Why we need to show that two affine subsets parallel to $U$ are equal or disjoint before defining addition and scalar multiplication on $V/U$ I am reading the textbook Linear Algebra Done Right Chapter 3 section E on Products and Quotients of Vectors Spaces.
The goal is to make $V/U$ into a vector space. So, my understanding is that we need to define addition and scalar multiplication of the quotient space, and show that it is a vector space.
I can understand the definition of addition and scalar multiplication on $V/U$:
$$(v+U) + (w+U) = (v+w)+U$$
$$\lambda (v+U) = (\lambda v) + U$$
But the author said to do this (have the above definition), we need the following result.

Suppose $U$ is a subspace of $V$ and $v,w \in V$. Then the following are equivalent:
  $$v-w \in U$$
$$v+U=w+U$$
$$(v+U) \cap (w+U) \neq \emptyset$$

Why the above result is needed to definethe addition and scalar multiplication?
 A: Further to J.W. Tanner's comment, let us consider an example which might help to illustrate why the above result is necessary. Specifically, why it allows for a well-defined notion of addition in $V/U.$ That is, a definition of addition on $V/U$ that 'makes sense.' 
Let us take the situation in Example 3.80 in Axler as our example. That is, let $U = \left\{ (x, 2x) \in \mathbb R^2 | x \in \mathbb R \right\}$. Then $U$ is the line through the origin in $\mathbb R^2$ with slope $2$. Thus $(17,20)+U$ is the line in $\mathbb R^2$ that contains the point $(17,20)$ and has slope $2$. By definition, $(17,20)+U$ is an affine subset that is parallel to $U$. Let us take another point on this line, say $(7,0)$ and consider the element $(7,0) + U$. Since $(7,0)$ is a point on the same line, we see that $(17,20)+U$ and $(7,0)+U$ represent the same element of $V/U$. 
Now suppose we were to add each of these affine subsets to some other affine subset, say $(3,10)+U$, using the definition of addition on $V/U$. Then we get $((17,20)+U) + ((3,10)+U) = (20,30)+U$ in the first case, and $((7,0)+U)+((3,10)+U) = (10,10)+U$ in the second case. The question is, how do we know that $(20,30)+U$ and $(10,10)+U$ represent the same element? Well, using the first equivalence in the result, we see that $(20,30)-(10,10)= (10,20) \in U,$ which implies that $(20,30)+U = (10,10)+U$. That is, they do in fact represent the same element. So hopefully this goes to show why the above result contributes to a well-defined notion of addition in $V/U$. A similar story holds for scalar multiplication.
Another thing you might like to do is carefully read the proof of 3.87 (Quotient Space is a Vector Space). The above result is critical in proving that the provided definitions of addition and scalar multiplication make sense. 
