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.

Past Exam Papers

The solutions to the 2007, 2009-2011 exams include details of the common mistakes made by students.

(Also, remember that the syllabus was different in 2007 so there were questions on polynomials that you cannot do.)

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
Counting Collections of Functions and Subsets Numbers of injections and bijections. Numbers of subsets and Binomial Numbers, Pascal's Triangle, Binomial Theorem.

Arithmetic - the study of the integers Division Theorem, Greatest Common Divisor, Euclid's Algorithm.

Bezout's Lemma, Linear Diophantine Equations.

Congruences Congruences, Modular Arithmetic.

Solving Linear Congruences.

Pairs of linear congruences, non-linear congruences.

Congruence Classes Congruence Classes, Multiplication Tables*, Invertible Elements, Reduced Systems of Classes*.

Partitions and Relations Partitions, Relations, generalizing Congruence Classes.


Appendix on Digraphs This is not covered in lectures but these notes may help you understand relations.

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, Cycles, The Permutation group Sn, Composition Tables.


Factorization, Order of cycles, Order of permutation.

Groups, Definition and very simple properties.


Examples from earlier in the course.

* 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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