6.2.6 Binary relations

Problem: Show that the relation \( x \equiv y(\bmod \mathbb{Z}) \), meaning that \( x-y \in \mathbb{Z} \), gives an equivalence on the set of real numbers \( R \), and construct an explicit bijection between the quotient set \( R / \mathbb{Z} \) and the unit circle \( S^{1} \stackrel{\text { def }}{=}\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2}=1\right\} \) in \( \mathbb{R}^{2} \). Does the standard order in \( \mathbb{R} \) induce any order in \( \mathbb{R} / \mathbb{Z} \) ? Construct an explicit bijection between the functions \( S^{1} \rightarrow \mathbb{R} \) and the periodic functions \( \mathbb{R} \rightarrow \mathbb{R} \) of period 1 .