Topics in Pure Mathematics module (MA12002)

This module provides an introduction to various areas of pure maths – logic, proof, group theory, and number theory.

Mathematical logic underpins almost all mathematics. Many people use it when they do mathematical calculations without realising it. Here you will study the basics of logical propositions. This will include negation, conjunction, disjunction, implication, and equivalence. You will analyse logical statements using truth tables. You will be able to apply this to justify methods of proof when you meet them later in the module.

Mathematical proofs are arguments that show that a mathematical statement is true. You will study rigorous proof techniques. These will include:

• constructive proofs
• disproof by counterexample
• proof by contradiction
• proof by contrapositive
• proof by induction

These methods are essential for validating mathematical concepts.

In group theory you will explore groups and their key properties. You'll learn to use Cayley tables to understand group structures. You will apply groups to understand permutations and cycles.

Number theory is an area of maths where we restrict the numbers used to whole numbers only, i.e. not fractions or decimals. In number theory you will discover the fascinating world of integers and divisibility. You will learn about solving equations to get integer solutions. You will find out how to factorise numbers into products of prime numbers and understand properties of divisibility. You'll also study continued fractions, and use modular arithmetic.

What you will learn

In this module you will:

• learn the structure of mathematical proofs and the rules of logic that underpin them
• develop an understanding of the structure of groups
• learn key results in number theory that can be applied to solve problems

By the end of this module you will be able to:

• use the rules of logic to prove logical relations
• prove mathematical results using a range of techniques
• show properties of groups
• use results in number theory to determine integer solutions of equations

Assignments / assessment

• coursework (50%)
• final exam (50%)

Teaching methods / timetable

• three one-hour lectures weekly
  • key points of the week's content will be discussed
  • lecture notes covering the full module content will be available before classes
  • in-class time will be prioritised for interactive discussion
• two hours of tutorials weekly
  • solve problems individually and in groups
  • support with difficulties will be provided by your lecturers and peers