Find proper subsets of a vector space 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?
 A: You're not defining a new operation, you are using the usual $+$ operation of $\mathbf{V}$.
If $\mathbf{V}$ is a vector space (or more generally, an abelian group), and $S$ and $T$ are subsets (not necessarily subspaces) of $\mathbf{V}$, then by definition 
$$S+T = \{s+t\mid s\in S,\ t\in T\}.$$
For $A$, you are looking for a subset with $A+A\subset A$, $A+A\neq A$. 


*

*Hint. If $\mathbf{v}\neq\mathbf{0}$, and $\mathbf{v}\in A$, then $2\mathbf{v}=\mathbf{v}+\mathbf{v}\in A+A\subset A$, so $3\mathbf{v}=2\mathbf{v}+\mathbf{v}\in A+A\subset A$, etc. So you will at least have $\{ n\mathbf{v}\mid n\text{ a positive integer}\}$ in $A$. Can that be all you have in $A$?


For $B$, you want a subset with $B\subset B+B$, $B\neq B+B$. 


*

*Hint. Well, if you have $\mathbf{v}\in B$ as above, but $2\mathbf{v}\notin B$, then that will guarantee they are not equal. But $B=\{\mathbf{v}\}$ will not work, because then $\mathbf{v}\notin B+B = \{\mathbf{v}+\mathbf{v}\} = \{2\mathbf{v}\}$. Maybe you can add stuff to $B$ to guarantee $\mathbf{v}\in B+B$? Careful, though, because now you'll need the thing you added to also be in $B+B$...


For $C$ you want something with $C+C=C$. 


*

*Hint. What happens if you have all of $\mathbf{V}$?

