APPROXIMATIONS IN THE SPACE (U,π)

Abstract

Introduction In [l] Z. Pawlak introduced the notion of an approximation apace as the pair A = (U,R), where U denotes an arbitrary non-empty set and S denotes some equivalence relation on U, called here indiscernibillty relation. Equivalence classes of R are called elementary sets in A. Every union of elementary sets in A and an empty set are called composed sets in A. If XcU, then the least composed set in A containing X will be called the best upper approximation of X in A, and will be denoted by AX. The greatest composed set in A contained in A will be called the best lower approximation of X in A, and will be denoted by AX. A definition of these two notions that we gave in [2] is based on a system of axioms for approximations and is different that given in [1]. In [1] and [3]-[4] Z. Pawlak introduced also the notions of rough equality, rough inclusion , rough relation and the notion of the approximation of function in the space A. Basic idea of all these notions is connected with the fact that in some applications we are unable to say for sure whether some element belongs to the set X or not. Theory of approximations in a sense of the papers [l]-[4] is a mathematical method for approximate classification of objects. In many branches of computer science these problems are of primary concern. This theory can be viewed as an alternative to the theory of fuzzy sets [5], and theory of tolerance space [6], however there are some essential differences between these three theories.