Intelligent Automation & Soft Computing

Another View of Weakly Open Sets Via DNA Recombination

Samirah Alzahrani1,*, A.I. El-Maghrabi2 and M.S. Badr3

1Department of Mathematics and Statistics, College of Science, Taif University, Taif, 21944, Saudi Arabia
2Department of Mathematics, Faculty of Science, Kafrelshikh University, Kafrelshikh, Egypt
3Department of Mathematics, Faculty of Science, New Valley University, El Kharga, Egypt
*Corresponding Author: Samirah Alzahrani. Email: mam_1420@hotmail.com
Received: 27 October 2021; Accepted: 17 January 2022

Abstract: The generalized structure of deoxyribonucleic acid (DNA) is based on the rules of topological spaces. DNA recombination is one of the most important processes within DNA, as it is essential in the pharmaceutical industry as well as in gene therapy. In this paper, we are discussing the relationship between rough sets, nano topological spaces (N τs ), nano Z open (N Z0 ) sets, and DNA recombination. We also created a new recombination mapping using the properties of the DNA recombination process. Further, by using the process of cutting and sticking of a sequence of genes, new topological structures are constructed and some of their properties and characterizations are investigated. Moreover, we study recombination operators in the statement “Sticky Ends”. Furthermore, we use nano topological structures to prove the validity of the mathematical model of the recombination process and the extent to which the topological mathematical properties correspond to the biological properties. Finally, we use nano Z-open sets to study many topological characteristics of the neighborhood, closure, interior, limit points, frontier, border and exterior.

Keywords: DNA recombination; sticky ends; nano Z-open; operator

1  Introduction

Liellis Thivagar [1] invented the notion of nano topological spaces (short for Nτs )), which define a subset of a universe using upper, lower approximations and a boundary region defined by an equivalence relation on it. Nano open (nano closed, nano-interior and nano-closure) sets (briefly, N0,Nc,Nint and Ncl ) as being defined by him. El-Maghrabi and Mubarki [2] defined Z-open sets in topological structures and investigated several of their features. The concept of nano Z-open sets plays a role in topological structures and their applications in domains such as mathematics, biology and other aspects of life. The goal of this research is to look at how mathematics is used in biological applications (DNA recombination) and prove the validity of the mathematical model of the recombination process using nano topology and NZ0 . The extent to which the topological mathematical properties correspond to the biological properties. We also use nano Z-open sets to study many topological characteristics of the neighborhood, closure, interior, limit points, frontier, border and exterior.

Proposition 1.1. [3] If (U,R) is an approximation space and X,YU , then

(i) LR(X)XUR(X) ,

(ii) LR(φ)=UR(φ)=φ and LR(U)=UR(U)=U ,

(iii) UR ( XY )= UR(X)UR(Y)) ,

(iv) UR ( XY ) UR(X)UR(Y)) ,

(v) LR ( XY ) LR(X)LR(Y) ,

(vi) LR ( XY )= LR(X)LR(Y) ,

(vii) IfXY then LR(X)LR(Y) and UR(X)UR(Y) ,

(viii) UR(Xc)=[LR(X)]c and LR(Xc)=[UR(X)]c ,

(ix) URUR(X)=LRUR(X)=UR(X) ,

(x) LRLR(X)=URLR(X)=LR(X) .

Definition 1.2 [1] Let HU and (U,τR(X)) be a Nts . Then H is a nano regular open ( Nro ) if H=Nint(Ncl(H)).

Definition 1.3 [4] If HU and (U,τR(X)) be a Nts . Then the nano θ -interior(resp. nano θ closure) of H is defined by NNintθ(H)={E:EisaNθ set and Ncl(E)H} (resp. Nintθ (H)= {xE:NclH , E is a Nθ set, xE} .

Definition 1.4 [4] A subset H of U is said to be a nano θ open ( Nθ0 )(resp. nano θ -closed( Nθ )) set if H= (resp. Hc is a nano θ open set ).

Definition 1.5 [5] If HU and (U,τR(X)) is Nts , then the nano δ -interior(resp. nano δ closure) of H is defined by Nintδ (H)= {xE:Nroset,EH} (resp. Nclδ (H)= {xU:Nint(Ncl(H))H , E is a No set, xE} .

Definition 1.6 [5] A subset H of U is said to be a nano δ open ( Nδo )(resp. nano δ -closed( Nδc )) set if H= Nintδ (H) (resp. Hc is a nano δ open set ).

Definition 1. 1 Let KU and (U,τR(X)) be a Nts . Then K is said to be:

(i) nano δ -preopen [5] (briefly, Nδ P0)set if KNint(Nclδ (K),

(ii) nano δ -semiopen [5] (briefly, Nδ S0)set if KNcl(Nintδ (K),

(iii) nano e-open [6] (briefly, N e0)set if KNint(Nclδ (K) Ncl(Nintδ (K),

(iv) nano θ -semiopen [6] (briefly, NθSO set if Ncl(Nintθ (K),

1.1 DNA Recombination

DNA recombination is one of the most important processes within DNA, as it is essential in the pharmaceutical industry as well as in gene therapy. Bacteria can acquire new genes by incorporating environmental DNA into their genomes [7], and the recombination process may be used for gene reproduction or tissue culture, and defects in homologous recombination may lead to a gastric cancer mutation [8]. However, recombination is not accurate [9]. Since the adoption of computer programmes depends on mathematics and the description of operations before their implementation, making a simulation of any problem requires a mathematical model of the problem, so we made a mathematical model of the recombination process and verified the validity of the mathematical model using nano-topological structure and nano Z-open and we also made a conclusion. Some mathematical results on nano Z-open.

Fig. 1 depicts the steps involved in creating recombinant DNA as:

•   Treat the DNA taken from both sources with the same restriction end nuclease.

•   The restriction enzyme is an enzyme that cleaves DNA into fragments at or near specific recognition sites, such as sticky ends (EcoRI recognizes the sequence G AATTC CTTAA G) and at blunt ends for example (HpaI restriction enzyme recognizes the sequence GTT AACCAA TTG.

•   Sticky ends are an overhanging segment of single-stranded DNA at the cut's ends.

•   These sticky ends can base pair with any complementary sticky end-containing DNA molecule.

•   When put together, complementary sticky ends can form a pair.

•   A DNA ligase is used to covalently join the two strands of recombinant DNA into a molecule.

•   Recombinant DNA must be copied several times before it can be used.


Figure 1: Recombinant DNA process

1.1.1 Mathematical of DNA Recombination

Mathematical modelling of biological processes is very useful in identifying the processes and designing programs that help in investigating possible solutions and avoiding errors. These models have helped in the development of other sciences and mathematics. Representing the problem of life or any process does not stop when building the mathematical model, but continues to prove the validity of the mathematical model and match the mathematical solution with the practical solution (to solve the problem of life). To prove the validity of the model, we use mathematical methods (topological, numerical analysis, differential equations, etc.). But during our work, we use topological methods more than any other branch of mathematics. Recombination is the production of offspring with combinations of hits that differ from those found in either parent. Stadler & Stadler [10,11] had defined the recombination process (most applications are inherited) and had problems with the definition, since the recombination of R(x, x) will not always give the x. Therefore, we will build on the definition of Stadler & Stadler definition of genetic re-synthesis, which is more commonly used in the manufacture of medicine and gene therapy. The DNA recombination process makes it possible to cut different strands of genotypes with a restriction enzyme (sticky ends) and join the genotypes together via complementary base pairing [1214]. In this study, we consists of mathematical modelling of the recombination process. Also, we consider a method for generating nano topologies by (recombination operator deduce the equivalence classes) rough set theory via one of the biological applications (DNA recombination processes). We will investigate the topological properties of nano Z-open sets using nano topological structures. Finally, we explore the extent of the match between mathematical and biological results.

1.2 The DNA Recombination Operator

“A new mathematical representation is proposed for the configuration structure induced by recombination. It consists of the mapping of pairs of objects to a power set of all objects in the search space. The mapping assigns to each pair of parental (genotypes) the set of all recombinant genotypes obtainable from the parental ones.”

Definition 1.3.1 [10] Let X be any set of types “strings of bites, vectors, DNA, ribonucleic acid (RNA) sequence …etc.” A recombination operator on X that is defined by F:A×AP(A) the following condition holds h,kA :





2  Topological Spaces of DNA Recombination

By constructing a new recombination mapping based on the properties of the recombination process, we aim to use topological concepts to build flexible mathematical models in biomathematics. In addition, we investigate the topological qualities of the newly formed map as well as the topological structures of DNA that are related to it. We study the properties of recombination mapping, new topological structures, and characterizations by using the new concepts “Cut and Sticks” for sequences of genotypes. Further, we define recombination mathematically by a matrix where enzymes can be cut and the integration of two “types” introduces the meaning of the process of recombination, and as a result of improved optimization of this definition, more than once, it’s a description of the recombination process is more accurate. The process of recombination consists of three elements: a gene, an enzyme, and plasma to form the mathematical model and then replace the enzyme with the slicing Boolean matrix. The functions (gene slicing and plasma slicing) and recombination composition were explained well by the way recombination between genes occurs. We have greater accuracy and better places to cut the injured part from the rest of the injured part.

Definition 2.1

Let X be a set of types “strings of bites, vectors, DNA, RNA sequence …etc.” and the span of X contains all the linear combination elements of X as well as recombinant. Then the topological DNA recombination operator Rs:X×XspanX , is defined by

Rs (x,y) =i=1n{cix53+cniy1,cjx35+cnjy2} , Since, Ci=(IIIII)


where “ I{0,1} and the matrix represents the unity of level i×n , O a zero matrix, Ci is called the matrix slicing and CiMn(f) [Boolean matrix] and the sense strand of DNA X53 , the antisense strand of DNA X35 .”

We will conduct a new topological study on DNA recombination in which recombination between types x and y occurs, where x is the first gene and y is the plasmid.

Proposition 2.2 Let Rs:X×XspanX be the recombination operator, defined by

Rs(x,y)=i=1n{cix53+cniy1,cjx35+cnjy2} , Rs(x,y) consists of (offspring) can be induced by x,y and it satisfies the following:

•    {x,y}Rs(x,y) .

•    Rs(x,y)Rs(y,x) .

•    zRs(x,y)Rs(x,z) Rs(x,y)x,y,zX .

•    Rs(x,x)=2x

•    Rs(x,y)Span{x,y} .

Proof. Obvious.

In this section, we present the definition of closure recombination space by using a recombination function.

Definition 2.3 Let AX . We take TRs(A) represents the closure recombination (“sticky end”) operator, since TRs(A)=x,yRs(x,y).

We work with the topological recombination (“sticky ends”) operator. The closure recombination space from the recombination operator Rs is denoted by (X,TRs) .

Theorem 2.4 The closure recombination structure (X,TRs) arising from recombination operator


•    TRs(φ)=φ ,

•   If H is a group of genes, then HTRs(H) ,

•   If HK , then TRs(H)TRs(K) ,

•    TRs(H)TRs(K)=TRs(HK) and TRs(H)TRs(K)TRs(HK) ,

•    TRs(TRs(H))TRs(K) .

Proof. Obvious.

A space (X,TRs) , is called Rs DNA recombination structure useful for DNA recombination.

2.1 The Upper and The Lower Approximation on DNA Recombination

In the following, we study the connection between rough sets [15] concepts and DNA recombination. Also, we investigate new definitions of the class of elements depend on definitions of recombination sets that result from the definition of recombination function, which identifier as follows, Rs:X×XspanX . Since Rs(x,y)=i=1n{cix53+cniy1,cjx35+cnjy2} .

Definition 2 .1.1 Set of all recombination products can be done or (offspring can be obtained) caused by x,y, let’s call it Rs(x,y) .

Definition 2.1.2 Let Rs (x,y) be a recombination set. Then the recombination class of x can be defined as [x]Rs={yU:yRs(x,y)} and the lower and the upper approximation of a subset X of U are defined as:



Example 2.1.3 Let the recombination process consists of three key pillars are: genes (g), plasmids(p), enzymes(e) and elements of recombination process (U) i.e. U = {g , e, p}, the recombination class of gene [g]R ={e},[p]R = {e} and [e]R = {g, p} and A any subsets of U (see, Tab. 1)


From Tab. 1, the set of all recombination open (closed) sets consists of the indiscrete recombination space when using the general topology, and this result applies to the nano topological structure defined on the set U (enter all items of DNA recombination). But when using the nano topology, more than one topology appears. For example, when using the enzyme, only group B appears, as the effect of the enzyme is practically on the gene and the plasm. When using the gene or the plasma only, the enzyme appears, and this applies as a cofactor. These results are consistent with the biological results.

Proposition 2.1.4 let (U,TRs) be an indiscrete recombination space, HU .Then the limit point of H is given by

H= { ifH=U{ p }ifH={P}UifHcontainsmorethenoneelement  

Proof. Obviously

These results mean that:

1.    That there is always an end point to the process.

2.    There is an output of the process of recombination, and this is applicable to the biological concept.

Remark 2.1.5 Every subset of an indiscrete recombination space is dense. The sense that it produces a very large number of plasma carrying the gene The aim of the following proposition is to describe any item in the recombination process.

Proposition 2.1.6 Every indiscrete recombination space is a regular space. We note that from Tab. 1, the appearance of more than nano topology structures, but it appears one topological structure.

3  Nano Z-Open Sets

“Throughout this paper (U,τR(X)) is a Nts with respect to X where XU and R is an equivalence relation on U. Then U/R denotes the family of equivalence classes of U by R.' [1,3,15]

Definition 3.1 [1] If HU and (U,τR(X)) is a Nts , then H is a nano Z -open(briefly, N Zo) (resp. nano Z-closed(briefly, N Zc)) set if HNint(Ncl (H) Ncl(Nintδ (H), (resp. H Nint(Nclδ (H) Ncl(Nint (H)). The family of all N Zo (resp. N Zc) sets are denoted by N Z O(U,τR(X) (resp. N Z C(U,τR(X)) .

Definition 3.2 If HU and (U,τR(X)) is a Nts , then H is a nano e -open(briefly Ne o) set if HNcl(Nint(Nclδ (H))). The family of all Neo sets are dented by NeO(U,τR(X)) .

Example 3.3 From Example 2.1.3.

From Tab. 2 the set of all nano Z-open sets to nano topological structure defined on the set U(enter all items of DNA recombination) is the discrete recombination space.


Definition 3.4 [16] If HU and (U,τR(X)) is a Nts , then a nano Z-interior of H is the union of all N Zo sets contained in H (briefly, NZint(H) .

Definition 3.5 [16] Let HU and (U,τR(X)) be a Nts . Then a nano Z-closure of H is the intersection of all N Zc sets containing H (briefly, NZcl(H)) .

Remark 3.6 Every N Z-open set is N b-open (resp. e-open and e -open ).

Lemma 3.7 Let G be a subset of a space U . Then the following statement are satisfied:

(1) N P intδ ( NPcl(G ))= NPcl(G )) Nint(Nclδ (G)),

(2) N P clδ ( NPint(G ))= NPint(G )) Ncl(Nintδ (G)).

Proof. (1) Since, N P intδ ( NPcl(G )) = NPcl(G)Nint(Nclδ(NPcl(G))=NPcl(G)Nint(Nclδ(GNint(Ncl(G))=NPcl(G)Nint(Nclδ(G))

(2) It follows from (1).

Definition 3.8 [17] A subset ZxU is called a nano Z (resp. nano e ) neighbourhood (briefly, N ZNbd (resp. NeNbd )) of a point xU if there exists HNZO(U,τR(X)) (resp. xHZx and a point x is called N ZNbd (resp. NeNbd ) point of the set H. The family of all N ZNbd (resp. NeNbd ) of a point xU is called N ZNbd (resp. NeNbd ) system of x (briefly, N ZNbd S(x)) (resp. NeNbd S (x)).

Theorem 3.9 An arbitrary union of N ZNbd (resp. NeNbd ) of a point xU is again NZNbd (resp. NeNbd ) of a point xU .

Proof. Let {Hi:iI} be an arbitrary collection of N ZNbd of xU . Since iI , Ai is N ZNbd of x, LiN ZO (U,τR(X)) such that. xLiHi , but for each iI , HiHi , therefore xHiAi which implies that Hi is again N ZNbd of x. The other cases are similar.

Lemma 3.10 The intersection of N ZNbd (resp. NeNbd ) of a point xU is not a N ZNbd (resp.) of the p NeNbd oint xU in general.

Example 3.11 From Example 3.3. and Tab. 2, let A={e} . Then the Nts τR(X) = {U,φ,{g,p}} . In the Nts , (U,τR(X)) , the sets {g,e} and {e,p} are N ZNbd (e) but {g,e} {e,p}={e} is not a N ZNbd (e).

Theorem 3.12 If (U,τR(Y)) is a Nts , then

1.     EveryNδNbd(y)isNNbd(y),yU,

2.     EveryNδNbd(y)isNZNbd(y),yU,

3.     EveryNδSNbd(y)isNZNbd(y),yU,

4.     EveryNPSNbd(y)isNZNbd(y),yU,

5.     EveryNZNbd(y)isNbNbd(y),yU,

6.     EveryNZNbd(y)isNeNbd(y),yU,

7.     EveryNZNbd(y)isNeNbd(y),yU,

Proof. (3) Let A be an arbitrary NδSNbd of yU. T henHNδSO(U,τR(X)) such that yHA . Since every NδSo is NZo , then H NZo , therefore yHA . Then A is NZNbd of y. The other cases are similar.

Definition 3.13 Let HU and (U,τR(X)) be a nano topological structure. Then H is called nano Z (briefly, NZ )-dense subset if NZ cl(H) = U.

Definition 3.14 A nano topological structure is said to be a nano Z-extremally disconnected space (briefly, NZEDS ) if the nano Z-closure of NZo set is NZo set for each NZo subset of U.

Definition 3.15 A nano topological space is called nano Z-submaximal (briefly, NZE ????) if each NZ -dense subset of U is NZo set.

Remark 3.16 A nano topological space is NZEM and NZEDS , then every NZNbd is NδSNbd and NPNbd of xU .

Theorem 3.17 For any point xU , NZNbd S(x) satisfies

(1) NZNbd S(x) φ .

(2) if HU and HNZNbd S(x), then xH .

(3) if HU and HNZNbd S(x), HC then CNZNbd S(x).

(4) if HU and HNZNbd S(x) then BNZNbd S(x)S(x) such that BH , HNZNbd S(y), yB .

Proof. (1) Since, xU and U is a NZo set, Then xU implies that U is NZNbd of x. Then UNZNbd S(x) NZNbd S(x) φ .

(2) Let HNZNbd S(x). Then H is a NZNbd of x implies that BNZO(U,τR(X)) xBH . Thus xH .

(3) Let HNZNbd S(x). Then that BNZO(U,τR(X)) xBH and HC xBHC . Therefore CNZNbd S(x).

(4) From (3) it is obvious.

Theorem 3.18 Let NZ O (U,τR(X)) be closed under finite intersection, H be NZ c subset of U and xUH . Then there exists a NZNbd A of x such that HA=φ .

Proof. Let H be a NZ c set. Then UH is a NZoNZo set. Therefore UH is NZNbd of each of its points. If xUH , implies that ANZo , since xAUH , then HA=φ .

Definition 3.19 [2] A point xU is called a nano Z- limit point of H , if for each KNZ O (U,τR(X)) containing x satisfies K (Hx) .

Definition 3.20 The set of all nano Z-limit points of H is a nano Z-derived set (briefly, NDZ(H) ).

Theorem 3.21 If H and K are subsets of a space U , then the following are hold.

(1) NDZ(ϕ)=ϕ ,

(2) if xNDZ(H) , therefore xNDZ(Hx) .

Proof. (1) Suppose that xU and GxNZ O (U,τR(X)) . Therefore (Gx)φ=φxNDZ(ϕ) , then xU , xNDZ(ϕ) . Thus NDZ(ϕ)=ϕ.

(2) If xNDZ(H) G(Hx)φ and GxNZO(U,τR(X)) and contains at least one point other than x of Hx . Thus xNDZ(Hx) .

Theorem 3.22 If HU and (U,τR(X)) is a N ts, then:

1.    If NZ ZC (U,τR(X)) is closed under arbitrary union, then HNDZ(H) is NZ c set,

2.     N Zcl(H) = HNDZ(H) .

Proof. To show that HNDZ(H) is a NZ c set, we want to prove U(HNDZ(H)) is a NZ o set,we have two cases:

Case 1: Let U(HNDZ(H)) . Then the result is clear.

Case 2: Let U(HNDZ(H))ϕ . Then xU(HNDZ(H)) implies that x(HNDZ(H)) and hence xH , xNDZ(H)GxNZO(U,τR(X)) . Since G(Hx)=φ such that xHGH=φ implies xGUH . Thus GNDZ(H) = φ implies xGUNDZ(H) . Then xG(UH)(UNDZ(H)) = U(HNDZ(H)) implies that xGU(HNDZ(H)) . Therefore U(HNDZ(H)) ) is a NZbd of each of its points. U(HNDZ(H)) is a NZ o set and then (HNDZ(H)) is NZ c.

(2) By (1), if HNDZ(H) is a NZ c set, then HNDZ(H) is a NZ c set containing H. Therefore NZ cl(H) HNDZ(H) and HNZ cl(H), implies that NDZ(H)NDZ(NZcl(H)NZ cl(H) because NZcl(H) is NZc . Hence HNDZ(H)NZ cl(H). Thus NZ cl(H)= HNDZ(H)

Theorem 3.23 If HU and (U,τR(X)) is a N ts, then the following holds:

1.     NZ cl(H) is the smallest NZc super set of H,

2.    H is a NZc set iff NZcl(H)=H .

Proof. (1) Let {Fi:iI} , FiU , Fi be a NZc set and HFi iI . Then NZ cl(H) ={Fi:iI} , hence {Fi:iI} is a NZc set. Therefore NZ cl(H) is a NZc set. Also, HFi , iIH{Fi:iI} = NZ cl(H). Then NZ cl(H) is a NZc set containing H such that NZ cl(H) = {Fi:iI} , therefore NZ cl(H) Fi , iI . Consequently, NZ cl(H) is the smallest NZc superset of H.

(2) Let H be a NZc set. Then NZc is the superset of H and hence NZ cl(H)= H. Thus H is a NZc set. Then H is a NZc set iff NZ cl(H)= H.

Proposition 3.24 If H and K are two subsets of a space U , then the following properties hold:

(1) If NZc(U,τR(X)) is closed under finite union, then NZ cl (HK)= NZcl (H) NZcl( K) for every H,KNZc(U,τR(X)) ,

(2) If Nec(U,τR(X)) is closed under finite union, then Necl(HK)=Necl(H)Necl(K) for every H,KNec(U,τR(X)) .

Proof. (1) Let H and K be N Zc sets in U. By hypothesis, HK is NZc . Thus NZcl(HK)=HK=NZcl(H)NZcl(K)

(2) Likewise (1).

Theorem 3.25 If H and K are two subsets of a space U , Then the following are holds:

1.     NZin(H) is the largest NZo set contained in H.

2.    H is a NZo set iff H = NZin(H) .

3.     NZint(φ)=φ and NZin(U)=U .

4.     NZint ( NZint(H)) ) = NZint(H) .

Proof. (1) Let BNZo(U,τR(X)) , BH . If xB , therefore xBH , BNZo(U,τR(X)) , Then H is NZbd of x and hence xBxNZint(H) . Therefore every NZo subset of H is contained in NZint(H) . Hence NZint(H) is the largest NZo set contained in H.

(2) Let HNZO(U,τR(X)) , HH and H be the largest NZo subset of H. By (1), NZint(H) is the largest NZo subset of H. Hence H = NZint(H) .

(3) It is clear.

(4) By (2), H is a NZo set iff H= NZint(H) and by (1), NZint(H) is the largest NZo set contained in H. Then NZint ( NZint(H)) ) = NZint(H) .

Theorem 3.26 If H and K are two subsets of a space U , then NZint(H)=HNDZ(UH).

Proof. Let x(HNDZ(UH))xHandxNDZ(UH). Then GxNZO(U,τR(X)) such that Gx(UH)=GxH. Hence xGHxNZin(H) . Then HNDZ(UH)NZin(H) .

If xNZin(H)xH and NZin(H) , xNZin(H)(UH) . Therefore xNDZ(UH) ). Then x(HNDZ(UH)) . Hence NZin(H)HNDZ(UH) . Then NZin(H) = HNDZ(UH).

Theorem 3.27 For H,TU , then NZin(HT)NZint(H)NZint(T) .

Proof. Let NZin(HT)=NZin(H(UT))NZint(H)NZint(UT)NZint(H)(UNZint(T))=NZint(H)NZint(T).

4  Nano Z-Exterior, Eorder and Frontier Sets

Definition 4.1 For HU , then a nano Z exterior (briefly, NZEr \x9D\x96\x97) of H is defined as N Zint\x9D\x96\x97(H) = N Zint (UH) .

Definition 4.2 [17] For HU , then a nano Z border (briefly, NZBr \x9D) of H is defined as NZBr \x9D\x96\x97 (H) = H - NZint(H) .

Theorem 4.3 If H and K are two subsets of a space U , then the following are satisfied:

1.     NE ???? δS(H)NZE ???? (H) ,

2.     NE ???? P(H)NZE ???? (H),

3.     NZE ???? (H)=UNZcl H),

4.     NZE ???? (HK)=NZE ???? (H)NZE ???? (K) ,

5.     NZint(H),NZE ???? (H)aremutuallydisjointandU=NZE ???? (H)NZint(H) ,

6.     HNZE ???? (H)= ,

7.     NZE ???? (H)UH

8.     NZE ???? (H)NZint(Hc) ,

Proof. (1) For KU , NSintδ(K)NZint(K) . Put K=UH , then NSintδ(UH)NZint(UH) . This implies NE ???? δS(H)NZE ???? (H) .

(2) For KU , NPint(K)NZint(K) Put K=UH , then NPint(UH)NZint(UH) . This implies NE ???? δS(H)NZE ???? (H) .

(3) By the definition of NZEr(H)=NZint(UH)=UNZcl(H).

(4) Consider NZEr(HK)=NZint(U(HK))=NZint((UH)(UK))NZint(UH)NZint(UK)=NZE ???? (H)NZE ???? (K) . That is, NZE ???? (HK)NZE ???? (H)NZE ???? (K).(1)

Also, we have HHK , KHK , then NZEr(HK)NZEr(H) and NZEr(HK)NZEr(K) . Hence,

NZEr(HK)NZEr(H)NZEr(K). (2)

(5) Assume that NZEr(H)NZint(H) . Then, xNZEr(H)NZint(H) therefore xNZEr(H) and x NZint(H)xUH and x H . contradiction, then NZEr(H)NZint(H)= . Similarly, U=NZE ???? (H)NZint(H).

(6) As NZEr(H)H=HNZint(UH)H(UH)=. Therefore HNZE ???? (H)= .

(7) By the definition of NZEr(H) = N Zint (UH)UH .

(8) This case is similar to (3).

Theorem 4.4 If H and K are two subsets of a space U, then the following are satisfied:

(i)NZB ????(H) NB ???? δS (H),

(ii) NZB ???? (H) NB ???? p(H),

(iii) H is NZ o set iff NZB ????(H)= ,

(iv) NZBr( H)=H NZint(H)=HNZcl(UH) ,

(v) If HK , Then NNZBr(K)NZBr(H),

(vi) NZBr(HK)NZBr( H) NZBr( K),

(vii) NZBr( H) NZBr( K) NZBr( H K ) ,

(viii) NZBr( H)= NDZ (U-H) and NDZ(H)=NZBr(U H).

Proof. (i) Since, NSintδ(H)NZint(H)UNZint(H)UNSintδ(H)

H(UNZint(H))H(UNSintδ(H))HNZint(H)HNSintδ(H). Therefore, NZBr (H) NBrδS (H).

(ii) Since, NPint(H)NZint(H)UNZint(H)UNPint(H)H(UNZint(H))H(UNPint(H))HNZint(H)HNPint(H) . Then NZB ???? (H) NB ???? p(H).

(iii) Let HU be a NZ o set iff H=NZint(H)HNZint(H)=NZBr (H)=

(iv) Since, NZBr (H)= HNZint(H)=H(UNZint(H))=HNZcl(UH).

(v)IfHK,thenNZint(H)NZint(K)UNZint(K)UNZint(H)H(UNZint(K))H(UNZint(H))KNZint(K)HNZint(H)NZBr(K)NZBr (H).

(vi) Since, HHK , KHK , hence by (ix) NZBr(H)NZBr(HK) , NZBr(K)NZBr(HK), then NZBr(HK)NZBr(K)NZBr (H).

(vii) As HKH,HKH by (ix) NZBr(H)NZBr (H K ) andNZBr(K)NZBr (H K ), therefore) NZBr( H) NZBr( K) NZBr( H K ).

(viii) By the definition of NZBr (H) =HNZint(H)=H(HNDZ(UH))=NDZ(UH) and NDZ (-H NZBr(UH) is obtained by replacing H by UH .

Definition 4.5 For HU , then a nano Z-frontier (briefly NZF ð) of H is defined as, NZF ð (H)= N Zcl(H)- N Zint(H).

Theorem 4.6 If H and K are two subsets of a space U, then the following are satisfied:

     i)   NZF ð (H) NF ð δS (H),

    ii)   NZF ð (H) NF ð p(H),

   iii)   NZB ð (H) NZF ð (H) ,

    iv)   NZcl(H)=NZint(H)NZF ð (H),

     v)   NZint(H)NZF ð (H) = ,

    vi)   NZF ð (H)= NZB ð (H) NDZ (H),

   vii)  H is N Zo set iff NZF ð (H)= NDZ (H),

  viii)   NZF ð (H)= NZcl(H)NZcl(UH) ,

    ix)   NZF ð (H)= NZF ð (U-H).

     x)   NZF ð (H) is a N Zc set ,

    xi)   NZint(H)=HNZF ð (H) ,

   xii)   NZF ð (H)= iff H is bothN Zo set andN Zc set,

  xiii)   NZF ð (NZint(H))NZF ð (H) ,

   xiv)   UNZF ð (H) =NZint(H)NZint(UH),

    xv)   NZF ð ( NZcl(H))NZF ð (H) ,

   xvi)   NZcl(H) =H NZF ð (H) NZF ð (H) ,

  xvii)   NZF ð ( NZF ð (H) )NZF ð (H) ,

 xviii)   NZF ð (H) NZE ð (H)=,

   xix)   NZF ð (H) NZE ð (H) = NZcl(Hc),

    xx)   NZE ð (H) , NZint(H) and NZF ð (H) are forms a partition.

Proof. (i) Since, Nintθ(H)NZint(H) implies that UNZint(H))UNZintθ(H) . Also, NZcl(H)NSclδ(H) . Therefore NZcl(H)(UNZint(H))NSclδ(H)UNZintθ(H) . This implies NZcl(H)NZint(H))NSclδ(H)NSintθ(H) . Hence, NZF ð (H) NF ð δS (H).

(ii) Since, NPint(H)NZint(H) implies UNZint(H)UNPint(H). Also, NZcl(H)NPcl(H) . Therefore NZcl(H)(UNZint(H))NPcl(H)(UNPint(H). This implies that NZcl(H)NZint(H)NPcl(H)NPint(H) . Hence NZF ð (H) NF ð p(H).

(iii) Since, HNZcl(H)H(UNZint(H))NZcl(H)(UNZint(H)). Therefore HNZint(H))NZcl(H)NZint(H). Thus NZB ð (H) NZF ð (H).

(iv) NZint(H)NZFr(H)=NZint(H)(NZcl(H)(UNZint(H))=(NZint(H)NZcl(H))(NZint(H)(UNZint(H))=NZcl(H)U=NZcl(H) .

(v) NZint(H)NZFr(H)=NZint(H)(NZcl(H)(UNZint(H))=(NZint(H)NZcl(H))(NZint(H)(UNZint(H))=NZint(H)=.

(vi) From (iv), NZcl(H)=NZint(H)NZF ð (H). Then H NDZ (H)= NZcl(H)=NZint(H)NZF ð (H). But H=NZE ð (H)NZint(H) by Theorem 3.22. Therefore NZE ð (H)NZint(H)NDZ (H)= NZint(H)NZFr( H). Hence NZF ð (H)= NZB ð (H) NDZ (H).

(vii) Let H be a NZ o set and by Theorem 4.4., and (iv), NZB ð (H) = φ From (vi) NZF ð (H)= NZB ð (H) NDZ (H)= NDZ (H). Therefore if A is NZ o set, NZF ð (H)= NDZ (H). Conversely. Suppose NZFr(H) = NDZ (H) from (iv), NZcl(H) = NZint(H)NZF ð(H). That is HNDZ (H) =NZint(H)NZF ð (H) by Theorem 3.22, implies HNDZ (H) = NZint(H)NDZ (H) by hypothesis. Therefore H = NZint(H) and hence H is a NZ o set.

(viii) NZFr(H)=NZcl(H)NZint(H)=NZcl(H)(UNZint(H))=NZcl(H)NZcl(UH) .

(ix) NZFr(UH) = NZcl(UH)NZint (U-H) = ( UNZint(H) ) ( UNZcl(H) )= NZcl(H)NZint(H) = NZFr(H)

(x) Since, a subset H of U is a NZcset iff H=NZcl(H). Consider, NZcl(NZFr(H))=NZcl(NZcl(H)NZint(H))=NZcl(NZcl(H)(UNZint(H)))=NZcl(NZcl(H))NZcl(UNZint(H))=NZcl(NZcl(H))NZcl(NZcl(UH))NZcl(H)NZcl(UH) = NZFr(H) by (iii), NZcl(NZFr(H))NZF ð (H). But NZFr(H) NZcl(NZFr(H)) is always true. Therefore NZcl(NZFr(H))=NZFr(H) and hence NZF ð (H) is a NZc set.

(xi) HNZFr(H)=H(UNZFr(H)) = H (U(NZcl(H)NZcl(UH) = H ((UNZcl(H))(UNZcl(UH))=(H(UNZcl(H))((H(UNZcl(UH)=(HNZint(H)=NZint(H).

(xii) If H is both NZ o and NZc sets, then H = NZint(H) and H = NZcl(H) . Now NZF ð (H) = NZcl(H) NZint(H) = HH=φ . Conversely, NZF ð (H)= φ implies, NZcl(H) NZint(H)= φ which implies, NZcl(H) NZint(H)H . That is, NZcl(H) H . But, H NZcl(H) is always true. Therefore H = NZcl(H) . Hence H is a NZ c set. Again NZF ð (H) = φ implies NZcl(H)NZint(H)= φ which implies NZcl(H)=NZint(H) implies HNDZ (H)= NZint(H) which implies H NZint(H) . But NZint(H)H is always true. Therefore NZint(H)= H. Hence H is NZ o set.

(xiii) Now, NZF ð ( NZint(H)) = NZcl(NZint(H)) NZint(NZint(H))NZcl(H) NZint(H) as NZint(H)H . This implies NZF ð ( NZint(H))NZF ð (H).

(xiv) Consider, UNZFr(H)=U(NZcl(H)NZint(H))=(UNZcl(H))NZint(H))=NZint(UH)NZint(H)).

(xv) NowNZF ð ( NZcl(H)) = NZcl(NZcl(H))NZint(NZcl(H))=NZcl(NZcl(H))(UNZint(NZcl(H)))=NZcl(H)NZcl(UNZcl(H)). Also, H NZcl(H) UNZcl(H)UHNZcl(UNZcl(H))NZcl(UH) . NZF ð( NZcl(H))NZcl(H) - NZcl(UH) = NZF ð (H). Thus NZFr(NZcl(H))NZF ð (H).

(xvi) From (iv), NZcl(H) = NZint(H))NZF ð (H)HNZF ð (H)as NZint(H))H . Also, from (iv), NZF ð (H)NZcl(H) and H NZcl(H) is always true. Therefore H NZF ð (H)NZcl(H) . It follows that, H NZF ð (H)=NZcl(H) .

(xviii) from (vii) NZF ð (H)=NZcl(H) NZcl(UH) and NZE ð (H) =U- NZcl(H) , then NZF ð (H)NZE ð (H) = NZcl(H) NZcl(UH) (UNZcl(H)) = φ .

(xix) It is clear from (xviii).

(xx) It is clear from (xix).

5  Conclusion

We consist of mathematical modelling of the recombination process and considering the method for generating nano topologies by (recombination operator deducing the equivalence classes) rough set theory via one of its biological applications (DNA recombination processes). Through the nano topological structure, we shall study the topological features of nano Z-open sets. Finally, we are exploring the extent of the match between mathematical and biological results.

Funding Statement: This research received funding from Taif University, Researchers supporting and Project Number (TURSP-2020/207) Taif University, Saudi Arabi.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.


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