Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 42 https://internationalpubls.com Open Maps via 𝜹-open Sets in Pythagorean Fuzzy Topological Spaces and its Applications A. Vadivel 𝟏 , G. Gavaskar 𝟐, G. Saravanakumar 𝟑 1PG and Research Department of Mathematics, Arignar Anna Government Arts College, Namakkal - 637 002, India. 1,2Department of Mathematics, Annamalai University, Annamalai Nagar - 608 002, India. 3Department of Mathematics, Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology (Deemed to be University), Avadi, Chennai-600062, India E-mail: 1avmaths@gmail.com,2gurugavaskar001@gmail.com, 3saravananguru2612@gmail.com Article History: Received: 12-09-2024 Revised: 17-11-2024 Accepted: 27-11-2024 Abstract: In this paper, we introduce the concept of Pythagorean fuzzy 𝛿 (resp. 𝛿𝛼, 𝛿𝒮, 𝛿𝒫 & 𝛿𝛽 or 𝑒∗)-open mappings are introduced and their properties are discussed. In current scenario people with symptom of Covid-19 like fever, cough, sneezing, sore throat, loss of taste and smell etc., were panic about the disease, and the diagnosis of Covid-19 takes many hours and people cannot go for the test frequently. Some other diseases like flu, pneumonia, cold etc., also has the same symptoms. Each patients has unique experience of that particular symptom and some time they may not experience that symptom even though they were affected by the Covid-19. Also, in this paper we tried to diagnosis Covid-19 with the help of picture fuzzy sets which helps to record all symptoms in précised manner. Introduction: This paper introduces the concept of Pythagorean fuzzy open mappings and their relevance to COVID-19 symptom diagnosis. Due to the overlapping symptoms of COVID-19 with other illnesses, a Pythagorean fuzzy set approach is utilized to more accurately capture and analyze symptom patterns. Objectives: The objective is to develop a precise mapping model that utilizes Pythagorean fuzzy sets to differentiate between COVID-19 symptoms and other similar conditions, aiming to improve diagnostic speed and accuracy. Methods: The method involves formulating fuzzy sets based on COVID-19 symptoms, computing distances (Hamming, Euclidean) between patients' symptoms and ideal symptom patterns, and identifying cases with higher COVID-19 risk. Results: The approach successfully identifies patients with symptoms closest to COVID- 19 indicators, offering a more systematic and timely diagnosis. Results suggest that this fuzzy set model can enhance early detection and intervention for affected individuals Conclusions: In this paper, Some new notions of strongly Pythagorean fuzzy open (closed) maps called Pythagorean fuzzy 𝛿-open and Pythagorean fuzzy 𝛿-closed maps are introduced and discussed their relationship between their near mappings with examples. Also, we have tried to diagnosis Covid-19 with the help of Pythagorean fuzzy sets which helps to record all symptoms in précised manner. In future, researchers can extend this model to other extensions of fuzzy sets such as rough sets and utilize the interdependency among the various evaluation criteria for better judgement. Keywords: Pythagorean fuzzy 𝛿-open mappings, distance between fuzzy sets, Covid-19 patient- record all symptoms in precise manner. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 43 https://internationalpubls.com 1. Introduction Traditional logic, which is interpreted as either true or false, found to be difficult to solve uncertain real-life problems. As a counter measure, Zadeh (1965) [35] invented fuzzy set theory, where the involvement of elements in a set is characterized by membership grade, which belongs to [0,1]. To handle much uncertainty, fuzzy sets were extended by the different researchers in different ways such as vague set (Gau and Buehrer 1993) [15], intuitionistic fuzzy set (IFS) (Atanassov 1986a, 1986b) [1, 2], fuzzy soft set (Das et al. 2018) [12], rough set (Pawlak 1982) [25], fuzzy interval theory (Gorzalczany 1987) [18], intuitionistic multi fuzzy set (Das et al. 2013) [11], interval-valued intuitionistic fuzzy set (Park et al. 2008) [26] , intuitionistic fuzzy soft set (Deng 1982) [14] and neutrosophic soft set (Das et al. 2019) [13] . Consequently, the application of fuzzy set theory and its extensions increased rapidly in the decision-making methods in various domains like medical diagnosis (Das et al. 2013) [11], pattern recognition (Wei and Lan 2008) [30], data analysis (Zou and Xiao 2008) [36], forecasting (Xiao et al. 2011) [31], optimization (Kov-kov et al. 2007) [20], simulation (Kalayathankal and Singh 2010) [19] and texture classification (Mushrif et al. 2006) [23]. Recently in 2014, Cuong (2014)[10] developed the picture fuzzy set (PFS) as the generalized form of fuzzy set and IFS. The PFS approaches are found to be more appropriate in those cases when the views of someone contain more option types like yes, abstain, no and refusal. The general election of a country is noted as a good example to describe PFS, where a voter can cast his vote in favour of the candidate (yes), against the candidate (no), may not cast his vote (abstain) or may refuse to cast his vote in favour of the given candidates and prefer for nota (refusal) (Cong and Son 2015) [9]. Nowadays, the whole world has become fully unbalanced and passing through an uncontrolled situation due to the dangerous and novel virus Covid-19. Most countries are totally stagnant and the people are quarantined to make themselves safe from Covid-19 (Ren et al. 2020) [27]. Many researchers are continuously contributing to developing various type of mathematical and hybrid models to predict the future trends, strength and transmission capability of Covid-19 virus, and have drawn some useful conclusions which assist the health department to take the necessary precaution to track and handle the Covid-19 situations. The authors in Melin et al. (2020) [22] introduced a novel hybrid prediction model that can mergethe ensemble architectures of fuzzy logic-based neural networks for response integration. The fundamental concept of the proposed model is to merge several fuzzy-based neural network predictors, control the uncertainty of the individual networks and try to reduce the uncertainty of the total predictions. This model was able to predict the future trends of Covid-19 up to some extent and help the authorities make the necessary decision to handle the health care system in a better manner. The authors in Sun and Wang (2020) [28] collected the Covid-19 data from a decided location within a specific time interval and trained through the ordinary differential equation model for fitting.Then, they modified the simulation by the trained model to realize the effect of the Covid-19 affected visitors. They found that the affected visitors have a great role in the newly introduced case of Covid-19. Stochastic simulations proved that the physical connections could be rapidly increased due to the affected visitors which are considered sufficient for the local outbreak of Covid-19. The confirmed case of asymptomatic patients was significantly less than the model predictions quantity. This indicated that a major portion of asymptomatic patients are not identified/found. Fuzzy-based hybrid approaches for forecasting the confirmed cases and deaths of the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 44 https://internationalpubls.com countries according to their time series are given in Castillo and Melin (2020) [6]. The fundamental concept of this proposed hybrid method (Castillo and Melin 2020) [6] is to combine the fractal dimension and fuzzy logic for enabling efficient and accurate forecasting of Covid-19 time series. The fractal dimension is provided to differentiate and categorize the object. They introduced a fuzzy rule- based system to represent the knowledge about the forecasting time series of the countries. The authors in Castillo and Melin (2021) [7] introduced the hybrid procedure for composing the fuzzy logic and fractal dimension which measured the uncommon activities of times series to classify countries according to their Covid-19 time series data. The proposed method generates an accurate classification of countries based on the complexity of the Covid-19 time series data. Editors (Boccaletti et al. 2020) [5] of the journal ‘‘Chaos, Solitons and Fractals’’ analysed the impact of Covid-19 pandemic throughout the world and felt the necessity to create a unique platform for the researchers to help the society to avoid the worst effects of future pandemics. Recently, Mishra et al. (2021) [27] proposed an extended fuzzy decision-making framework using hesitant fuzzy sets for the drug selection to treat the mild symptoms of Covid-19. Although the researchers are working hard, they are still struggling to recover from this unwanted situation. The scientists from different domains are consistently trying to apply their knowledge in different perspectives such as dominating the virus, identifying the virus, isolating from the virus, protecting from the virus, and finding the treatment of the virus affected patients, to manage the superfluous situation (Kumar et al., 2020 [21], Ghosh et al. , 2020) [17], which are considered to be the long term project. As an intermediate solution, the most important aspect is to provide suitable medical service to the affected patients and recover those who are critically ill due to perilous virus Covid-19. The health department of India has classified the Covid- 19 affected patients into some categories according to the patients physical condition. The extreme condition is called severe cases, and this type of patient requires quality treatment (Clinical Management Protocol 2020) [8]. In current scenario people with symptom of Covid-19 like fever, cough, sneezing, sore throat, loss of taste and smell etc., were panic about the disease, and the diagnosis of Covid-19 takes many hours and people cannot go for the test frequently. Some other diseases like flu, pneumonia, cold etc., also has the same symptoms. Each patients has unique experience of that particular symptom and some time they may not experience that symptom even though they were affected by the Covid-19. To fill up this research gap, this paper proposes Pythagorean fuzzy 𝛿 (resp. 𝛿𝛼, 𝛿𝒮, 𝛿𝒫 & 𝛿𝛽 or 𝑒∗)- open, closed mappings and an alternative Pythagorean fuzzy set based approach, here we tried to diagnosis Covid-19 with the help of Pythagorean fuzzy sets which helps to record all symptoms in preccised manner. 2 Preliminaries We recall some basic notions of fuzzy sets, 𝐼𝐹𝑆’s and 𝑝𝑓𝑠’s . Definition 2.1 [35] Let 𝑋 be a nonempty set. A fuzzy set 𝐴 in 𝑋 is characterized by a membership function 𝜇𝐴: 𝑋 → [0,1]. That is: 𝜇𝐴(𝑥) = { 1, if 𝑥 ∈ 𝑋 0, if 𝑥 ∉ 𝑋 (0,1) if 𝑥 ispartlyin 𝑋. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 45 https://internationalpubls.com Alternatively, a fuzzy set 𝐴 in 𝑋 is an object having the form 𝐴 = {< 𝑥, 𝜇𝐴(𝑥) > |𝑥 ∈ 𝑋} or 𝐴 = {⟨ 𝜇𝐴(𝑥) 𝑥 ⟩ |𝑥 ∈ 𝑋}, where the function 𝜇𝐴(𝑥): 𝑋 → [0,1] defines the degree of membership of the element, 𝑥 ∈ 𝑋. The closer the membership value 𝜇𝐴(𝑥) to 1, the more 𝑥 belongs to 𝐴, where the grades 1 and 0 represent full membership and full nonmembership. Fuzzy set is a collection of objects with graded membership, that is, having degree of membership. Fuzzy set is an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in a binary terms according to a bivalent condition; an element either belongs or does not belong to the set. Classical bivalent sets are in fuzzy set theory called crisp sets. Fuzzy sets are generalized classical sets, since the indicator function of classical sets is special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. Fuzzy sets theory permits the gradual assessment of the membership of element in a set; this is described with the aid of a membership function valued in the real unit interval [0,1]. Let us consider two examples: (i) all employees of 𝑋𝑌𝑍 who are over 1.8𝑚 in height; (ii) all employees of 𝑋𝑌𝑍 who are tall. The first example is a classical set with a universe (all 𝑋𝑌𝑍 employees) and a membership rule that divides the universe into members (those over 1.8𝑚) and nonmembers. The second example is a fuzzy set, because some employees are definitely in the set and some are definitely not in the set, but some are borderline. This distinction between the ins, the outs, and the borderline is made more exact by the membership function, 𝜇. If we return to our second example and let 𝐴 represent the fuzzy set of all tall employees and 𝑥 represent a member of the universe 𝑋 (i.e. all employees), then 𝜇𝐴(𝑥) would be 𝜇𝐴(𝑥) = 1 if 𝑥 is definitely tall or 𝜇𝐴(𝑥) = 0 if 𝑥 is definitely not tall or 0 < 𝜇𝐴(𝑥) < 1 for borderline cases. Definition 2.2 [1, 2, 3, 4] Let a nonempty set X be fixed. An IFS A in X is an object having the form: A = {< x, μ A (x), νA(x) > |x ∈ X} or A = {⟨ μA(x),νA(x) x ⟩ |x ∈ X}, where the functions μ A (x): X → [0,1] and νA(x): X → [0,1] define the degree of membership and the degree of nonmembership, respectively, of the element x ∈ X to A, which is a subset of X, and for every x ∈ X: 0 ≤ μ A (x) + νA(x) ≤ 1. For each A in X: πA(x) = 1 − μ A (x) − νA(x) is the intuitionistic fuzzy set index or hesitation margin of x in X. The hesitation margin πA(x) is the degree of nondeterminacy of x ∈ X to the set A and πA(x) ∈ [0,1]. The hesitation margin is the function that expresses lack of knowledge of whether x ∈ X or x ∉ X. Thus: μ A (x) + νA(x) + πA(x) = 1. Example 2.1 Let X = {x, y, z} be a fixed universe of discourse and A = {⟨ 0.6,0.1 x ⟩ , ⟨ 0.8,0.1 y ⟩ , ⟨ 0.5,0.3 z ⟩}, be the intuitionistic fuzzy set in X. The hesitation margins of the elements x, y, z to A are as follows: πA(x) = 0.3, πA(y) = 0.1 and πA(z) = 0.2. Definition 2.3 [32, 33, 34] Let X be a universal set. Then, a Pythagorean fuzzy set A, which is a set of ordered pairs over X, is defined by the following: A = {< x, μ A (x), νA(x)|x ∈ X} or A = {⟨ μA(x),νA(x) x ⟩ |x ∈ X}, where the functions μ A (x): X → [0,1] and νA(x): X → [0,1] define the degree of membership and the degree of nonmembership, respectively, of the element x ∈ X to A, which is a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 46 https://internationalpubls.com subset of X, and for every x ∈ X, 0 ≤ (μ A (x))2 + (νA(x))2 ≤ 1. Supposing (μ A (x))2 + (νA(x))2 ≤ 1, then there is a degree of indeterminacy of x ∈ X to A defined by πA(x) = √1 − [(μ A (x))2 + (νA(x))2] and πA(x) ∈ [0,1]. In what follows, (μ A (x))2 + (νA(x))2 + (πA(x))2 = 1. Otherwise, πA(x) = 0 whenever (μ A (x))2 + (νA(x))2 = 1. We denote the set of all PFS’s over X by pfs(X). Definition 2.4 [34] Let A and B be pfs’s of the forms A = {< a, λA(a), μ A (a) > |a ∈ X} and B = {< a, λB(a), μ B (a) > |a ∈ X}. Then 1. A ⊆ B if and only if λA(a) ≤ λB(a) and μ A (a) ≥ μ B (a) for all a ∈ X. 2. A = B if and only if A ⊆ B and B ⊆ A. 3. A̅ = {< a, μ A (a), λA(a) > |a ∈ X}. 4. A ∩ B = {< a, λA(a) ∧ λB(a), μ A (a) ∨ μ B (a) > |a ∈ X}. 5. A ∪ B = {< a, λA(a) ∨ λB(a), μ A (a) ∧ μ B (a) > |a ∈ X}. 6. ϕ = {< a, ϕ, X > |a ∈ X} and X = {< a, X, ϕ > |a ∈ X}. 7. X̅ = ϕ and ϕ̅ = X. Definition 2.5 [24] An Pythagorean fuzzy topology by subsets of a non-empty set X is a family τ of pfs’s satisfying the following axioms. [(i)] 1. ϕ, X ∈ τ. 2. G1 ∩ G2 ∈ τ for every G1, G2 ∈ τ and 3. ⋃ Gi ∈ τ for any arbitrary family {Gi|i ∈ j} ⊆ τ. The pair (X, τ) is called an Pythagorean fuzzy topological space (pfts in short) and any pfs G in τ is called an Pythagorean fuzzy open set (pfos in short) in X. The complement A̅ of an Pythagorean fuzzy open set A in an pfts(X, τ) is called an Pythagorean fuzzy closed set (pfcs in short). Definition 2.6 [24] Let (X, τ) be an pfts and A = {< a, λA(a), μ A (a) > |a ∈ X} be an pfs in X. Then the interior and the closure of A are denoted by pfint(A) and pfcl(A) and are defined as follows: pfcl(A) =∩ {K|K isan pfcs and A ⊆ K} and pfint(A) =∪ {G|G isan pfos and G ⊆ A}. Also, it can be established that pfcl(A) is an pfcs and pfint(A) is an pfos, A is an pfcs if and only if pfcl(A) = A and A is an pfos if and only if pfint(A) = A. We say that A is pf-dense if pfcl(A) = X. Lemma 2.1 [29] For any Pythagorean fuzzy set A in (X, τ), we have X − pfint(A) = pfcl(X − A) and X − pfcl(A) = pfint(X − A). Definition 2.7 [29] Let (X, τ) be an pfts and A be an pfs. Then A is said to be an Pythagorean fuzzy (i) regular open set (pfros in short) if A = pfint(pfcl(A)). (ii) regular closed set (pfrcs in short) if A = pfcl(pfint(A)). By Lemma 2.1, it follows that A is an pfros iff A̅ is an pfrcs. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 47 https://internationalpubls.com Definition 2.8 [16] Let (X1, ΓP) & (X2, ΨP) be a pfts’s. A mapping hP: (X1, ΓP) → (X2, ΨP) is said to be a Pythagorean fuzzy continuous (briefly, pfCts ) if the inverse image of every pfos in (X2, ΨP) is a pfos. 3 Pythagorean fuzzy 𝜹-open mapping Definition 3.1 Let (X, τ) be an pfts and A = {< a, λA(a), μ A (a) > |a ∈ X} be an pfs in X. Then the δ-interior and the δ-closure of A are denoted by pfδint(A) and pfδcl(A) and are defined as follows. pfδcl(A) =∩ {K|K is an pfrcs and A ⊆ K}, (pfδint(A) =∪ {G|G is an pfros and G ⊆ A}. Definition 3.2 Let (X, τ) be an pfts and A = {< a, λA(a), μ A (a) > |a ∈ X} be an pfs in X. A set A is said to be pf 1. δ-open set (briefly, pfδos) if A = pfδint(A), 2. δ-pre open set (briefly, pfδ𝒫os) if A ⊆ pfint(pfδcl(A)). 3. δ-semi open set (briefly, pfδ𝒮os) if A ⊆ pfcl(pfδint(A)). 4. δ-α open set or a-open set (briefly, pfδαos or pfaos) if A ⊆ pfint(pfcl(pfδint(A))). 5. δ-β open set or e∗-open set (briefly, pfδβos or pfe∗os) if A ⊆ pfcl(pfint(pfδcl(A))). 6. δ (resp. δ-pre, δ-semi, δ-α and δ-β) dense if pfδcl(A) (resp. pfδpcl(A), pfδ𝒮cl(A), pfδαcl(A) and pfδβcl(A)) = X. The complement of an pfδos (resp. pfδ𝒫os, pfδ𝒮os, pfδαos and pfδβos) is called an pfδ (resp. pfδ𝒫, pfδ𝒮, pfδα and pfδβ) closed set (briefly, pfδcs (resp. pfδ𝒫cs, pfδ𝒮cs, pfδαcs and pfδβcs in X. The family of all pfδos (resp. pfδcs, pfδ𝒫os, pfδ𝒫cs, pfδ𝒮os, pfδ𝒮cs, pfδαos, pfδαcs, pfδβos and pfδβcs) of X is denoted by pfδOS(X), (resp. pfδCS(X), pfδ𝒫OS(X), pfδ𝒫CS(X), pfδ𝒮OS(X), pfδ𝒮CS(X), pfδαOS(X), pfδαCS(X), pfδβOS(X) and pfδβCS(X)). Definition 3.3 Let (X, τ) be an pfts and A = {< a, λA(a), μ A (a) > |a ∈ X} be an pfs in X. Then the pfδ-pre (resp. pfδ-semi, pfδα and pfδβ)-interior and the pfδ-pre (resp. pfδ-semi, pfδα and pfδβ)-closure of A are denoted by pfδ𝒫int(A) (resp. pfδ𝒮int(A), pfδαint(A) and pfδβint(A)) and the pfδ𝒫cl(A) (resp. pfδ𝒮cl(A), pfδαcl(A) and pfδβcl(A) and are defined as follows: pfδ𝒫int(A) (resp. pfδ𝒮int(A), pfδαint(A) and pfδβint(A) =∪ {G|G in a pfδ𝒫os (resp. pfδ𝒮os, pfδαos and pfδβos) and G ⊆ A} and pfδ𝒫cl(A) (resp. pfδ𝒮cl(A), pfδαcl(A) and pfδβcl(A) =∩ {K|K is an pfδ𝒫cs (resp. pfδ𝒮cs, pfδαcs, pfδβcs) and A ⊆ K}. Definition 3.4 Let (X1, ΓP) & (X2, ΨP) be a pfts’s. A mapping hP: (X1, ΓP) → (X2, ΨP) is said to be a Pythagorean fuzzy δ (resp. δα, δ𝒮, δ𝒫 & δβ or e∗)-continuous (briefly, pfδCts (resp. pfδαCts, pfδ𝒮Cts, pfδ𝒫Cts & pfδβCts or pfe∗Cts)) if the inverse image of every pfos in (X2, ΨP) is a pfδos (resp. pfδαos, pfδ𝒮os, pfδ𝒫os & pfδβos or pfe∗os) in (X1, ΓP). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 48 https://internationalpubls.com Definition 3.5 Let (X1, ΓP) & (X2, ΨP) be a pfts’s. A mapping hP: (X1, ΓP) → (X2, ΨP) is said to be a Pythagorean fuzzy (resp. δ, δα, δ𝒮, δ𝒫 & δβ or e∗)-open map (briefly, pfO (resp. pfδO, pfδαO, pfδ𝒮O, pfδ𝒫O & pfδβO or pfe∗O)) if the image of every pfos in (X1, ΓP) is a pfos (resp. pfδos, pfδαos, pfδ𝒮os, pfδ𝒫os & pfδβos or pfe∗os) in (X2, ΨP). Theorem 3.1 Let (X1, ΓP) & (X2, ΨP) be a pfts’s. Let hP: (X1, ΓP) → (X2, ΨP) be a mapping. Then the following statements are hold for pfts, but not conversely. 1. Every pfδO mapping is a pfO mapping. 2. Every pfδO mapping is a pfδ𝒮O mapping. 3. Every pfδO mapping is a pfδ𝒫O mapping. 4. Every pfδ𝒮O mapping is a pfδβO mapping. 5. Every pfδ𝒫O mapping is a pfδβO mapping. 6. Every pfδαO mapping is a pfδ𝒮O mapping. 7. Every pfδαO mapping is a pfδ𝒫O mapping. Proof. (i) Let M be a pfos in X1. Since hP is pfδO map, hP(M) is a pfδos in X2. Since every pfδos is a pfos, hP(M) is a pfos in X2. Hence hP is a pfO. (ii) Let M be a pfos in X1. Since hP is pfO map, hP(M) is a pfos in X2. Since every pfos is a pfδ𝒮os, hP(M) is a pfδ𝒮os in X2. Hence hP is a pfδ𝒮O. (iii) Let M be a pfos in X1. Since hP is pfO map, hP(M) is a pfos in X2. Since every pfos is a pfδ𝒫os, hP(M) is a pfδ𝒫os in X2. Hence hP is a pfδ𝒫O. (iv) Let M be a pfos in X1. Since hP is pfδ𝒮O map, hP(M) is a pfδ𝒮os in X2. Since every pfδ𝒮os is a pfδβos, hP(M) is a pfδβos in X2. Hence hP is a pfδβO. (v) Let M be a pfos in X1. Since hP is pfδ𝒫O map, hP(M) is a pfδ𝒫os in X2. Since every pfδ𝒫os is a pfδβos, hP(M) is a pfδβos in X2. Hence hP is a pfδβO. (vi) Let M be a pfos in X1. Since hP is pfδαO map, hP(M) is a pfδαos in X2. Since every pfδαos is a pfδ𝒮os, hP(M) is a pfδ𝒮os in X2. Hence hP is a pfδ𝒮O. (vii) Let M be a pfos in X1. Since hP is pfδαO map, hP(M) is a pfδαos in X2. Since every pfδαos is a pfδ𝒫os, hP(M) is a pfδ𝒫os in X2. Hence hP is a pfδ𝒫O. Remark 3.1 We obtain the following diagram from the results we discussed above. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 49 https://internationalpubls.com Fig. 1 : 𝑝𝑓𝛿𝑂 mappings in 𝑝𝑓𝑡𝑠. Note: 𝐴 → 𝐵 denotes 𝐴 implies 𝐵. But not conversely. Example 3.1 Let X = X1 = X2 = X3 = X4 = X5 = {x1, x2} and the pfs’s A1, A2 and A3 are defined as A1 = {< x1, 0.020,0.040 >, < x2, 0.050,0.050 >} A2 = {< x1, 0.010,0.040 >, < x2, 0.050,0.050 >} A3 = {< x1, 0.020,0.030 >, < x2, 0.050,0.050 >} Here we have τ1 = {0X1 , 1X1 , A1, A2}, τ2 = {0X2 , 1X2 , A2}, τ3 = {0X3 , 1X3 , A1 c}, τ4 = {0X4 , 1X4 , A2 c} and τ5 = {0X5 , 1X5 , A3} be a pfts’s on X. Let h1P: (X2, τ2) → (X1, τ1), h2P: (X3, τ3) → (X1, τ1), h3P: (X4, τ4) → (X1, τ1), h4P: (X5, τ5) → (X1, τ1) be an identity mapping. Then 1. h1P is pfO (resp. pfδβO and pfδ𝒫O) but not pfδO (resp. pfδ𝒮O and pfδαO), because the set A2 is a pfos in X2 but h1P(A2) = A2 is not pfδos (resp. pfδ𝒮os and pfδαos) in X1. 2. h2P is pfδ𝒮O but not pfδO, because the set A1 c is a pfos X3 but h2P(A1 c) = A1 c is not pfδos in X1. 3. h3P is pfδ𝒫O but not pfδO, because the set A2 c is a pfos X4 but h3P(A2 c) = A2 c is not pfδ𝒫os in X1. 4. h4P is pfδβO (resp. pfδ𝒮O) but not pfδ𝒫O (resp. pfδαO), because the set A3 is a pfos X5 but h4(A3) = A3 is not pfδ𝒫os (resp. pfδαos) in X1. Theorem 3.2 Let (X1, ΓP) & (X2, ΨP) be any pfts’s. A mapping hP: (X1, ΓP) → (X2, ΨP) is pfδβO iff for every pfs M of (X1, ΓP), hP(pfint(M)) ⊆ pfδβint(hP(M)). Necessity: Let hP be a pfδβO and M be a pfos in (X1, ΓP). Now, pfint(M) ⊆ M implies hP(pfint(M)) ⊆ hP(M). Since hP is a pfδβO, hP(pfint(M)) is pfδβos in (X2, ΨP) such that hP(pfint(M)) ⊆ hP(M) therefore hP(pfint(M)) ⊆ pfδβint(hP(M)). Sufficiency: Assume M is a pfos of (X1, ΓP). Then hP(M) = hP(pfint(M)) ⊆ pfδβint(hP(M)). But pfδβint (hP(M)) ⊆ hP(M). So hP(M) = pfδβint(M) which implies hP(M) is a pfδβos of (X2, ΨP) and hence hP is a pfδβO. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 50 https://internationalpubls.com Theorem 3.3 Let (X1, ΓP) & (X2, ΨP) be any pfts’s. Let hP: (X1, ΓP) → (X2, ΨP) be a mapping. If hP: (X1, ΓP) → (X2, ΨP) is a pfδβO, then pfint(hP −1(M)) ⊆ hP −1(pfδβint(M)) for every pfs M of (X2, ΨP). Proof. Let M be a pfs of (X2, ΨP). Then pfint(hP −1(M)) is a pfos in (X1, ΓP). Since hP is pfδβO, hP(pfint (hP −1(M)) is pfδβo in (X2, ΨP) and hence hP(pfint(hP −1( λ))) ⊆ pfδβint(hP(hP −1(M))) ⊆ pfδβint(M). Thus pfint(hP −1(M)) ⊆ hP −1(pfδβint(M)). Theorem 3.4 Let (X1, ΓP) & (X2, ΨP) be any pfts’s. A mapping hP: (X1, ΓP) → (X2, ΨP) is pfδβO iff for each pfs μ 1 of (X2, ΨP) and for each pfs μ 2 of (X1, ΓP) containing hP −1(μ) there is an pfδβcs ν of (X2, ΨP) such that μ 1 ⊆ μ 2 and hP −1(ν) ⊆ μ 2 . Necessity: Assume hP is a pfδβO. Let μ 1 be the pfcs of (X2, ΨP) and μ 2 is a pfcs of (X1, ΓP) such that hP −1(μ 1 ) ⊆ μ 2 . Then ν = (hP −1(μ 2 c))c is pfδβcs of (X2, ΨP) such that hP −1(ν) ⊆ μ 2 . Sufficiency: Assume ω is a pfos of (X1, ΓP). Then hP −1((hP(ω))c ⊆ ωc and ωc is pfcs in (X1, ΓP). By hypothesis there is a pfδβcs ν of (X2, ΨP) such that (hP(ω))c ⊆ ν and hP −1(ν) ⊆ ωc. Therefore ω ⊆ (hP −1(ν))c. Hence νc ⊆ hP(ω) ⊆ hP((hP −1(ν))c) ⊆ νc which implies hP(ω) = νc. Since νc is pfδβos of (X2, ΨP). Hence hP(ω) is pfδβo in (X2, ΨP) and thus hP is pfδβO. Theorem 3.5 Let (X1, ΓP) & (X2, ΨP) be any pfts’s. A mapping hP: (X1, ΓP) → (X2, ΨP) is pfδβO iff hP −1(pfδβcl(M) ⊆ pfcl(hP −1(M)) for every pfs M of (X2, ΨP). Proof. Necessity: Assume hP is a pfδβO. For any pfs M of (X2, ΨP), hP −1(M) ⊆ pfcl(hP −1(M)). Therefore by Theorem 3.4, there exists a pfδβcs μ in (X2, ΨP) such that λ ⊆ μ and hP −1(μ) ⊆ pfcl(hP −1(M)). Therefore we obtain that hP −1(pfδβcl(M)) ⊆ hP −1(μ) ⊆ pfcl(hP −1(M)). Sufficiency: Assume M is a pfs of (X2, ΨP) and μ is a pfos of (X1, ΓP) containing hP −1(M). Put ζ = cl(M), then M ⊆ ζ and ζ is pfδβc and hP −1(ζ) ⊊ cl(hP −1(M)) ⊆ μ. Then by Theorem 3.4, hP is pfδβO map. Theorem 3.6 Let (X1, ΓP), (X2, ΨP) & (X3, ΦP) be any pfts’s. If hP: (X1, ΓP) → (X2, ΨP) and g P : (X2, ΨP) → (X3, ΦP) are mappings and g P ∘ hP: (X1, ΓP) → (X3, ΦP) is pfδβO. If g P : (X2, ΨP) → (X3, ΦP) is pfδβIrr then hP: (X1, ΓP) → (X2, ΨP) is pfδβO. Proof. Let ν be a pfos in (X1, ΓP). Then g P ∘ hP(ν) is pfδβos of (X3, ΦP) because g P ∘ hP is pfδβO. Since g P is pfδβIrr and g P ∘ hP(ψ) is pfδβos of (X3, ΦP), g P −1(g P ∘ hP(ψ)) = hP(ψ) is pfδβos in (X2, ΨP). Hence hP is pfδβO. Theorem 3.7 Let (X1, ΓP), (X2, ΨP) & (X3, ΦP) be any pfts’s. If hP: (X1, ΓP) → (X2, ΨP) is pfO and g P : (X2, ΨP) → (X3, ΦP) is pfδβO, then g P ∘ hP: (X1, ΓP) → (X3, ΦP) is pfδβO. Proof. Let ψ be a pfos in (X1, ΓP). Then hP(ψ) is a pfos of (X2, ΨP) because hP is a pfO map. Since g P is pfδβO, g P (hP(ψ)) = (g P ∘ h)(ψ) is pfδβos of (X3, ΦP). Hence g P ∘ hP is pfδβO map. Remark 3.2 Theorems 3.2 to 3.7 is also holds for pfO (resp. pfδO, pfδ𝒮O and pfδ𝒫O) mappings. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 51 https://internationalpubls.com 4 Pythagorean fuzzy 𝜹-closed mappings In this section, Pythagorean fuzzy δ-closed mappings are introduced and studied their properties. Definition 4.1 Let (X1, ΓP) & (X2, ΨP) be any two pfts’s. A mapping hP: (X1, ΓP) → (X2, ΨP) is said to be Pythagorean fuzzy (resp. δ, δ𝒮, δ𝒫 and δβ) closed map (briefly, pfC (resp. pfδC, pfδ𝒮C, pfδ𝒫C and pfδβC)) if the image of every pfcs in (X1, ΓP) is a pfcs (resp. pfδcs, pfδ𝒮cs, pfδ𝒫cs and pfδβcs) in (X2, ΨP). Theorem 4.1 Let (X1, ΓP) & (X2, ΨP) be any pfts’s. Let hP: (X1, ΓP) → (X2, ΨP) be a mapping. Then the following statements are hold. 1. Every pfδC map is a pfC map. 2. Every pfC map is a pfδ𝒮C map. 3. Every pfC map is a pfδ𝒫C map. 4. Every pfδ𝒮C map is a pfδβC map. 5. Every pfδ𝒫C map is a pfδβC map. 6. Every pfδαC map is a pfδ𝒮C map. 7. Every pfδαC map is a pfδ𝒫C map. Proof. (i) Let M be a pfcs in X1. Since hP is pfδC map, hP(M) is a pfδcs in X2. Since every pfδcs is a pfcs, hP(M) is a pfcs in X2. Hence hP is a pfC. (ii) Let M be a pfcs in X1. Since hP is pfC map, hP(M) is a pfcs in X2. Since every pfcs is a pfδ𝒮cs, hP(M) is a pfδ𝒮cs in X2. Hence hP is a pfδ𝒮C. (iii) Let M be a pfcs in X1. Since hP is pfC map, hP(M) is a pfcs in X2. Since every pfcs is a pfδ𝒫cs, hP(M) is a pfδ𝒫cs in X2. Hence hP is a pfδ𝒫C. (iv) Let M be a pfcs in X1. Since hP is pfδ𝒮C map, hP(M) is a pfδ𝒮cs in X2. Since every pfδ𝒮cs is a pfδβcs, hP(M) is a pfδβcs in X2. Hence hP is a pfδβC. (v) Let M be a pfcs in X1. Since hP is pfδ𝒫C map, hP(M) is a pfδ𝒫cs in X2. Since every pfδ𝒫cs is a pfδβcs, hP(M) is a pfδβcs in X2. Hence hP is a pfδβC. (vi) Let M be a pfcs in X1. Since hP is pfδαC map, hP(M) is a pfδαcs in X2. Since every pfδαcs is a pfδ𝒮cs, hP(M) is a pfδ𝒮cs in X2. Hence hP is a pfδ𝒮C. (vii) Let M be a pfcs in X1. Since hP is pfδαC map, hP(M) is a pfδαcs in X2. Since every pfδαcs is a pfδ𝒫cs, hP(M) is a pfδ𝒫cs in X2. Hence hP is a pfδ𝒫C. Example 4.1 Let X = X1 = X2 = X3 = X4 = X5 = {x1, x2} and the pfs’s A1, A2 and A3 are defined as A1 = {< x1, 0.020,0.040 >, < x2, 0.050,0.050 >} A2 = {< x1, 0.010,0.040 >, < x2, 0.050,0.050 >} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 52 https://internationalpubls.com A3 = {< x1, 0.020,0.030 >, < x2, 0.050,0.050 >} Here we have τ1 = {0X1 , 1X1 , A1, A2}, τ2 = {0X2 , 1X2 , A2}, τ3 = {0X3 , 1X3 , A1 c}, τ4 = {0X4 , 1X4 , A2 c} and τ5 = {0X5 , 1X5 , A3} be a pfts’s on X. Let h1P: (X2, τ2) → (X1, τ1), h2P: (X3, τ3) → (X1, τ1), h3P: (X4, τ4) → (X1, τ1), h4P: (X5, τ5) → (X1, τ1) be an identity mapping. Then [(i)] 1. h1P is pfC (resp. pfδβC and pfδ𝒫C) but not pfδC (resp. pfδ𝒮C and pfδαC), because the set A2 c is a pfcs in X2 but h1P(A2 c) = A2 c is not pfδcs (resp. pfδ𝒮cs and pfδαcs) in X1. 2. h2P is pfδ𝒮C but not pfδC, because the set A1 is a pfcs X3 but h2P(A1) = A1 is not pfδcs in X1. 3. h3P is pfδ𝒫C but not pfδC, because the set A2 is a pfcs X4 but h3P(A2) = A2 is not pfδ𝒫cs in X1. 4. h4P is pfδβC (resp. pfδ𝒮C) but not pfδ𝒫C (resp. pfδαC), because the set A3 c is a pfcs in X5 but h4(A3 c) = A3 c is not pfδ𝒫cs (resp. pfδαcs) in X1. Fig. 2: 𝑝𝑓𝛿𝐶 mappings in 𝑝𝑓𝑡𝑠. Theorem 4.2 Let (X1, ΓP) & (X2, ΨP) be any pfts’s. A mapping hP: (X1, ΓP) → (X2, ΨP) is pfδβC iff for each pfs μ of (X2, ΨP) and for each pfos M of (X1, ΓP) containing hP −1(μ) there is an pfδβos ψ of (X2, ΨP) such that μ ⊆ ψ and hP −1(ψ) ⊆ M. Proof. Necessity: Assume hP is a pfδβC. Let μ be the pfcs of (X2, ΨP) and M is a pfos of (X1, ΓP) such that hP −1(μ) ⊆ M. Then ψ = Y − hP −1(Mc) is pfδβos of (X2, ΨP) such that hP −1(ψ) ⊆ M. Sufficiency: Assume ψ is a pfcs of (X1, ΓP). Then (hP(ψ))c is a pfs of (X2, ΨP) and ψc is pfos in (X1, ΓP) such that hP −1((hP(ψ))c) ⊆ ψc. By hypothesis there is a pfδβos ψ of (X2, ΨP) such that (hP(ψ))c ⊆ ψ and hP −1(ψ) ⊆ ψc. Therefore ψ ⊆ (hP −1(ψ))c. Hence ψc ⊆ hP(ψ) ⊆ hP((hP −1(ψ))c) ⊆ ψc which implies hP(ψ) = ψc. Since ψc is pfδβcs of (X2, ΨP). Hence hP(ψ) is pfδβc in (X2, ΨP) and thus hP is pfδβC. Theorem 4.3 Let (X1, ΓP), (X2, ΨP) & (X3, ΦP) be any pfts’s. If hP: (X1, ΓP) → (X2, ΨP) is pfC and g P : (X2, ΨP) → (X3, ΦP) is pfδβC, then g P ∘ hP: (X1, ΓP) → (X3, ΦP) is pfδβC. Proof. Let ψ be a pfcs in (X1, ΓP). Then hP(ψ) is pfcs of (X2, ΨP) because hP is pfC. Now (g P ∘ hP)(ψ) = g P (hP(ψ)) is pfδβcs in (X3, ΦP) because g P is pfδβC. Thus g P ∘ hP is pfδβC. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 53 https://internationalpubls.com Theorem 4.4 Let (X1, ΓP) & (X2, ΨP) be any pfts’s. If hP: (X1, ΓP) → (X2, ΨP) is pfδβC, then pfδβcl(hP(ψ)) ⊊ hP(pfcl(ψ)). Proof. Necessity: Let hP be a pfδβC and K be a pfcs in (X1, ΓP). Now, K ⊆ pfcl(K) implies hP(K) ⊆ hP(pfcl(K)). Since hP is a pfδβC, (pfδβcl(hP(K)) is pfδβcs in (X2, ΨP) such that hP(K) ⊆ pfδβcl(hP(K)) therefore pfδβcl(hP(K)) ⊆ hP(pfcl(K)). Sufficiency: Assume K is a pfcs of (X1, ΓP). Then hP(K) = pfδβcl(hP(K)) ⊆ hP(pfcl(K)). But hP(K) ⊆ pfδβcl(hP(K)). So hP(K) = pfδβcl(K) which implies hP(K) is a pfδβcs of (X2, ΨP) and hence hP is a pfδβC. Theorem 4.5 Let hP: (X1, ΓP) → (X2, ΨP) and g P : (X2, ΨP) → (X3, ΦP) be pfδβC mappings. If every pfδβcs of (X2, ΨP) is pfc then, g P ∘ hP: (X1, ΓP) → (X3, ΦP) is pfδβC. Proof. Let ψ be a pfcs in (X1, ΓP). Then hP(ψ) is pfδβcs of (X2, ΨP) because hP is pfδβC. By hypothesis hP(ψ) is pfcs of (X2, ΨP). Now g P (hP(ψ)) = (g P ∘ h)(ψ) is pfδβcs in (X3, ΦP) because g P is pfδβC. Thus g P ∘ hP is pfδβC. Theorem 4.6 Let (X1, ΓP) & (X2, ΨP) be any pfts’s. Let hP: (X1, ΓP) → (X2, ΨP) be a map, then the following statements are equivalent: 1. hP is a pfδβO. 2. hP is a pfδβC. 3. hP −1 is pfδβCts. Proof. (i) ⇒ (ii): Let us assume that hP is a pfδβO. By definition, ψ is a pfos in (X1, ΓP), then hP(ψ) is a pfδβos in (X2, ΨP). Here, ψ is pfcs in (X1, ΓP), then X − ψ is a pfos in (X1, ΓP). By assumption, hP(X − ψ) is a pfδβos in (X2, ΨP). Hence, Y − hP(X − ψ) is a pfδβcs in (X2, ΨP). Therefore, hP is a pfδβC. (ii) ⇒ (iii): Let ψ be a pfcs in (X1, ΓP) By (ii), hP(ψ) is a pfδβcs in (X2, ΨP). Hence, hP(ψ) = (hP −1)−1(ψ), so hP −1 is a pfδβcs in (X2, ΨP). Hence, hP −1 is pfδβCts. (iii) ⇒ (i): Let ψ be a pfos in (X1, ΓP). By (iii), (hP −1)−1(ψ) = hP(ψ) is a pfδβO. 5 Application In current scenario people with symptom of Covid-19 like fever, cough, sneezing, sore throat, loss of taste and smell etc., were panic about the disease, and the diagnosis of Covid-19 takes many hours and people cannot go for the test frequently. Some other diseases like flu, pneumonia, cold etc., also has the same symptoms. Each patients has unique experience of that particular symptom and some time they may not experience that symptom even though they were affected by the Covid-19. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 54 https://internationalpubls.com Here we tried to diagnosis Covid-19 with the help of Pythagorean fuzzy sets (in short pfs’s) which helps to record all symptoms in précised manner. 5.1 Algorithm and flow chart This section includes the algorithm based on the computation of the Hamming distance, Normalized Hamming distance, Euclidean distance and Normalized Euclidean distance between the pfs’s. Step:1 Identify the universe set with most common symptoms of the Covid-19 patients. Step:2 Formulates the pfs of each patient based on their experience of each symptom of universe set. Step:3 Formulates the ideal pfs from the patients who affected by Covid-19 based on their experience of each symptom of covid-19. Step:4 Compute the various distances between the ideal pfs of Covid-19 affected patients and the pfs of the patient who experiences the symptoms of Covid-19. Step:5 Compare the distance between the pfs sets and also between the various distances. Step:6 Conclude, the patient with minimum distance from the pfs of Covid-19 patient has the huge chance to affected by Covid-19 virus. 5.2 Flow chart Identify universe set of Covid-19 symptoms ↓ Formulates the 𝑝𝑓𝑠 of each patient ↓ Formulates the ideal 𝑝𝑓𝑠 from the covid-19 patient ↓ Compute distance between the ideal 𝑝𝑓𝑠 the 𝑝𝑓𝑠 of the patient ↓ Conclude, huge chance for affected by covid-19 by who has minimum distance. 5.3 Example Let 𝐴, 𝐵, 𝐶 denote the patients who has the symptoms of Covid-19. Now their symptoms can be represented by the 𝑝𝑓𝑠 and their members are taken from the universe set 𝑋 which includes the all possible symptoms of Covid-19 are 𝑓 denotes fever, 𝑡 denotes tiredness, 𝑑 denotes dry cough, 𝑠 denotes shortness of breath, 𝑝 denotes body pain / chest pain, 𝑎 denotes diarrhea, 𝑙 denotes loss of taste or smell, 𝑠𝑡 denotes sore throat, 𝑏 denotes difficulty in breathing and 𝑟 denotes rhinorrhea. We convert the frequency of experience of the symptoms of the patients from last three days as Pythagorean fuzzy set by considering the severe symptom as the degree of positive membership, mild symptom as the degree of nutral membership and no experience of that symptom as the degree of negative membership. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 55 https://internationalpubls.com Now, the ideal Pythagorean fuzzy set 𝐼 = {(𝑥, 𝜇𝑖(𝑥), 𝜈𝑖(𝑥))/𝑥 ∈ 𝑋} denotes the model set for the Covid-19 patient which is framed by the data collected from the hospital resources. The membership values of the elements of 𝐼 are (𝑓, 𝜇𝑖(𝑓), 𝜈𝑖(𝑓)), where 𝜇𝑖(𝑓) ≥ 0.8, 𝜈𝑖(𝑓) ≥ 0.1 (𝑡, 𝜇𝑖(𝑡), 𝜈𝑖(𝑡)), where 𝜇𝑖(𝑡) ≥ 0.5, 𝜈𝑖(𝑡) ≥ 0.02 (𝑑, 𝜇𝑖(𝑑), 𝜈𝑖(𝑑)), where 𝜇𝑖(𝑑) ≥ 0.7, 𝜈𝑖(𝑑) ≥ 0.02 (𝑠, 𝜇𝑖(𝑠), 𝜈𝑖(𝑠)), where 𝜇𝑖(𝑠) ≥ 0.6, 𝜈𝑖(𝑠) ≥ 0.04 (𝑝, 𝜇𝑖(𝑝), 𝜈𝑖(𝑝)), where 𝜇𝑖(𝑝) ≥ 0.4, 𝜈𝑖(𝑝) ≥ 0.05 (𝑎, 𝜇𝑖(𝑎), 𝜈𝑖(𝑎)), where 𝜇𝑖(𝑎) ≥ 0.2, 𝜈𝑖(𝑎) ≥ 0.4 (𝑙, 𝜇𝑖(𝑙), 𝜈𝑖(𝑙)), where 𝜇𝑖(𝑙) ≥ 0.6, 𝜈𝑖(𝑙) ≥ 0.06 (𝑠𝑡, 𝜇𝑖(𝑠𝑡), 𝜈𝑖(𝑠𝑡)), where 𝜇𝑖(𝑠𝑡) ≥ 0.7, 𝜈𝑖(𝑠𝑡) ≥ 0.02 (𝑏, 𝜇𝑖(𝑏), 𝜈𝑖(𝑏)), where 𝜇𝑖(𝑏) ≥ 0.5, 𝜈𝑖(𝑏) ≥ 0.16 (𝑟, 𝜇𝑖(𝑟), 𝜈𝑖(𝑟)), where 𝜇𝑖(𝑟) ≥ 0.78, 𝜈𝑖(𝑟) ≥ 0.01 𝜇𝑖(𝑥) - severe symptom, 𝜈𝑖(𝑥) - no symptom and 0 ≤ (𝜇𝑖(𝑥))2 + (𝜈𝑖(𝑥))2 ≤ 1. The decision can be made, which patient have the more chances to have the Covid-19 by finding the distance between the ideal 𝑝𝑓𝑠 and the 𝑝𝑓𝑠’s of the patients 𝐴, 𝐵, 𝐶. In the following table symptom, membership values, Ideal 𝑝𝑓𝑠, Patient 𝐴, Patient 𝐵, Patient 𝐶, Hamming Distance (𝐼, 𝐴), Hamming Distance (𝐼, 𝐵), Hamming Distance (𝐼, 𝐶), Euclidean distance (𝐼, 𝐴), Euclidean distance (𝐼, 𝐵) and Euclidean distance (𝐼, 𝐶) are briefly denoted as sym, mval, 𝐼𝑃𝐹𝑆, 𝑃𝐴, 𝑃𝐵, 𝑃𝐶, 𝐻𝐷(𝐼, 𝐴), 𝐻𝐷(𝐼, 𝐵), 𝐻𝐷(𝐼, 𝐶), 𝐸𝐷(𝐼, 𝐴), 𝐸𝐷(𝐼, 𝐵) and 𝐸𝐷(𝐼, 𝐶). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 56 https://internationalpubls.com Calculation: Hamming distance: 𝑑𝐻𝐷(𝐼, 𝐴) = 1.075 𝑑𝐻𝐷(𝐼, 𝐵) = 1.660 𝑑𝐻𝐷(𝐼, 𝐶) = 3.220 Normalized Hamming distance: 𝑑𝑁𝐻𝐷(𝐼, 𝐴) = 0.108 𝑑𝑁𝐻𝐷(𝐼, 𝐵) = 0.166 𝑑𝑁𝐻𝐷(𝐼, 𝐶) = 0.322 Euclidean distance: 𝐸𝐷(𝐼, 𝐴) = 0.550 𝐸𝐷(𝐼, 𝐵) = 0.677 𝐸𝐷(𝐼, 𝐶) = 1.128 Normalized Euclidean distance: 𝑁𝐸𝐷(𝐼, 𝐴) = 0.174 𝑁𝐸𝐷(𝐼, 𝐵) = 0.214 𝑁𝐸𝐷(𝐼, 𝐶) = 0.357 Threfore, from the above table we observe that 𝑑𝐻𝐷(𝐼, 𝐴) < 𝑑𝐻𝐷(𝐼, 𝐵) < 𝑑𝐻𝐷(𝐼, 𝐶) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 57 https://internationalpubls.com 𝑑𝑁𝐻𝐷(𝐼, 𝐴) < 𝑑𝑁𝐻𝐷(𝐼, 𝐵) < 𝑑𝑁𝐻𝐷(𝐼, 𝐶) 𝐸𝐷(𝐼, 𝐴) < 𝐸𝐷(𝐼, 𝐵) < 𝐸𝐷(𝐼, 𝐶) and 𝑁𝐸𝐷(𝐼, 𝐴) < 𝑁𝐸𝐷(𝐼, 𝐵) < 𝑁𝐸𝐷(𝐼, 𝐶) with this evidence we may conclude that the patient 𝐴 have the more chance to affected by the Covid- 19 among these three patients. 6 Conclusion In this paper, Some new notions of strongly Pythagorean fuzzy open (closed) maps called Pythagorean fuzzy 𝛿-open and Pythagorean fuzzy 𝛿-closed maps are introduced and discussed their relationship between their near mappings with examples. 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