Consider a finite-dimensional vector space, $V$ over a generic field $F$.  

Suppose $v\in V$ and $U$ is a subspace of $V$. Define an affine subset of $V$ as:

> 
> $v+U=\{v+u:u\in U\}$
> 
> 
> 

$\textbf{My Question:}$ I am little confused about the affine subset of a vector space. So, is an affine subset $always$ parallel to $U$? The way the below reference defines the quotient space is that it is the set of all affine subsets of $V$ parallel to $U$. Isn't this tautology? In other words, is there a set of affine subsets of $V$ $not$ parallel to U?

Reference:
Axler, Sheldon J. $\textit{Linear Algebra Done Right}$, New York: Springer, 2015.