MA13210 Foundations
Introductory description
University mathematics introduces progressively more and more abstract ideas and structures, and demands more and more in the way of proof, until by the end of a mathematics degree most of the student's time is occupied with understanding proofs and creating his or her own. This is not because university mathematicians are more pedantic than schoolteachers, but because proof is how one knows things in mathematics, and it is in its proofs that the strength and richness of mathematics is to be found.
Module aims
But learning to deal with abstraction and with proofs takes time. This module aims to bridge the gap between school and university mathematics, by beginning with some rather concrete techniques where the emphasis is on calculation, and gradually moving towards abstraction and proof.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
Number systems
Number systems: Natural numbers, integers. Rationals and real numbers. Existence of irrational numbers. Complex Numbers.
Polar and exponential form of complex numbers. De Moivre's Theorem, n'th roots and roots of unity.
Euclidean algorithm; greatest common divisor and least common multiple.
Prime numbers, existence and uniqueness of prime factorisation. Infiniteness of the set of primes.
Modular arithmetic. Congruence, addition and multiplication modulo
n.
Language and Proof
Proof by induction.
Wellordering Principle.
Proof by contradiction.
Basic set theory: ∩,∪, Venn diagrams and de Morgan's Laws. Cartesian product of sets, power set.
Logical connectives∧, ∨, ⇒ and their relation with ∩, ∪and ⊆. Quantifiers ∀ and ∃.
Sets, functions and relations
Injective, surjective and bijective functions.
Inverse functions.
Relations: equivalence relations, order relations.
Polynomials
Multiplication and long division of polynomials.
Euclidean algorithm for polynomials.
Remainder theorem; a degree n polynomial has at most n roots.
Algebraic and transcendental numbers. Fundamental theorem of
Algebra (statement only).
Counting
Cardinalities, including infinite cardinalities.
Cardinality of the power set of X is greater than cardinality of X.
Russell's paradox.
Countability of the rational numbers, uncountability of the reals.
Transcendental numbers exist!
Learning outcomes
By the end of the module, students should be able to:
 Students will work with number systems and develop fluency with their properties;
 they will learn the language of sets and quantifiers, of functions and relations, and will become familiar with various methods and styles of proof.
Indicative reading list
None of these is the course text, but each would be useful, especially the first:
A.F.Beardon, Algebra and Geometry, CUP, 2005.
I.N. Stewart and D.O. Tall, Foundations of Mathematics, OUP, 1977.
J. A. Green, Sets and Groups; First Course in Algebra, Chapman and Hall, 1995.
Subject specific skills
After taking this module, students will be familiar with the concept of a rigorous mathematical proof and most of commonly used notations.
Transferable skills
Students will work with number systems and develop fluency with their properties; they will learn the language of sets and quantifiers, of functions and relations, and will become familiar with various methods and styles of proof.
Study time
Type  Required 

Lectures  28 sessions of 1 hour (28%) 
Tutorials  8 sessions of 30 minutes (4%) 
Private study  68 hours (68%) 
Total  100 hours 
Private study description
Reviewing lectured material and revising for 10 weekly assignments with 5 fortnightly tests based on them.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

4 Tests  15%  
Multiple choice tests 

Inperson Examination  85%  
exam

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  
exam

Feedback on assessment
Tests will be marked and feedback given after exam.
Courses
This module is Core for:
 Year 1 of UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)

UMAAG100 Undergraduate Mathematics (BSc)
 Year 1 of G100 Mathematics
 Year 1 of G100 Mathematics
 Year 1 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 1 of G100 Mathematics
 Year 1 of G103 Mathematics (MMath)
 Year 1 of G103 Mathematics (MMath)
 Year 1 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 1 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 1 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 1 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 1 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)
 Year 1 of UMAAG101 Undergraduate Mathematics with Intercalated Year