This module provides a foundation for all subsequent modules in Pure Mathematics. All of Pure Mathematics is written in the language of sets, functions and relations, and a large part of the module is devoted to gaining familiarity with both reading and writing this language.There will be an introduction to some basic counting principles, and the most important number systems will be introduced. Along the way, a variety of useful and interesting facts will be discussed. The module will include formal proofs and students will be given practice in writing proofs themselves. Topics to be covered will include:• counting problems, binomial coefficients;• the language of set theory;• relations and functions;• countable and uncountable sets.This module is aimed at first-year honours mathematics university students. However, pre-university students (including keen GCSE students) may find that much of the material is accessible. Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area. See also Dr Feinstein's blog http://explainingmaths.wordpress.com/