# Aluffi exercise 1.6

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?

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.