# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2016/2017

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# Undergraduate Course: Fundamentals of Pure Mathematics (MATH08064)

 School School of Mathematics College College of Science and Engineering Credit level (Normal year taken) SCQF Level 8 (Year 2 Undergraduate) Availability Available to all students SCQF Credits 20 ECTS Credits 10 Summary This is a first course in real analysis and a concrete introduction to group theory and the mathematics of symmetry. Course description Analysis: Week 1-2: Real numbers and sets (including inequalities, supremum, and countability) Week 3-4: Real sequences (from limits to Bolzano-Weierstrass theorem including Cauchy sequences) Week 5-6: Continuity (from limits for real-valued functions to continuity, including extreme value and intermediate value theorems) Week 7-9: Differentiability (from the definition to the mean value theorem and inverse function theorem) Week 10-11: Series (including the definition, integral (without proof), comparison, and ratio tests). Group theory: Week 1: Symmetries of squares and circles (Chapter 1) Week 2: Permutations (Chapter 2) Weeks 3-4: Linear transformations and matrices. The group axioms. Subgroups. (Chapters 3-5) Week 5: Cyclic groups (Chapter 6) Week 6: Group actions (Chapter 7) Week 7: Equivalence relations and modular arithmetic (Chapter 8) Week 8: Homomorphisms and isomorphisms (Chapter 9) Week 9: Cosets and Lagrange's Theorem (Chapter 10) Week 10: The orbit-stabiliser theorem (Chapter 11) Week 11: Colouring problems (Chapter 12)
 Pre-requisites Students MUST have passed: ( Introduction to Linear Algebra (MATH08057) AND Calculus and its Applications (MATH08058) AND Proofs and Problem Solving (MATH08059)) OR ( Accelerated Algebra and Calculus for Direct Entry (MATH08062) AND Accelerated Proofs and Problem Solving (MATH08071)) Co-requisites Prohibited Combinations Other requirements None
 Pre-requisites Visiting students are advised to check that they have studied the material covered in the syllabus of each pre-requisite course before enrolling. High Demand Course? Yes
 Academic year 2016/17, Available to all students (SV1) Quota:  None Course Start Semester 2 Timetable Timetable Learning and Teaching activities (Further Info) Total Hours: 200 ( Lecture Hours 44, Seminar/Tutorial Hours 11, Summative Assessment Hours 3, Programme Level Learning and Teaching Hours 4, Directed Learning and Independent Learning Hours 138 ) Additional Information (Learning and Teaching) Students must pass exam and course overall. Assessment (Further Info) Written Exam 85 %, Coursework 15 %, Practical Exam 0 % Additional Information (Assessment) Coursework 15%, Examination 85% Feedback Not entered Exam Information Exam Diet Paper Name Hours & Minutes Main Exam Diet S2 (April/May) Fundamentals of Pure Mathematics 3:00 Resit Exam Diet (August) Fundamentals of Pure Mathematics 3:00
 1. Perform basic set manipulation and to distinguish between common countable and uncountable sets 2. Using straightforward epsilon methods to establish convergence/non convergence of sequences and determine whether a given sequence is Cauchy. 3. Verifying limits of functions and check continuity using the epsilon-delta method. 4. Computing derivatives from first principles, and by manipulation rules. 5. Performing simple proofs using epsilon-delta techniques. 6. Using the following tests to check convergence/non-convergence of series: comparison, ratio, root, integral, alternating series and understand absolute convergence. 7. Familiarity with the language and ideas of basic group theory. 8. Ability to calculate in several different sorts of group. 9. Familiarity with the language and ideas of group actions. 10. A knowledge of the basic theorems in group theory mentioned in the syllabus 11. Ability to apply these theorems to solve combinatorial problems involving symmetry.
 Analysis: Students are expected to have a personal copy of: An Introduction to Analysis by W. R. Wade. (This book is also relevant for Y3 courses.) Group theory: Students are expected to have a personal copy of: Groups, by C. R. Jordan and D. A. Jordan
 Graduate Attributes and Skills Not entered Keywords FPM
 Course organiser Dr Nikolaos Bournaveas Tel: (0131 6)50 5063 Email: N.Bournaveas@ed.ac.uk Course secretary Mr Martin Delaney Tel: (0131 6)50 6427 Email: Martin.Delaney@ed.ac.uk
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