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Module Title

Foundation and Abstraction

School

Mathematics

Department

Mathematics

Module Code

06 22512

Module Lead

Richard Kaye

Level

Intermediate Level

Credits

20

Semester

Pre-requisites

Co-requisites

Restrictions

none

Contact Hours

Lecture-46 hours
Seminar-0 hours
Tutorial-10 hours
Project supervision-0 hours
Demonstration-0 hours
Practical Classes and workshops-0 hours
Supervised time in studio/workshop-0 hours
Fieldwork-0 hours
External Visits-0 hours
Work based learning-0 hours
Guided independent study-0 hours
Placement-0 hours
Year Abroad-0 hours

Exclusions

Description

Beginning with elementary number theory students move on to a study of symmetries, permutations and their cycle structure. There then follows a more abstract treatment of functions and relations as subsets of the Cartesian product, equiv relations and partitions, which are applied eg to the construction of the rational numbers. This leads naturally in to group theory - including the axioms for a group, subgroups, proof of uniqueness of unit and inverse, examples of groups, corsets, Lagrange's Theorem and its applications. The second half is devoted to an introduction to analysis - covering real numbers: axioms for the field of real numbers: revision of modulus and inequalities, harder inequalties; completeness axiom and simple consequences. Sequences: convergence; theorems on limits; monotonic sequences; Bolzano-Weirstrass Theorem. These ideas culminate in a study of infinite series - including convergence; series of positive terms; geometric series; comparison test; absolute convergence; ratio test

Learning Outcomes

By the end of the module the student will be able to:
- Understand the need for mathematical proof, elementary number theory and equivalence relations and how to use them;
- Know what group is and be able to prove elementary results about groups;
- Know what a convergent sequence is and be able to prove elementary theorems concerning sequences;
- Apply standard tests to determine convergence of series, and be able to prove elementary theorems concerning such series;
- Tackle wekly-posed problems in collaboration with others and/or produce a written or oral report on the outcome of an investigationBy the end of the module the student will be able to:

Assessment

22512-01 : CA Sem 1 : Coursework (10%)
22512-02 : CA Sem 2 : Coursework (10%)
22512-04 : Exam : Exam (Centrally Timetabled) - Written Unseen (80%)

Assessment Methods & Exceptions

80% on one three-hour examination 20% from other work done during the two terms