Towards a theory of granular sets

1. INTRODUCTION: Set theory as a branch of human endeavour was developed by the efforts of many mathematicians [ Kam ]. Such a theory found many applications in science, technology and other fields. In an effort to capture uncertainity in human reasoning, Zadeh formulated and studied the theory of fuzzy sets. The theory of fuzzy sets found applications in many branches of science and technology. Pawlak, a computer scientist proposed and studied the concept of rough set in an effort to capture other aspects of uncertainity arising in applications such as database design. The theory of rough sets is found to have complemented the theory of fuzzy sets. The author, in his research efforts related to the fusion problem in Wireless Sensor Networks (WSN) discovered the idea of " graded set " (discussed in Section 2) as a generalization of the idea of rough set. When understanding the details of rough set theory, the author discovered the idea of " granular set ". The basic motivation for such sets is discussed below. Motivation for Granular Sets: In biological systems such as trees, when the tissue is examined under a microscope at different resolutions, different cells/parts are observed. It is very clear that as the resolution increases, finer granular structure is observed. Our goal is to arrive at a mathematical abstraction of such sets observed in biological systems (physical, chemical, biological etc) as well as artificial systems (such as databases). It is expected that a detailed theory of such sets will find many applications as in the case of rough sets, fractal sets etc.