Computers, Materials & Continua DOI:10.32604/cmc.2022.019345 | |

Article |

Soft

1Department of Mathematics and Statistics, College of Science, Taif University, Taif, 21944, Saudi Arabia

2Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

3Department of Mathematics, College of Science and Arts, Najran University, Najran, 66445, Saudi Arabia

*Corresponding Author: M. A. El Safty. Email: m.elsafty@tu.edu.sa

Received: 9 April 2021; Accepted: 10 May 2021

Abstract: In this paper, we present a proposed method for generating a soft rough approximation as a modification and generalization of Zhaowen et al. approach. Comparisons were obtained between our approach and the previous study and also. Eventually, an application on Coronavirus (COVID-19) has been presented, illustrated using our proposed concept, and some influencing results for symptoms of Coronavirus patients have been deduced. Moreover, following these concepts, we construct an algorithm and apply it to a decision-making problem to demonstrate the applicability of our proposed approach. Finally, a proposed approach that competes with others has been obtained, as well as realistic results for patients with Coronavirus. Moreover, we used MATLAB programming to obtain the results; these results are consistent with those of the World Health Organization and an accurate proposal competing with the method of Zhaowen et al. has been studied. Therefore, it is recommended that our proposed concept be used in future decision making.

Keywords: Soft set; soft rough set; soft ζ rough set; COVID-19; intelligence discovery; decision making

In 1999 Molodtsov [1] have introduced the soft set notion and progressing basics of this theory as a new diverse for modeling roughness and uncertainties. Diverse fields of applications of his approach were used in solving many practical problems in economics, engineering, social science, medical science

Often, the right decision making for many real-life issues is very difficult in our daily lives, which is highly essential for choosing the best solution to our discussions. Therefore, we have to consider various features in order to produce the highest practical solution to these problems. For this cause, we use the chosen mathematical instrument in the current article, namely soft rough set theory, in decision making. Decision making application was applied by Maji et al. [10,11]. Using soft set approach and accordingly they expand this approach to fuzzy soft set theory in [13]. Soft rough model was defined by [15].

Coronavirus emerged in 2019, in Wuhan, China. This virus is a new strain that has not been previously identified in humans. It was believed that Coronaviruses spread from dirty, dry surfaces, like automatic mucous membrane pollination in the nose, eyes, or mouth, reinforcing the importance of a clear understanding of the persistence of coronaviruses on inanimate surfaces [16]. Therefore, two factors which are in contact with infected surfaces and encounters with infected viruses, affect the transmission. As a result, many scientific papers have been published and many researchers have studied this virus, such as ([16–22]).

As a generalization to Pawlak’s rough models [23]. Based on this structure, they defined soft rough approximations, soft rough sets and some related concepts, such as ([23–27]).

The main objective of our belief is to have a certain influence on the continuous approximation of such basic mathematical principles and to provide a modern method for computational mathematics of real-life problems. In fact, it considers latest generalized soft, rough approximations, called soft

Several examples are provided to illustrate the links between topologies and relationships of the soft set. Finally, we are added three applications. in making decisions regarding our strategy. One of them represents a beginning point for apply soft rough approach to solve the problem of Coronavirus contagion. At the end of the paper, we give two an algorithm which can be used to have a decision making for information system in terms of soft

The main programming for this paper is as follows:

Step 1: Input the set

Step 2: Compute the rough neighborhood from the information table.

Step 3: Compute the soft

Step 4: Remove a feature a1 from the condition’s features (A) and then find the rough neighborhood

Step 5: Comparing

Step 6: Repeat Steps 4 and 5 for all attributes in A.

Step 7: Those attributes in A for which

Finally, we explain the importance of the proposed method in the medical sciences for application in decision-making problems. In fact, a medical application has been introduced in the decision-making process of COVID-19 Medical Diagnostic Information System with the algorithm. This application may help the world to reduce the spread of Coronavirus.

The paper is structured as follows: The basic concepts of the rough set and soft set were explored in section two and three. The implementation of COVID-19 for each subclass of attributes in the information systems and comparative analysis was presented in section four and five. Section six concludes and highlights future scope.

In this section, we give some basic definitions and results that used in sequel are mentioned.

In 1982, Pawlak [23] introduced the theory of rough set as a new mathematical methodology or easy tools in order to deal with the vagueness in knowledge-based systems, information systems and data dissection. This theory has many applications in many fields that are used to process control, economics, such as medical diagnosis, chemistry, psychology, finance, marketing, biochemistry, environmental science, intelligent agents, image analysis, biology, conflict analysis, telecommunication, and other fields (See: [23–27], and the bibliography in these papers).

Definition 2.1 [23] Assuming that

According to Pawlak’s definition, M it’s called a rough set if

Proposition 2.1 [23] Let

2.2 Soft Set Theory and Soft Rough Set

Let us recall now the soft set notion, which is a newly-emerging mathematical approach to vagueness. Let

Definition 2.3 [12] Let

i)

ii) For each

Definition 2.4 [15] Let

Proposition 2.2 [15] Assuming that

i)

ii)

iii)

iv)

v)

vi)

vii) If

viii)

Proposition 2.3 [15] Let

i) If

ii) If

Proposition 2.4 [15] Let

i)

ii)

Definition 2.5 [2] Let

It is clear that if

Proposition 2.5 [15] Let

i)

ii)

3 Generalized Soft Rough Approximations

In this section, we define new generalized soft, rough approximations so-called soft

Definition 3.1 Let

Definition 3.2 Let

Clearly, if

The main goal of the following results is to introduce and studied the basic properties of soft

Example 3.1 Let

Proposition 3.1 Let

i) If

ii) If

iii)

iv)

v)

vi)

Proof

i) Since

ii) Since

iii) Since

We shall prove that

iv) Since

v) By similar way as (iv).

vi) By using (iv)–(v), the proof is obvious

Remark 3.1 The inclusion in the above Proposition part (iv) is not instead of to equal the following example shows this remark.

Example 3.2 Let

Proposition 3.2 Assuming that

i)

ii)

Proof

i) Since

ii) Since

Proposition 3.3 Assuming that

i)

ii)

iii)

iv)

Proof

i) Since

ii) Obvious.

iii) Since

iv) Obvious.

Remark 3.2 Note that the inclusion relations in Proposition 3.3 may be strict, as shown in Examples 3.1 and 3.2.

Example 3.3 From Example 3.1 let

Proposition 3.4 Assuming that

i)

ii)

Proposition 3.5 Assuming that

i)

ii)

Proof

i) Let

(a)

(b)

ii) Obvious

Corollary 3.1 Assuming that

i)

ii)

Corollary 3.2 If

Remark 3.3 The converse of the above results is not true in general as Example 3.3 illustrated.

Example 3.4 Consider Ex..3.1 Let

Example 3.5 From Example 3.1 Let

From the above Tab. 1, we deduce our method is better than Zhaowen method [15]. Also, from the above Tab. 1, we get the following Tab. 2,

Proposition 3.6 Assuming that

i) Firstly, by Proposition 3.3, we get

let

Remark 3.4 Assuming that

i)

ii)

Proposition 3.7 Assuming that

i)

ii)

Proposition 3.8 Let

i)

ii)

Proposition 3.9 Assuming that

Proof

Assume that M is

4 Relationship Between Our Method and the Pawlak Approximation

In this section, we shall compare between current method and the method of Pawlak.

Definition 4.1 [15] If

Theorem 4.1 Let

i)

ii)

Remark 4.1 Propositions 3.1, 3.2 and 3.4 represent one of the deviations between our approach and in [15] approach. By this proposition, our approximations satisfied most of Pawlak’s properties and then Tab. 3, summarize these properties and give first comparison among our method and [15] method. We then list codes in Tab. 3 to show whether these approximations satisfy the properties (L1) to (U9). In Tab. 3, the number 1 denotes yes and 0 denotes not.

The main goal of the following results is to illustrate the relationship between soft rough approximations (that given by Wang et al. [16]) and soft pre-rough approximations (that given by our approach in the present paper).

Definition 4.2 Assuming that

i) M is roughly soft

ii) M is internally soft

iii) M is externally soft

iv) M is totally soft

The intuitive meaning of this classification is as follows:

–-If M is roughly soft

–-If M is internally soft

–-If M is externally soft

–-If M is totally soft

Theorem 4.2 Let

i) If M is roughly soft

ii) If M is internally soft

iii) If M is externally soft

iv) If M is totally soft

Proof: By Proposition 3.5, the proof is obvious.

Remark 4.2 Theorem 4.2 represents a one of differences between soft rough approximations (that given by [15]) and soft