Question

着急、数学集合！！！！！ 已知集合A={a+b√2|a∈Q,b∈Q},m∈A,n∈A 求证:m*n∈A

已知集合A={a+b√2|a∈Q,b∈Q},m∈A,n∈A
求证:m*n∈A

Answer 1

任取 a1,a2,b1,b2∈Q
并设m=(a1)+(b1)√2 n=(a2)+(b2)√2

因为m*n=[(a1)+(b1)√2]*[(a2)+(b2)√2]
=(a1a2+2b1b2)+(a1b2+a2b1)√2

而a1a2+2b1b2∈Q ,且 a1b2+a2b1属于Q

所以m*n∈A

Answer 2

设m=(a1)+(b1)√2 n=(a2)+(b2)√2
∴m*n=[(a1)+(b1)√2]*[(a2)+(b2)√2]=(a1a2+2b1b2)+(a1b2+a2b1)√2
∴m*n∈A