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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 $B_q\ne\varnothing$.

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$B$ 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 $n$.

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  • $\begingroup$ Yes, i understand that it is a translation, but i am having a hard time figuring out how to put that into words to prove that it is nonempty $\endgroup$ – H.B Jan 30 at 18:16
  • $\begingroup$ If you have an element, and you translate that element, it is still an element. It can't disappear. $\endgroup$ – Nathaniel Mayer Jan 30 at 18:21
  • $\begingroup$ so this would just be a straight forward proof? $\endgroup$ – H.B Jan 30 at 18:30
  • $\begingroup$ it just follows from since B is not empty then B_q is nonempty since B_q=x+q? $\endgroup$ – H.B Jan 30 at 18:37
  • $\begingroup$ Yeah. It's really immediate $\endgroup$ – Nathaniel Mayer Jan 30 at 18:44

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