The Pythagorean fuzzy models deal with graphical and algebraic structures in case of vague information related to membership and non-membership grades. Here, we use Pythagorean fuzzy sets to generalize the concept of vector spaces and discuss their basis and dimensions. We also highlight the concept of Pythagorean fuzzy matroids and examine some of their fundamental characteristics like circuits, basis, dimensions, and rank functions. Additionally, we explore the concept of Pythagorean fuzzy matroids in linear algebra, graph theory, and combinatorics. Finally, we demonstrate the use of Pythagorean fuzzy matroids for minimizing the time taken by a salesman in delivering given products.
Background: Drug-resistant tuberculosis (DR-TB) is the major cause of mortality worldwide. Our objectives were to determine the distribution of DR-TB by sex, age groups, occupation, province, division, district, type of disease, type of drug resistance, treatment regimen and outcome of treatment in DR-TB population in D.I.Khan Division, Pakistan.Materials Methods: This cross-sectional study was conducted in Department of Community Medicine, Gomal Medical College, D.I.Khan, Pakistan. A sample of 286 DR-TB patients was selected consecutively from population at risk. Sex, age groups, occupation, province, division and district were demographic while type of disease, type of drug resistance, treatment regimen and outcome of treatment were research variables. All variables being nominal were described by count, percentage cumulative percentage with 95% confidence interval for proportion. Distribution of DR-TB patients by all the ten variables were substantiated by chi-square goodness-of-fit test.Results: Out of 286 DR-TB patients, 123 (43%) were men and 163 (57%) women. DR-TB cases were most prevalent in age group 15-44 years 172 (60.14%), housewife 140 (48.95%), Khyber Pakhtunkhwa 175 (61.19%), D.I.Khan Division 178 (62.24%) and district 121 (42.31%). Most common type of disease, drug resistance and treatment regimen was pulmonary TB 282 (98.60%), MDR 273 (95.45%) and longer treatment (n=273 MDR-TB) 246 (90.11%) respectively. Treatment success rate was 161 (56.29%). The observed prevalence by occupation, province, division, district and type of disease in our sample was similar to expected prevalence in population (p.05 for all), while it was different from population by sex, age groups, type of drug resistance, regimen and treatment outcome (p.05 for all).Conclusion: The prevalence of DR-TB was higher in women, age group 15-44 years, housewife, Khyber Pakhtunkhwa and D.I.Khan Division and District. Most common type of disease, drug resistance and treatment regimen was pulmonary TB, MDR and longer treatment respectively. Treatment success rate was 56.29%. The observed prevalence by occupation, province, division, district and type of disease in sample was similar to population, while it was different by sex, age groups, type of drug resistance, regimen and treatment outcome.
The theory of q -rung orthopair fuzzy sets ( q -ROFSs) is emerging for the provision of more comprehensive and useful information in comparison to their counterparts like intuitionistic and Pythagorean fuzzy sets, especially when responding to the models of vague data with membership and non-membership grades of elements. In this study, a significant generalized model q -ROFS is used to introduce the concept of q -rung orthopair fuzzy vector spaces ( q -ROFVSs) and illustrated by an example. We further elaborate the q -rung orthopair fuzzy linearly independent vectors. The study also involves the results regarding q -rung orthopair fuzzy basis and dimensions of q -ROFVSs. The main focus of this study is to define the concepts of q -rung orthopair fuzzy matroids ( q -ROFMs) and apply them to explore the characteristics of their basis, dimensions, and rank function. Ultimately, to show the significance of our proposed work, we combine these ideas and offer an application. We provide an algorithm to solve the numerical problems related to human flow between particular regions to ensure the increased government response action against frequently used path (heavy path) for the countries involved via directed q -rung orthopair fuzzy graph ( q -ROFG). At last, a comparative study of the proposed work with the existing theory of Pythagorean fuzzy matroids is also presented.
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