Chapter 1: Sets, Functions, and Logic
In this chapter, we lay down some of the fundamental concepts regarding discrete mathematics – sets, functions and logic; without a rigorous treatment of these ideas, it would be hard for us to continue. First, we define the concept of a set, and look at properties of sets and operations between sets. We also explore how to count the size of various finite sets with operations applied to them. We then define exactly what a function is, and discuss various classes of functions (injections, bijections and surjections).
We will then look at the common sets of numbers we deal with in the real world in the order we are introduced to them – the natural numbers, the integers, rational (and irrational) numbers, real numbers and complex numbers, and investigate the formal relationship between these sets. We’ll conclude with introducting propositional logic, which will segue nicely into proof techniques in the second chapter.