Summary: A relative reduct can be considered as a minimum set of attributes that preserves a certain classification property. This paper investigates three different classification properties, and suggests three distinct definitions accordingly. In the Pawlak rough set model, while the three definitions yield the same set of relative reducts in consistent decision tables, they may result in different sets in inconsistent tables.
Relative reduct construction can be carried out based on a discernibility matrix. The study explicitly stresses a fact, that the definition of a discernibility matrix should be tied to a certain property. Regarding the three classification properties, we can define three distinct definitions accordingly.
Based on the common structure of the specific definitions of relative reducts and discernibility matrices, general definitions of relative reducts and discernibility matrices are suggested.