Medical applications of Couroupita guianensis Abul plant and Covid-19 best Safety measure by using Mathematical Nano topological spaces

2.1 Definition

Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relation. Then U is divided into disjoint equivalence classes. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (U, R) is said to be the approximation space Let X  $\subseteq$ U.

(i) The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as X with respect to R and it is denoted by ${\mathrm{L}}_{\mathrm{R}}$ (X).

That is ${\mathrm{L}}_{\mathrm{R}}$ (X) =  $\left\{\genfrac{}{}{0.0pt}{}{\cup }{\mathrm{x}\in \mathrm{U}}\{\mathrm{R}\left(\mathrm{x}\right):\mathrm{R}(\mathrm{x})\subseteq \mathrm{X}\right\}$ .

Where R(x) denotes the equivalence class determined by ×  $\subseteq$ U.

(ii) The upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X with respect to R and it is denoted by ${\bigcup }_{\mathrm{R}}(\mathrm{X})$ =. $\genfrac{}{}{0.0pt}{}{\cup }{\mathrm{x}\in \mathrm{U}}\{\mathrm{R}\left(\mathrm{x}\right):\mathrm{R}(\mathrm{x})\cap \mathrm{X}\ne \mathrm{\varphi }\}$

(iii) The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor as not-X with respect to R and it is denoted by ${\mathrm{B}}_{\mathrm{R}}(\mathrm{X})$ . That is ${\mathrm{B}}_{\mathrm{R}}(\mathrm{X})$ = ${{\mathrm{U}}^p}_{\mathrm{R}}(\mathrm{X})$ - ${{\mathrm{L}}^o}_{\mathrm{R}}(\mathrm{X})$ .

2.2 Property (Lellis Thivagar, 2012)

If (U, R) is an approximation space and X, Y $\subseteq$ U, then.

1. ${{\mathrm{L}}^o}_{\mathrm{R}}\left(\mathrm{X}\right)\subseteq {\mathrm{L}}_{\mathrm{R}}\left(\mathrm{Y}\right)$ and ${{\mathrm{U}}^p}_{\mathrm{R}}\left(\mathrm{X}\right)\subseteq {{\mathrm{U}}^p}_{\mathrm{R}}\left(\mathrm{Y}\right)$ whenever X  ⊆ Y.

2. ${{\mathrm{U}}^p}_{\mathrm{R}}\left({\mathrm{X}}^{\mathrm{C}}\right)={\left[{{\mathrm{L}}^o}_{\mathrm{R}}(\mathrm{X})\right]}^{\mathrm{C}}\mathrm{a}\mathrm{n}\mathrm{d}{{\mathrm{L}}^o}_{\mathrm{R}}\left({\mathrm{X}}^{\mathrm{C}}\right)={\left[{{\mathrm{U}}^p}_{\mathrm{R}}(\mathrm{X})\right]}^{\mathrm{C}}$

2.3 Definition (Lellis Thivagar, 2012)

Let U be a non-empty, finite universe of objects and R be an equivalence relation on U. Let X ⊆ U.

Let ${\mathrm{\tau }}_{\mathrm{R}}\left(\mathrm{X}\right)=\mathrm{N}\mathrm{\tau }=\{\mathrm{U},\mathrm{\varphi },{{\mathrm{L}}^o}_{\mathrm{R}}\left(\mathrm{X}\right),{{\mathrm{U}}^p}_{\mathrm{R}}\left(\mathrm{X}\right),{\mathrm{B}}_{\mathrm{R}}\left(\mathrm{X}\right)\}$ .

Then ${\mathrm{\tau }}_{\mathrm{R}}\left(\mathrm{X}\right)$ is a topology on U, called as the Nano topology with respect to X. Elements of the Nano topology are known as the Nano open sets in U and (U, $\mathrm{N}\mathrm{\tau })$ is called the Nano topological space. Elements of ${\left[\mathrm{N}\mathrm{\tau }\right]}^{\mathrm{C}}$ are called as Nano closed sets.

3.1 Medical applications in Nano topological spaces

Algorithm

Step 1: Considering a deterministic universe U, a finite set $\mathcal{H}$ of attributes split into two classes, ${\mathcal{H}}_1$ for conditional attributes as well as ${\mathcal{H}}_2$ for decision attributes, an equivalence relation R on U correlating to ${\mathcal{H}}_1$ , and a subset X of U, represent data as an information table with columns labelled with attributes, rows for objects, and entries for attribute values (Table 2).

Step 2: Determine the lower and upper approximations, as well as the boundary region of X in relation to R.

Step 3: Create the nanotopology ${\tau }_R(X)$ on U, as well as and its basis. ${\beta }_R(X)$

Step 4: Remove an attribute x from ${\mathcal{H}}_1$ and determine the lower and upper approximations and the boundary region of X with respect to the equivalence relation on ${\mathcal{H}}_1-\{x\}.$

Step 5: Create a nanotopology ${\tau }_R-(x)(X)$ on $U$ .

Step 6: Repeat steps 3 and 4 for all attributes in $A_1$ .

Step 7: Those attributes in $A_1$ for which ${\beta }_R-(x)(X)\ne {\beta }_R(X)$ form the Core (R).

3.2 Medical applications of Guianensis couroupita Abul using Nano topological spa

Nanoparticles have become part of our life in cosmetics, drug delivery systems; Therapeutic, biosensor, and pharmaceutical materials applications can be harnessed to dramatically enhance important material properties (Chandrasekar et al., 2022; Perumal et al., 2022; Panimalar et al., 2022). Couroupita guianensis, often known as cannonball tree, is just a deciduous tree with in Lecythidaceae flowering plant family. It's indeed native to the tropical woods of South and Central America and is grown in a variety of other tropical places worldwide for its lovely, fragrant blossoms and huge, intriguing fruits. The fruits have a brownish grey colour numerous components of Couroupita guianensis may have therapeutic use. Couroupita guianensis is also a kind of medicinal plant which contains antibacterial, antimycobacterial, antimicrobial, antioxidant, anticancer, antiulcer, antinociceptive, anthelmintic, antifertility, and antifungal activities (Majumder et al., 2014). The chemical ingredient of C-guianensis such as indirubin acts as an antibacterial and antifungal agent since it notably heals fungal illnesses. It is active for the treatment of chronic myelocytic leukaemia. The extract of isatin from the flower of C-guianensis is also a chemical component that has been utilised as preventative agent, protects free radial-induced cancer,

The rows represent various human diseases are Wounds $\Rightarrow W$ , Tumor $\Rightarrow T$ , Cold $\Rightarrow C$ , Snake Bite $\Rightarrow SB$ , Stomach Ache $\Rightarrow$ SA Skin Infection $\Rightarrow SI$ , and Hypertension. $\Rightarrow H$

The entries in the table are the attribute values

The given information system is incomplete and is given by (U, A)

where $U=\{W,T,C,SB,P,SA,SI,H\}$ , and

H = {Bark (B), Fruit(Fr), Flower(Fl), Leaves(L), Seeds(S)}

First Case

Guianensis couroupita Abul’s parts of plant contains Antibacterial, Antifungal component.

Initial step: 1.

Let $X=\{W,T,C,SI\}$ denote the set of human health issues associated with Guianensis couroupita Abul’s parts of plant contain Antibacterial, Antifungal component, and R denote the equivalence relation between U and the set of all condition qualities (Kryszkiewicz, 1998; Molodtsov, 1999; Pawlak, 1982; Pawlak, 1991; Qian, 2010).

The set of equivalence classes for R is denoted by. $$U/R=\{\{H\},\{C,SI\},\{W,P\},\{T\},\{SB\},\{SA\}\},$$ $$L_R(\mathrm{X})=\{T,C,SI\}$$ $$U_R^P(\mathrm{X})=\{T,C,SI,W,P$$ $$B_R^d(\mathrm{X})=\{W,P\}$$

Then nano topology ${\tau }_R(X)=\{U,\varphi ,\{T,C,SI\},\{T,C,SI,W,P\},\{W,P\}\}.$

The basis of nano topology is ${\beta }_R(X)=\{U,\{T,C,SI\},\{W,P\}\}.$

Step: 1.

If “Bark” is removed from the set of condition attributes, then. $$U/R-(B)=\left\{\left\{T\right\},\left\{W,P\right\},\left\{C,SI\right\},\left\{SB\right\},\left\{SA\right\},\left\{H\right\}\right\}.$$ $$L_R^O-(B)(\mathrm{X})=\{T,C,SI\}$$ $$U_R^P-(B)(\mathrm{X})=\{T,C,SI,W,P$$ $$B_R^d-(B)(\mathrm{X})=\{W,P\}$$

Hence $,{\tau }_R-(B)(X)=\{U,\varphi ,\{T,C,SI\},\{T,C,SI,W,P\},\{W,P\}\}.$

Step: 2.

If “Fruit” is removed from the set of condition attributes, then. $$U/R-(Fru)=\{\{W,P\},\{T,SB\},\{C,SI\},\{SA\},\{H\}\}.$$ $$L_R^O-(Fru)(\mathrm{X})=\{C,SI\}$$ $$U_R^P-(Fru)(\mathrm{X})=\{W,T,C,SB,P,SI\}$$ $$B_R^d-(Fru)(\mathrm{X})=\{W,T,SB,P\}$$

Hence, ${\tau }_R-(Fru)(X)=\{U,\varphi ,\{C,SI\},\{W,T,C,SB,P,SI\},\{W,T,SB,P\}\}.$

Step: 3.

If “Flower” is removed from the set of condition attributes, then

$U/R-(Flo)=\{\{H\},\{C,SI\},\{W,P\},\{T\},\{SB\},\{SA\}\},$ and

$L_R^O-(Flo)(\mathrm{X})=\{T,C,SI\}$

$U_R^P-(Flo)(\mathrm{X})=\{T,C,SI,W,P\}$

$B_R^d-(Flo)(\mathrm{X})=\{W,T,SB,P\}$ .

Hence ${\tau }_R-\left(Flo\right)\left(X\right)=\{U,\varphi ,\{T,C,SI\},\{T,C,SI,W,P\},\{W,P\}\}.$

Step: 4

If “Leaves” is removed from the set of condition attributes, then $$U/R-(L)=\{\{W,P\},\{T\},\{C,SA,SI\},\{SB\},\{H\}\}.$$ $$L_R^O-(L)(\mathrm{X})=\{T\}{U}_R^P-(L)(\mathrm{X})=\{W,T,C,P,SA,SI\}$$ $$B_R^d-(L)(\mathrm{X})=\{W,C,P,SA,SI\}$$

Hence, ${\tau }_R-\left(L\right)\left(X\right)=\left\{U,\varphi ,\left\{T\right\},\left\{W,T,C,P,SA,SI\right\},\left\{W,C,P,SA,SI\right\}\right\}.$

Step: 5

If “Seed” is removed from the set of condition attributes, then

$U/R-(S)=\{\{H\},\{C,SI\},\{W,P\},\{T\},\{SB\},\{SA\}\},$ and $$L_R^O-(S)(\mathrm{X})=\{T,C,SI\}$$ $$U_R^P-(S)(\mathrm{X})=\{\{T,C,SI,W,P\}$$ $$B_R^d-(S)(\mathrm{X})=\{W,P\}$$ $${\tau }_R-\left(S\right)\left(X\right)=\{U,\varphi ,\{T,C,SI\},\{T,C,SI,W,P\},\{W,P\}\}$$

Second Case II:

Guianensis couroupita Abul’s parts of plant not contain Antibacterial, Antifungal component

Initial step: 2

Let $X=\{SB,P,SA,H\}$ denotes the set of human health issues associated with Guianensis couroupita Abul’s parts of plant contain Antibacterial, Antifungal component.

Then $U/R=\{\{H\},\{C,SI\},\{W,P\},\{T\},\{SB\},\{SA\}\}$ , $$L_R(\mathrm{X})=\{SB,SA,H\}$$ $$U_R^P(\mathrm{X})=\{W,SB,P,SA,H$$ $$B_R^d(\mathrm{X})=\{W,P\}$$

Then nano topology ${\tau }_H(X)=\{U,\varphi ,\{SB,SA,H\},\{W,SB,P,SA,H\},\{W,P\}\}.$

Step: 1

If the attribute “Bark” is removed, then $$U/R-(B)=\left\{\left\{T\right\},\left\{W,P\right\},\left\{C,SI\right\},\left\{SB\right\},\left\{SA\right\},\left\{H\right\}\right\}.$$ $$L_R^O-(B)(\mathrm{X})=\{SB,SA,H\}$$ $$U_R^P-(B)(\mathrm{X})=\{W,SB,P,SA,H$$ $$B_R^d-(B)(\mathrm{X})=\{W,P\}$$

Thus, ${\tau }_R-(B)(X)=\{U,\varphi ,\{SB,SA,H\},\{W,SB,P,SA,H\},\{W,P\}\}$

Step: 2

If the attribute “Fruit” is removed, then

$U/R-(Fru)=\{\{W,P\},\{T,SB\},\{C,SI\},\{SA\},\{H\}\}.$ Thus $$L_R^O-(Fru)(\mathrm{X})=\{SA,H\}$$ $$U_R^P-(Fru)(\mathrm{X})=\{W,T,SB,P,SA,H\}$$ $$B_R^d-(Fru)(\mathrm{X})=\{W,T,SB,P\}$$ $${\tau }_R-(W)(X)=\{U,\varphi ,\{SA,H\},\{W,T,SB,P,SA,H\},\{W,T,SB,P\}\}$$

${\tau }_R-\left(W\right)\left(X\right)=\left\{U,\varphi ,\left\{P,R\right\},\left\{W,L,B,T,P,R\right\},\left\{W,L,B,T\right\}\right\}\ne {\tau }_R\left(X\right)$

Step: 3

If “Flower” is removed from the set of condition attributes, the $$U/R-(Flo)=\left\{\left\{T\right\},\left\{W,P\right\},\left\{C,SI\right\},\left\{SB\right\},\left\{SA\right\},\left\{H\right\}\right\}.$$ $$L_R^O-(Flo)(\mathrm{X})=\{SB,SA,H\},$$ $$U_R^P-(Flo)(\mathrm{X})=\{W,SB,P,SA,H$$ $$B_R^d-(Flo)(\mathrm{X})=\{W,P\}$$

Thus, ${\tau }_R-\left(Flo\right)\left(X\right)=\left\{U,\varphi ,\left\{SB,SA,H\right\},\left\{W,SB,P,SA,H\right\},\left\{W,P\right\}\right\}$

Step: 4

If the attribute “Leaves” is removed, then $$U/R-(L)=\{\{W,P\},\{T\},\{C,SA,SI\},\{SB\},\{H\}\}.$$ $$L_R^O-(L)(\mathrm{X})=\{SB,H\}$$ $$U_R^P-(L)(\mathrm{X})=\{W,C,SB,P,SA,SI,H\}$$ $$B_R^d-(L)(\mathrm{X})=\{W,C,P,SA,SI\}$$ $${\tau }_R-(L)(X)=\{U,\varphi ,\{SB,H\},\{W,C,SB,P,SA,SI,H\},\{W,C,P,SA,SI\}.$$

Step: 5

If the attribute “Seeds” is removed, then $U/R-(S)=U/R$ and

$L_R^O-(S)$ (X)= $\{SB,H\}$

$U_R^P-(S)$ (X $)=\{W,C,SB,P,SA,SI,H\}$ $$B_R^d-(S)(\mathrm{X})=\{W,C,P,SA,SI\}$$

Thus, ${\tau }_R-\left(S\right)\left(X\right)=\left\{U,\varphi ,\left\{SB,SA,H\right\},\left\{W,SB,P,SA,H\right\},\left\{W,P\right\}\right\}$

3.3 Results

From Case-I and II, We get $\mathit{Core}(R)=\{Fruit,Leaves\}$ .

3.4 Observation

From the Core of R, we conclude that “Fruits” and “Leaves” of Guianensis Couroupita Abul’s contain maximum Antibacterial, Antifungal component (Table 3).

3.5 Applications-II

Coronavirus disease 2019 (COVID-19) is first diagnosed in December 2019 in Wuhan, Hubei Province, China as an unidentified pneumonia. Later that year, the international committee on viral taxonomy (ICTV) identified COVID-19′s causal agent as a new coronavirus called acute respiratory syndrome sars syndrome coronavirus 2 (SARS–CoV2). COVID-19 is quickly spreading not just in China, but also globally, and as a result, the World Health Organization (WHO) declared it a pandemic on March 12, 2020. As of August 25, 2020, individuals infected with COVID-19 have reported a broad variety of symptoms, ranging from moderate to severe sickness. Symptoms often manifest 2–14 days following viral contact. Anyone might have symptoms ranging from mild to severe.

Individuals WHO exhibit these symptoms may have COVID-19 Fibrosis or chills, Cough, Breathlessness or trouble breathing, Fatigue, Aches in the muscles or throughout the body, Headache. Experiencing a new loss of taste or scent. Throat irritation, Excessive congestion or a runny nose. Vomiting or nausea. Diarrhea the total number of reported cases and fatalities in 216 countries is 23,491,520 and 809,970, correspondingly. Numerous government initiatives have been implemented to mitigate the danger of disease transmission. These tactics include travel restrictions, obligatory quarantines for travellers, social isolation, prohibitions on public gatherings, closure of schools and colleges, company closures, self-isolation, requiring individuals to work from home, curfews, and lockdowns. Authorities in a number of nations have imposed a lockdown or curfew in order to halt the rapid spread of viral infection. These policies have a detrimental influence on the global economy, education, health, and tourism. The epidemic of COVID-19 has impacted all levels of schooling (9). Worldwide (in 192 countries), educational institutions have either temporarily shuttered or enacted localised closures, impacting around 1.7 billion students. Numerous institutions worldwide either postponed or cancelled all campus events in order to reduce crowding and thereby viral spread. These measurements, however, have a greater economic, medical, and social impact on both undergraduate and graduate populations (Akerele, 1993; Xu et al., 2020; Acter et al., 2019; Kahler and Hain, 2020).

CASE I

COLLEGE STUDENTS ARE AWARE OF COVID-19 SAFTY MEASURE

Assume X = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ } be the set of college students are aware of covid-19 safty measure then

U/R(C) = {{ ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_7$ }, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_{15}$ ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_4$ , ${\mathit{CS}}_8$ }, { ${\mathit{CS}}_5$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{13}$ }, { ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ }, { ${\mathit{CS}}_{11}$ , ${\mathit{CS}}_{14}$ }, { ${\mathit{CS}}_{12}$ }} and

3.6 Nano topology

${\mathrm{\tau }}_R(C)(X)$ = {U, ϕ, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ . ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ }}

Step 1. When the attribute “ Wearing Mask ” is removed from C then

U/R(C-WM) = {{ ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_{12}$ }, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_4$ , ${\mathit{CS}}_8$ }, { ${\mathit{CS}}_5$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{13}$ }, { ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ }, { ${\mathit{CS}}_{11}$ , ${\mathit{CS}}_{14}$ }}, here

$L_R^O$ (C- WM) (X) = { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$U_R^P$ (C- WM) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ . ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{12}$ , ${\mathit{CS}}_{15}$ },

$B_R^d$ (C- WM) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{12}$ }.

Therefore Nano Topology

${\mathrm{\tau }}_R(C-WM)(X)$ = {U, ϕ, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ . ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{12}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , H3, ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{12}$ }}.

Step 2. When the attribute “ Social Distancing ” is removed from C then

U/R(C-SD) ={{ ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_7$ }, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_4$ , ${\mathit{CS}}_8$ }, { ${\mathit{CS}}_5$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{13}$ }, { ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ }, { ${\mathit{CS}}_{11}$ , ${\mathit{CS}}_{14}$ }, { ${\mathit{CS}}_{12}$ }}, here

$L_R^O$ (C-SD) (X) = { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$U_R^P$ (C-SD) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$B_R^d$ (C-SD) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ }.

Therefore Nano Topology ${\mathrm{\tau }}_R(C-SD)(X)=$ {U, ϕ, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ }}.

Hence ${\mathrm{\tau }}_R$ $(C-SD)X={\mathrm{\tau }}_R(C)X.$

Step 3.

When the attribute “ Wash Hands Often ” is removed from C then U/R(C-V) ={{ ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_7$ }, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_4$ , ${\mathit{CS}}_8$ }, { ${\mathit{CS}}_5$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{13}$ }, { ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ }, { ${\mathit{CS}}_{11}$ , ${\mathit{CS}}_{14}$ }, { ${\mathit{CS}}_{12}$ }},

$L_R^O$ (C-WHO) (X) = { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$U_R^P$ (C- WHO) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$B_R^d$ (C- WHO) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ }.

Therefore Nano Topology ${\mathrm{\tau }}_R(C-WHO)(X)$ = {U, ϕ, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ }}.

Hence ${\mathrm{\tau }}_R(C-WHO)X={\mathrm{\tau }}_R(C)(X$ ).

Step 4.

When the attribute “ Covering Cough/Sneeze ” is removed from C

then U/R(C- CCS) ={{ ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_7$ }, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_4$ , ${\mathit{CS}}_8$ }, { ${\mathit{CS}}_5$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{13}$ }, { ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ }, { ${\mathit{CS}}_{11}$ , ${\mathit{CS}}_{14}$ }, { ${\mathit{CS}}_{12}$ }}, here

$L_R^O$ (C- CCS) (X) = { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$U_R^P$ (C- CCS) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$B_R^d$ (C-CCS) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ }.

Therefore Nano Topology ${\mathrm{\tau }}_R$ (C − CCS) (X) = {U, ϕ, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },{ ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ }}. Hence ${\mathrm{\tau }}_R$ (C − CCS(X) = ${\mathrm{\tau }}_R$ (C) (X).

Step 5.

When the attribute “ Avoid Touching Surfaces ” is removed from C then

U/R(C- ATS) ={{ ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_7$ }, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_4$ , ${\mathit{CS}}_8$ }, { ${\mathit{CS}}_5$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{13}$ }, { ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ }, { ${\mathit{CS}}_{11}$ , ${\mathit{CS}}_{14}$ }, { ${\mathit{CS}}_{12}$ }}, here

$L_R^O$ (C- ATS) (X) = { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$U_R^P$ (C- ATS) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$B_R^d$ (C- ATS) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ }.

Therefore Nano Topology ${\mathrm{\tau }}_R$ (C − ATS) (X) = {U, ϕ, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ }}. Hence ${\mathrm{\tau }}_R$ (C − ATS)(X) = ${\mathrm{\tau }}_R$ (C)(X).

Step 6. When the attribute “Covid-19 Vaccines “ is removed from C then

U/R(C- CV) ={{ ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_7$ }, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_4$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_5$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{13}$ }, { ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ }, { ${\mathit{CS}}_{11}$ , ${\mathit{CS}}_{14}$ }, { ${\mathit{CS}}_{12}$ }}, here

$L_R^O$ (C- CV) (X) = { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$U_R^P$ (C- CV) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_5$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{13}$ , ${\mathit{CS}}_{15}$ },

$B_R^d$ (C- CV) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_5$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{13}$ }.

Therefore Nano Topology ${\mathrm{\tau }}_R$ (C − CV) (X) = {U, ϕ, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_5$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{13}$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_5$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{13}$ }}. Hence ${\mathrm{\tau }}_R$ (C − CV)(X) ≠ ${\mathrm{\tau }}_R$ (C)(X).

Step 7. When the attribute “Hand Sanitizer” is removed from C then

U/R(C- B) ={{ ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_7$ }, { ${\mathit{CS}}_2$ , ${\mathit{CS}}_{15}$ }, { ${\mathit{CS}}_4$ , ${\mathit{CS}}_8$ }, { ${\mathit{CS}}_5$ , ${\mathit{CS}}_9$ , ${\mathit{CS}}_{13}$ }, { ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ }, { ${\mathit{CS}}_{11}$ , ${\mathit{CS}}_{14}$ }, { ${\mathit{CS}}_{12}$ }}, here

$L_R^O$ (C- HS) (X) = { ${\mathit{CS}}_2$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$U_R^P$ (C- HS) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_2$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_6$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ , ${\mathit{CS}}_{10}$ , ${\mathit{CS}}_{15}$ },

$B_R^d$ (C- HS) (X) = { ${\mathit{CS}}_1$ , ${\mathit{CS}}_3$ , ${\mathit{CS}}_4$ , ${\mathit{CS}}_7$ , ${\mathit{CS}}_8$ }.