Ratio Mathematica Weaker Forms of Nano Irresolute and Its Contra Functions A.Yuvarani * S. Vijaya† P. Santhi‡ Abstract In this paper the concept of some weaker forms of irresolute and con- tra irresolute functions in Nano Topological spaces are studied and its related characteristics are discussed. Also we introduced the notion called contra nano alpha irresolute function, contra nano semi irres- olute function, contra nano pre irresolute function and its properties are examined. Finally, we have revealed some applications related to recent scenario of online teaching and COVID-19 which can be ex- pressed as nano irresolute functions and contra irresolute functions respectively. Keywords: Ns-irresolute function, Np-irresolute function, contra Nα- irresolute function, contra Ns-irresolute function, contra Np-irresolute function. 2020 AMS subject classifications: 54B05 1 *The American College, Madurai, India; yuvamaths2003@gmail.com. †Thiagarajar College, Madurai, India; viviphd.11@gmail.com. ‡The Standard Fireworks Rajaratnam College for Women, Sivakasi, saayphd.11@gmail.com. 1 India; Received on April 18, 2022. Accepted on August 25, 2022. Published on September 25, 2022. doi: 10.23755/rm.v43i0.764. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. Volume 43, 2022 A. Yuvarani, S. Vijaya, P. Santhi 1 Introduction In 1982 Pawlak [Pawlak, 1982] investigated about approximate operations, equality and inclusion on sets. In [Crossley and Hildebrand, 1972], irresolute functions was introduced and analysed by Crossley and Hildebrand in topological spaces. Weak and Strong forms of irresolute functions in topology were discussed by Maio and Noiri [Maio and Noiri, 1988]. The conception of nano-topology was initiated by Lellis Thivagar [Thivagar and Richard, 2013b],[M. Lellis Thivagar and Richard, 2013] and [M. Lellis Thivagar and Devi, 2017]. Also in [Thiva- gar and Richard, 2013a], nano continuous functions, nano interior and nano clo- sure was look over by Lellis and Carmel Richard. Bhuvaneshwari and Ezhila- rasi[Bhuvaneshwari and Ezhilarasi, 2016] introduced irresolute maps and semi- generalized irresolute maps in nano topological spaces. New functions called Ns- irresolute and Np-irresolute functions are originated and look into its behaviour in this article. Further the notions called contra Nα-irresolute function, contra Ns-irresolute function, contra Np-irresolute function were introduced and exam- ined their properties. Throughout this article we use the notation NTS, N-open, Nα-open, Ns-open, Np-open, Nα-continuous, Ns-continuous, Np-continuous for ”Nano Topological spaces, Nano open, Nano α-open, Nano semi-open, Nano Pre- open sets, Nano α-continuous, Nano semi-continuous, Nano pre-continuous” re- spectively. Similar notation is used for their respective closed sets. 2 Nano Irresolute Functions Definition 2.1. Let U1 and U2 be NTS with respect to τR(X) and τR′ (Y). Then h : U1 → U2 is called 1. Ns-irresolute if h−1(S) is Ns-open set in U1 for each Ns-open set S in U2, 2. Np-irresolute if h−1(S) is Np-open set in U1 for each Np-open set S in U2. Example 2.1. Take U1 = {w,x,y,z} with U1/R = {{x,z},{y,w}} and X = {x,z}. Then τR(X) = {U1,φ,{x,z}}. Let U2 = {q,r,s,t} with U2/R′ = {{q},{r,s},{t}} and Y = {q,t}. Then τR′ (Y) = {U2,φ,{q,t}}. We define h : (U1, τR(X)) → (U2, τR′ (Y)) as h(x) = q, h(y) = r, h(z) = t, h(w) = s. Then the inverse image of any Ns-open in U2 is Ns-open in U1 and the inverse image of any Np-open in U2 is Np-open in U1. Therefore h is Ns-irresolute and Np-irresolute. Theorem 2.1. Let U1 and U2 be the NTS with reference to τR(X) and τR′ (Y) and h : U1 → U2 be a mapping. Then the statements given below are equivalent. 1. h is Nα-irresolute. Weaker Forms of Nano Irresolute and Its Contra Functions 2. h−1(S) is Nα-closed in U1, for each Nα-closed set S in U2. 3. h(Nαcl(S)) ⊆ Nαcl(h(S)) for each S ⊆ U1. 4. Nαcl(h−1(S)) ⊆ h−1(Nαcl(S)) for each S ⊆ U2. 5. h−1(Nαint(S)) ⊆ (Nαint(h−1(S)) for each S ⊆ U2. 6. h is Nα-irresolute for each x ∈ U1. Proof. (i) =⇒ (ii). It is obvious. (ii) =⇒ (iii). Let S ⊆ U1. Then, Nαcl(h(S)) is a Nα-closed set of U2. By (ii), h−1(Nαcl(h(S))) is a Nα-closed set in U1 and Nαcl(S) ⊆ Nαcl(h−1h(S)) ⊆ Nαcl(h−1(Nαcl((h(S)))) = h−1(Nαcl(h(S))). So h(Nαcl(h(S)) ⊆ Nαcl(h(S)). (iii) =⇒ (iv). Let S be a subset of U2. By (iii) h(Nαcl(h−1(S))) ⊆ Nαcl(hh−1(S)) ⊆ Nαcl(S). So Nαcl(h−1(S)) ⊆ h−1h(Nαcl(h−1(S))) ⊆ h−1(Nαcl(S)). (iv) =⇒ (v). Let S be a subset of U2. By (iv), h−1(Nαcl(U2-S)) ⊇ Nαcl(h−1(U2− S)) = Nαcl(U1−h−1(S)). Since U1−Nαcl(U1−S) = Nαint(S), then h−1(Nαint(S)) = h−1(U2−Nαcl(U2−S)) = U1−h−1(Nαcl(U2−S)) ⊆ U1−Nαcl(U1−h−1(S)) = Nαint(h−1(S)). (v) =⇒ (vi). Let S be a Nα-open set of U2, then S = Nαint(S). By (v), h−1(S)= h−1(Nαint(S)) ⊆ Nαint(h−1(S)) ⊆ h−1(S). So, h−1(S) = Nαint(h−1(S)). Thus, h−1(S) is a Nα-open set of U. Therefore h is Nα-irresolute. (i) =⇒ (vi). Let h be Nα-irresolute, x ∈ U1 and any Nα-open set S of U2, such that h(x) ⊆ S. Then x ∈ h−1(S) = Nαint(h−1(S)). Let B = h−1(S), then B is a Nα-open set of U1 and so h(B) = hh−1(S) ⊆ S. Thus h is Nα-irresolute for each x ∈ U1. (vi) =⇒ (i). Let S be a Nα-open set of U2, x ∈ h−1(S). Then h(x) ∈ S. By hy- pothesis there exists a Nα-open set B of U1 such that x ∈ B and h(B) ⊆ S. Hence x ∈ B ⊆h−1(h(B)) ⊆h−1(S) and x ∈ B = Nαint(B) ⊆ Nαint(h−1(S)). So, h−1(S) ⊆ Nαint(h−1((S)). Hence h−1(S) = Nαint(h−1(S)). Thus h is Nα-irresolute.2 Theorem 2.2. Let U1 and U2 be the NTS with respect to τR(X) and τR′ (Y) and h : U1 → U2 be a 1-1 and onto function. Then h is Nα-irresolute iff Nαint(h(S)) ⊆ h(Nαint(S)) for each S ⊆ of U1. Proof. Let S be any subset of U1. By Theorem 2.1 and since h is 1-1 and onto, h−1(Nαint(h(S))) ⊆ Nαint(h−1(h(S))) = Nαint(S). So, hh−1(Nαint(h(S))) ⊆h(N αint(S)). Thus Nαint(h(S)) ⊆ h(Nαint(S)). Conversely, Let S be a Nα-open set of U2. Then S = Nαint(S). By hypothesis, h(Nαint(h−1(S))) ⊇ Nαint(h(h−1(S))) = Nαint(S) = S. Thus we get h−1h(Nαint (h−1(S))) ⊇h−1(S). Since h is 1-1 and onto, Nαint(h−1(S))=h−1h(Nαint(h−1(S))) ⊇h−1(S). Hence h−1(S) = Nαint(h−1(S)). So h−1(S) is Nα-open set of U. Thus h is Nα-irresolute.2 A. Yuvarani, S. Vijaya, P. Santhi Lemma 2.1. Let U1 be a NTS with respect to τR(X) then 1. Nαcl(S) ⊆ Ncl(S) for every subset S of U1, 2. Ncl(S) = Nαcl(S) for every α-open subset S of U1. Theorem 2.3. Let U1 and U2 be the NTS with respect to τR(X) and τR′ (Y) and h : U1 → U2 be a Nα-irresolute. Then Ncl(h−1(S)) ⊆h−1(Ncl(S)) for every S ⊆ U2. Proof. Let S be any N-open subset of U2. Since h is Nα-irresolute and Nαcl(h−1 (S)) is equal to Ncl(h−1(A)). By Theorem 2.1, Nαcl(h−1(S)) ⊆h−1(Nαcl(S)) and by Lemma 2.1 h−1(Nαcl(S) ⊆ h−1(Ncl(S)). Then Nαcl(h−1(S)) ⊆ h−1(Ncl(S)). Therefore Ncl(h−1(S)) ⊆ h−1(Ncl(S)).2 Theorem 2.4. Let U1 and U2 be the NTS with respect to τR(X) and τR′ (Y).Then h : U1 → U2 is a Ns- irresolute iff for each Ns-closed subset h−1(S) is Ns-closed in U1. Proof. If h is Ns-irresolute, then h−1(B) is Ns-open in U1 for each Ns-open set B ⊆ U2. If S is any Ns-closed subset of U2, then U2−S is Ns-open. Thus h−1(U2−S) is Ns-open in U1, but h−1(U2−S) = U1−h−1(S) so that h−1(S) is Ns-closed in U1. Conversely, if for all Ns-closed set S ⊆ U2, h−1(S) is Ns-closed in U1 and if B is any Ns-open subset of U2, then U2−B is Ns-closed. Also h−1(U2−B) = U1−h−1(B) which is Ns-closed in U1. Therefore h−1(B) is Ns-open set in U1. Hence h is Ns-irresolute.2 Theorem 2.5. If h : U1 → U2 and g : U2 → U3 is Ns-irresolute(Np-irresolute) then g◦h : U1 → U3 is Ns-irresolute(Np-irresolute). Proof. (i) If A ⊆ U3 is Ns-open(Np-open), then g−1(S) is Ns-open(Np-open) set in U2 because g is Ns-irresolute(Np-irresolute). Consequently since h is Ns- irresolute(Np-irresolute), h−1(g−1(S))= (g◦h)−1(S) is Ns-open(Np-open) set in U1. Hence g◦h is Ns-irresolute(Np-irresolute). Theorem 2.6. If h : U1 → U2 is Nα-irresolute(Ns-irresolute, Np-irresolute) and g : U2 → U3 is Nα-continuous(Ns-continuous, Np-continuous) then g◦h : U1 → U3 is Nα-continuous(Ns-continuous, Np-continuous). Proof. Let S ⊆ U3 is N-open. Since g is Nα-continuous(Ns-continuous, Np- continuous), g−1(S) is Nα-open(Ns-open, Np-open)set in U2. Consequently since h is Nα-irresolute(Ns-irresolute, Np-irresolute), h−1(g−1(S)) = (g◦h)−1(S) is Nα- open(Ns-open, Np-open) set in U1. Hence g◦h is Nα-continuous(Ns-continuous, Np-continuous).2 Weaker Forms of Nano Irresolute and Its Contra Functions Theorem 2.7. Let U1 and U2 be the NTS with respect to τR(X) and τR′ (Y). A function h : U1 → U2 is 1. Ns-irresolute and Np-irresolute then h is Nα-irresolute, 2. Nα-continuous iff it is Ns-continuous and Np-continuous. Proof. It is obvious. 3 Nano Contra Irresolute Functions Here we introduce contra irresolute functions and its characteristics are dis- cussed. The notations used are NCα-open, NCs-open, NCp-open for ”Nano con- tra α-open, Nano contra semi-open, Nano contra pre-open functions” respectively. Definition 3.1. Let U1 and U2 be the NTS with respect to τR(X) and τR′ (Y). Then h : U1 → U2 is said to be 1. NCα-open if h(S) is Nα-closed in U2 for each N-open S in U1, 2. NCs-open if h(S) is Ns-closed in U2 for each N-open S in U1, 3. NCp-open if h(S) is Np-closed in U2 for each N-open S in U1. Example 3.1. 1. Let U1 = {j,k,l} with U1/R = {{l},{j,k}} and X = {k,l}. Then τR(X) = {U1,φ,{l},{j,k}}. Let U2 = {x,y,z}, U2/R’ = {{y},{x,z}} and Y = {y,z}. Subsequently τR′ (Y) = {VU2,φ,{y},{x,z}}. We label h : (U1, τR(X)) → (U2, τR′ (Y)) as h(j) = x, h(k) = z, h(l) = y. Subsequently h(S) is Nα-closed in U2 for every N-open set S in U1. Hence h is NCα-open. 2. Let U1 = {j,k,l,m} with U1/R = {{j},{l},{k,m}} and X = {j,k}. Subse- quently τR(X) = {U1,φ,{j},{k,m},{j,k,m}}. Let U2 = {p,q,r,s} with U2/R’ = {{p},{s},{q,r}} and Y = {p,r}. Subsequently τR′ (Y) = {U2,φ,{p},{q,r},{p, q,r}}. We label h : (U1, τR(X)) → (U2, τR′ (Y)) as h(j) = s, h(k) = r, h(l) = p, h(m) = q. Then h(S) is Ns-closed in U2 for every N-open set S in U1. Hence h is NCs-open. 3. Let U1 = {j,k,l,m} with U1/R = {{l},{m},{j,k}} and X = {j,l}. Subse- quently τR(X) = {U1,φ,{l},{j,k},{j,k,l}}. Let U2 = {p,q,r,s} with U2/R’ = {{q},{r},{p,s}} and Y = {p,r}. Subsequently τR′ (Y) = {U2,φ,{r},{p,s},{p,r, s}}. We define h : (U1, τR(X)) → (U2, τR′ (Y)) as h(j) = q, h(k) = s, h(l) = p, h(m) = r. Then h(S) is Np-closed in U2 for every N-open set S in U1. Hence h is NCp-open. A. Yuvarani, S. Vijaya, P. Santhi Definition 3.2. Let U1 and U2 be the NTS with respect to τR(X) and τR′ (Y). Then h : U1 → U2 is said to be CNα-irresolute(CNs-irresolute, CNp-irresolute) if h−1(S) is Nα-closed(Ns-closed, Np-closed)set in U1 for every Nα-open set(Ns- open, Np-open) in U2 . Example 3.2. 1. Let U1 = {j,k,l,m} with U1/R = {{j},{k},{l},{m}} and X = {j}. Then τR(X) = {U1,φ,{j}}. Let U2 = {w,x,y,z} with U2/R’={{w},{x},{y },{z}} and Y = {x,y,z}. Then τR′ (Y) = {V,φ,{x,y,z}}. We label h : (U1, τR(X)) → (U2, τR′ (Y)) as h(j) = w, h(k) = x, h(l) = y, h(m) = z. Then h−1(S) is Ns-closed in U1 for every Ns-open set S in U2 . Therefore h is CNα-irresolute and CNs-irresolute. 2. Let U1 = {p,q,r} with U1/R = {{p},{q,r}}and X={q,r}. Then τR(X) = {U1,φ,{q,r}}. Let U2 = {j,k,l} with U2/R’ = {{j},{k,l}} and Y = {j}. Then τR′ (Y) = {U2,φ,{j}}. We define h : (U1,τR(X)) → (U2, τR′ (Y)) as h(p) = j, h(q) = k, h(r) = l. Then h−1(S) is Np-closed in U1 for every Np-open set S in U2 . So h is CNp-irresolute. Theorem 3.1. Consider U1 and U2 be the NTS with respect to τR(X) and τR′ (Y). Then h : U1 → U2 is CNα-irresolute iff for each Nα-closed subset S of U2, h−1(S) is Nα-open in U1. Proof. If h is CNα-irresolute, then for each Nα-open subset B in U2, h−1(B) is Nα-closed in U1. If S is any Nα-closed subset in U2, then U2− S is Nα-open. Thus h−1(U2− S) is Nα-closed but h−1(U2− S) = U1 − h−1(S) so that h−1(S) is Nα-open in U1. Conversely, if, for all Nα-closed subsets S of U2, h−1(S) is Nα-open in U1 and if B is any Nα-open subset of U2, then U2 − B is Nα-closed. Also h−1(U2− B) = U1 − h−1(B) is Nα-open. Thus h−1(B) is Nα-closed in U1. Hence h is CNα-irresolute.2 Corolary 3.1. Let U1 and U2 be the NTS with respect to τR(X) and τR′ (Y). Then h : U1 → U2 is CNs-irresolute(CNp-irresolute) if and only if for each Ns-closed subset(Np-closed subset) S of U2, h−1(S) is Ns-open(Np-open) in U1. Theorem 3.2. If the functions h : U1 → U2 and g : U2 → U3 are CNα-irresolute then g◦h : U1 → U3 is Nα-irresolute. Proof. If S ⊆ U3 is Nα-open, then g−1(S) is Nα-closed in U2 because g is CNα- irresolute. Consequently since h is CNα-irresolute, h−1(g−1(S))= (g◦h)−1(S) is Nα-open set in U1, by corollary 4.6. Hence g◦h is Nα-irresolute.2 Corolary 3.2. If the functions h : U1 → U2 and g : U2 → U3 are CNs-irresolute (CNp-irresolute) then g◦h : U1 → U3 is Ns-irresolute(Np-irresolute). Weaker Forms of Nano Irresolute and Its Contra Functions Theorem 3.3. If the function h : U1 → U2 is CNα-irresolute and the function g : U2 → U3 is NCα-continuous then g◦h : U1 → U3 is Nα-continuous. Proof. Let S ⊆ U3 is N-open. Since g is NCα-continuous, g−1(S) is Nα-closed in U2. Consequently since h is CNα-irresolute, h−1(g−1(S))= (g◦h)−1(S) is Nα-open set in U1, by theorem 4.5. Hence g◦h is Nα-continuous.2 Corolary 3.3. If the function h : U1 → U2 is CNs-irresolute(CNp-irresolute) and the function g : U2 → U3 is NCs-continuous(NCp-continuous) then g◦h : U1 → U3 is Ns-continuous(Np-continuous). Theorem 3.4. Let U1 and U2 be the NTS with respect to τR(X) and τR′ (Y). Then h : U1 → U2 is CNs-irresolute and CNp-irresolute then h is CNα-irresolute. Proof. It is obvious. Theorem 3.5. Let U1 and U2 be the NTS with respect to τR(X) and τR′ (Y). Then h : U1 → U2 is CNα-irresolute then it is NCα-continuous. Proof. Consider the N-open set T ⊆ U2. Which implies T is a Nα-open set in U2. But h is CNα-irresolute So h−1(T) is a Nα-closed set in U1. It shows that h is NCα-continuous function.2 4 Applications Finally, we discuss the application of nano irresolute functions and its contra functions. Example 4.1. Advances in technology and some pandemic situations allow stu- dents to study entirely online. Consider the impact of e-learning on students char- acteristics, as a function of, the innovative strategies used in online teaching. Let us consider some of the strategies used in online teaching are powerpoint presentation (P), videos (V), mind map (M), Online Quiz (Q), Group discussion (G) and its impact on students characteristics are Intellectually curious (I), Good time management (T), Self-driven (S), Enhanced Communication skills (C). Let U1={P,V,M,Q,G} be the universe of the innovative strategies used in online teach- ing with U1/R ={{P,V},{M,G}, {Q}} and X1 ={P,Q}. Subsequently τR(X1)={U1,φ,{Q},{P,V},{P,V,Q}}. Let U2 ={I,T,S,C} be the universe on students characteristics with U2/R’ = {{I,S},{T,C}} and X2 = {T,C}. Then τR′ (X2) = {U2, φ, {T,C}}. We define h : (U1, τR(X1)) → (U2, τR′ (X2)) as h(P) = C, h(V) = C, h(M) = I, h(Q) = T and h(G) = S. Then for every Nα-open set in U2, inverse image is Nα-open set in U1 and also for every Ns-open set in U2, inverse image is Ns-open set in U1. Hence h is Nα-irresolute A. Yuvarani, S. Vijaya, P. Santhi and Ns-irresolute. Thus, the impact of e-learning on students characteristics, as a function of the innovative strategies used in online teaching, are Nα-irresolute and Ns-irresolute function. Example 4.2. The main cause of illness is the infectious diseases. However, some initial precautions may help to prevent infections. If not, it leads to serious med- ical conditions and sometimes to death. Consider the precautionary measures to be adopted to prevent affecting from COVID-19, as a function of, its symp- toms. Let the symptoms of COVID-19 are Dry cough (K), Fever (F), Shortness of Breath (B), Loss of Taste/Smell (L) and the precautionary measures to be adopted are Sanitizing (S), Social distancing (D), Wearing mask (M), Boosting Immunity power (I). Let U1 = {K,F,B,L} be the universe of symptoms of COVID-19 with U1/R = {{K},{F},{B},{L}} and X1 = {K}. Then τR(X1) = {U1, φ, {K}}. Let U2 = {S,D,M,I} be the universe of the precautionary measures to be adopted with U2/R’ = {{S},{D},{M},{I}} and X2 = {D,M,I}. Then τR′ (X2) = {U2, φ, {D,M,I}}. We define h : (U1, τR(X1)) → (U2, τR′ (X2)) as h(K) = S, h(F) = D, h(B) = M and h(L) = I. Then for every Nα-open set in U2, inverse image is Nα-closed set in U1 and also for every Ns-open set in U2, inverse image is Ns-closed set in U1. Thus h is contra Nα-irresolute and contra Ns-irresolute. Thus, the precautionary measures to be adopted to prevent affecting from COVID- 19, as a function of its symptoms, are contra Nα-irresolute and contra Ns-irresolute function. 5 Conclusions Through the above discussions we have summarized the conceptulation of irresolute functions and contra irresolute functions in NTS along with examples. 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