Rough set

In mathematical logic, a rough set is an imprecise representation of a crisp set (conventional set) in terms of two subsets, a lower approximation and upper approximation. The approximations themselves can further be imprecise or fuzzy.

The idea of rough set was proposed by Pawlak as a new mathematical tool to deal with vague concepts. Comer, Grzymala-Busse, Iwinski, Nieminen, Novotny and Pawlak, Obtulowicz, and Pomykala and Pomykala have studied algebraic properties of rough sets. Rough sets can be used to represent ambiguity, vagueness and general uncertainty.

A rough set S is a set defined by the relation

S = (U, R)

here U is the "universe" and R is the "indiscernibility relation" that partitions the "universe". Thus, S partitions the universe into equivalence sets.

Rough set can be used as a theoretical basis for some problems in machine learning. The concept of rough set has also inspired some logical research.

See also

• Alternative set theory
• Fuzzy logic
• Fuzzy set

References

• Chanas, S. and D. Kuchta. "Further remarks on the relation between rough and fuzzy sets." Fuzzy Sets and Systems (1991).
• Comer, S. D. "An algebraic approach to the approximation of information." Fundamenta Informaticae (1991).
• Dubois, D. and H. Prade. "Rough fuzzy sets and fuzzy rough sets." International Journal of General Systems (1988).