Mathematical Structures

Module title	Mathematical Structures
Module code	INT1201
Academic year	2022/3
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 language of sets and functions, the theory of groups and vector spaces, and you will learn how to construct clear and logically correct proofs. 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 MTH2008, MTH2010 and MTH 2011 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.

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
6. Read and write formal proofs using an interactive theorem prover

ILO: Discipline-specific skills

On successfully completing the module you will be able to...

7. Evaluate the importance of abstract algebraic structures in unifying and generalising disparate situations exhibiting similar mathematical properties
8. 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...

9. Formulate and express precise and rigorous arguments, based on explicitly stated assumptions
10. Reason using abstract ideas and communicate reasoning effectively in writing
11. Use learning resources appropriately
12. Exhibit self-management and time management skills

Syllabus plan

Writing proofs using the Lean interactive theorem prover;

Sets; relations; functions; countability; logic; proof.

Primes; elementary number theory.

Limits of sequences; convergence of series;

Groups; examples; basic proofs; homomorphisms & isomorphisms;

Vector spaces; linear independence; spanning; bases; linear maps; isomorphisms; n-dimensional spaces over R are isomorphic to 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
120	180

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	44	Small group lessons
Scheduled learning and teaching activities	10	Tutorials
Guided independent study	180	Reading lecture notes; working exercises

Assessment

Formative assessment

Form of assessment	Size of the assessment (eg length / duration)	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

Summative assessment (% of credit)

Coursework	Written exams	Practical exams
10	90

Details of summative assessment

Form of assessment	% of credit	Size of the assessment (eg length / duration)	ILOs assessed	Feedback method
Written Exam A - Closed book (Jan)	45	2 Hours	All	Via SRS
Written Exam B - Closed book (May)	45	2 Hours	All	Via SRS
Computer-based proof A	5	5 x 1 hour	All	Online tools
Computer-based proof B	5	5 x 1 hour	All	Online tools

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 Examination - closed book	Ref/Def Examination	All	During next exam period
Computer-based proof	Computer-based proof		During next exam period

Re-assessment notes

Deferral– if you miss an assessment for reasons judged legitimate by the Mitigation Committee, the applicable assessment will normally be deferred. Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the assessment mark will be uncapped.

Referral– if you have failed the module overall (i.e. a final overall module mark of less than 40%) you will be required to take a referral exam. Only your performance in this exam will count towards your final module grade. A grade of 40% will be awarded if the examination is passed.

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
Houston, K	How to think like a mathematician: A companion to undergraduate mathematics	1st	Cambridge University Press	2009	978-0521719780

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 – https://vle.exeter.ac.uk/course/view.php?id=7506

Module has an active ELE page

Key words search

proof; logic; number systems; symmetries; groups; vectors; matrices; geometry; linear algebra.

Credit value	30
Module ECTS	15
Module pre-requisites	None
Module co-requisites	None
NQF level (module)	4
Available as distance learning?	No
Origin date	14/08/2017
Last revision date	19/07/2022