Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem
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Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem M. E. El-Shafei1 · T. M. Al-shami1,2 Received: 15 November 2019 / Revised: 11 February 2020 / Accepted: 9 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This study introduces a new family of soft separation axioms and a real-life application utilizing partial belong and natural non-belong relations. First, we initiate the concepts of w-soft Ti -spaces (i = 0, 1, 2, 3, 4) with respect to distinct ordinary points. These concepts generate a wider family of soft spaces compared with soft Ti -spaces, p-soft Ti -spaces and e-soft Ti -spaces. We illustrate the relationships between w-soft Ti -spaces with the help of examples and discuss some sufficient conditions of soft topological spaces to be w-soft Ti spaces. Additionally, we point out that stable or soft regular spaces are sufficient conditions for the equivalence among the concepts of soft Ti , p-soft Ti and w-soft Ti . We highlight on explaining the links between w-soft Ti -spaces and their parametric topological spaces and studying the role of enriched spaces in these links. Furthermore, we prove that w-soft Ti spaces are hereditary and topological properties, and they are preserved under finite product soft spaces. Finally, we propose an algorithm to bring out the optimal choices. This algorithm is based on dividing the whole parameters set into parameter sets and then apply a partial belong relation in the favorite soft sets. This application is supported with an interesting example to show how to implement this algorithm. Keywords Partial belong relation · Natural non-belong relation · w-soft Ti -space · Enriched soft topology · Decision-making problem Mathematics Subject Classification 54B05 · 54B10 · 54D10 · 54D15
Communicated by Anibal Tavares de Azevedo.
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T. M. Al-shami [email protected] M. E. El-Shafei [email protected]
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Department of Mathematics, Mansoura University, Mansoura, Egypt
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Department of Mathematics, Sana’a University, Sana’a, Yemen 0123456789().: V,-vol
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M. E. El-Shafei, T. M. Al-shami
1 Introduction Molodtsov (1999) came up the brilliant idea of soft sets to handle intricate problems which suffer from uncertainty and vagueness. He demonstrated that soft set is free from the obstacles existed in the previous tools such as fuzzy set and probability theory. He also described a rich potential of soft sets for applications in many disciplines such as game theory, operations research, Riemann integration and theory of measurement. At recent days, the theory of soft set becomes very widespread among scientists around the world and one of the most developing tool to handle uncertainty in various fields such as information theory (Ali et al 2018), computer sciences (Caˇgman and Enginoˇglu 2010), engineering (Karaaslan 2016), medical sciences (Kharal and Ahmad 2011; Yuksel et al 2013), and economy (Caˇgman and Enginoˇglu 2010; Maji et al 2002). T
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