Soft rational line integral

Abstract

Soft set theory is a new area of mathematics that deals with uncertainties. Applications of soft set theory are widely spread in various areas of science and social science viz. decision making, computer science, pattern recognition, artificial intelligence, etc. The importance of soft set-theoretical versions of mathematical analysis has been felt in several areas of computer science. This paper suggests some concepts of a soft gradient of a function and a soft integral, an analogue of a line integral in classical analysis. The fundamental properties of soft gradients are established. A necessary and sufficient condition is found so that a set can be a subset of the soft gradient of some function. The inclusion of a soft gradient in a soft integral is proved. Semi-additivity and positive uniformity of a soft integral are established. Estimates are obtained for a soft integral and the size of its segment. Semi-additivity with respect to the upper limit of integration is proved. Moreover, this paper enriches the theoretical development of a soft rational line integral and associated areas for better functionality in terms of computing systems.