Module description

The module has three main aims:

1. to introduce the basic objects of mathematics (numbers, sets and functions), and their properties;
2. to emphasize the fact that mathematics is concerned with proofs, which establish results beyond doubt, and to show you how to construct proofs, how to spot false "proofs", how to use definitions, etc.;
3. to get you involved in the excitement of doing mathematics.

The module description, with the syllabus and learning outcomes, can be found here in the list of modules on the School's web page.

Course information

The course information sheet is here.

Lecture notes

There are ten chapters, on the following topics:

• Introduction: notes, supplementary material
• Sets: notes, supplementary material
• Infinity: notes, supplementary material
• Functions and relations: notes, supplementary material
• Natural numbers: notes, supplementary material
• Integers and rational numbers: notes, supplementary material
• Real numbers: notes, supplementary material
• Complex numbers: notes, supplementary material
• Proofs: notes, supplementary material
• Constructing and debugging proofs: notes, supplementary material

Here is an index to the notes, which might help you to find your way around; and here are the study skills collected into a single document.

Problem sheets

The timing for problem sheets has been changed, starting from Sheet 4. The dates on the sheet will be the week in which the material will be discussed in tutorials; you can hand it in any time between the end of the tutorial and the beginning of the following week's tutorial.

• Sheet 1 (due 1–5 October 2012) and solutions
• Sheet 2 (due 8–12 October 2012) and solutions
• Sheet 3 (due 15–19 October 2012) and solutions
• Sheet 4 (for discussion 22–26 October 2012) and solutions
• Sheet 5 (for discussion 29 October–2 November 2012) and solutions
• Sheet 6 (for discussion 12–16 November 2012) and solutions
• Mark the test! (for discussion 19–23 November 2012) and my comments
• Sheet 7 (for discussion 19–23 November 2012) and solutions
• Sheet 8 (for discussion 26–30 November 2012) and solutions
• Sheet 9 (for discussion 3–7 December 2012) and solutions
• Sheet 10 (for discussion 10–14 December 2012) and solutions

Administrative matters

My response to the questionnaires is here.

Extra questions

A sample exam paper can be found here. Solutions to the sample exam will not be published, but the lecturer is happy to go through your attempts at this paper with you; please email for an appointment.

I have put the sample test and test papers here. I have also posted a list of statements made on test papers; you are asked to explain what is wrong with each, and to try to clear up the misunderstanding of the person who wrote it.

Here are some questions on sets, functions and relations and some questions on numbers that you might like to practise on.

Resources

There is no compulsory textbook for the module. The recommended books are:

• Timothy Gowers, Mathematics: A very short introduction, Oxford University Press, Oxford (2002)
• Kevin Houston, How to think like a mathematician, Cambridge University Press, Cambridge (2009)

I gave the LMS-Gresham lecture about the Mathematical Structures module on 14 May. Slides of the talk are available: screen version, printer version.

Some websites you might find useful:

• Jobs Rated and Oxford College, Emory University: two views on why a mathematics degree is a good thing
• 10 ways to think like a mathematician by Kevin Houston
• Book of Proof by Richard Hammack
• Some Theorems of the Day by Robin Whitty
• Sudoku, logic and proof by Catherine Greenhill
• How big is infinity? by Dennis Wildfogel
• Proofs that all odd numbers are prime
• Logicomix on Amazon
• Cantor, a movie from the University of Miami School of Communication

Peter J. Cameron
17 January 2013