Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2022.020066

ARTICLE

On Soft Pre-Rough Approximation Space with Applications in Decision Making

M. El Sayed1,*, Wadia Faid Hassan Al-shameri1 and M. A. El Safty2

1Department of Mathematics, College of Science and Arts, Najran University, Najran, 66445, Saudi Arabia
2Department of Mathematics and Statistics, College of Science, Taif University, Taif, 21944, Saudi Arabia
*Corresponding Author: M. El Sayed. Email: mebadria@nu.edu.sa
Received: 02 November 2021; Accepted: 20 January 2022

1  Introduction

The chikungunya virus is transmitted to humans by the bite of an infected mosquito. Fever and joint discomfort are the most typical symptoms of infection. Headache, muscle soreness, joint swelling, and rash are some of the other symptoms. The chikungunya virus was first discovered in the Americas in late 2013 on the Caribbean islands. The number of research articles published has exploded at a quick pace, particularly in mathematics. Several proposals were given for solving real-world problems with mathematical methodologies and relevant formulas to assist decision-makers in making the best decisions possible. To deal with challenges that are uncertain ([1–3]).

To reduce the uncertainty and vagueness of knowledge, Molodtsov [4] developed soft sets; as a novel technique for modeling uncertainty, creating the fundamentals of the corresponding theory. He has demonstrated how this theory may be used to solve a variety of practical issues in economics, engineering, social science, and medicine, among other fields. Many researchers introduced several applications of soft set theory ([5–7]). We are implementing new classes of concepts in this paper based on soft pre-rough set [8–11].

In our everyday lives, we are often constantly faced with challenges that necessarily require rational decision-making. Yet, we get uncertain about the correct answer in several of these situations. We must consider different criteria related to the solution in order to arrive at the best possible solution to these problems. For this, in our paper we can use the best mathematical tool namely soft, rough set theory in decision making. The classical soft sets were also applied to fuzzy soft sets by the same authors [12–14]. Maji et al. [15,16] discussed the application of soft set theory to a problem of decision-making, and the implementation of soft set theory was explored. They created soft, rough approximations, soft, rough sets, and several related concepts based on this granulation structure. The soft rough sets model established by Feng et al. [17] is generalized in this paper. The goal was to exert some influence on the ongoing issue. This approximation is a generalization to Feng et al. [17]; we have demonstrated that our approach is more accurate and comprehensive than that of Feng et al. [17] defined the soft rough model as a generalization of the Pawlak rough models (SRs) [18]. Since then, many researchers have further studied RS as in the following published articles [19–21]. The authors in [22–24] introduced a new approach coupled with applications based on relation in soft generalized topological spaces and they studied their properties. Mathematical modeling of vagueness and ambiguity is becoming an increasingly important in a variety of fields of study. Another statistical tool that has been used in many aspects of life is regression analysis [25,26].

In this paper, we used this approximation to define many new concepts based on it, namely soft pre-rough belonging, soft pre-rough inclusion, soft pre-rough definability, soft pre-rough equality, and we studied the properties of these concepts. The present approximations are significant not just because they reduce or eliminate border areas. Finally, we will introduce an application in decision making of these concepts. At the end of the paper, we will present an algorithm that can be used to decide on an information system to show the importance of this approximation.

Here's how the document goes: Originality starts from Sections 2 and 3, and the preliminary basic concepts are covered. Sections 4 and 5 discuss both the rough and soft sets, as well as the use of soft pre-rough for each subclass of characteristics in information systems and applications. In Section 6, a few concluding notes provide a discussion and recommendations for future scope.

2  Preliminaries

We offer some fundamental concepts and outcomes that are utilized in the paper:

Definition 2.1 [3] Let S=(Γ,A)be a soft set over upon Uand A=(U,S)be a space. Consequently, the soft “pre-lower” and “pre-upper” approximations of any subset X⊆U are defined respectively, by: the soft pre-lower B−s(X)=X∩N_(N¯(X)) and the soft pre-upper B−s(X)=X∪N¯(N_(X)).

We refer to (B−s(X),B−s(X)) as “soft pre-rough approximations” with respect to As.

Definition 2.2 [3] Assuming that As=(U,S)be a space and X⊆U. Consequently, the soft “pre- positive, pre-negative, pre-boundary” regions and the “pre-accuracy” of the soft pre-approximations are defined respectively by: POS(X)=B−S(X), NEG(X)=U−B−S(X), BND(X)=B−S(X)−B−S(X) and μ=|B−S(X)B−S(X)|, where B−S(X)≠0.

Clearly, if B−S(X)=B−S(X),BND(X)=φ and μ=1. Then X⊆U it is called “soft pre-definable” or “soft pre-exact” set; otherwise X it is called a “soft pre-rough” set.

The principal purpose of the following outcomes is to present and superimpose the fundamental features of soft pre-rough approximations B−S(X) and B−S(X).

Proposition 2.1 [3] Let S=(Γ,A) be a soft set upon U and As=(U,S) be a space. Thus, the soft pre-lower and pre-upper approximations of X⊆U satisfy the following properties:

(i) B−s(φ)=B−s(φ)=φ

(ii) B−s(U)=∪e∈Af(e),B−s(U)=U

(iii) IX⊆Y then B−s(X)⊆B−s(Y)

(iv) If X⊆Y, then B−s(X)⊆B−s(Y)

(v) B−s(X∩Y)=B−s(X)∩B−s(Y)

(vi) B−s(X∪Y)⊇B−s(X)∪B−s(Y)

(vii) B−s(X∩Y)⊆B−s(X)∩B−s(Y)

(viii) B−s(X∪Y)=B−s(X)∪B−s(Y)

3  Approaches on Soft Set

3.1 Soft Pre-Definability of Sets

In this section, we presented the definitions of definability of sets by using soft pre-rough approximation, namely, soft pre-internal upper, soft pre-external lower of a set A, and we denote to the soft pre-internal upper, soft pre-external lower of a set A by Inte−S(A) and Exte−S(A), respectively.

The definition that follows introduces new concepts of definability for a subset A⊆X in soft pre-rough approximation space.

Definition 3.1 Assuming S=(Γ,A) be a soft set upon U and As=(U,S) be a space. Then, the soft pre-lower and pre-upper approximations of and A⊆X,A≠φ. Then A is called.

i.   Soft pre-internally definable if and only if Exte−S(A)=φ and Inte−S(A)≠φ⇒A=app−S(A).

ii.  Soft pre-externally definable if and only if Inte−S(A)=φ and Exte−S(A)≠φ⇒B−S(A)=A.

iii. 6Soft pre-roughly undefinable if and only if B−S(A)≠B−S(A)≠A. This means that some elements of X belong to A and some elements belongs to AC.

iv.  Soft pre-exact (briefly Soft pre-exact) set if and only if A=B−S(A)=B−S(A) and hence bS(A)=φ.

Definition 3.2 Let S=(Γ,A) be a soft set upon Uand As=(U,S) a soft approximation space. and A⊆U,A≠φ. Consequently A is called:

i.   Soft pre-internally definable (resp. Soft pre-externally definable and soft pre-exact).

ii.  Soft pre-internally definable set if and only if A=B−S(A), i.e., ExteS(A)=φ.

iii. Soft pre-externally definable set if and only if A=app−S(A), i.e., Inte−S(A)=φ.

Definition 3.3 Let S=(Γ,A)be a soft set upon, As=(U,S) be a soft space. Then, the soft pre-lower and pre-upper approximations of and A⊆U,A≠φ. Then A is called:

i.   Soft pre-external lower (briefly Exste−p(A)) if Exste−p(A)=A−B−S(A).

ii.  Soft pre-external lower (briefly Ixste−p(A) ) if Ixste−p(A)=B−S(A)−A.

iii. Soft pre-exterior (briefly extp(A) ) if extp(A)=U−B−S(A).

Proposition 3.1 Let S=(Γ,A)be a soft set upon U, As=(U,S) be a soft space, A,B⊆U,A,B≠φ. Then, the next announcements are held:

i.   If A⊆B, then extp(B)⊆extp(A).

ii.  extp(A∪B)=extp(A)∪extp(B).

iii. extp(A∩B)⊇extp(A)∪extp(B).

Proof.

i.   Since B−S(A)⊆B−S(B) by taken the complement for both sides, we get X−B−S(B)⊆X−B−S(A) and hence extp(B)⊆extp(A).

ii.  Since extp(A∪B)=U−B−S(A∪B)=(U−B−S(A))∩(U−B−S(B))=extp(A)∪extp(B) then extp(A∪B)=extp(A)∪extp(B).

iii. Since extp(A∩B)=U−B−S(A∩B)⊇(U−B−S(A))∪(U−B−S(B)=extp(A)∪extp(B). Then extp(A∩B)⊇extp(A)∪extp(B).

The following example shows the equality in (iii) of the above proposition.

Example 3.1 Suppose that S=(Γ,A) be a soft set upon U, As=(U,S) a soft approximation space, where, U={χ1,χ2,χ3,χ4,χ5}, E={δ1,δ2,δ3,…,δ6} and A={δ1,δ2,δ3,δ4}⊆E, so that (Γ,A)={(δ1,{χ1}),(δ2,{χ2,χ5}),(δ3,{χ3}),(δ4,{χ5})}. Now let X={χ1,χ2,χ3},Y={χ3,χ4,χ5}. Then, we get B−s(X)={χ1,χ2,χ3} and B−s(Y)={χ2,χ3,χ4,χ5} which implies extpX={χ4,χ5} and ext(Y)={χ1}. Hence, ext(X)∪ext(Y)={χ1,χ4,χ5}…(i)since appr−s(X∩Y)={χ3}. Then ext(X∩Y)={χ1,χ2,χ4,χ5}…(ii) thus from (i), (ii) we get extp(A∩B)≠extp(A)∪extp(B).

Proposition 3.2 Let be full soft set upon U, As=(U,S) be a soft space, X⊆U,X≠φ. Then,

i.  (B−S(X))C⊆B−S(XC)

ii. NEG(X)=(B−S(X))C⊆B−S(X))C

Proof.

i.  Since B−S(X)=X∩B−S(B−S(X)) then B−S(X))c=(X∩N−⁡(N−⁡(X)))c=XC∪(N−⁡(N−⁡(X)))c⊆XC∪N−⁡((N−⁡(X))c)⊆XC∪N−⁡(N−⁡(X)C)=B−S(X)C Then (B−S(X))C⊆B−S(XC).

ii. Since NEG(X)=(B−S(X))Cand since B−S(X)=X∩N−⁡(N−⁡(X) thus (B−S(X))C=(X∪N−⁡(N−⁡(X)))C=XC∩(N−⁡(N−⁡(X))C⊆XC∩(N−⁡(N−⁡(X))C⊆XC∩N−⁡(N−⁡(XC)=B−S(XC).

Proposition 3.3 Assumption that S=(Γ,A) be a soft set upon, As=(U,S) be a soft space, A⊆X,A≠φ. Then:

i.   A is soft pre-exact set then A is soft pre-internally and soft pre-externally definable.

ii.   A is soft pre-roughly un-definable set if and only if A is neither soft pre-internally nor soft pre-externally definable.

Proof.

i.   Let A be a soft pre-exact. Then, A=B−S(A)=B−S(A). Since A=B−S(A). Then ExtS(A)=φ. And hence A is soft pre-internally definable since A=B−S(A). Then Int−S(A)=φ therefore A is soft pre-externally definable.

ii.   Let A be a soft pre-rough undefinable set hence B−S(A)≠B−S(A)≠A. Then A is neither soft pre-internally and soft pre-externally definable. Conversely, it is obvious.

Proposition 3.4 Assumption that S=(Γ,A) be a soft set over upon U, As=(U,S) be a space, A⊆X,A≠φ. If A is soft pre-roughly undefinable the B−S(A)≠B−S(A)≠A.

Proof. Let A be a soft pre-roughly undefinable. Therefore A is not soft pre-exact and hence B−S(A)≠B−S(A)≠A.

Proposition 3.5 Assumption that S=(Γ,A) be a soft set over upon U, As=(U,S) be a space, A⊆X,A≠φ. Thence A is soft pre-roughly undefinable if and only if B−S(A)≠B−S(A)≠A.

Proof. Let A be a soft pre-roughly undefinable. Thus, is not soft pre-exact then B−S(A)≠B−S(A)≠A. Conversely, let B−S(A)≠B−S(A)≠A. Thus, A is not soft pre-exact. Therefore, A is not simply open set and B−S(A)≠B−S(A)≠A. Hence, A is soft pre-roughly undefinable.

3.2 Soft Pre-Rough Belonging

In this section, we introduce new definitions on rough membership relation which indicates belonging to the elements of the set by using soft pre lower and soft pre upper approximations and we are studying some of their properties.

Definition 3.4 Assumption that S=(Γ,A) be a soft set over upon U, As=(U,S) be a space and upper belong as follows:

i.   a is soft pre-lower belonging to A (briefly a∈−p⁡A ) iff a∈B−S(A).

ii.   a is soft pre-upper belonging to A (briefly a∈−p⁡A ) iff a∈B−S(A).

Remark 3.1 Assumption that S=(Γ,A) be a soft set over upon U and As=(U,S) be a space. and A⊆U,A≠φ. Then, for each a∈U we have:

i.   If a∈−p⁡A, then a∈−p⁡A.

ii.   If a∈−R⁡A, then a∈−p⁡A.

iii.   If a∈−p⁡A, then a∈−R⁡A.

The next example clarifies the above remark. Also, this example shows the concepts of soft pre-lower belong and soft pre lower belong which we shall use in the following application.

Example 3.2 Assumption that S=(Γ,A) be a soft set upon U, As=(U,S) is a space, where U={χ1,χ2,χ3,χ4,χ5,χ6},E={δ1,δ2,….,δ6} and A={e1,e2,e3,e4}⊆E such that (Γ,A)={(δ1,{χ1,χ6}),(δ2,{χ3}),(δ3,φ),(δ4,{χ1,χ2,χ5})}. Now let X={χ3,χ4,χ5} but B−s(X)={χ3,χ5} and B−s(X)={χ3,χ4,χ5}. It is clear that an element χ4∈X, but χ4∉−pX hence χ4∉B−s(X). Also, the same element χ4∈B−s(X) and hence χ4∈p−X.

Proposition 3.6 Assumption that S=(Γ,A) be a soft set upon U, As=(U,S) is a space, and N,H^⊆U. Consequently the next assumptions are valid:

i.   If N⊆H^, then (z∈−P⁡N→z∈−P⁡H^ and z∈−P⁡N→z∈−P⁡H^).

ii.   If z∈−P⁡(N∩H) then z∈−P⁡N and z∈−P⁡H^.

iii.   z∈−P⁡(N∪H^)⇔z∈−P⁡N or z∈−P⁡H^.

iv.   If z∈−P⁡N or z∈−P⁡H^ Then z∈−P⁡(N∪H^).

v.   z∈−P⁡(N∩H^)⇔z∈−P⁡N and z∈−P⁡H^.

vi.   z∈−P⁡NC⇔z∉−PN.

vii.   z∈−P⁡NC⇔z∉−PN.

Proof.

i.   Let N⊆H^. Thus B−S(N)⊆B−S(H^) and B−S(N)⊆B−S(H^) if z∈−p⁡N. Then z∈B−S(N) but B−S(N)⊆B−S(H^). Thus z∈B−S(H^) and hence z∈−p⁡H^. Similarly we can prove the second part.

ii.   Let z∈−P⁡(N∩H^). Thus z∈B−S(N∩H^) and hence z∈(B−S(N)∩B−S(H^)). This tends to z∈B−S(N) and z∈B−S(H^). Then z∈−P⁡A and z∈−P⁡H^.

iii.   Let z∈−P⁡(N∪H^). Thus z∈B−S(N∪H^)⇔z∈B−S(N) or z∈B−S(H^)⇔z∈−P⁡N or z∈−P⁡H^.

iv.   Let z∈−p⁡N or z∈−p⁡H^. Then z∈B−S(N) or z∈B−S(H^). Hence z∈B−S(N)∪B−S(H^)⊆B−S(N∪H^) thus z∈B−S(N∪H^). Therefore z∈−p⁡(N∪H^).

v.   Let z∈−p⁡(N∩H^). Thus z∈B−S(N∩H^) but B−S(N∩H^)=B−S(N)∩B−S(H^) . Then z∈B−S(N) and z∈B−S(H^)⇔z∈−p⁡N and z∈−p⁡H^.

3.3 Soft Pre-Rough Equality

We will introduce in this section a new class of equality by using soft pre-lower and soft pre -upper approximations namely, soft pre lower equal, soft pre upper equal and approximations of any two sets and we study some of their properties.

Definition 3.5 Assuming that S=(Γ,A) be a soft set upon U, AS=(U,S) is a space and A,H^⊆U, then:

i.   The sets A and H^ are soft pre lower equal (briefly A∼−p⁡H^ ) if B−S(A)=B−S(H^)

ii.   The sets A and H^ are soft upper equal (briefly A≃pH^) if B−S(A)=B−S(H^).

iii.   The sets A and H^ are soft approximations (briefly A≈pH^) if A∼−p⁡H^ and A≃pH^.

Proposition 3.7 Assumption that S=(Γ,A) be a soft set upon U,As=(U,S) is a space, Consequently and W^,J^,C and D⊆X Then:

i.   If W^≃PC and J≃PD subsequently, W^∪J^≃PC∪D.

ii.   If W^≃PJ^ and W^∪J^≃PW^ subsequently, W^∪J^≃PW^≃PJ^.

Proof.

i.   Since W^≃PC and J≃PD Then B−S(W^)=B−S(C) and B−S(J^)=B−S(D) Hence B−S(W^∪J^)=B−S(W^)∪B−S(J^)=B−S(C)∪B−S(D)=B−S(C∪D). Therefore W^∪J^≃PC∪D.

ii.   Since W^≃pJ^ then B−S(W^)=B−S(J^). But B−S(W^∪J^)=B−S(W^)∪B−S(J^). Then B−S(W^∪J^)=B−S(W^) and B−S(W^∪J^)=B−S(J^). Then W^∪J^≃pW^. and W^∪J≃PJ^. Conversely, since W^∪J^≃PW^ and W^∪J^≃PJ^. Then B−S(W^∪J^)=B−S(W^)∪B−S(H^)=B−S(W^) and B−S(W^∪J^)=B−S(W^)∪B−S(J^)=B−S(J^). Therefore B−S(J^)⊆B−S(W^) and B−S(W^)⊆B−S(J^) thus B−S(W^)=B−S(J^) and hence W^≃PJ^

Definition 3.6 Assumption that S=(Γ,A) be a soft set upon U and AS=(U,S) is a soft space. and A⊆U Then, A is called:

i.   Soft pre dense in AS=(U,S) if and only if A≃pU.

ii.   Soft pre co-dense in AS=(U,S) if and only if A∼−p⁡φ.

Proposition 3.8 Assumption that S=(Γ,A) be a soft set upon U, AS=(U,S) is a space and A⊆U. Then we have:

i.   Any set which contains soft pre dense is also soft pre dense set AS=(U,S).

ii.   Any subset of soft pre condense set is soft pre co-dense set AS=(U,S).

Proof.

i.   Let A⊆B⊆U and since A is soft pre dense then A≃pU and hence B−S(A)=B−S(U)., B−S(A)⊆B−S(B). This implies B−S(A)=B−S(U)=U⊆B−S(B). But B−S(B)⊆U thus B−S(B)=U=B−S(U) and hence B is soft pre dense set in AS=(U,S).

ii.   Let A⊆B, since B is soft pre co-dense set then B∼−p⁡φ then B−S(φ)⊆B−S(A)⊆B−S(B)=B−S(φ) hence B−S(A)=B−S(φ). Then A∼−p⁡φ therefore A is soft pre co-dense set AS=(U,S).

3.4 Soft Pre-Rough Inclusion

In this section, we present a new type of inclusion based on the soft pre rough approximation space called soft pre-upper inclusion, soft pre-lower inclusion, and soft pre-upper inclusion and we studied some of their results.

Definition 3.7 Assumption that S=(Γ,A) be a soft set over upon U, AS=(U,S) be a space and A,H^⊆X. Then, we called:

i.   A is soft pre-upper included in H^ (rough upper subset of H^ ) (briefly, A⊆∼P⁡H^ ) if and only if B−S(A)⊆B−S(H^)

ii.   A is soft pre-lower included in H^ (rough lower subset of H^ ) (briefly, A⊆∼p⁡H^) if and only if B−S(A)⊆B−S(H^)

iii.   A is soft pre roughly included in H^ (rough subset of H^ ) (briefly, A⊆∼P∼H^) if and only if B−S(A)⊆B−S(H^) and B−S(A)⊆B−S(H^)

In the following example we show that the rough inclusion of sets does not imply to the inclusion of the ordinary sets.

Example 3.3 Assumption that S=(Γ,A) is a soft set upon, AS=(U,S) is a space, where, U={χ1,χ2,χ3,χ4,χ5,χ6},E={δ1,δ2,….,δ6}, A={δ1,δ2,δ3,δ4}⊆E and (Γ,A)={(δ1,{χ1,χ6}),(δ2,{χ3}),(δ3,φ),(δ4,{χ1,χ2,χ5})}. Let us now X={χ6} and Y={χ3,χ5}. But B−s(X)=φ,B−s(Y)={χ3,χ5}. It is clear that X⊆∼PY.

Example 3.4 Assumption that S=(Γ,A) be a soft set upon, AS=(U,S) is a space, where U={χ1,χ2,χ3,χ4},E={δ1,δ2,….,δ6} and A={δ1,δ2,δ3}⊆E such that (Γ,A)={(δ1,{χ1}),(δ2,{χ1,χ3}),(δ3,{χ2,χ3})}. Now let X={χ3,χ4} and Y={x2,x3}. But B−s(X)={x3},B−s(Y)={x2,x3}. It is clear that X⊆∼p⁡Y.

Proposition 3.9 Assumption that S=(Γ,A) be a soft set upon, AS=(U,S) is a space and A,H^⊆U. Then we have:

i.   If A⊆H^, then A⊆∼p⁡H^, A⊆∼PH^ and A⊆∼∼pH^.

ii.  If A⊆∼p⁡H^ and H^⊆∼p⁡A⇔A∼−p⁡H^.

iii. If A⊆∼PH^ and H^⊆∼PA⇔A≃pH^

Proof.

i.   Obvious.

ii.   Since A⊆∼p⁡H^ and H^⊆∼p⁡A Then B−S(A)⊆B−S(H^) and B−S(H^)⊆B−S(A) which implies that B−S(A)=B−S(H^) and hence A∼−p⁡H^. Conversely, let A∼−p⁡H^ Then B−S(A)=B−S(H^) and this means that B−S(A)⊆B−S(H^) and B−S(H^)⊆B−S(A). This implies that A⊆∼p⁡H^ and H^⊆∼p⁡A. If A⊆∼P∼H^ and hence A∼−p⁡H^.

iii.   Similarly, as (ii).

iv.   Similarly, as (ii).

Proposition 3.10 Assumption that S=(Γ,A) be a soft set upon U, AS=(U,S) is a space, W^,N,C and D⊆U. Consequently:

i.   W^⊆∼PN⇔W^∪N≃pN.

ii.   W^∩N⊆∼p⁡W^⊂P∼W^∪N

iii.   W^⊆N,W^∼−p⁡C, and N∼−p⁡D⇒C⊆∼⁡D

iv.   W^⊆N,W^≃pC, and N≃pD⇒C⊆∼PD.

v.   W^⊆N,W^≈pC, and N≈pD⇒C⊆∼∼pD.

vi.   C⊆∼⁡W^, D⊆∼PN⇒C∪D⊆∼PW^∪N.

vii.   C⊆∼pW^ and D⊆∼pN⇒C∩D⊆∼pW^∩N.

Proof.

i.   Since W^⊆∼p⁡N. Then B−S(W^)⊆B−S(N). Since B−S(W^∪N)=B−S(W^)∪B−S(N). Then B−S(W^∪N)=B−S(W^). Therefore W^∪N∼−p⁡W^. Conversely, let W^∪N∼−p⁡W^. Thus B−S(W^∪N)=B−S(W^)∪B−S(N)=B−S(W^), where B−S(W^)⊇B−S(N). Therefore W^⊆∼⁡N.

ii.   Since W^∩N⊂W^. Then B−S(W^∩N)⊆B−S(W^) hence W^∩N⊆∼p⁡W^ and since W^⊆W^∪N. Thus B−S(W^)⊆B−S(W^∪N) then W^⊆∼PW^∪N. Hence W^∩N⊆∼P⁡W^⊆∼PW^∪N.

iii.   Since W^⊆N and W^∼−P⁡C,N≃PD. Then B−S(W^)=B−S(C) and B−S(N)=B−S(D) since B−S(W^)⊆B−S(N) therefore B−S(C)⊆B−S(D). Thus C⊆∼P⁡D.