MTH4113 / MTH4213 - Numbers, Sets and Functions - 2023/24
Topic outline
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General news and announcements from the module organisers.
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Please use this forum to ask questions about the module (either mathematical or administrative), and we'll try to answer as quickly as possible.
You should also feel free to answer each other's questions.
If you have a question but you don't want other students to see that it was you who asked it (because it's embarrassingly simple), then please ask the lecturer by email (m.jerrum@qmul.ac.uk or m.fayers@qmul.ac.uk). We may then answer it here (saying "a student has asked ...") so that everyone can benefit from the answer.
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What do a mathematician and Sherlock Holmes have in common? They both use reasoning to solve problems.
By the end of the module you will be able to apply reasoning and write short proofs accurately. You will be able to assess whether or not a simple mathematical statement is true or false and formally show that the square root of two is not a rational number.
The module cover the fundamental building blocks of mathematics (sets, sequences, functions, relations and numbers). It introduces the main number systems (natural numbers, integers, rational, real and complex numbers), outlining their construction and main properties. Through these mathematical objects, it also introduces the concepts of definition, theorem, proof and counterexample.
The material covered here is fundamental to most if not all second and third year modules.
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Syllabus
For the exam, you are expected to learn and understand all the material taught in lectures (except for any advanced bits explicitly labelled "non-examinable"). But here's a more detailed list of what we'll see in the module.
- Logic and proofs: statements, truth tables, quantifiers, implications, converse and contrapositive. Proof structure, proving implications, proof by contradiction, proof by induction.
- Integers: divisibility, primes, greatest common divisor, lowest common multiple, Euclid's algorithm.
- Sets: set notation, subsets, union and intersection, Cartesian product.
- Sets and counting: cardinality of a set, cardinality of the power set, number of subsets of a given size, binomial coefficients.
- Functions: definition, injective, surjective and bijective functions, restriction and composition of functions, inverse functions, images and inverse images of subsets.
- Relations: definition, reflexive, symmetric, antisymmetric and transitive relations.
- Sequences: definition, subsequences, increasing, decreasing and constant sequences.
- Rational and real numbers: definition of the rationals, irrationality of √2, definition of real numbers, upper bound and supremum.
- Complex numbers: definition and operations, real and imaginary parts, complex conjugate, polar form, the complex plane, de Moivre's Theorem, roots of unity.
Learning outcomes
At the end of this module, students should be able to:
- understand mathematical statements, combine them using "and", "or" and "not", and use quantifiers;
- use different methods of proof, including contradiction, contraposition and induction;
- define the relation of divisibility for integers, and understand its basic properties;
- understand greatest common divisor and least common multiple, and calculate them using Euclid's algorithm;
- specify sets in various ways;
- define and apply basic concepts for sets (union, intersection, set difference, symetric difference, power set, cardinality);
- find the number of subsets of a given set, and the number of subsets of a given size;
- understand the definition of functions, and determine whether a function is injective, surjective or bijective;
- understand the relationship between bijectivity and invertibility;
- define restriction and composition of functions, and images and inverse images of subsets;
- define what is meant by a relation on a set, and determine whether a given relation is reflexive, symmetric, anti-symmetric or transitive;
- define sequences and basic properties they may have;
- define rational and real numbers, and prove that √2 is irrational;
- define maximum, upper bounds and supremum, and find the supremum of a given set;
- define complex numbers and perform basic arithmetic, and convert complex numbers between Cartesian and polar representations;
- state de Moivre's Theorem, and find the nth roots of 1.
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Full lecture notes for the module appear below. We'll be modifying the notes as we go along, so please check for updates. The notes appear as a single document - they should be regarded as a continuous body of material, and the breaks between lectures are unimportant.
These notes are to supplement your own notes, not to replace them. You should take your own notes and use them as your main learning aid.
We may also add some supplementary notes to help explain things better, or to give further (non-examinable) information on various topics for students who are interested.
Please let the module organisers know of any mistakes you find, or things that you think could be explained better.
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Here's a brief introduction to what university maths will be like, how it differs from school maths, and what you should do to get the best out of your studies.
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As you go through your maths studies, you'll encounter the Greek alphabet, and you're expected to be familiar with it. Here it is; you don't need to memorise it now, but you should practise writing these letters.
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The assessment for this module will involve two components:
- Four in-term coursework assignments, worth 5% each;
- A final exam worth 80%.
The final exam will take place in January. It will be an in-person invigilated exam. You will not be allowed a calculator or any notes.
The exam will last for three hours.
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There will be four assessed courseworks that count towards your final mark (5% each). These courseworks will appear here, with instructions on how and when to submit your work.
Late submission: Solutions to the in-term coursework will be posted as soon as the deadline passes, so we won't be able to accept late submissions. If you are prevented from submitting by circumstances beyond your control, then you can submit an extenuating circumstances claim (talk to the Maths office about this). If this is accepted, then this coursework can be excluded from your final module mark.
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This is the first assessed coursework, which counts for 5% of your module mark. Please either type your solutions, or write and scan/photograph them, and upload here (as a single PDF file), by 11pm on Friday 20th October.
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This is the first assessed coursework, which counts for 5% of your module mark. Please either type your solutions, or write and scan/photograph them, and upload here (as a single PDF file), by 11pm on Friday 3rd November.
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This is the third assessed coursework, which counts for 5% of your module mark. Please either type your solutions, or write and scan/photograph them, and upload here (as a single PDF file), by 11pm on Friday 24th November.
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This is the fourth assessed coursework, which counts for 5% of your module mark. Please either type your solutions, or write and scan/photograph them, and upload here (as a single PDF file), by 11pm on Friday 8th December.
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Here are some exercises for your weekly tutorials. You should attempt these before the tutorial, and then your tutor may talk about them.
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Each week we'll set a quiz to help you test your understanding of that week's lectures. The quiz will be quite short, with short simple answers, and it will be marked automatically straight away. You can have as many attempts as you want, so keep trying until you get all the questions right! If you're stuck or you can't see why your answer is wrong, please ask your tutor or the module organisers.
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These questions are designed to give you some extra practice, if you like doing lots of exercises. They're not compulsory, but should be helpful for you in learning and consolidating the material. Please ask your tutor or lecturer if you'd like any hints.
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Here are some extra exercises to give you some more practice.
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The exam papers from last year appear below to give you an idea of what to expect. This year's exam will be very similar in style.
The exam papers from 2018, 2019 and 2020 are also included below. There have been some very slight syllabus changes since then, so there may be some questions you can't answer with what you've learnt – please ask if you're unsure.
The exams from 2021 and 2022 were online; they were quizzes, with a variety of question formats: som0e multiple-choice or short numerical answers where you could just fill in answers in the quiz, and some longer written answers where you write your answer on paper, scan and upload. The 2022 exam is available below for you to practise.
As you'll find with most lecturers, it is my policy not to provide written solutions to past exam papers. This is because experience has shown that many students just look at the solutions rather than actually trying to do the questions. If you're stuck on any question from a past paper, please ask me and I'll give you some hints.
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This is the final exam that students in 2021-22 had; it was an online exam because of the Covid-19 pandemic. I've copied it here and you should be able to attempt it whenever you like.
The exam was available for 24 hours, but students had to submit within 3 hours of starting their attempt.
As you'll see, the first part of the exam was short questions which were automatically marked, and part required uploaded written solutions. You can attempt the automatically-marked parts here. If you want any hints for the written questions, or if you'd like me to have a look at your attempts, let me know.
You can attempt it as many times as you like. Most of the questions had different versions, with a random one being chosen for each student to reduce cheating. So if you attempt the quiz more than once, you should see different questions coming up.
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