Learning Functional Programming

Chapter 2. Mathematical Preliminaries

This chapter presents some material that you will need to know (and may well have learned at some point in the past). It will mostly cover basic facts about functions both from a mathematical and a computer science point of view. Feel free to skip this chapter and move on to Chapter 3 if you don’t need a mathematical refresher.

Set Theory

Set theory is a huge field. For our purposes, we just need a few basic definitions from set theory. You would think that set theory provides a definition of set. Actually, set is taken as an undefined term in set theory and we informally think of it as a collection of objects. We can talk about the set of even numbers, the set that consists of 0, 1, and 2, the set that contains my mother and my father; anything we can conceive of as a collection is a set.¹ We generally denote sets by capital letters A, B, C, etc. We call the things in a set, elements. If every element of A is also an element of B, we say that A is a subset of B or A is contained in B, as you’ll see in Figure 2-1.

Figure 2-1. A is a subset of B

The intersection of two sets is the set that contains all the elements that are in both A and B, as you will see in Figure 2-2.

Figure 2-2. The intersection of A and B

The union of A and B is the ...