Last updated 8^{th} September 2015
The first half of this course will be given by Prof. N. Ray.
From September 2015 my half of the course is being taught by Dr Jonathan Fraser. Because this course is still being given I have removed my lecture notes.
The official source of material for this course is the syllabus page, but here I will give more details of the course and its examination procedures. This page is a place of resources for students on the course.
The Solutions & feedback on my half of the exam paper can be found here
In this document you will find the solutions along with a description of some of the commonly seen mistakes.
Overall I think that most errors were made due to lack of practice.
Many times as I read the scripts I felt that students were writing definitions for the first time.
To learn the material in the course it does not suffice to simply read your notes, you have to do mathematics.
Revise by reading the definitions, statements of results and their proofs. And then write them out again. If you fail, read your notes again, close your file and try to write them out again. Continue until you get it right. This is how you put in the hours to get a good result.
Too many students feel that the study techniques that got them successfully through Alevels will work at University. The exam results of first year students prove this is false for many. For example, if you leave revising too late there won't be hours enough in the day to do the practice described above. As soon as you have any notes you have something you can revise.
Also, if you revise only from past papers you won't be able to deal with any changes in the exam paper, and it is not written down anywhere that an exam paper need look like the previous year's paper.
You get out what you put in, so if you don't go to lectures and supervision groups you will not do as well as you are capable of.
Many students wasted time by attempting too many B part questions. If you are asked to answer 3 part B questions what is the point of answering 4 and getting 9 marks in each? You'll get 27 marks yet if you only answered 3 and took the time that you previously put into the fourth B question to find an extra mark or two than you would get 28 or 29 marks towards the exam.
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.
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.
Notes  Contents 

Week 7 Numbers of injections and bijections. Numbers of subsets and Binomial Numbers, Pascal's Triangle, Binomial Theorem.  
Week 8 Division Theorem, Greatest Common Divisor, Euclid's Algorithm, Bezout's Lemma, 

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

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 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 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.