Description
Mathematical Structures
Module title | Mathematical Structures |
---|---|
Module code | INT1201 |
Academic year | 2018/9 |
Credits | 30 |
Module staff |
Duration: Term | 1 | 2 | 3 |
---|---|---|---|
Duration: Weeks | 11 | 11 |
Number students taking module (anticipated) | 20 |
---|
Description - summary of the module content
Module description
A key success of mathematics throughout its history has been its ability to unify and generalise disparate situations exhibiting similar mathematical properties through the use of abstract algebraic structures. In this module, you will explore key developments along that journey, including the theory of groups and vector spaces, and you will learn how to develop proofs and present your reasoning clearly.
Module 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. This module is equivalent to MTH1001 and students will join MTH1001 for lectures.
Intended Learning Outcomes (ILOs)
ILO: Module-specific skills
On successfully completing the module you will be able to...
- 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
ILO: Discipline-specific skills
On successfully completing the module you will be able to...
- 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
ILO: Personal and key skills
On successfully completing the module you will be able to...
- 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
Syllabus plan
Sets; relations; functions; countability; logic; proof.
Primes; elementary number theory.
Groups; examples; basic proofs; homomorphisms & isomorphisms; Cayley's Theorem.
Topology of the real and complex numbers; limits of sequences; power series; radius of convergence.
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 and Teaching Activities | Guided independent study | Placement / study abroad |
---|---|---|
106 | 194 |
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 | 40 | Small group lessons |
Guided independent study | 194 | Reading lecture notes; working exercises |
Assessment
Formative assessment
Form of assessment | Size of the assessment (eg length / duration) | ILOs assessed | Feedback method |
---|---|---|---|
Exercise sheets | 10x10 Hours | All ILOs | Tutorial; model answers provided on ELE and discussed in class |
Summative assessment (% of credit)
Coursework | Written exams | Practical exams |
---|---|---|
0 | 100 | 0 |
Details of summative assessment
Form of assessment | % of credit | Size of the assessment (eg length / duration) | ILOs assessed | Feedback method |
---|---|---|---|---|
Written exam - Closed book (Jan) | 30 | 2 hours | All ILOs | Via SRS |
Written exam - Closed book (May) | 70 | 2 Hours | All ILOs | Via SRS |
Re-assessment
Details of re-assessment (where required by referral or deferral)
Original form of assessment | Form of re-assessment | ILOs re-assessed | Timescale for re-assessment |
---|---|---|---|
Written exam - closed book | Ref/Def exam | All | During next exam period |
Re-assessment notes
Referred and deferred assessment will normally be by examination. The module mark is calculated solely from the mark on the referred/deferred exam. For referred candidates, this mark is capped at 40%.
Resources
Indicative learning resources - Basic reading
Core Text:
Author |
Title |
Edition |
Publisher |
Year |
ISBN |
Thomas, G, Weir, M, Hass, J |
Thomas' Calculus |
13th |
Pearson |
2016 |
978-1292089799 |
Additional Recommended Reading for this module:
Author |
Title |
Edition |
Publisher |
Year |
ISBN |
Liebeck M. |
A Concise Introduction to Pure Mathematics |
3rd |
Chapman & Hall/CRC Press |
2010 |
978-1439835982 |
Allenby R.B.J.T. |
Numbers and Proofs |
|
Arnold |
1997 |
000-0-340-67653-1 |
Stewart J. |
Calculus |
5th |
Brooks/Cole |
2003 |
000-0-534-27408-0 |
McGregor C., Nimmo J. & Stothers W. |
Fundamentals of University Mathematics |
2nd |
Horwood, Chichester |
2000 |
000-1-898-56310-1 |
Allenby R.B. |
Linear Algebra, Modular Mathematics |
|
Arnold |
1995 |
000-0-340-61044-1 |
Hamilton A.G. |
Linear Algebra: an introduction with concurrent examples |
|
Cambridge University Press |
1989 |
000-0-521-32517-X |
Jordan, C. and Jordan, D A. |
Groups |
|
Arnold |
1994 |
0-340-61045-X |
Lipschutz, S, Lipson, M |
Schaum's outlines: linear algebra |
4th |
Mc-Graw-Hill |
2008 |
978-0071543521 |
Indicative learning resources - Web based and electronic resources
ELE – http://vle.exeter.ac.uk
Module has an active ELE page
Credit value | 30 |
---|---|
Module ECTS | 15 |
NQF level (module) | 4 |
Available as distance learning? | No |
Origin date | 14/08/2017 |