# NUMBERS AND SETS - 2020/1

Module code: MAT2051

Module Overview

This module introduces students to some aspects of number theory, combinatorics and set theory. It complements other pure mathematics modules by considering a variety of topics not encountered elsewhere.

Module provider

Mathematics

Module Leader

FISHER David (Maths)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 5

JACs code: G130

Module cap (Maximum number of students): N/A

Module Availability

Semester 2

Prerequisites / Co-requisites

MAT1031 Algebra

Module content

Indicative content includes:

Review of basics:Euclidean algorithm, prime factorization, congruences, Euler’s totient function.

Chinese Remainder Theorem, Fermat's Little Theorem, Wilson's Theorem.

RSA cryptography.

Arithmetic functions. Möbius inversion.

Quadratic residues. Quadratic reciprocity. Euler's criterion. The Legendre symbol.

Simple Diophantine equations.

Recurrence relations, generating functions, partitions.

Binomial identities and their application to combinatorial problems.

Axioms of set theory. Relations on sets.Ordered sets.The axiom of choice.

Countability. Cardinal and ordinal numbers.

Assessment pattern

Assessment type | Unit of assessment | Weighting |
---|---|---|

Examination | EXAMINATION | 80 |

School-timetabled exam/test | CLASS TESTS (50 MINS) | 20 |

Alternative Assessment

N/A

Assessment Strategy

The __assessment strategy__ is designed to provide students with the opportunity to demonstrate their ability to

construct and interpret mathematical arguments in the context of this module;

display subject knowledge by recalling key definitions and results;

apply the techniques learnt to both routine and unfamiliar problems.

Thus, the

__summative assessment__for this module consists of:

One two-hour examination at the end of Semester 2, worth 80% of the module mark.

One class test, worth 20% of the module mark.

__Formative assessment and feedback__

Students receive written comments on their marked coursework assignments. Verbal feedback is provided in lectures and office hours.

Module aims

- provide an insight into classical number theory and its applications, and to some elementary combinatorics
- introduce axiomatic set theory and investigate properties of infinite sets
- establish a basis for further study in pure mathematics and logic

Learning outcomes

Attributes Developed | ||
---|---|---|

001 | Demonstrate an awareness of some standard results and methods in number theory and set theory | K |

002 | Apply the basic techniques of number theory to congruences, arithmetic functions and public-key cryptography | KC |

003 | Calculate Legendre symbols using quadratic reciprocity | KC |

004 | Use binomial coefficients to solve simple problems in combinatorics | KC |

005 | Be familiar with the axioms of set theory, have a basic understanding of cardinal and ordinal numbers, know the axiom of choice and its equivalents | KC |

006 | Construct simple proofs similar to those encountered in the module. | KCT |

Attributes Developed

**C** - Cognitive/analytical

**K** - Subject knowledge

**T** - Transferable skills

**P** - Professional/Practical skills

Overall student workload

Independent Study Hours: 117

Lecture Hours: 33

Methods of Teaching / Learning

The __learning and teaching__ strategy is designed to provide:

Knowledge of the theory and practice of the topics covered.

Experience of the methods used to interpret, understand and solve problems in these areas.

The

__learning and teaching__methods include:

Three 50-minute lectures per week for eleven weeks, some being used as problem classes.

Online notes supplemented by additional examples in lectures.

Two unassessed coursework assignments, marked and returned.

Personal assistance given to individuals and small groups in office hours.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Programmes this module appears in

Programme | Semester | Classification | Qualifying conditions |
---|---|---|---|

Mathematics and Physics BSc (Hons) | 2 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics and Physics MPhys | 2 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics and Physics MMath | 2 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics with Statistics MMath | 2 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics with Statistics BSc (Hons) | 2 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics BSc (Hons) | 2 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics MMath | 2 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics with Music BSc (Hons) | 2 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Financial Mathematics BSc (Hons) | 2 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2020/1 academic year.