I'm having a bit of trouble understanding this exercise:
Find proper subsets $A$, $B$, $C$ of a vector space $V$ over the field $\mathbb{R}^2$, so that $A + A \subset A$, $B \subset B + B$ and $C + C = C$.
Questions: Can I define the operation $+$ myself (f.e difference, or union etc) as long as $V$ still remains a vector space?
If yes: Since $A$, $B$, $C$ are subsets of the same vector space $V$, I can not define 3 different operations, right? One operation must be able to handle all those cases?
If yes: How is this possible?