Dr Marianne Johnson (marianne.johnson@manchester.ac.uk)
Office hours: Thursdays 10:30-12:30, 2.143 (Alan Turing Building).

 Welcome to MATH10111!

This course forms part of the common core for the first year of all Mathematics joint honours programmes. You can find all the key information about the course on this page. Lecture notes, slides, exercises and solutions will be added to the 'Resources' section as the course progresses.

 Timetable

Lectures: Tuesdays 11:00-12:50, Thursdays 15:00-15:50. (No lectures in week 6 - but don't forget the midterm!) Weekly feedback supervisions: Various times - check your personalised timetable. (No supervisions in week 6.) The Midterm test will take place at 15:00 on Thursday 1st November (week 6) in Simon Lecture Theatre B .

  Syllabus

0. Introduction

1. The language of mathematics. Mathematical statements, quantifiers, truth tables, proof.

2. Number theory I. Prime numbers, proof by contradiction

3. Proof by induction. Method and examples.

4. Set Theory. Sets, subsets, well known sets such as the integers, rational numbers, real numbers.Set Theoretic constructions such as unions, intersections, power sets, Cartesian products.

5. Functions. Definition of functions,examples, injective and surjective functions, bijective functions, composition of functions, inverse functions.

6. Cardinality of sets. Counting of (mostly) finite sets, inclusion-exclusion principle, pigeonhole principle, binomial theorem.

7. Euclidean Algorithm. Greatest common divisor,proof of the Euclidean Algorithm and some consequences, using the Algorithm.

8. Congruence of Integers. Arithmetic properties of congruences,solving certain equations in integers.

9. Relations. Examples of various relations,reflexive, symmetric and transitive relations. Equivalence relations and equivalence classes. Partitions.

10. Number Theory II. Fundamental theorem of Arithmetic, Fermat's little theorem.

11. Binary Operations. Definition and examples of binary operations. Definition of groups and fields with examples. Proving that integers mod p ( p a prime) give a finite field.


  Resources


  Assessment

Weekly feedback supervisions Mid-term test: Thursday 1st November, 15:00-15:40 in Simon Building, Theatre B. Final exam: Date, time and location to be announced by the exam office.
If you use these papers as part of your revision, there are ways to check that your own answers are correct. You can find solutions to many of these questions in the lecture notes -- particularly, those questions that ask you to give a definition, or state a result, but also some of the proofs. The remaining questions are similar to questions that have appeared on the exercise sheets, and so by the time of the exam you should have plenty of practice with similar questions. The course text provides a resource for further exercises (with answers provided). When preparing for assessments, it is useful to keep in mind the intended learning outcomes of this course.

Intended Learning outcomes.
On successful completion of this module students will be able to:

ILO1: Analyse the meaning of mathematical statements involving quantifiers and logical connectives, and construct the negation of a given statement.

ILO2: Construct truth tables of simple mathematical statements and use these to determine whether two given statements are equivalent.

ILO3: Construct elementary proofs of mathematical statements using a range of fundamental proof techniques (direct argumentation, induction, contradiction, use of contrapositive).

ILO4: Use basic set theoretic language and constructions, and be able to determine whether two given sets are equal.

ILO5: Use elementary counting arguments (pigeonhole principle, inclusion-exclusion, binomial theorem) to compute cardinalities of finite sets.

ILO6: Describe and apply basic number theoretic concepts to compute greatest common divisors and to solve linear congruences.

ILO7: Recall formal definitions and apply these to give examples and non-examples of bijections, equivalence relations, binary operations and (abelian) groups.

ILO8: Compose and invert given permutations, expressing the result in two-line notation and in cycle notation.

  Feedback

Weekly feedback supervisions

Office hour Mid-term test Opportunities for students to give feedback on the course


  FAQ

If you have questions about the course that are not answered on this page, then let me know. If the answer is of interest to everyone, then I'll post it here.

Here are some common concerns to get us started:

What to do in lectures?

What not to do in lectures?

This list is not exhaustive, but here are a few things to bear in mind:

What to do after lectures?

How to approach the homework?