Abstract
We introduce a new type of functions from a soft set to a soft set and study their properties under soft real number setting. Firstly, we investigate some properties of soft real sets. Considering the partial order relation of soft real numbers, we introduce concept of soft intervals. Boundedness of soft real sets is defined, and the celebrated theorems like nested intervals theorem and Bolzano-Weierstrass theorem are extended in this setting. Next, we introduce the concepts of limit, continuity, and differentiability of functions of soft sets. It has been possible for us to study some fundamental results of continuity of functions of soft sets such as Bolzano’s theorem, intermediate value property, and fixed point theorem. Because the soft real numbers are not linearly ordered, several twists in the arguments are required for proving those results. In the context of differentiability of functions, some basic theorems like Rolle’s theorem and Lagrange’s mean value theorem are also extended in soft setting.
Highlights
Following the seminal work of Zadeh [1] on fuzzy set theory, development of mathematical theory and their applications in handling the problems under uncertain environment have been gaining momentum day by day
After that Maji et al [3, 4] defined some operations on soft sets based on which Shabir and Naz [5] introduced soft topologies, Aktas and Cagman [6] soft group, and Nazmul and Samanta [7] soft topological group
[10] Das and Samanta introduced the concept of soft real numbers
Summary
Following the seminal work of Zadeh [1] on fuzzy set theory, development of mathematical theory and their applications in handling the problems under uncertain environment have been gaining momentum day by day. Considering some difficulties in the parametrization process in fuzzy set theory, Molodtsov in 1999 [2] introduced an idea of soft set as a parametrized family of sets where parameter set takes values from an arbitrary set. He showed the applications of soft sets in fields like smoothness of functions, probability theory, measure theory, and game theory. In this paper we have considered such type of functions and have studied some fundamental properties of continuous functions, like Bolzano’s property, intermediate value property, and fixed point property. We have introduced the concept of differentiation of such functions and have extended Rolle’s theorem and Lagrange’s theorem in soft settings
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