I am looking at a proposition on page 95 of the below reference.
It states "two affine subsets parallel to $U$ are equivalent or disjoint".
I think I understand what this statement means. But I don't understand how we make this English statement from the proposition below:
Suppose $U$ is a subspace of $V$ and $v,w,\in V$. Then the following are equivalent.
(1) $v-w\in U$
(2) $v+U=w+U$
(3) $(v+U)\cap(w+U)\neq\emptyset$
$\textbf{My Question:}$ I understood the proof of showing the equivalence among (a), (b), and (c). But how does showing this equate to the above quoted statement?
Reference:
Axler, Sheldon J. $\textit{Linear Algebra Done Right}$, New York: Springer, 2015.
Best Answer
3) implies 2) says if $v+U$ and $w+U$ are not disjoint then they are equal. This is precisely the assertion you are asked to prove.