# Mathematics

## MTH1001 - Mathematical Structures (2019)

MODULE TITLE CREDIT VALUE Mathematical Structures 30 MTH1001 Prof Nigel Byott (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 11 0
 Number of Students Taking Module (anticipated) 180
DESCRIPTION - summary of the module content

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. You will use these methods to prove rigorous general results about the convergence of sequences and series, thereby justifying the techniques developed in MTH1002 and laying the foundations for a deeper study of Analysis in MTH2001. You will also learn the definitions and properties of abstract algebraic structures such as groups and vector spaces. These ideas will be developed further in MTH2002. 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.

AIMS - intentions of the module

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 finite-dimensional 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 second-year streams Analysis and Algebra, and subsequent pure mathematics modules in years 3 and 4.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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 open-ended 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 self-management and time management skills.

SYLLABUS PLAN - summary of the structure and academic content of the module

- 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; n-dimensional spaces over C (resp. R) are isomorphic to C^n (resp. R^n).

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 76 224
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled Learning and Teaching Activities 66 Lectures Scheduled Learning and Teaching Activities 10 Tutorials Guided Independent Study 224 Reading lecture notes; working exercises

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
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
Mid-Term Tests 2 x 1 hour All Feedback on marked sheets, class feedback

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 0 100 0
DETAILS OF SUMMATIVE ASSESSMENT
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

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
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

RE-ASSESSMENT NOTES

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%.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Type Author Title Edition Publisher Year ISBN Search
Set Liebeck, M. A Concise Introduction to Pure Mathematics 3rd Chapman & Hall/CRC Press 2010 978-1439835982 [Library]
Set Allenby, R.B.J.T. Numbers and Proofs Arnold 1997 000-0-340-67653-1 [Library]
Set Stewart, J. Calculus 5th Brooks/Cole 2003 000-0-534-27408-0 [Library]
Set McGregor, C., Nimmo, J. & Stothers, W. Fundamentals of University Mathematics 2nd Horwood, Chichester 2000 000-1-898-56310-1 [Library]
Set Allenby, R.B. Linear Algebra, Modular Mathematics Arnold 1995 000-0-340-61044-1 [Library]
Set Hamilton, A.G. Linear Algebra: An Introduction with Concurrent Examples Cambridge University Press 1989 000-0-521-32517-X [Library]
Set Jordan, C. and Jordan, D. A. Groups Arnold 1994 0-340-61045-X [Library]
Set Houston, K. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics 1st Cambridge University Press 2009 978-0521719780 [Library]
Set Thomas, G., Weir, M., Hass, J. Thomas' Calculus 12th Pearson 2010 978-0321643636 [Library]
Set Lipschutz, S., Lipson, M. Schaum's Outlines: Linear Algebra 4th McGraw-Hill 2008 978-0071543521 [Library]
CREDIT VALUE ECTS VALUE 30 15
PRE-REQUISITE MODULES None None
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 4 No Tuesday 10 July 2018 Thursday 14 November 2019
KEY WORDS SEARCH Proof; Logic; Number Systems; Symmetries; Groups; Vectors; Matrices; Geometry; Linear Algebra