设集合M={a|a=x^2-y^2,x,y∈z}求两个属于M的数之积还属于M错了是设集合M={a|a=x^2+y^2,x,y∈z}

Question

设集合M={a|a=x^2-y^2,x,y∈z} 求两个属于M的数之积还属于M
错了 是设集合M={a|a=x^2+y^2,x,y∈z}

Answer 1

证明：设a1、a2∈M,则
a1a2=(x1^2-y1^2)(x2^2-y2^2)=(x1x2)^2+(y1y2)^2-[(x1y2)^2+(x2y1)^2]
=(x1x2)^2+(y1y2)^2+2x1x2y1y2-[(x1y2)^2+(x2y1)^2+2x1x2y1y2]
=(x1x2+y1y2)^2-(x1y2+x2y1)^2∈M