We aim through this paper to achieve two goals: first, we define some types of belong and nonbelong relations between ordinary points and double-framed soft sets. These relations are one of the distinguishing characteristics of double-framed soft sets and are somewhat expression of the degrees of membership and nonmembership. We explore their main properties and determine the conditions under which some of them are equivalent. Also, we introduce the concept of soft mappings between two classes of double-framed soft sets and investigate the relationship between an ordinary point and its image and preimage with respect to the different types of belong and nonbelong relations. By the notions presented herein, many concepts can be studied on double-framed soft topology such as soft separation axioms and cover properties. Second, we give an educational application of optimal choices using the idea of double-framed soft sets. We provide an algorithm of this application with an example to show how this algorithm is carried out.

中文翻译：

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双帧软集上的属于和非属于关系及其应用

通过本文，我们旨在实现两个目标：首先，定义普通点与双框软集之间的某些类型的归属和非归属关系。这些关系是双框架软集的显着特征之一，在某种程度上表达了隶属度和非隶属度。我们探索了它们的主要特性，并确定了其中一些等效的条件。此外，我们介绍了两类双帧软集之间的软映射的概念，并针对不同类型的归属和非归属关系研究了普通点及其图像和原像之间的关系。通过本文提出的概念，可以在双帧软拓扑上研究许多概念，例如软分离公理和覆盖属性。第二，我们使用双重框架软集的思想在教育上应用最佳选择。我们提供此应用程序的算法和一个示例，以演示如何执行此算法。