What are $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb Z_n$?

Question

Is this correct?

$$\mathbb{Z}/n\mathbb{Z}=\{0,1,2,...,n-1\}$$

And what about, for example,

$$\mathbb Z_2=\left\{[0],[1]\right\}=\left\{\{...,-2,0,2,...\},\{...,-3,-1,1,3,...\}\right\}$$

Answer 1

As for the first formula, typically, no. By definition, $\mathbb{Z}/n\mathbb{Z}$ is a factor group. The set underlying this group is $\{i+n\mathbb{Z}\mid i\in \mathbb{Z}\}$ $=$ $\{[0],[1],...,[n-1]\}$.

The second notation $\mathbb{Z}_2$ is a matter of definition. Authors can define it as $\{0,1\}$, or as $\{[0],[1]\}$, or as something completely different.