證明 由交集的定義r1∩r2={(a,b)|(a,b)Îr1且(a,b)Îr2}.
對任意一個aÎA,因為r1和r2都是自反的,所以有(a,a)Îr1且(a,a)Îr2,因而有(a,a)Îr1∩r2,故r1∩r2是自反的.
對任意a,bÎA,若(a,b)Îr1∩r2,則有(a,b)Îr1且(a,b)Îr2,由r1和r2的對稱性有(b,a)Îr1且(b,a)Îr2,因而有(b,a)Îr1∩r2,故r1∩r2是對稱的.
對任意a,b,cÎA,若(a,b)Îr1∩r2,(b,c)Îr1∩r2,則有(a,b)Îr1,(b,c)Îr1；(a,b)Îr2,(b,c)Îr2.由r1和r2的傳遞性有(a,c)Îr1,(a,c)Îr2,因而有(a,c)Îr1∩r2,故r1∩r2是傳遞的.
由以上三方面知r1∩r2是A上的等價關(guān)系.證畢