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This module aims to introduce students to ideas and topics in discrete mathematics, for example set theory, group theory and number theory. These topics are not covered elsewhere in Mathematics courses for such students but are of current importance, in computer science, for example. The module is designed to provide prerequisite knowledge to enable students to take level 6 (subsequently level 7) mathematics courses in these areas.

One of the most powerful ideas in mathematics is abstraction. In this course you will see the importance of the ability to understand and work with abstract concepts and constructions. You will be introduced to the basic language of mathematics, some important algebraic structures, and the idea of a proof. After taking this course, a successful student should be able to take the following Mathematics module options in their third year:

1. Algebra 4: Groups and Rings;
2. Number Theory;
3. Graph Theory and Combinatorics.

The course will be divided into four sections with a balance between abstract ideas and more concrete methods, particularly in number theory. In section 1 we will cover some ideas from set theory and we will introduce the concept of mathematical proof. In section 2 we will study an important algebraic structure called a group. We aim to give a thorough grounding in the basics of group theory and to build confidence working with abstract definitions and concepts. In section 3 we will focus on number theory. In particular, we will introduce Z/(n), the integers modulo n, and we will learn methods to solve equations in this new setting. Finally, in section 4 we will study another important algebraic structure called a field which is a generalisation of the real numbers. We will learn some new examples of fields and some general theory which generalises results learnt from linear algebra. In particular, sections 2 and 4 will provide a good basis for studying Algebra 4.