APPROXIMATIONS IN THE SPACE (U,π)

Introduction In [l] Z. Pawlak introduced the notion of an approximation apace as the pair A = (U,R), where U denotes an a rb i t ra ry non-empty set and S denotes some equivalence r e l a t i o n on U, ca l led here i n d i s c e r n i b i l l t y r e l a t i o n . Equivalence c l a s s e s of R are ca l l ed elementary s e t s in A. Every union of elementary s e t s in A and an empty set are ca l l ed composed s e t s in A. I f X c U , then the l e a s t composed set in A containing X w i l l be ca l led the best upper approximation of X in A, and w i l l be denoted by AX. The g rea te s t composed set in A contained in A w i l l be ca l l ed the best lower approximation of X in A, and w i l l be denoted by AX. A d e f i n i t i o n of these two notions that we gave in [2] i s based on a system of axioms f o r approximations and i s d i f f e r e n t that given in [1]. In [1] and [3] [4] Z. Pawlak introduced a l so the notions of rough equa l i ty , rough inc lus ion , rough r e l a t i o n and the notion of the approximation of funct ion in the space A. Basic idea of a l l these notions i s connected with the f a c t that in some app l i ca t ions we are unable to say f o r sure whether some element belongs to the set X or not . Theory of approximations in a sense of the papers [ l ] [4 ] i s a mathematical method f o r approximate c l a s s i f i c a t i o n of o b j e c t s . In many branches of computer science these problems are of primary concern. This theory can be viewed as an a l t e r native to the theory of fuzzy s e t s [ 5 ] , and theory of to le rance space [6] , however there are some e s s e n t i a l d i f f e r e n c e s between these three theor ie s .