Prove intersection of these two sets is empty. Define $x\sim y$ iff $x-y\in\mathbb{Q}$.
Let $S = [0, 1)$. Define $A\subset S$ such that $A$ contains exactly one element of each class of equivalence of the relation $x\sim y$.
Let $Q = \mathbb{Q}\cap S$. Chose $q\in Q$ and build the set $A_q$ from it: cut the set $A$ in two sets $1$ and $2$, in the position $1-q$. Translate $1$ to where was $2$, and translate $2$ to where was $1$. Then you will glue such pieces in the position $q$.
In other words, define: $A_q = \{x+q : x\in A\cap[0, 1-q)\}\quad\cup\quad \{x+q-1 : x\in A\cap[1 - q, 1)\}$.
If $p\neq q$, how does one proves $A_q\cap A_p = \emptyset$?

The place I was reading proved in the following way. But I was unable to follow it. Maybe it helps you...
Let $x\in S$ and $x\sim y$ which is in $A$, then:
$$
x\in A_q,\quad\mbox{where }q = \begin{cases}
x-y &  \mbox{if }x\ge y \\
x-y+1 & \mbox{if }x < y 
\end{cases}
$$
Thus if existed $x\in A_q\cap A_p$ then $x-q$ and $x-p$ would be distinct and equivalent elements of $A$, which is a contradiction of the definition of $A$.
 A: The argument that you found is not entirely correct. Here’s a corrected and expanded version.
Suppose that $x\in A_q$ for some $q\in Q$. There are two cases to be considered.


*

*If $x\ge q$, then $x$ must be of the form $y+q$ for some $y\in A\cap[0,1-q)$, so $x-q\in A$. 

*If $x<q$, then $x$ must be of the form $y-1+q$ for some $y\in A\cap[1-q,1)$, so $x-q+1\in A$.
Thus, either $x-q\in A$, or $x-q+1\in A$. Note that $x-q\sim x\sim x-q+1$.
Now suppose that $x\in A_q\cap A_p$ for some $p,q\in Q$. There are four possibilities:


*

*$x-q\in A$ and $x-p\in A$: then $x-q\sim x\sim x-p$, so $x-q\sim x-p$. But $x-q$ and $x-p$ are both in $A$, which contains only one member of each $\sim$-class, so $x-q=x-p$, and therefore $q=p$.  

*$x-q\in A$ and $x-p+1\in A$: then $x-q\sim x\sim x-p+1$, so $x-q\sim x-p+1$. It follows as in the previous case that $x-q=x-p+1$ and hence that $q=p-1$. This, however, is impossible, since $0\le q,p<1$, so this case cannot actually occur. (Recall that $Q\subseteq S\subseteq[0,1)$.)  

*$x-q+1\in A$ and $x-p\in A$: this is just like the previous case.  

*$x-q+1\in A$ and $x-p+1\in A$: this is like the first case.


We conclude that $q=p$, and hence that $A_q\cap A_p=\varnothing$ whenever $q,p\in Q$ and $q\ne p$.
This shows that $\{A_q:q\in Q\}$ is a pairwise disjoint family.
The first part of the proof that you found does not seem to me to be needed in order to show that the sets $A_q$ are pairwise disjoint. It does show, however, that they cover $S$; I’ll expand on that.
Let $x\in S$. $A$ contains a member of each $\sim$-equivalence class, so there is some $y\in $ such that $x\sim y$. Now there are two possibilities: either $x\ge y$, or $x<y$.


*

*If $x\ge y$, let $q=x-y$. Then $y=x-q<1-q$, since $x<1$, so $y\in A\cap[0,1-q)$, and therefore $x=y+q\in A_q$ by definition. 

*If $x<y$, let $q=x-y+1$. Then $y=x+1-q\ge 1-q$, since $x\ge 0$, and therefore $y\in A\cap[1-q),1)$, so that again $x=y+q-1\in A_q$ by definition. 
The two parts together show that $\{A_q:q\in A\}$ is a partition of $S$.
