General Tool-Based Approximation Framework Based on Partial Approximation of Sets

Let us assume that we observe a class of objects and have some well-defined features with which an observed object possesses or not. In real life, two relevant groups of objects can be established determined by our current and necessarily constrained knowledge. In particular, a group whose elements really possess a feature in question, and another group whose elements substantially do not possess the same feature. In practice, as a rule, we can observe a feature of objects via only tools with which we are able to judge easily whether an object possesses a property or not. Of course, a property ascertained by tools does not coincide with a feature completely. To manage this problem, we propose a general tool-based approximation framework based on partial approximation of sets in which a positive feature and its negative one of any proportion of the observed objects can simultaneously be approximated.