Undergraduate Programme and Module Handbook 2024-2025

# Module MATH1031: DISCRETE MATHEMATICS

## Department: Mathematical Sciences

### MATH1031: DISCRETE MATHEMATICS

Type | Open | Level | 1 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | Location | Durham |
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#### Prerequisites

- A level Mathematics at grade A or better, or equivalent.

#### Corequisites

- None.

#### Excluded Combination of Modules

- None.

#### Aims

- To provide students with a range of tools for counting discrete mathematical objects.
- To provide experience in problem-solving, presentation, mathematical writing and group working skills through guided self-study and seminars in topics in combinatorics and graph theory.

#### Content

- Principles of counting: arrangements and permutations, selections and combinations, mathematical induction, combinatorial vs. computational proof; pigeon-hole principle, inclusion-exclusion formula.
- Recurrence relations and generating functions: recurrence relations, generating functions, partitions.
- Graphs: basic concepts (paths, circuits, connectedness, trees, etc.).
- Seminar topics: students will work through guided group-study and self-study on one of a set of graph theoretic or combinatorial topics. Topics may include partitions, graph colouring, trees, finite-state automata, or network flows.
- The seminar topics will be assessed by a short report and group presentation.

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will:
- be able to solve a range of predictable or less predictable problems in Discrete Mathematics.
- have a broad knowledge and basic understanding of Discrete Mathematics.

Subject-specific Skills:

- Students will have basic mathematical skills in the following areas: Spatial awareness, Abstract reasoning, Modelling. Students will develop the ability to study mathematical topics via guided reading, and to write mathematical reports with clarity, rigour, and precision.

Key Skills:

- Problem solving, guided study, independent study, presentation skills, technical writing skills.

#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

- Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
- Tutorials provide the practice and support in applying the methods to relevant situations as well as active engagement and feedback to the learning process.
- Continuous assessment provides an opportunity for students to consolidate the learning of material as the module progresses (there are no higher level modules in the department of Mathematical Sciences which build on this module). It serves as a guide in the correct development of students' knowledge and skills, as well as an aid in developing their awareness of standards required.
- Seminars in term 2 will develop the students’ ability for self-study on an extended and open-ended programme, and enhance their group working skills.
- Presentations will develop students’ oral communication skills.
- The written report will train students to write an extended report with precision and rigour of expression.
- The end-of-year written examination provides a rigorous assessment of the mastery of the lecture material.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 27 | 2 per week in weeks 1-10, 11-13, 21 (excluding collection examination) | 1 Hour | 27 | |

Tutorials | 7 | Weeks 3, 5, 7, 9 (Term 1) and 12, 13 (Term 2) plus 1 revision tutorial in Easter term. | 1 Hour | 7 | ■ |

Seminars | 6 | Weekly in weeks 14-19 | 2 Hours | 12 | ■ |

Preparation and Reading | 154 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 70% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

Written examination | 2 hours | 100% | Yes |

Component: Coursework | Component Weighting: 30% | ||

Element | Length / duration | Element Weighting | Resit Opportunity |

Presentation | 33% | Yes | |

Written Report | 67% | Yes |

#### Formative Assessment:

Term 1: 8 formative assessments (e-assessments or written assignments). Students will have about one week to complete each assignment. Term 2: 45 minute collection paper in the beginning of Epiphany term. Term 2: Submission of sample of mathematical writing for feedback.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University