Introduction to Pure Mathematics
Module code: G5087
15 credits in autumn teaching
Teaching method: Workshop, Lecture
Assessment modes: Coursework, Unseen examination
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.
Module learning outcomes
- Basic rigorous proofs, with Number Theory as the main source of applications.
- Manipulation of sets and basic counting arguments (using Induction), usage of Binomial Theorem, Exclusion-Inclusion and Pigeon Hole.
- Using the Induction principle in intuitive and rigorous reasoning.
- Elementary logical manipulations: contraposition, contradiction, use of quantifiers, with sets as the main application area.