Similarity measures on three kinds of fuzzy sets

Intuitionistic fuzzy sets (IFSs) proposed by Atanassov, fuzzy rough sets (FRSs) proposed by Nanda and Majumdar, rough fuzzy sets (RFSs) proposed by Banerjee and Pal, have gained attention from researchers for their applications in various fields. Then similarity measures between three fuzzy sets were developed. In this paper, we first point out: IFSs, FRSs and RFSs are L-fuzzy sets with L being a special fuzzy lattice. At the same time, we suggest some rules which is considered when we give a similarity measure for measuring the degree of similarity between elements and between some fuzzy sets. After that, some existing measures of similarity are reviewed, some examples are applied to show that some existing similarity measures are not always effective in some cases. We propose some new similarity measures for measuring the degree of similarity between three fuzzy sets under an unifying form and between IFSs. Finally, we illustrate the problem in the context of colorectal cancer diagnosis by similarity measure between fuzzy rough sets. Therefore, the proposed similarity measures can provide a useful way for measuring three fuzzy sets more effectively.