# Introduction to Pure Mathematics (G5087)

15 credits, Level 4

Autumn teaching

In this module, the topics you will cover will include:

• numbers; introduction of mathematical symbols, natural numbers, integers, rationals, real numbers, basic number algebra
• ordering, inequalities, absolute value (modulus), homogeneity, triangle inequality
• concept of algebraic structure, groups
• sequences, induction principle, well-ordering principle, sums, products, factorials, Fibonacci numbers, fractions
• irrational roots of integers, divisibility, prime numbers, Euclidean Division, highest common factor, Euclidean algorithm, number theory, atomic property of primes, coprime factorisation, fundamental theorem of arithmetic, square-free numbers
• logic; concept of proof, logical argument, direct proof, propositional manipulation, basic logic, and, or, not, implication, contraposition, contradiction, logical equivalence, quantifiers
• operations with sets; equality, intersection, difference, union, empty set, ordered pairs, cartesian products, power set
• counting; maps and functions, distinguished functions, injections, surjections, bijections, one-to-one correspondences, pigeonhole principle, counting the power set, counting subsets of the power set, cherry picking, binomial coefficients, binomial formula, combinatorics, inclusion-exclusion formula, permutations, counting maps
• functions and maps; formal definition, finite and infinite sets, pigeonhole principle revisited, counterimage, inverse functions, partial inverses
• relations; relations, equivalence relations, modular arithmetic and quotient sets, order relations, partial ordering, total ordering, linear ordering
• rigorous extension of N to Z and Q
• rings, fields.

### Teaching and assessment

We’re currently reviewing teaching and assessment of our modules in light of the COVID-19 situation. We’ll publish the latest information as soon as possible.

This module is approximately 150 hours of work. This breaks down into about 33 hours of contact time and about 117 hours of independent study. The University may make minor variations to the contact hours for operational reasons, including timetabling requirements.

This module is running in the academic year 2020/21. We also plan to offer it in future academic years. However, there may be changes to this module in response to COVID-19, or due to staff availability, student demand or updates to our curriculum. We’ll make sure to let our applicants know of material changes to modules at the earliest opportunity.

It may not be possible to take some module combinations due to timetabling constraints. The structure of some courses means that the modules you choose first may determine whether later modules are core or optional.