1Introduction

This is a quick primer intended to introduce philosophy students to a few basic ways of manipulating sets. It also serves as an introduction to constructing mathematical proofs in general.

Mathematics at university is a bit different to what you are likely used to from high school. There is less of an emphasis on calculation and much more of an emphasis on understanding precise definitions and producing rigorous proofs.

The idea of constructing a proof might sound intimidating. Don’t worry—it isn’t. A proof is just a persuasive argument without gaps (or where it is easy to see how the gaps can be completely filled in). We generally proceed one step at a time, and each step can be one of two things:

• (i) An assumption that we have made, explicitly or implicitly.
• (ii) A logical deduction we can make from the previous steps.

An assumption will generally be something you have been explicitly told, or something that is so obvious that nobody could reasonably doubt it. Exactly what counts as “sufficiently obvious” depends on the context. For example, you might appeal to the fact that between any distinct real numbers x and z, there is a number y which is strictly between them. In any philosophy course, this is obvious enough to need no further argument. In an analysis course in a mathematics department, it might be something that needs its own proof (for example, by letting y = x + z2 and showing that this number is in between x and z).

A logical deduction sounds like a complicated thing, but as you will see, it’s actually usually a very simple step. For example, I might have a particular shape, and I’ve already shown that it must be either a triangle or a square. If I can then also show that the shape is not a triangle, then I can conclude that it must be a square. Logical deductions aren’t scary, they’re usually simple, obvious steps like this.

Sometimes, students worry that they won’t be able to do mathematics. But the truth is that anyone can do mathematics with a bit of patience and hard work. That’s not to say that everyone can do really difficult mathematics, the kind that research mathematicians do. I can’t do that kind of thing myself. But anyone who can reason logically can do the basics, and yes, that includes you, the reader.

I’m not teaching you a mathematics course. But I am (most likely) teaching you a philosophy course which involves some proofs or formal results. This primer is designed to get you up to speed with some of the basics which will be useful for understanding these formal results. Specifically, we’re going to learn what sets are and how to manipulate them. The reason we’re learning this is that manipulating sets is essential to many formal results across philosophy and economics (and many other areas besides).

Along the way, I will give you some examples of mathematical proofs so you can see how they work. There are also exercises which ask you to complete a few proofs yourself. Please attempt the exercises. The best way to learn anything in mathematics thoroughly is to do it yourself. Learning mathematics without proving things yourself is like learning to eat without bringing the food to your mouth with your own hands. It’s suitable for children, but you’ll never eat independently that way.