So I have really been struggling with this question. The original question said: The map [tex]\varphi[/tex]:Z->Z defined by [tex]\varphi[/tex](n)=n+1 for n in Z is one to one and onto Z. For (Z, . ) onto (Z,*) (i am using . for usual multiplication) define * and show that * makes phi into an isomorphism. I know that the operation must be m*n=mn-m-n+2. But I get stuck in proving that the operations are preserved. When I do [tex]\varphi[/tex](m.n) i get mn+1. and i can't get [tex]\varphi[/tex](m). [tex]\varphi[/tex](n) to work. I think I am doing something wrong. Can any one help?