$\mathbb{Z}_{2}G$ as the class of finite subsets of $G$

Question

Let $G$ be a group. Then the group ring $RG$ is defined as $\sum_{g\in G} R$ (a copy of $R$ for every $g\in G$) and a typical element is of the form $ \sum_{i=1}^{n} r_{g_{i}}g_{i} $. Now if $R= \mathbb{Z}_{2}$ we can think $\mathbb{Z}_{2}G$ as the class of finite subsets of $G$. Why is that? I'm having trouble understanding it.

Answer 1

Because each element in $\Bbb Z_2$ is either $0$ or $1$.

If you interpret $0$ as "not in the subset" and $1$ as "in the subset", then a finite sum $\sum_{g\in G} r_g g$ defines a finite subset and this gives a bijection between elements in $\Bbb Z_2 G$ and finite subsets of $G$.

Note that this gives a structure of group ring to the set of finite subsets of $G$.