Let $A$, $B$ and $C$ be subspaces of a vector space $V$. Prove or falsify the following:
a) If $A$ $+$ $B$ $=$ $A$ $+$ $C$, then $B$ $=$ $C$.
b) If $A$ $⊕$ $B$ $=$ $A$ $⊕$ $C$ then $B$ $=$ $C$.
I thought that a) was false since it is possible that B is a subset of $A$ and $C$ is a subset of $B$. This means that $B$ and $C$ need not be the exact same subspaces. In this case, either of the two subspaces ($B$ and $C$) will lead to a sum of $A$ with $A$.
For b), I thought it would be true because since the intersection of $A$ and $B$ and $A$ and $C$ will be $0$, then the only way to add the two subspaces and get the same result would be if $B$ equaled $C$.
But I am a little skeptical of my reasoning as I am not sure if the ideas of subspaces and subsets can be as easily interchanged as I have done above. I am also not sure if I have fully understood the idea of sums and direct sums of subspaces in this case.
Any help?
a) and b) are both false: take $A=\{(x,0): x\in \mathbb R\}$, $B=\{(0,y): y\in \mathbb R\}$, and $C=\{(x,x): x\in \mathbb R\}$. Then $A+B=\mathbb R^{2}=B+C$ but $B \neq C$. Also $A \cap B= A\cap C=\{0\}$. Remark: if the sum in b) denotes orthogonal sum then b) is true, but I don't think you are referring to orthogonal sums.