# How do I prove $B_{q} \neq \emptyset$.

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 $$n$$.