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Module Title Foundation and Abstraction
Department Mathematics
Module Code 06 22512
Module Lead Richard Kaye
Level Intermediate Level
Credits 20
Semester Full Term
Restrictions none
Contact Hours Tutorial-10 hours
Lecture-46 hours
Total: 56 hours
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
Reading List