MA4030 - Mathematical Proofs and Structure (2017/18)
|Module specification||Module approved to run in 2017/18|
|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|
|Running in 2017/18||
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.
The module aims to give the students an understanding of origins and history of modern mathematics and develop the idea of mathematical proof techniques via different branches of mathematics, namely Sets, Numbers and Graphs.
- 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).
- 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.
- Set Theory topics. Definition, representation, operations, power sets.
- Relations. Properties, equivalence, partial order.
- Functions Injection, surjection and bijection.
- Cardinal Numbers. Cardinal Arithmetic, countable and uncountable sets.
- Definitions, adjacency and incidence matrices
- 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.
Learning and teaching
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.
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.
The assessment for this module consists of coursework (50%) 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;
solve problem sheets on different proof techniques covering. LO1-2
There will be a progression test covering LO3 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.
Houston, K., How to Think Like a Mathematician, Cambridge Unidversity Press, 2009.
Johnson, D. L., Elements of Logic via Numbers and Sets, Springer-Verlag, 1998
R.J.Wilson, Introduction to Graph Theory (4-th edition) Longman, 1996.
M.Behzad, G.Chartrand, L.Lesniak-Forster, Graphs and Digraphs, CRC Press, 1996.
A.Dolan, J.Aldous, Networks and Algorithms, Wiley, 1993.
G.Chartrand, O.R.Oellermann, Applied and Algorithmic Graph Theory, McGraw-Hill, 1993.