Set coverings and tolerance relations
- PDF / 164,111 Bytes
- 8 Pages / 595.276 x 793.701 pts Page_size
- 53 Downloads / 177 Views
SET COVERINGS AND TOLERANCE RELATIONS S. N. Gerasin,a V. V. Shlyakhov,a and S. V. Yakovlev b
UDC 51.001.57
Properties of tolerance relations and tolerance classes are studied. Tolerance classes are demonstrated to form carrier set coverings. The conditions are proved under which tolerance and equivalence classes coincide. Keywords: tolerance relation, adjacent classes and preclasses, set covering, functional relation. INTRODUCTION The traditional approach to the investigation of similarity or indistinguishability between objects is as follows: a similarity measure is first defined and then the relative positions of similar objects are investigated. E. Zieman, an English mathematician, studied models of the visual apparatus and proposed an axiomatic definition of similarity [1]. Based on this, the possibility arose to investigate properties of similarity irrespective of the concreteness of its specification in some situation or other, i.e., irrespective of the distance between objects, coincidence of some features, or a subjective opinion of an observer. If only the similarity between objects is specified, then they cannot be partitioned into crisp classes so that objects are similar within one class and the similarity between objects of different classes is absent. In the case of similarity, a fuzzy situation without crisp boundaries arises. However, the accumulation of insignificant distinctions between similar objects can lead to absolutely dissimilar objects. Earlier investigations of tolerance relations [2–4] were basically related with topological aspects and some applications to semantics. A considerably smaller attention was paid to the importance of this relation in recognition and classification problems. We note that an equivalence relation generates a partitioning of a set into disjoint adjacent classes, and a tolerance relation induces classes forming a covering of the initial set. A natural connection between a tolerance relation and an equivalence relation leads to the conclusion that the corresponding adjacent classes of these relations can coincide under some additional conditions. The solution of this problem is obtained in [5]. STRUCTURE OF TOLERANCE CLASSES Let us consider a rectangular domain D Ì R 2 containing n points at points of a grid of size n1 ´ n 2 = n, i.e., consisting of n1 points along the vertical direction and n 2 points along the horizontal direction (Fig. 1). After an ordered numbering, these points form a set A = {1,2, K , n}. We assume that, on the set A, an integer function f is given whose image consists of discrete points from R that form a set B = J m f ( A ) = {n + 1, K , n + m} after an ordered numbering. Let there be a covering P B = {P1B , K , P Bq } on the set , i.e., we have P Bi Ì B, P Bi ¹ P Bj , and
q
U P Bi
= B , i ¹ j, i = 1, s, j = 1, s. Then it is easy to see
i =1
that the function f and the covering P B induce the following binary relation on A:
a
Kharkov National University of Radioelectronics, Kharkov, Ukraine, [email protected]. bKharkov National Uni
Data Loading...
