Hello I need help solving this exercise, I just want a path not a solution.
Consider the equivalence relation $\sim$ on $\mathbb{R}^2$ generated by $(x, y) \sim (x + 1, y) $ and $(x, y) \sim (-x, y + 1).$ Show that there are two real numbers $a$ and $b$ such that the subspace $ [0, a[×[0, b[ $ of $\mathbb{R}$ contains one and only one representative of each equivalence class.
Hint: Since $(x, y) \sim (x + 1, y)$, what upper bound on $a$ can you deduce? More to the point, what points is $(0, 0)$ equivalent to? Now look at what $(0, 1)$ and $(1, 0)$ are equivalent to. What pattern can you see and can you prove it?
In general, first find $a$ and $b$ such that $[0, a) \times [0, b)$ contains one representative of each class and then make $a$ and $b$ smaller, if this is necessary, for the "and only one" condition.