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!)
• Each lecture will be a mixture of pre-prepared material on slides and board presentation. It's important to attend and take notes.
• The slides contain statements of definitions and of results:
• Most proofs and examples will be written on the board.
Weekly feedback supervisions: Various times - check your personalised timetable. (No supervisions in week 6.)
• Attendance is taken in these classes, and homework will be assigned.
• Supervision classes begin in week 1. In the first session your supervisor will provide details of when and where to hand in work.
• From week two onwards you will receive feedback on your submitted work.
• Homework and attendance at feedback supervisions form 10% of the assessment of this course.
The Midterm test will take place at 15:00 on Thursday 1st November (week 6) in Simon Lecture Theatre B .
• The midterm accounts for 15% of the assessment of this course.

•  You can find a copy of last year's test, together with solutions and some feedback (listing the most common mistakes) below. (This is meant as a rough guide only. The questions on this year's paper will, of course, be different!) Midterm Test Questions Midterm Test Answers Midterm Test Feedback.

### 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

•  L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12
• A set of lecture notes for each section will be made available below at the end of that section. There are eleven sections in total.
• (We have not finished Section 4 yet.)
• A full set of exercises for the course can be found below. Download and print the exercises, and work through them as you cover the material in lectures. Your supervisor will ask you to hand in a selection of exercises each week, to be discussed in the weekly feedback supervisions.
•  Ex. 0 Ex. 1 Ex. 2 Ex. 3 Ex. 4 Ex. 5 Ex. 6 Ex. 7 Ex. 8 Ex. 9 Ex. 10 Ex. 11 (Note: Exercise sheets correspond to syllabus topics, not weeks.)
• Solutions to exercises will appear below.
•  Sol. 0 Sol. 1 Sol. 2
(Solutions to sheet 3 will appear after all supervision classes relating to that sheet have taken place. The last class is on Thursdays.)
• The course is based on the following text:
• You will each have access to a personal online copy of the text via Kortext. Physical copies of this book can also be found in the library.  P.J. Eccles, An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, Cambridge University Press, 1997. [IMR]
• Further supplementary materials: From time to time I may post some extra resources here.

### Assessment

Weekly feedback supervisions
• Homework and attendance at feedback supervisions form 10% of the assessment of this course.
Mid-term test: Thursday 1st November, 15:00-15:40 in Simon Building, Theatre B.
• The mid-term test forms 15% of the assessment of this course.
Final exam: Date, time and location to be announced by the exam office.
• The final exam accounts for the remaining 75%.
• Format: There will be a Section A (contributing 40 marks out of a total of 100 for the exam), in which all questions are compulsory.
There will be a Section B (contributing 60 marks), in which you should complete four of the six questions, with the best four questions contributing if more than four questions are attempted.

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

• These classes will provide a weekly opportunity for students' work to be discussed and provide feedback on their understanding.
Office hour
• Students can also get feedback on their understanding directly from during my office hours (Thursdays 10:30-12:30 in Alan Turing 2.143).
• Do bear in mind that this is a large class, and so there is a limit to the amount of time I can spend discussing with individuals.
Mid-term test
• Provides an opportunity for students to receive feedback from the lecturer and to gauge their progress.
Opportunities for students to give feedback on the course
• If at any time you experience a problem with this course, please let me know.
• In week 3 you had the opportunity to leave feedback on the course. A summary of your comments together with my response is here
• Towards the end of the course you will have another opportunity to leave feedback on the course. I will post a response to the feedback received here.

### 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?

• Prepare your own set of notes. These should at least include what is written on the board during the lecture, but you may find it helpful to include occasional annotations from the lecturer's comments and notes for yourself. Making notes from a board is an important skill which you must learn (you will get better at it with practice), as it is the most common form of mathematical communication - its power lies in its immediacy and flexibility.
• Don't panic if you don't understand something straight away! It is not expected that you will understand everything during lectures, but you should try to keep engaged with the topic by taking notes, including any questions you have about the material, and participating in any class exercises.
• Remember that maths is really cool. Some of our time this semester will be devoted to carefully proving things that you may feel are stating the obvious. I absolutely promise that all of this is useful (it will lay the foundation for all of the really cool stuff you'll study in later years) and that you'll thank me later for being a pedant (maybe by then, you'll be one too).
What not to do in lectures?

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

• Don't be passive: If you only listen to the lecture, your mind is likely to wander. Keep yourself occupied and on topic by taking your own notes!
• Don't get distracted: It is fine to use electronic devices to look at resources, but you may find you are tempted to look at other things.
• Don't be disruptive: Be considerate to those around you - turn the sound off on any electronic devices, and keep noise to an absolute minimum.

What to do after lectures?

• After each lecture you will need to spend some time reading through your notes, thinking about the topics and doing the exercises.
• If you have any questions relating to the material that you can't answer by yourself, make a note of these. You may find it useful discuss any questions arising from the lectures and exercise sheets in your supervision class.
How to approach the homework?

• Don't know where to begin? Get your notes out! Make sure that you understand exactly what the question is asking. Look up the definition of any new terms. You may find it helpful to rewrite the question in your own words. This process can often help to answer the basic questions.
• Check through your notes to see if you can find a similar question, example or proof. Some questions are variations on questions and examples discussed in the lectures.
• Don't panic if your first idea didn't work. Getting things a bit wrong and working out how to correct and improve them is all part of learning. You may need to try a few different ideas before you hit upon a correct approach.
• Doing examples can help to generate ideas. Looking at a single example is unlikely to solve the problem immediately, but it might give you an idea of how to proceed. Think of it as a warm-up exercise for your brain.
• Keep a record of everything you tried, even if it didn't solve the problem. One good reason for doing so is that writing out your ideas can help to clarify your thinking. Another good reason is that you can get feedback on your written work via the supervision classes.
(Yet another good reason: the grade you receive for your homework is based upon engagement, rather than correctness. A written record demonstrates your efforts much better than a blank page.)