The first half of this course will be given by Prof. N. Ray.

I will be teaching the course in weeks 7-12.

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.

Feedback on the second half of the January 2015 exam

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 A-levels 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.

Past Exam Papers

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.

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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.

The questions of Problem Sheet 6 are based on material presented in the last week of the semester. For this reason there will be no supervision in which these questions will be discussed. Nonetheless if you look at the past papers you will see that you may well be examined on this material. Thus it is important that you attempt these questions and compare your answers to those that will be given on this web site.

Solution Sheets

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Additional Questions

If you need additional practice, and I believe you only learn mathematics by doing mathematics, then here is a collection of further problems.
Additional Questions Solutions

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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.

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

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Additional Notes

In the previous year there was a Chapter on Polynomials. It was shown that congruences could be defined, and the division theorem held, for polynomials. With this we could define addition and multiplication on congruence classes of polynomials. Thus we had another example of what we did for integers, i.e. define congruences, congruence classes and then put algebraic structure on the set of classes. Also, polynomials gave yet another source of multiplication tables. This material has been removed from the course (due to lack of time), but since you may find it interesting, the notes are still available here.

Additional Notes

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