MATH10101

Sets, Numbers and Functions

CONTACT: Dr. M D Coleman. Room 1.109 Alan Turing Building

Last updated 8^th September 2015

Course Information.

The main reference for this course is An Introduction to Mathematical Reasoning by P.J.Eccles. There are, though, a few topics in this second half of the course that are not covered in the book. I hope that the notes you find on this site will be adequate for your needs. Even for those topics that do appear in the book my notes will contain alternative examples.

Question Sheets

There are a lot of questions on my question sheets - more than were on Prof. Ray's sheets. This is because I believe that practice makes perfect. Also a lot of my questions are straightforward and involve only calculations. Finally I give a lot because you are expected to study on your own for at least 7 hours a week on this course.

Solution Sheets

Lecture Notes

Notes	Contents
Week 7 Appendix 7	Week 7 Numbers of injections and bijections. Numbers of subsets and Binomial Numbers, Pascal's Triangle, Binomial Theorem.
Week 8 Appendix 8	Week 8 Division Theorem, Greatest Common Divisor, Euclid's Algorithm, Bezout's Lemma,
Week 9 Appendix 9	Week 9 Linear Diophantine Equations, Congruences, Modular Arithmetic, Solving Linear Congruences, Multiplicative inverses, Pairs of congruences, Triplets of congruences, Method of Successive Squaring, non-linear Diophantine equations,
Week 10 Part A Week 10 Part B Appendix 10	Week 10 Congruence Classes, Multiplication Tables, Invertible Elements, Reduced Systems of Classes. Partitions, Relations, generalizing Congruence Classes, from relations to partitions, from partitions to relations.
Week 11 Appendix 11	Week 11 Prime Numbers, Sieve of Eratosthenes, Infinitude of Primes, Conjectures about Primes, Euler's Theorem*, Fermat's Little Theorem. Applications of Euler's and Fermat's Theorem. Permutations, Bijections, two row notation, Composition.
Week 12 Appendix 12	Week 12 Permutations continued, cycles, factoring, orders. Groups.

Notes

Contents

Week 7
Appendix 7

Week 7 Numbers of injections and bijections. Numbers of subsets and Binomial Numbers, Pascal's Triangle, Binomial Theorem.

Week 8
Appendix 8

Week 8 Division Theorem, Greatest Common Divisor, Euclid's Algorithm, Bezout's Lemma,

Week 9
Appendix 9

Week 9 Linear Diophantine Equations, Congruences, Modular Arithmetic, Solving Linear Congruences, Multiplicative inverses, Pairs of congruences, Triplets of congruences, Method of Successive Squaring, non-linear Diophantine equations,

Week 10 Part A Week 10 Part B

Appendix 10

Week 10 Congruence Classes, Multiplication Tables*, Invertible Elements, Reduced Systems of Classes*. Partitions, Relations, generalizing Congruence Classes, from relations to partitions, from partitions to relations.

Week 11

Appendix 11

Week 11 Prime Numbers, Sieve of Eratosthenes, Infinitude of Primes, Conjectures about Primes, Euler's Theorem*, Fermat's Little Theorem. Applications of Euler's and Fermat's Theorem.

Permutations, Bijections, two row notation, Composition.

Week 12

Appendix 12

Week 12 Permutations continued, cycles, factoring, orders. Groups.

* means that the material does not appear in P. J. Eccles book. Topics surrounded by [...] will be covered if there is time.