ON A CONCEPT OF ROUGH SEIS

In t roduc t ion In [ 1 ] Z.Pjawlak introduced the not ion of a rough se t used to def ine approximate operat ions on s e t s , approximate e q u a l i ty of s e t s and approximate inc lus ion of s e t s . The notion of a rough se t can be viewed as an a l t e r n a t i v e to fuzzy set [ 2 ] , 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 two no t ions . Theory of fuzzy s e t s , theory of to le rance spaces £3] and theory of ro> <jh s e t s are a mathematical methods i n approximate c l a s s i f i c a t i o n of ob j ec t s . In many branches of computer app l i ca t ions these problems are of primary conotorn. Let us quote here some no ta t ions and d e f i n i t i o n s from [1]» • Let U denote some non empty s^t (universum) which i s f ixed i n t h i s paper and l e t R denote some f ixed equivalence r e l a t i o n on U. The pai r A =<U,R>.. w i l l be cal led an approximation space. Equivalence c l a s se s of the r e l a t i o n R w i l l be r e f e r r e d to as elementary s e t s in A ( s h o r t : elementary s e t s ) . Every union of elementary s e t s in A and an empty se t w i l l be ca l led a composed set i n A ( sho r t : composed s e t ) . Let X c U. The l e a s t composed set i n A containing X w i l l be ca l l ed the best upper approximation of X in A, denoted as AX. The g r e a t e s t composed set in A contained i n X w i l l be cal led the beet lower approximation of X in A, denoted as AX. In [1] the fol lowing condi t ions are the axioms f o r approximations val id in every approximation space