Study information

Mathematical Structures

Module title Mathematical Structures INT1201 2023/4 30
 Duration: Term Duration: Weeks 1 2 3 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 the language of logic, sets, and functions, which underpins all of modern pure mathematics, and will learn how to use it to construct clear and logically correct mathematical proofsThe 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 MTH2008You will also learn the definition and properties of abstract algebraic structures such as groups and vector spacesThese ideas are developed further in MTH2010 and MTH2011The 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 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 activities and teaching methods (given in hours of study time)

Scheduled Learning and Teaching ActivitiesGuided independent studyPlacement / study abroad
120180

Details of learning activities and teaching methods

CategoryHours of study timeDescription
Scheduled learning and teaching activities66Lectures
Scheduled learning and teaching activities44Small group lessons
Scheduled learning and teaching activities10Tutorials
Guided independent study180Reading lecture notes; working exercises

Formative assessment

Form of assessmentSize of the assessment (eg length / duration)ILOs assessedFeedback method
Exercise sheets10 x 10 Hours 1-5, 7-12Tutorial; model answers provided on ELE and discussed in class
Computer-based proofs10 x 1 hourAllOnline Tools

Summative assessment (% of credit)

CourseworkWritten examsPractical exams
100

Details of summative assessment

Form of assessment% of creditSize of the assessment (eg length / duration)ILOs assessedFeedback method
Written Exam A - Closed book (Jan) 452 hours1-5, 7-12Via SRS
Written Exam B - Closed book (May) 452 Hours1-5, 7-12Via SRS
Mid-term Test 151 hour 1-5, 7-12 Written and class feedback
Mid-term Test 2 51 hour1-5, 7-12 Written and class feedback

Details of re-assessment (where required by referral or deferral)

Original form of assessmentForm of re-assessmentILOs re-assessedTimescale for re-assessment
Written Examination - closed bookRef/Def ExaminationAllDuring next exam period
Mid-term test 1 or 2 (deferral only) Written test (1 hour)AllDuring 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 onlyFor 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.

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