Topological connections between sets, rough sets and fuzzy sets

Abstract EN

The notion of a set is not only basic for the whole mathematics but it also plays an important role in natural language.

This notion leads to antinomies, i.e., it is contradictory.

Another issue discussed in connection with this notion is vagueness.

Mathematics requires that all mathematical notions (including set) must be exact.

However, philosophers and recently computer scientists got interested in vague concepts.

Theories of rough sets and fuzzy sets are related and complementary methodologies to handle uncertainty of vagueness and coarseness, respectively.

They are generalizations of classical set theory for modeling vagueness and uncertainty.

Both theories represent two different approaches to vagueness.

Fuzzy set theory addresses gradualness of knowledge, expressed by the fuzzy membership – whereas rough set theory addresses granularity of knowledge, expressed by the indiscernibility relation.

A fundamental question concerning both theories is their connections and differences.

There have been many studies on this topic.

Topology is a branch of Libyan bulletin for studies – eighteen issue mathematics, whose ideas exist not only in almost all branches of mathematics but also in many real life applications.

The topological structure on an abstract set is used as the base, which used to extract knowledge from data.

In this paper, topological structure is used to study the relation and connection between sets, rough sets and fuzzy sets.

Membership function is used to convert from rough set to fuzzy set and vice versa.

This conversion will achieve the advantages of these theories.

In addition, we will give the basic deviations between rough set theory "RST" and ordinary set theory "OST”.

Some examples and theories are introduced to indicate the importance of using general binary relations in the construction of these concepts, and indicate the relation between sets, rough sets and fuzzy sets according to the topological spaces.-