This module will familiarise you with the basic structures of mathematics: Sets, numbers, basic algebraic structures, and the notion of a limit. Moreover you will learn what a mathematical proof is and how to prove a mathematical statement.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

• Know and apply the definitions of basic structures like sets and fields and understand their basic properties.
• Have a working knowledge of the properties of the real and complex numbers
• Understand the concept of supremum/infimum
• Understand the concept of a limit and other basic properties of a sequence
• Know and apply convergence criteria for sequences and series
• Find and write short proofs of basic mathematical statements

54 fifty-minute lectures

10 fifty-minute tutorial classes

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Problem solving

Use of mathematical notation and subject-specific language

Written communication of mathematical ideas

Reading and understanding subject-specific literature

Transferable Skills:
Problem solving

Formative assessment is by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length.  Section B has a choice of three from four equally weighted questions.

• Numbers:  Natural numbers and proof by induction. Review of properties of rational and  real numbers. Introduction of complex numbers and rules for addition, subtraction, multiplication and division. Real and imaginary parts, conjugation, modulus and argument. Geometric interpretation in the complex plane. Statement of the fundamental theorem of algebra.
• Logic, sets and functions: Logic operations, quantifiers and negations.
  Non-axiomatic set theory (operations with sets, power set, Cartesian products, set-theoretic definition of functions).  Injective, surjective and bijective functions, composition of functions. Axiom of choice. (Optional) Finite, countable and uncountable sets.
• Algebraic structures: Group Axioms.  Examples for groups, in particular the symmetric group.   Proof of basic properties. Sub-groups. Equivalence classes and cosets. Homomorphisms.
• Order and Distance: Review of inequalities. The absolute value and its properties
  Supremum and Infimum. Completeness by supremum property.
• Sequences of real and complex numbers and their limits: Definition, proof of convergence/divergence by using the definition, examples, algebra of limits, sandwich theorem, monotone sequences, sub-sequences and the theorem of Bolzano-Weierstrass.  Cauchy-sequences. Completeness by Cauchy sequences.
• Series of real and complex numbers: Convergence and absolute convergence. A selection of convergence criteria.  Product of series. Re-ordering theorem (Optional, no proof). Complex power series Radius of convergence. (Optional) The exponential function as series.  Statement that set of real numbers is not countable with sketch of proof.

Smith, G. 1998. Introductory mathematics: algebra and analysis. Springer. 

Spivak. M. 2008. Calculus. 4th ed. Publish or Perish.

Howie, J.M. 2000. Real analysis. Springer