Different core attributes´s comparison and analysis

The key of attribute reduction based on rough set is find the core attributes. Most existing works are mainly based on Hu´s discernibility matrix. Till now, there are three kinds of core attributes: Hu´s core based on discernibility matrix (denoted by Core1(C)), core based on positive region (denoted by Core2(C)), and core based on information entropy (denoted by Core3(C)). Some researchers have been pointed out that these three kinds of cores are not equivalent to each other. Based on the above three kinds of core attributes, we at first propose three kinds of simplified discernibility matrices and their corresponding cores, which are denoted by SDCore1(C), SDCore2(C), and SDCore3(C) respectively. And then it is proved that Core1(C)=SDCore1(C), Core2(C)= SDCore2(C), and Core3(C)=SDCore3(C). Finally, based on three proposed simplified discernibility matrices and their corresponding cores, it is proved that Core2(C)subeCore3(C)subeCore1(C).