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Define a relation $\sim$ on the set $\mathbb{R}$ of real numbers by setting $a \sim b \iff b - a \in \mathbb{Z}$. Prove that this is an equivalence relation, and find a `compelling' description for $\mathbb{R}/\sim$. Do the same for the relation $\approx$ on the plane $\mathbb{R} \times \mathbb{R}$ defined by declaring $(a_1, a_2) \approx (b_1, b_2) \iff b_1 - a_1 \in \mathbb{Z}$ and $b_2 - a_2 \in \mathbb{Z}$.

Here is my attempt.

Given $a \in \mathbb{R}$, we have $a - a = 0 \in \mathbb{Z}$, so $a \sim a$. If $a \sim b$, then $b - a \in \mathbb{Z}$, so $- (b-a) = a - b \in \mathbb{Z}$, so $b \sim a$. Finally, if $a \sim b$ and $b \sim c$, then $b-a, c- b \in \mathbb{Z}$, and hence $a - b, b - c \in \mathbb{Z}$, so we have $a - c = (a - b) + (b - c)$, which is a sum of integers and hence an integer, so $a \sim c$. So $\sim$ is an equivalence relation. $R/\sim$ consists of equivalence classes of the form $[\epsilon] \in R/\sim$ \begin{align*} [\epsilon] = \{x + \epsilon \mid x \in \mathbb{Z}\}, \end{align*} where $\epsilon \in [0,1)$. Similarly, given $(a_1, a_2) \in \mathbb{R} \times \mathbb{R}$, we have $a_1 - a_1 = 0 = a_2 - a_2 \in \mathbb{Z}$, so $(a_1, a_2) \approx (a_1, a_2)$. Suppose $(a_1, a_2) \approx (b_1, b_2)$, so $b_1 - a_1, b_2 - a_2 \in \mathbb{Z}$. But then $- (b_1 - a_1) = a_1 - b_1$ and $-(b_2 - a_2) = a_2 - b_2$ are integers, so $(b_1, b_2) \approx (a_1, a_2)$. Finally, suppose $(a_1, a_2) \approx (b_1, b_2)$ and $(b_1, b_2) \approx (c_1, c_2)$. So $b_1 - a_1 \in \mathbb{Z}$, $b_2 - a_2 \in \mathbb{Z}$, $c_2 - b_2 \in \mathbb{Z}$, and $c_1 - b_1 \in \mathbb{Z}$. We then have \begin{align*} c_1 - a_1 = (c_1 - b_1) + (b_1 - a_1), \; c_2 - a_2 = (c_2 - b_2) + (b_2 - a_2), \end{align*} so both $c_1 - a_1$ and $c_2 - a_2$ are sums of integers and hence integers, so $(a_1, a_2) \approx (c_1, c_2)$. So $\approx$ is an equivalence relation, as desired. The equivalence classes then take the form $[(\epsilon, \delta)]$ where \begin{align*} [(\epsilon, \delta)] = \{(x,y) \mid \mathbb{R} \times \mathbb{R} \mid x = z + \epsilon, \; y = r + \delta, \; r,z \in \mathbb{Z} \}, \end{align*} where $\epsilon, \delta \in [0,1)$.

How does this look? Is there a better characterization for the equivalence classes?

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1 Answer 1

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It looks fine to me. Good job!

The only thing I could fault you on is starting a sentence with a mathematical symbol. But this is just a matter of style.

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