使用环论 整数环 利用 封闭性 单位元素 反元素 分配律 交换律 大概是代数二的范围
一般当做习题
by ring thm in Z
-1 is the inverse element of 1 such that 1+(-1)=0
by closure -1*-1 exist in Z
suppose -1*-1=a
a=-1*-1
=(0+(-1))*-1
=(1+(-1)+(-1))*-1 by distributive law
=(1*-1)+(-1*-1)+(-1*-1) since 1 is identity of *
=-1+a+a
we have
a=-1+a+a since a in Z, -a exists and in Z
=>
-a+a=-a+(-1)+a+a by comutative
=>
0=-1+a
=>
a is the inverse emelent of -1
=>a=1
so we got -1*-1=1
for m n in Z
-m is the inverse element of m
-1*m+m=(-1+1)*m=0
we have -1*m is also the inverse element of m
so -1*m=-m
generally since <Z,*> is commutative
-m*-n=-1*-1*m*n=1*m*n=m*n
Q.E.D