Generalized rough approximations in image Original Research Article

In this paper we will treat a generalization of inner and outer approximations of fuzzy sets, which we will call image-inner and image-outer approximations respectively (image being any finite set of rational numbers in image). In particular we will discuss the case of those fuzzy sets which are definable in the logic image by means of step functions from the hypercube image and taking value in an arbitrary (finite) subset of image. Then, we will show that if a fuzzy set is definable as truth table of a formula of image, then both its image-inner and image-outer approximation are definable as truth table of formulas of image. Finally, we will introduce a generalization of abstract approximation spaces and compare our approach with the notion of fuzzy rough set.