Suppose that $\mathbb P$ is the uniform distribution on $[0,1)$. Partition the interval $[0,1)$ into an equivalence class such that $x\sim y$ ($x$ is equivalent to $y$) if $x-y\in\mathbb Q$, the set of rational numbers.
(a) Show that $\sim$ is an equivalence relation.
(b) Show that $\mathbb{Q}\cap[0,1)$ is an equivalence class.
(c) Find $(\pi/10 + 5/6) \pmod 1$ up to $10$ decimal places.
(d) Given a subset of $A$ of $[0,1)$ and $x\in[0,1)$, define $A_x = x + A = \{(x+a) \pmod{1} \;|\;a\in A\}$. Then $A_x\subset[0,1)$.
(i) Show that ${\pi\over10} + \mathbb{Q}\cap[0,1)$ is an equivalence class.
(ii) Show that $x+\mathbb{Q}\cap[0,1)$ is an equivalence class for any $x\in[0,1)\setminus\mathbb{Q}$.
Ive already answered problem 1 a-d. Now i must prove 2 part a) I'm stuck.
Given $1$, by the Axiom of Choice, there exists a nonempty set $B\subset[0,1)$ such that $B$ contains exactly one member of each equivalence class. Prove each of the following:
(a) Suppose that $q\in\mathbb{Q}\cap[0,1)$. Show that $B_q\ne\varnothing$.
$B$ is nonempty, and $B_q$ is just a translation of $B$ by $q$.
Choose $x \in B$. Then $x + q \mod 1 = x+q - n \in B_q$ for some integer $n$.