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MTH1001  Mathematical Structures (2020)
MODULE TITLE  Mathematical Structures  CREDIT VALUE  30 

MODULE CODE  MTH1001  MODULE CONVENER  Prof Nigel Byott (Coordinator) 
DURATION: TERM  1  2  3 

DURATION: WEEKS  11  11  0 
Number of Students Taking Module (anticipated)  200 

A key aspect of mathematics is its ability to unify and generalise disparate situations exhibiting similar properties by developing the concepts and language to describe the common features abstractly and reason about them rigorously. In this module, you will be introduced to the language of sets and functions which underpins of all modern pure mathematics, and will learn how to use it to construct clear and logically correct mathematical proofs. The content goes beyond mathematics taught at Alevel: you will learn and use methods to prove rigorous general results about the convergence of sequences and series, justifying the techniques developed in MTH1002 and laying the foundations for a deeper study of Analysis in MTH2008. You will also learn the definitions and properties of abstract algebraic structures such as groups and vector spaces. These ideas are developed further in MTH2010 and MTH2011. The material in this module is fundamental to many other modules in the mathematics degree programmes. It underpins the topics you will see in more advanced modules in pure mathematics and enables a deeper understanding and rigorous justification of the mathematical tools you will meet in more applied mathematics modules and which are widely used in physics, economics, and many other disciplines.
The purpose of this module is to provide you with an introduction to axiomatic reasoning in mathematics, particularly in relation to the perspective adopted by modern algebra and analysis. The building blocks of mathematics will be developed, from sets and functions through to proving key properties of the standard number systems. We will introduce and explore the abstract definition of a group, and rigorously prove standard results in the theory of groups, before progressing to consider vector spaces, both in the abstract and with a specific focus on finitedimensional vector spaces over the real and complex numbers. The ideas and techniques of this module are essential to the further development of these themes in the two secondyear streams Analysis and Algebra, and subsequent pure mathematics modules in years 3 and 4.
On successful completion of this module, you should be able to:
Module Specific Skills and Knowledge:
1 read, write and evaluate expressions in formal logic relating to a wide variety of mathematical contexts;
2 use accurately the abstract language of sets, relations, functions and their mathematical properties;
3 identify and use common methods of proof and understand their foundations in the logical and axiomatic basis of modern mathematics;
4 state and apply properties of familiar number systems (N, Z, Z/nZ, Q, R, C) and the logical relationships between these properties;
5 recall key definitions, theorems and proofs in the theory of groups and vector spaces;
Discipline Specific Skills and Knowledge:
6 evaluate the importance of abstract algebraic structures in unifying and generalising disparate situations exhibiting similar mathematical properties;
7 explore openended problems independently and clearly state their findings with appropriate justification;
Personal and Key Transferable/ Employment Skills and Knowledge:
8 formulate and express precise and rigorous arguments, based on explicitly stated assumptions;
9 reason using abstract ideas and communicate reasoning effectively in writing;
10 use learning resources appropriately;
11 exhibit selfmanagement and time management skills.
 Sets; relations; functions; countability; logic; proof;
 Primes; elementary number theory;
 Topology of the real and complex numbers; limits of sequences; power series; radius of convergence;
 Groups; examples; basic proofs; homomorphisms & isomorphisms;
 Vector spaces; linear independence; spanning; bases; linear maps; isomorphisms; ndimensional spaces over C (resp. R) are isomorphic to C^n (resp. R^n).
Scheduled Learning & Teaching Activities  76.00  Guided Independent Study  224.00  Placement / Study Abroad 

Category  Hours of study time  Description 
Scheduled Learning and Teaching Activities  66  Lectures 
Scheduled Learning and Teaching Activities  10  Tutorials 
Guided Independent Study  224 
Studying additional recordings complementing lectures, and reading material, example sheets and revision.

Form of Assessment  Size of Assessment (e.g. duration/length)  ILOs Assessed  Feedback Method 

Exercise Sheets  10 x 10 hours  All  Tutorial; model answers provided on ELE and discussed in class 
MidTerm Tests  2 x 1 hour  All  Feedback on marked sheets, class feedback 
Coursework  0  Written Exams  100  Practical Exams  0 

Form of Assessment  % of Credit  Size of Assessment (e.g. duration/length)  ILOs Assessed  Feedback Method 

Written Exam A  Closed Book (Jan)  50  2 hours  All  Via SRS 
Written Exam B  Closed Book (May)  50  2 hours  All  Via SRS 
Original Form of Assessment  Form of Reassessment  ILOs Reassessed  Time Scale for Rereassessment 

Written Exam A  Closed Book  Ref/Def Exam A (50%, 2hr)  All  August Ref/Def Period 
Written Exam B  Closed Book  Written Exam B (50%, 2hr)  All  August Ref/Def Period 
In the case of module referral, the higher of the original assessment and the reassessment will be recorded for each component mark. In the case of module referral, the final mark for the module reassessment will be capped at 40%.
information that you are expected to consult. Further guidance will be provided by the Module Convener
ELE – http://vle.exeter.ac.uk
Reading list for this module:
Type  Author  Title  Edition  Publisher  Year  ISBN  Search 

Set  Liebeck, M.  A Concise Introduction to Pure Mathematics  3rd  Chapman & Hall/CRC Press  2010  9781439835982  [Library] 
Set  Allenby, R.B.J.T.  Numbers and Proofs  Arnold  1997  0000340676531  [Library]  
Set  Stewart, J.  Calculus  5th  Brooks/Cole  2003  0000534274080  [Library] 
Set  McGregor, C., Nimmo, J. & Stothers, W.  Fundamentals of University Mathematics  2nd  Horwood, Chichester  2000  0001898563101  [Library] 
Set  Allenby, R.B.  Linear Algebra, Modular Mathematics  Arnold  1995  0000340610441  [Library]  
Set  Hamilton, A.G.  Linear Algebra: An Introduction with Concurrent Examples  Cambridge University Press  1989  000052132517X  [Library]  
Set  Jordan, C. and Jordan, D. A.  Groups  Arnold  1994  034061045X  [Library]  
Set  Houston, K.  How to Think Like a Mathematician: A Companion to Undergraduate Mathematics  1st  Cambridge University Press  2009  9780521719780  [Library] 
Set  Thomas, G., Weir, M., Hass, J.  Thomas' Calculus  12th  Pearson  2010  9780321643636  [Library] 
Set  Lipschutz, S., Lipson, M.  Schaum's Outlines: Linear Algebra  4th  McGrawHill  2008  9780071543521  [Library] 
CREDIT VALUE  30  ECTS VALUE  15 

PREREQUISITE MODULES  None 

COREQUISITE MODULES  None 
NQF LEVEL (FHEQ)  4  AVAILABLE AS DISTANCE LEARNING  No 

ORIGIN DATE  Tuesday 10 July 2018  LAST REVISION DATE  Wednesday 15 July 2020 
KEY WORDS SEARCH  Proof; Logic; Number Systems; Symmetries; Groups; Vectors; Matrices; Geometry; Linear Algebra 
