MATH1051
Lecturer : Norbert Peyerimhoff
Term : Michaelmas 2014
Literature
The following is a list of books on which the course is based. Although we will not follow a book strictly, most of the material can be found in them and they may sometimes offer a different approach to the material.Resources
Tutorial and Homework problems
| Week | Tutorial Problems | Extra Tutorial Problems | Homework Problems |
| Week 1 | --- | --- | 1, 5b, 8, 11 |
| Week 2 | 3, 4a, 7, 10ac | 2, 6, 9, 12 | 16, 18, 23, 26 |
| Week 3 | 17, 19, 20, 22 | 15, 21, 24, 25 | 27ag, 28ehk, 42 |
| Week 4 | 27de, 28fgi, 33 | 37c, 47 | 28d, 35, 40, 43 |
| Week 5 | 28ab, 30ab, 36, 37b | 27bf, 31a, 32a, 34 | 49, 51, 58 |
| Week 6 | 48, 50, 52, 55ii | 53, 54, 56, 57 | 59, 63a, 65a, 70 |
| Week 7 | 60, 63b, 64, 66 | 61, 62, 65b, 68 | 73, 74, 79a, 83 |
| Week 8 | 56, 72, 75, 76b | 71 | 77, 84, 86, 87a |
| Week 9 | 81, 82, 85, 87b | 80 | 92, 93, 97, 101 |
| Week 10 | 88cd, 89, 95, 96a | 88ab, 90, 91, 94, 96b, 98 | 99, 100, 106, 109 |
Content of Lectures
| Date | Content |
| Monday, 6 October 2014 (Week 1) | Some logic: Statements, truth tables, De Morgan and other laws |
| Tuesday, 7 October 2014 (Week 1) | Sets, Venn diagrams, set operations and their laws |
| Monday, 13 October 2014 (Week 2) | Number sets, axioms of ordering, finding solution sets of inequalities |
| Tuesday, 14 October 2014 (Week 2) | Absolute value, triangle inequality, inequalities involving the absolute value |
| Monday, 20 October 2014 (Week 3) | Sequences of real numbers, the notion of a limit of a sequence, examples of convergent and divergent sequences |
| Tuesday, 21 October 2014 (Week 3) | Fundamental Theorems on sequences (e.g. Squeezing Theorem, COLT) |
| Monday, 27 October 2014 (Week 4) | Euler's number e as a limit, rules "exponentials beat polynomials" and "powers beat logarithms", using continuity of certain functions to calculate limits of sequences |
| Tuesday, 28 October 2014 (Week 4) | More fundamental facts about limits, Limit behavior of a^n, complex sequences |
| Monday, 3 November 2014 (Week 5) | Quantifiers, Induction proof, negation of statements with quantifiers |
| Tuesday, 4 November 2014 (Week 5) | "If...then" connective, negation of "If A then B", indirect proof, contrapositive statement and contrapositive proof, intersection and union of infinite sets |
| Monday, 10 November 2014 (Week 6) | Minimum/maximum and infimum/supremum of sets of real numbers, examples |
| Tuesday, 11 November 2014 (Week 6) | Completeness Axiom for the real numbers, image of a set, infimum/supremum of a real-valued function, examples |
| Monday, 17 November 2014 (Week 7) | Subsequences and facts about them, preparing for Bolzano-Weierstrass |
| Tuesday, 18 November 2014 (Week 7) | Bolzano-Weierstrass and introduction into Cauchy sequences |
| Monday, 24 November 2014 (Week 8) | Real/complex Cauchy sequences are bounded and are convergent, Example of Cauchy sequence based on Newton method |
| Tuesday, 25 November 2014 (Week 8) | Newton method in a nutshell, basic notions related to functions like injectivity, surjectivity, image and preimage of a function |
| Monday, 1 December 2014 (Week 9) | Compatibility between preimages and set operations, epsilon/delta-definition of limit "as x to c" and "as x to infinity" of a function, COLT for such limits |
| Tuesday, 2 December 2014 (Week 9) | Examples of limits, limits of functions via sequences, further examples, definition of one-sided limits, epsilon/delta definition of continuity |
| Monday, 8 December 2014 (Week 10) | definition of continuity via sequences, example, continuity of sum/product/quotient/composition of continuous functions, definition of closed/open, bounded and compact intervals, fundamental theorems about continuous functions on compact intervals |
| Tuesday, 9 December 2014 (Week 10) | Proofs of the fundamental theorems about continuous functions on compact intervals: Intermediate Value Theorem, Continuous functions are bounded, Continuous functions assume maximum and minimum |