已知集合A={x‖x=a+根号2b,a∈Z,b∈Z}.设x1∈A,x2∈A.求证x1乘以x2∈A..
问题描述:
已知集合A={x‖x=a+根号2b,a∈Z,b∈Z}.设x1∈A,x2∈A.求证x1乘以x2∈A..
答
x1∈A,x2∈A.
则x1=a1+根号2b1
x2=a2+根号2b2
x1*x2=(a1+根号2b1)(a2+根号2b2)
=(a1a2)+根号2*(b1b2+a1^2*b2+a2^2*b1)
设a1a2=a3,b1b2+a1^2*b2+a2^2*b1=b3
x1x2=a3+根号2*b3,∈A