Prove operation on binary relation

Question

There is a relation $\sim$ on $\mathbb{Z}\times\mathbb{Z}$ such that $(a,b)\sim(c,d)$ when $a+b=c+d$. Let $R={[(a,b)]:(a,b)\in\mathbb{Z}\times\mathbb{Z}}$ (i.e $R$ is the set of all equivalence classes of $\mathbb{Z}\times\mathbb{Z}$ under the equivalence relation $\sim$). Is $[(x,y)]*[(w,z)]=[(x+w,y+z)]$ well defined? ----After several examples, I think it is well defined. $[(2,3)]*[(4,5)]=[(6,8)]$, $[(1,4)]*[(2,7)]=[(3,11)]$ and $(6,8)\sim(3,11)$... But I am not sure how to prove it in a general way. Please help!

Answer 1

You have to prove that for any $a,b,c,d,e,f,g,h$ such that $(a,b)\sim(c,d)$ and $(e,f)\sim(g,h)$ then $(a,b)*(e,f)\sim(c,d)*(g,h)$.

Answer 2

Let $ (a,b)\sim(c,d)$ and $(e,f)\sim(g,h)$. Thus $a+b=c+d, e+f=g+h$. This implies $a+b+e+f=c+d+g+h$. Thus $(a+e)+(b+f)=(c+g)+(d+h)$. Hence $(a+e,b+f)\sim (c+d,g+h)$.