# Functions and Equivalence Relations

## Abstract

Functions and relations are used throughout mathematics, but the focus in discrete mathematics is a bit different than it was in elementary algebra and calculus. Here we’re more interested in exploring their structural properties than in using them for some other purpose. The chapter opens by looking at properties of functions (one-to-one, onto), at composition of functions, and at identity and inverse functions. It then focuses on relations, particularly equivalence relations and their associated equivalence classes. The chapter ends by looking at how these ideas are used in number extensions—in the construction of \(\mathbb {Z}\) as a set of equivalence classes of \(\mathbb {N}\times \mathbb {N}\), and in the construction of \(\mathbb {Z}_n\), the integers modulo *n*. The next chapter continues the discussion of relations but considers other types that are also of importance to discrete mathematics and computer science.