# module specification

## MA4030 - Mathematical Proofs and Structure (2018/19)

Module specification Module approved to run in 2018/19
Module title Mathematical Proofs and Structure
Module level Certificate (04)
Credit rating for module 30
School School of Computing and Digital Media
Total study hours 300

 210 hours Guided independent study 90 hours Scheduled learning & teaching activities
Assessment components
Type Weighting Qualifying mark Description
Coursework 20%   Reflective Learning (Individual)
Coursework 30%   Problem sheets (Individual)
In-Course Test 10%   Unseen Test (1 hour)
In-Course Test 40%   Unseen Exam (2 hours)
Running in 2018/19
Period Campus Day Time Module Leader
Year North Thursday Afternoon

## Module summary

This module develops the skills necessary to support academic study at degree level. It will also develop reflective learning and action planning via the Personal Development Planning (PDP) process. The first term topics will look into history of mathematics , development of modern number system and introduce idea of mathematical proofs. Different proof techniques will be covered using  examples from Set Theory and Number Theory.
The topics covered in the second term part of this module is to introduces the main ideas of graph theory and includes a variety of algorithms.

## Syllabus

• Origins of mathematics (Babylonian Mathematics; Egyptian Mathematics; the ancient Greeks and the introduction of proof);
•  Development of the modern number system; non traditional approaches to mathematics (e.g. Vedic mathematics). LO1
• Proof Techniques in Mathematics. Various proof methods will be introduced: deduction, induction, direct proof, contradiction, contrapositive, well-ordered principle via topics in Set and Number Theory below.
•  Number theory topics. Divisibility, Prime decomposition, Quotient- Remainder Theorem, Euclid’s Algorithm, Modular arithmetic.
• Definitions, adjacency and incidence matrices LO2,LO3
• Particular graphs, graphic sequences, walks, paths, trails, cycles.
• Disconnecting sets, separating sets, edge-connectivity, vertex connectivity
• Counting walks, Eulerian graphs, Hamiltonian graphs
• Planar graphs– Euler’s formula, Kuratowski’s  theorem, dual graphs
• Independent sets,  Independence number
• Vertex colourings,  Chromatic number and  Chromatic polynomials. LO4

## Balance of independent study and scheduled teaching activity

The module will be taught by a mixture of lectures, workshops and self study practical exercises. The lectures will be used to introduce the various concepts and principles of the module and their strengths in applications. Lectures will be followed by workshops.

The workshops will afford students the opportunity to work in small groups on exercises related to previously taught material. The students will be able to present previously completed exercises for comment from the lecturer and other students.  In this class time students will also be encouraged to explore and experiment with the concepts and techniques to encourage their own sense of mathematical creativity.

Students will be expected to spend time on unsupervised work, for example, private study of problem sheets and in the preparation of coursework (219 hours).  A framework will be put in place to encourage disciplined learning through student self-awareness of progress in volume of work, understanding, attendance and punctuality.

In addition to standard VLE presence there will be links available for further readings and discussion groups.

## Learning outcomes

LO1: Be familiar with the historical developments of mathematics from the origin to modern times.
LO2: Understand different proof techniques and be able to apply them.
LO3: Be familiar with the basic results from the Number Theory, Set Theory.
LO4: Be able to identify the main properties of given graphs and be able to interpret and evaluate the outcomes of algorithms.
LO5: Reflecting on the formative feedbacks received on all level 3 modules.

## Assessment strategy

The assessment for this module consists of two courseworks (50% combined) and two tests (50% combined).
The coursework will require students to:

produce an account of students’ learning so far, experience on the course, the feedback received in modules and reflection on formative feedback; LO5

solve problem sheets on different proof techniques covering. LO1-2

There will be a progression test covering LO3, LO4 which will give students opportunity to demonstrate their understanding of selection of topics.

The final assessment will be an exam where students will be tested on LO4.

## Bibliography

Core Text:

Johnson, D. L.(1998), Elements of Logic via Numbers and Sets, Springer-Verlag
R.J.Wilson(1996), Introduction to Graph Theory (4-th edition) Longman