A Research On The Lower Approximation Fuzzy Mapping And The Neighbourhood-assignment Reduction

Both fuzzy set theory and rough set theory are mathematics tools for dealing withuncertain information. Extension principle presented by L. A. Zadeh is one of the basicprinciples of fuzzy mathematics and an important tool of fuzzy set theory. Rough settheory that doesn’t need priori knowledge in dealing with uncertain information issuperior to fuzzy set theory which needs it. In this paper, corresponding work is donebased on fuzzy set theory and rough set theory, which results are as follows:(1) For fuzzy set theory, basic concepts and properties of fuzzy sets are given. Then,a new fuzzy mapping, lower approximation fuzzy mapping, is presented. And the lowerand upper fuzzy approximation operators based on lower approximation fuzzy mappingare constructed. Finally, some invariant properties of lower approximation fuzzymapping and lower approximation inverse fuzzy mapping are investigated, and they areturned out to be true.(2) For rough set theory, the concept of neighbourhood-assignment consistent setbased on binary relations is presented and its some properties are discussed. Then,neighbourhood-assignment reduction, a new attribute reduction method based onneighbourhood-assignment discernibility matrix is proposed. Finally, by analyzing somefactors which are related to water pollutions, we obtain main factors polluting waterquality. The practical example shows that the proposed method is effective.