Introduction

You should check the website regularly for updates to these notes. You can tell if there have been any updates by looking at the date on the front cover.

The formal side of mathematics - that of theorems and proofs - is a major part of the subject and is the main focus of Paper 2. These notes are intended to be a brief introduction to the ideas involved, for the benefit of candidates who have not yet met them within their mathematics classes or within their wider mathematical reading.

Mathematics, in part, is about working out the relationships between (mathematical) statements. And it is very important that everyone (that is, all mathematicians) write and talk about these relationships in the same standard way. It is important because mathematics is expressed formally using a rigorous language and if different people mean different things when they use the same words, then it would be difficult to ensure that everyone is talking about (and agreeing about) the very same things. Learning the basics of the rules and terms used by mathematicians is essential if you are to be able to understand and contribute to mathematics. These notes are written to help introduce you to some of the terms used regularly by mathematicians – specifically the terms we have included in Section 2 of the TMUA specification.

Before you launch into reading through what we have written, there are a few things to keep in mind:

1. This guide is designed to be a brief overview. It is not designed to be an extensive textbook, but it should be enough to allow you to get a good understanding of the topics in Section 2 of the specification. It should also be sufficient to enable you to understand and tackle the questions we will ask in the admissions test.
2. As you read through the guide, make sure you take the time to think through everything very carefully. Many of the ideas set out here are quite subtle and take some time to grasp, so skimming through everything is certainly not enough to ensure you have a good understanding. Rather, you should play around with the ideas as you meet them, and try to come up with your own examples. In other words, read through things actively with a pencil and paper to hand; think carefully about everything, draw your own pictures, write out your own examples, and so on.
3. Another thing you should do is to try to read more widely on the topics set out here. The internet has some good explanations and examples of the ideas we outline and there are some good books available from libraries that might also help.
4. A good way to test if you have understood something is to see if you would be able to explain the ideas to a friend or a class of students. If you get a chance, it is always useful to study the ideas and talk about them with other people – again, maybe with others in your maths classes or with your maths teacher.
5. As you work through these notes, make sure you are aware that the language used by mathematicians has very precise meanings and these do not always coincide with the way words are used in everyday casual contexts. For instance, if I am told that I may have “jelly or cake” for pudding, I would probably assume it meant I could have jelly or cake but not both. However, in mathematics, ‘or’ means ‘one, or the other, or both’, so if I were being offered pudding by a mathematician then I could have both jelly and cake if I wanted.
6. Throughout, we have tried to explain the ideas in at least two different ways: one using the formal notion of truth tables and one using a more intuitive diagrammatic approach. It is worth making sure you understand both approaches and how they relate to each other.
7. There are a few footnotes scattered throughout this guide. Almost all the footnotes concern topics and ideas that are outside the scope of the TMUA but which some readers might find interesting to explore. You can ignore almost all the footnotes if you wish.
8. Finally, a small note of caution: in this specification we deliberately adopt a simple and slightly naïve view of the ideas we are trying to explain rather than one that sets out all the deep subtleties that abound in mathematics and the philosophy of logic/mathematics. We have been as rigorous as necessary to achieve our aims, but you should not take what we have written as the perfect and final word on things and we have deliberately avoided some issues as they would only complicate matters unnecessarily. For instance, some mathematicians or philosophers might take issue with our examples of statements or our notion of truth and so on. For what we aim to achieve, these issues are not relevant, but that is not to say they aren’t interesting.

A final note

We have used boxes throughout these notes to help you navigate.

The relevant part of the specification is found in these sorts of boxes:

Examples in these sorts of boxes

And exercises [we have not given any answers to our exercises] in these sorts of boxes.

We hope to be able to update and, if necessary, correct these notes now and again. Look at the date on the front page to see when these notes were last edited.