For subsets $U,W,W'$ of $V$ is $U \oplus W = U \oplus W' \Rightarrow W=W'$ true? I have the following math homework quesion. I have found a solution saying this is false. However I can't seem to find an example or a reason for this to be false. 
The question is:
For subsets $U,W,W'$ of $V$ if $U \oplus W = U \oplus W'$ then is $W=W'$?
Thanks for any help.

No and in $\Bbb R^2$ we have a simple counterexample:
$$\Bbb R^2=\operatorname{span}((1,0))\oplus\operatorname{span}((0,1))=\operatorname{span}((1,0))\oplus\operatorname{span}((1,1))$$
and 
$$\operatorname{span}((0,1))\ne\operatorname{span}((1,1))$$
