Question

已知集合A={t^2+s^2〡t,s∈Z},且x∈A,y∈A,则下列结论正确的是

A.x+y∈A B,x-y∈A C,xy∈A D,x/y∈A
答案是 C 为什么（不用排除法）

Answer 1

x∈A，y∈A，则有
x=t₁²+s₁²，y=t₂²+s₂²
于是
xy=(t₁²+s₁²)(t₂²+s₂²)
=(t₁t₂)²+(s₁s₂)²+(t₁s₂)²+(s₁t₂)²
=(t₁t₂)²+2t₁t₂s₁s₂+(s₁s₂)²+(t₁s₂)²-2t₁t₂s₁s₂+(s₁t₂)²
=(t₁t₂+s₁s₂)²+(t₁s₂-s₁t₂)²∈A

Answer 2

x=t1^2+s1^2,
y=t2^2+s2^2,
xy=(t1t2)^2+(t1s2)^2+(s1t2)^2+(s1s2)^2