Aims

To introduce the basic concepts, notation, and techniques of discrete mathematics, particularly group theory, graph theory and the theory of algorithms, and the use of these methods in the representation and solutions of problems coming from the real world as well as other parts of mathematics.

Module Summary

Groups will be introduced, and their basic properties will be studied via permutation groups, building on knowledge of mappings and permutations. The concept of an algorithm will be introduced and formally defined, with some discussion on how the complexity of an algorithm can be measured. The basic notation of graphs will be introduced, some well-known problems will be formulated in that language. Solutions to the problem will be discussed, as well as their complexity.

Outline Of Syllabus

Permutations. Groups: definition, properties, examples. Symmetric, alternating and dihedral groups. Subgroups, conjugation, homomorphisms, isomorphisms, cosets, Lagrange's Theorem. Algorithms and growth of functions, comparison of algorithms, P vs NP. Basic concepts of graph theory: examples and problem-solving algorithms such as finite-state automata.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.