Let $A=P(\Bbb Z)$
Prove that $R=\{(S,T)\in A{\times}A: \exists n\in \Bbb Z~\forall x\in \Bbb Z~(x\in S\leftrightarrow x+n\in T)\}$ is an equivalence relation on $A$ (15 points).
Write down all elements in $A/R$ that consist of a single element (7 point)
I already proved that $R$ is an equivalence relation, but I am unsure about the second question regarding the equivalence class. If $S$ and $T$ were both empty sets, would that be the equivalence class that contained only one element?
Well, $\emptyset\, \mathcal{R}\, \emptyset$ and $[\emptyset]_{\mathcal{R}}=\{\emptyset\}$ hence the equivalence class of the empty set indeed contains only one element (in fact, it's the only one). Also, notice that if $S\,\mathcal{R}\,T,$ then $\#(S)=\#(T)$ - in fact, there is a bijection between the elements of $S$ and the elements of $T$ (and this is why the equivalence class of the empty set contains only the empty set itself). How can you use this fact to conclude?
Hint: for any non-empty set $S,$ consider the sets $T_n=\{s+n: s\in S\}$ and prove that $S\,\mathcal{R}\,T_n$ for any $n\in \mathbb{Z}.$