of (
CSδ-
SA), it is not necessarily true to be (
CSδ-
SA). Furthermore, we present the required conditions to prove that the R-union of two members is (
CSδ-
SA) if each one of them is (
CSδ-
SA). To illustrate our notations, the applied (
CSδ-
SA) to study the effectiveness of medications for COVID-19.
2. Preliminary
In this section, we will present some definitions that are necessary for our work.
Definition 1. ([
17])
We denote as -algebra (briefly, ) if , and the following assumptions are fulfilled:- (i)
- (ii)
- (iii)
and → , for all .
- (iv)
For all → .
- (v)
For all → .
Definition 2. ([
8])
Let . A mapping is called fuzzy set (FS) of . We denote the family of all (FSs) in by . Let ≤ be a relation on specified by:Let () and () be operations on , specified by: For each , we denote its complement as , specified by Let be a collection of (FSs), where is an index set. Therefore, () and () are specified by: Definition 3. ([
18])
Let be a closed subinterval of ; Z is said to be an interval number (IN), where . The family of all interval numbers (INs) is symbolized by Some operations on
like
r min (refined minimum),
r max (refined maximum), “
”, “
” and “=”, are specified by:
and
Moreover,
,
, if
, is a collection of INs. Then,
We refer to the complement of any
by
, where
Let
. Then,
is called an interval-valued fuzzy set (IVFS) in
. We refer to the family of all interval-valued fuzzy sets (IVFSs) in
by
. On the other side, if
and
, we refer to the degree of membership of
to
by
or
, where
is the lower fuzzy set (LFS), and
is the upper fuzzy set (UFS) in
. The definitions of the symbols “
” and “=” on any
can be given as follows:
We refer to the complement of any
by
, where
,
. That means
If
is a family of (IVFSs), then “
and “
” are defined in
as follows:
Definition 4. ([
12])
Let be a universal set, with parameter set ; is said to be a soft set (over ), where , and is the power set of with . Definition 5. ([
16])
We define a cubic set (CS) in byWe can also write it as , where is IVFS, and is FS.
Definition 6. ([
16])
Let and be a pair of cubic sets (CSs) in . We define “”, “”, and “=” by the following:- (i)
(P-order) and .
- (ii)
(R-order) and .
- (iii)
(Equality) and .
Definition 7. ([
16])
Let be a collection of (CSs) in . The symbol “” (resp., “”, “” and “”) is said to be (P-union) (resp., P-intersection, R-union, and R-intersection) and is obtained as follows:- (1)
,
- (2)
,
- (3)
,
- (4)
,
Remark 1. The complement of is defined as: If is a collection of (CSs) in , then we have the followng; Therefore, a (CS) is denoted by . The family of all (CSs) in is referred to as .
Definition 8. ([
19])
Let be a universal set with the parameter set ; is said to be a cubic soft set (CSS) over , where , and is a mapping. We write as:The set of all cubic soft sets (CSSs) is symbolized by .
Definition 9. ([
19])
Let , . The R-union of and is a (CSS) symbolized by = , where and Definition 10. ([
19])
Let , . The p-union of and is a (CSS) symbolized by , where and Definition 11. ([
19])
Let , . The p-intersection of and is a (CSS) symbolized by , where and Definition 12. ([
19])
Let , . We say is an R-cubic soft subset of if Definition 13. ([
19])
Let , . We say is a P-cubic soft subset of if Example 1. Let the set of students under consideration be . Let {pleasing personality (); conduct (); good result (); sincerity ()} be the set of parameters used to choose the best student. Suppose that the soft set describing Mr. ’s opinion about the best student in an academic year is defined by The description of Mr.
’s opinion is explained see
Figure 1.
However, if we define {{, 〈[0.3, 0.5], 0.3〉, 〈[0.4, 0.6], 0.7〉}, {〈[0.4, 0.5], 0.8〉, 〈[0.3, 0.4], 0.8〉, 〈[0.2, 0.5], 0.6〉}}, then is a cubic soft set over {, }, dependent on (FS) to describe the best student by the rates of some activities of ; each rate ranges between 0 and 1 and approaches 0 when an activity is low, while it approaches 1 when an activity is high).
3. Cubic Soft -Subalgebras in -Algebras and Its Application for COVID-19
In this section, we will consider several new forms of cubic soft algebras and see how they can be used to study the effectiveness of medications for COVID-19.
Definition 14. Let be (CSS) over ; is , if there exists a parameter that satisfies the following: is said to be a cubic soft -subalgebra over based on a parameter (briefly, (λ-CSδ-SA) over ) and is called a cubic soft -subalgebra (CSδ-SA) over , if is an (λ-CSδ-SA) over .
Theorem 1. If , with and are disjoint, then their P-union is a (CSδ-SA) over .
Proof. From Definition (10), we have
, where
and
Therefore, either or (since ). If , then is a -subalgebra over . In addition, if , then is a (CSδ-SA) over . So, is a (CSδ-SA) over . □
Remark 2. The above theorem is not true in general when and are not disjoint.
Example 2. Let be a universal set of some medications for (COVID-19), as follows Chloroquine, Arbidol, = Tamiflu, = Kaletra, = Remdesivir. These medications were chosen because they have been tried and discussed by researchers, for example, Chloroquine in [20], Arbidol in [21], Tamiflu in [22], Kaletra in [23], and Remdesivir in [24]. We used virtual reality to introduce a mathematical method where the composition of the members forms an system; we determined how to find the cubic soft set over , when it is dependent on (FS) to describe the best medication in the basis of its activity evaluated by rates, with each rate confined between 0 and 1. If a rate appr2oaches 0, then activity is low, whereas if the rate is closer to 1, the activity is high. Suppose that for any two members in , their composition under operation is defined by the python program as follows:from numpy import array |
X = ['f','v','w','σ','τ'] |
i = 0 |
lst = array (range (25), dtype = str). reshape (5,5) |
for a in X: |
j = 0 |
for b in X: |
# print (a, ' ', b) |
if ((a == 'f') or (a == b) : |
m = 'f' |
elif ((b = 'f')): |
m = a |
elif ((a! = 'f') and (b! = 'f') ): |
m = 'v' |
lst [i,j] = m |
j = j + 1 |
I = I + 1 |
print(lst) |
Using this program, let us consider
Figure 2, where rows are placed in a table.
This algorithm makes the members distributive for any set that has members inside a matrix of degree (), where for some and all . By this matrix, our table can have the structure of -algebra.
Therefore, the binary operation
is described in a
Table 1.
Then,
is a
-algebra.
Figure 3 explains that the member
does not change and retains more than 50% of its properties if
is entered from pipe 1, and any member
h in
is entered from pipe 2.
Moreover, the same engineering device in
Figure 4 explains that
will lose more than 50% of its properties if it is entered from pipe 2 and any member
h in
is entered from pipe 1; the member
will chang and get the same properties of the member
h. In
-algebra, the member
f is called the fixed member.
Now, let
“body temperature” (
), “cough with chest congestion” (
), “body ache” (
), “cough with no chest congestion” (
), “breathing trouble”, (
)} be a parameter set. Here,
give us the effectiveness for these medications that help somebody want to select one of them based on his opinion of what he prefers of these attributives. Take
,
,
and
,
,
,
, then from
Table 2 and
Table 3, we consider that
and
are
-subalgebras over
.
Here,
and
are not disjoint. The
R-union
is given in
Table 4.
Remark 3. - (1)
For any R-cubic soft subset of (CSδ-SA), it is not necessary that each one is (CSδ-SA).
- (2)
For any P-cubic soft subset of (CSδ-SA), it is not necessary that each one is (CSδ-SA) too.
Example 3. In Example 2, let and be an R-cubic soft subset of as shown in Table 5. Then, we have
[0.2, 0.3] ≤ [0.3, 0.4] = r min {[0.3, 0.4], [0.4, 0.5]} = r min {, }. is not a (CSδ-SA) over .
Here, we consider that it is not necessary that any P-cubic soft subset of (CSδ-SA) is (CSδ-SA) too.
Example 4. In Example 2, let and be a P-cubic soft subset of , as defined in Table 6: = [0.2, 0.3] [0.3, 0.4] = r min {[0.3, 0.4], [0.4, 0.5]} = r min {, }. is not a (CSδ-SA) over . Here, we consider that for any R-cubic soft subset of (CSδ-SA), it is not necessary to be (CSδ-SA) too.
Proposition 1. Let with is and . Then, and , , if is (λ-CSδ-SA) over .
Proof. , we consider that:
= r min {, } = r min{[, ], [, ]} = [, ] = and = max {, } = . □
Theorem 2. Assume is (λ-CSδ-SA) over . Then, = [1, 1] and , if is a sequence in with = [1, 1] and .
Proof. Since
,
,
, we have
However, [1, 1] = [1, 1]. Also, 0 . Therefore = [1, 1] and . □
Theorem 3. If each of is a (CSδ-SA), then their R-intersection is also (CSδ-SA).
Proof. Let
are
and
, where
and
Now, , we consider three states: (i) , (ii) , (iii) .
In state (ii), we obtain;
In state (iii), we obtain;
Hence is a (CSδ-SA) over . □
Corollary 1. If is a family of cubic soft -subalgebras over , then the R-intersection is a (CSδ-SA) over .
Proof. From Definition (7) and Theorem (3), the proof is straightforward. □