Introduction to Pure Mathematics

Module code: G5087
Level 4
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.