MA1005: Foundations of Mathematics I

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA1005
External Subject Code 100405
Number of Credits 20
Level L4
Language of Delivery English
Module Leader Dr Ulrich Pennig
Semester Autumn Semester
Academic Year 2022/3

Outline Description of Module

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

On completion of the module a student should be able to

  • 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

How the module will be delivered

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

  • Weekly face to face classes (e.g. labs, lectures, exercise classes)
  • Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills that will be practised and developed

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

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 85 Foundations Of Mathematics I 3
Class Test 15 Foundations Of Mathematics I N/A

Syllabus content

  • 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.

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