集合A仅含有3个元素,可以定义几种不同的对称关系

对称关系的个数为64种
设A={1,2,3}
元素仅由0,1构成的3阶矩阵有多少种对称矩阵就有多少种对称关系.这种由0,1构成的3阶矩阵共2^9=512种,对称关系的个数为64种
不含序偶的零关系
含1个序偶的有3个对称关系:
R1={(1,1)},R2={(2,2)},R3={(3,3)}
含2个序偶的有6个对称关系:
R1={(1,1),(2,2)},R2={(1,2),(2,1)},R3={(1,1),(3,3)},R4={(1,3),(2,3)},
R5={(2,2),(3,3)},R6={(2,3),(3,2)},
含3个序偶的有10个对称关系:
R0={(1,1),(2,2),(3,3)},
R1={(1,1),(1,2),(2,1)},R2={(1,1),(1,3),(3,1)},R3={(1,1),(2,3),(3,2)},
R4={(2,2),(1,2),(2,1)},R5={(2,2),(1,3),(3,1)},R6={(2,2),(2,3),(3,2)},
R7={(3,3),(1,2),(2,1)},R8={(3,3),(1,3),(3,1)},R9={(3,3),(2,3),(3,2)},
含有4个序偶有12个对称关系:
R1={(1,1),(2,2),(1,2),(2,1)},R2={(1,1),(3,3),(1,2),(2,1)},
R3={(2,2),(2,2),(1,2),(2,1)},
R4={(1,1),(2,2),(2,3),(3,2)},R5={(1,1),(3,3),(2,3),(3,2)},
R6={(2,2),(2,2),(2,3),（3,3)},
R7={(1,1),(2,2),(1,3),(3,1)},R8={(1,1),(3,3),(1,3),(3,1)},
R9={(2,2),(2,2),(1,3),(3,1)},
R10={(1,2),(2,1),(2,3),(3,2)},R11={(1,3),(3,1),(2,3),(3,2)},
R12={(1,2),(2,1),(1,3),(3,1)},
有5个序偶有12个对称关系:
R1={(1,1),(1,2),(2,1),(2,3),(3,2)},
R2={(1,1),(1,3),(3,1),(2,3),(3,2)},
R3={(1,1),(1,2),(2,1),(1,3),(3,1)},
R4={(2,2),(1,2),(2,1),(2,3),(3,2)},
R5={(2,2),(1,3),(3,1),(2,3),(3,2)},
R6={(2,2),(1,2),(2,1),(1,3),(3,1)},
R7={(3,3),(1,2),(2,1),(2,3),(3,2)},
R8={(3,3),(1,3),(3,1),(2,3),(3,2)},
R9={(3,3),(1,2),(2,1),(1,3),(3,1)},
R10={(1,1),(2,2),(3,3),(1,2),(2,1)},
R11={(1,1),(2,2),(3,3),(2,3),(3,2)},
R12={(1,1),(2,2),(3,3),(1,3),(3,1)},
有6个序偶有10个对称关系:
R1={(1,1),(2,2),(1,2),(2,1)},(1,3) ,(3,1)},
R2={(1,1),(2,2),(1,2),(2,1),(2,3),(3,2)},
R3={(1,1),(2,2),(1,3),(3,1),(2,3),(3,2)},
R4={(1,1),(3,3),(1,2),(2,1)},(1,3) ,(3,1)},
R5={(1,1),(3,3),(1,2),(2,1),(2,3),(3,2)},
R6={(1,1),(3,3),(1,3),(3,1),(2,3),(3,2)},
R7={(2,2),(3,3),(1,2),(2,1)},(1,3) ,(3,1)},
R8={(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)},
R9={(2,2),(3,3),(1,3),(3,1),(2,3),(3,2)},
R10={(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)},
有7个序偶有6个
R1={(1,1),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)},
R2={(2,2),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)},
R3={(3,3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)},
R4={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)},
R5={(1,1),(2,2),(3,3),(1,3),(3,1),(2,3),(3,2)},
R6={(1,1,),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1)},
有8个序偶有3个对称关系:
R1={(1,1),(2,2),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)},
R2={(1,1),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)},
R3={(2,2),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)},
有9个序偶有1个
R={(1,1,)(2,2),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)},
对称关系总共有
1+3+6+10+1212+10+6+3+1=64
统计十分繁琐,我没有找到好的办法.