4. Sets, Functions, and Relations

We deal with the complexity of the world by putting things into categories. There are not just hordes of individual creatures. There are dogs, cats, elephants and mice. There are mammals, insects and fish. Animals, vegetables and minerals. Solids, liquids and gases. Things that are red. Big cities. Pleasant memories… . Categories build on categories. They are the subject and the substance of thought.

In mathematics, which operates in its own abstract and rigorous world, categories are modelled by sets. A set is just a collection of elements. Along with logic, sets form the ‘foundation’ of mathematics, just as categories are part of the foundation of day-to-day thought. In this chapter, we study sets and relationships among sets. And, yes, that means we’ll prove theorems about sets!