Mathematical Structures

Module title	Mathematical Structures
Module code	INT1201
Academic year	2025/6
Credits	30
Module staff

Duration: Term	1	2	3
Duration: Weeks	11	11

Number students taking module (anticipated)	20

Module description

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 logic, sets, and functions, which underpins 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 A-level: you will learn and use methods to prove rigorous general results about the convergence of sequences and series, justifying the techniques developed in INT1202 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 fundamental 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 logic, 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 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 fundamental 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 mathematical arguments, and express them in a clear and logically correct form.
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

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 activities and teaching methods (given in hours of study time)

Scheduled Learning and Teaching Activities	Guided independent study	Placement / study abroad
110	190	0

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
Guided independent study	190	Reading lecture notes; working exercises

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
Formative tests	8 x 40 minutes	All	Formative scripts marked by the tutor

Summative assessment (% of credit)

Coursework	Written exams	Practical exams
20	80

Details of summative assessment

Form of assessment	% of credit	Size of the assessment (eg length / duration)	ILOs assessed	Feedback method
Written Exam A (Jan)	40	2 hours	All	Via SRS
Written Exam B (May)	40	2 Hours	All	Via SRS
Mid-term Test 1	10	40 minutes	All	Via SRS
Mid-term Test 2	10	40 minutes	All	Via SRS

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 A or B - Closed book	Exam (deferral)	All	Next assessment opportunity
Mid-term test 1 or 2	Test (deferral)	All	Next assessment opportunity
N/A	Referral Exam	All	Next assessment opportunity

Re-assessment notes

Deferral – if you miss an assessment for reasons judged legitimate by the Mitigation Committee, the applicable assessment will normally be deferred. See ‘Details of reassessment’ for the form that assessment usually takes. When deferral occurs, there is ordinarily no change to the overall weighting of that assessment.

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.

Indicative learning resources - Basic reading

Core Text:

Author	Title	Edition	Publisher	Year	ISBN
Thomas, G, Weir, M, Hass, J	Thomas' Calculus	14th	Pearson	2020	978-1292253220
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

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	12/06/2025