# 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

77%: Lecture
23%: Practical (Workshop)

### Assessment

20%: Coursework (Problem Set, Test)
80%: Examination (Unseen examination)

This module is 150 hours of work. This breaks down into 48 hours of contact time and 102 hours of independent study.

This module is running in the academic year 2017/18. We also plan to offer it in future academic years. It may become unavailable due to staff availability, student demand or updates to our curriculum. We’ll make sure to let our applicants know of such changes to modules at the earliest opportunity.

### Courses

This module is offered on the following courses: