# Mathematics

## Introduction to Pure Mathematics

Module code: G5087
Level 4
15 credits in autumn teaching
Teaching method: Workshop, Lecture
Assessment modes: Unseen examination, Coursework

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

• Numbers: introduction of mathematical symbols, natural numbers, integers, rationales, 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.
• Functions and maps: formal definition, finite and infinite sets, Pigeon Hole 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. Examples.
• Real numbers: construction of the set of real numbers as equivalence classes of Cauchy sequences of rational numbers; basic properties.