Chapter 6 Equivalence relations and their applications

In this chapter we change direction somewhat; instead of confining ourselves to either $\mathbb{Z}$ or $\mathbb{N}$ as in the two previous chapters, we shall work in an arbitrary non-empty set for most of the time. You will learn what a relation is, what it means for a relation to be reflexive, symmetric and transitive, and what an equivalence relation is. Moreover, for an integer $m\geqslant 2$, you will find out what a congruence class modulo $m$ is and how to add and multiply two such classes. Finally, you will see how to build $\mathbb{Z}$ from $\mathbb{N}$, $\mathbb{Q}$ from $\mathbb{Z}$, and $\mathbb{C}$ from $\mathbb{R}$.