Mathematical Structures - 2023 entry

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 of 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 MTH1002 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 fundemental 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.

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.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

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 interactive theorem prover

Discipline Specific Skills and Knowledge:

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;

Personal and Key Transferable/ Employment Skills and Knowledge:

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.

- Writing proffs using the Lean interactive theorem prover

- Sets; relations; functions; countability; logic; proof;

- Primes; elementary number theory;

- Limits of sequences; convergenence 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.

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only.  For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a written exam (worth 50% of the module mark) for each term where the mark for the term is less than 40%. As it is a referral, the module mark will be capped at 40%.

ELE – http://vle.exeter.ac.uk

Reading list for this module: