Covering Based Pessimistic Multigranular Rough Equalities and their Properties

The basic rough set theory introduced by\udPawlak as a model to capture imprecision in data has\udbeen extended in many directions and covering based\udrough set models are among them. Again from the\udgranular computing point of view, the basic rough sets\udare unigranular by nature. Two types of extensions to the\udcontext of multigranular computing are done; called the\udoptimistic and pessimistic multigranulation by Qian et al\udin 2006 and 2010 respectively. Combining these two\udconcepts of covering and multigranulation, covering\udbased multigranular models have been introduced by Liu\udet al in 2012. Extending the stringent concept of\udmathematical equality of sets rough equalities were\udintroduced by Novotny and Pawlak in 1985. Three more\udtypes of such approximate equalities were introduced by\udTripathy in 2011. In this paper we study the approximate\udequalities introduced by Novotny and Pawlak from the\udpessimistic multigranular computing point of view and\udestablish several of their properties. These concepts and\udproperties are shown to be useful in approximate\udreasoning