I have a question on this proof given by Furstenberg proof on the infinitude primes. I am a non-mathematician with some basic knowledge on set theory and topology.
Define for $a,b\in\mathbb{Z}$ where $a\neq0$ the set $$S(a,b)=\{an+b:n\in\mathbb{Z}\}.$$
The Wikipedia entry says that the identify $$S(a,b)=\mathbb{Z}\setminus\bigcup_{j=1}^{a-1}S(a,b+j)$$ holds for all $a,b\in\mathbb{Z}$ where $a\neq0$. I do not see why this is the case. Can someone help me out?