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Courses in Mathematics and Statistics Level 1 Modules Level 2 Modules Level 3 Modules Level 4 Modules Level 5 Modules Honours timetable 2011/2012 Sem. 1 2011/2012 Sem. 2 2012/2013 Sem. 1 & Sem. 2 2013/2014 Sem. 1 & Sem. 2

MT2002 ALGEBRA AND ANALYSIS Aims The overall aim of this module is to provide an introduction to two of the cornerstones of modern Mathematics, namely Algebra and Analysis. Algebra provides a common framework for studying basic mathematical concepts such as number systems, polynomials, matrices, permutations, geometrical symmetries, etc. This framework are abstract algebraic structures, three fundamental examples of which - namely groups, rings and fields - will be introduced and studied in this module. Analysis is the study of the limiting behaviour of infinite processes. Many mathematical notions, such as the derivative, the integral and even the real numbers themselves, are best defined as limits of infinite processes. The analysis part of the module will introduce the fundamental concepts in studying sequences and functions of real numbers, namely convergence and continuity. The abstract concepts from both parts will be illustrated by numerous examples, ranging from the amusing (the 15 puzzle and the Rubik's cube) to results that play a central role in contemporary Mathematics. The module will strengthen students' appreciation of the logical structure of Mathematics, and enable them to make full use of a wide choice of Pure Mathematics honours modules. Objectives By the end of the course the students should be familiar with: - the definition and basic properties of groups; - examples of groups: modular arithmetic, permutations, symmetries; - subgroups, normal subgroups, quotient groups and homomorphisms; - the definitions, basic properties and examples of rings and fields; - the definition and basic properties of the real numbers; - sequences: convergence, Cauchy sequences and the General Principle of Convergence; - continuity of real functions. Textbooks The teaching material for this module will be based on lecture notes.  The following selection of books contain material that is covered in the module, and may serve as a suggestion for supplementary and/or further reading. Algebra: T.S. Blyth & E.F. Robertson, Sets and Groups: a First Course in Algebra, Routlege 1988. R.B.J.T. Allenby, Rings, Fields and Groups, Arnold 1991. D.A.R. Wallace, Groups, Rings and Fields, Springer-Verlag, 1998. Analysis: R. Bartle & D. Sherbert, Introduction to Real Analysis, Wiley, 1982. R. Haggerty, Fundamentals of Mathematical Analysis, Addison-Wesley, 1992. J.M. Howie, Real Analysis, Springer-Verlag, 2001. P.E. Kopp, Analysis, Arnold, 1996. W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976. Lectures, practicals and tutorials The average load, in hours per week, is as follows: Lectures: 5 Practicals: 1 Tutorials: 1 These figures do not include the revision & exam period at the end of each semester. Assessment 30% of the assessment mark is from continuous assessment during the semester. The remainder is from a 3 hour exam at the end of the semester. Re-assessment is entirely by a 3 hour exam in September. Prerequisites MT1002 Availability This module is taught every year in Semester 1 at 11.00. Lecturers Prof L Olsen (Module coordinator), Dr J D Mitchell Click here for access to past examination papers for this module. Click here to see the University Course Catalogue entry. Revised: PMH (August 2011)