Description

Mathematical Structures

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)

Details of learning activities and teaching methods

Assessment

Formative assessment

Summative assessment (% of credit)

Details of summative assessment

Re-assessment

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

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.

Resources

Indicative learning resources - Basic reading

Core Text:

 Additional Recommended Reading for this module:

Indicative learning resources - Web based and electronic resources

 ELE – http://vle.exeter.ac.uk

Module has an active ELE page

Key words search

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