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  • Proofs and Mathematical Reasoning

    University of Birmingham

    Author:

    Agata Stefanowicz

    Supervisors:

    Joe KyleMichael Grove

    September 2014

    cUniversity of Birmingham 2014

    http://creativecommons.org/licenses/by-nc-nd/4.0/

  • Contents

    1 Introduction 6

    2 Mathematical language and symbols 62.1 Mathematics is a language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Greek alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Words in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    3 What is a proof? 93.1 Writer versus reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Methods of proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Implications and if and only if statements . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    4 Direct proof 114.1 Description of method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Hard parts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Fallacious proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    5 Proof by cases 175.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 Hard parts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.3 Examples of proof by cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    6 Mathematical Induction 196.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Versions of induction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3 Hard parts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.4 Examples of mathematical induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    7 Contradiction 267.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 Hard parts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.3 Examples of proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    8 Contrapositive 298.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.2 Hard parts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    9 Tips 319.1 What common mistakes do students make when trying to present the proofs? . . . . . 319.2 What are the reasons for mistakes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.3 Advice to students for writing good proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 329.4 Friendly reminder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    cUniversity of Birmingham 2014

    http://creativecommons.org/licenses/by-nc-nd/4.0/

  • 10 Sets 3410.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.2 Subsets and power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3 Cardinality and equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.4 Common sets of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.5 How to describe a set? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.6 More on cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.7 Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.8 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    11 Functions 4111.1 Image and preimage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.2 Composition of the functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.3 Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.4 Injectivity, surjectivity, bijectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.5 Inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.6 Even and odd functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    12 Appendix 47

    cUniversity of Birmingham 2014

    http://creativecommons.org/licenses/by-nc-nd/4.0/

  • Foreword

    Talk to any group of lecturers about how their students handle proof and reasoning whenpresenting mathematics and you will soon hear a long list of improvements they would wish for.And yet, if no one has ever explained clearly, in simple but rigorous terms, what is expected it ishardly a surprise that this is a regular comment. The project that Agata Stefanowicz worked onat the University of Birmingham over the summer of 2014 had as its aim, clarifying and codifyingviews of staff on these matters and then using these as the basis of an introduction to the basicmethods of proof and reasoning in a single document that might help new (and indeed continuing)students to gain a deeper understanding of how we write good proofs and present clear and logicalmathematics. Through a judicious selection of examples and techniques, students are presentedwith instructive examples and straightforward advice on how to improve the way they produceand present good mathematics. An added feature that further enhances the written text is theuse of linked videos files that offer the reader the experience of live mathematics developed byan expert. And Chapter 9, that looks at common mistakes that are made when students presentproofs, should be compulsory reading for every student of mathematics. We are confident that,regardless of ability, all students will find something to improve their study of mathematics withinthe pages that follow. But this will be doubly true if they engage with the problems by tryingthem as they go through this guide.

    Michael Grove & Joe KyleSeptember 2014

    cUniversity of Birmingham 2014

    http://creativecommons.org/licenses/by-nc-nd/4.0/

  • Acknowledgements

    I would like to say a big thank you to the Mathematics Support Centre team for the opportunityto work on an interesting project and for the help and advice from the very first day. Specialgratitude goes to Dr Joe Kyle for his detailed comments on my work and tips on creating thedocument. Thank you also to Michael Grove for his cheerful supervision, fruitful brainstormingconversations and many ideas on improving the document. I cannot forget to mention Dr SimonGoodwin and Dr Corneliu Hoffman; thank you for your time and friendly advice. The documentwould not be the same without help from the lecturers at the University of Birmingham who tookpart in my survey - thank you all.

    Finally, thank you to my fellow interns, Heather Collis, Allan Cunningham, Mano Sivanthara-jah and Rory Whelan for making the internship an excellent experience.

    cUniversity of Birmingham 2014

    http://creativecommons.org/licenses/by-nc-nd/4.0/

  • 1 Introduction

    From the first day at university you will hear mention of writing Mathematics in a good style andusing proper English. You will probably start wondering what is the whole deal with words, whenyou just wanted to work with numbers. If, on top of this scary welcome talk, you get a number ofdefinitions and theorems thrown at you in your first week, where most of them include strange notionsthat you cannot completely make sense of - do not worry! It is important to notice how big differencethere is between mathematics at school and at the university. Before the start of the course, many ofus visualise really hard differential equations, long calculations and x-long digit numbers. Most of uswill be struck seeing theorems like a0 = 0. Now, while it is obvious to everybody, mathematiciansare the ones who will not take things for granted and would like to see the proof.

    This booklet is intended to give the gist of mathematics at university, present the language used andthe methods of proofs. A number of examples will be given, which should be a good resource for furtherstudy and an extra exercise in constructing your own arguments. We will start with introducing themathematical language and symbols before moving onto the serious matter of writing the mathematicalproofs. Each theorem is followed by the notes, which are the thoughts on the topic, intended to givea deeper idea of the statement. You will find that some proofs are missing the steps and the purplenotes will hopefully guide you to complete the proof yourself. If stuck, you can watch the videos whichshould explain the argument step by step. Most of the theorems presented, some easier and othersmore complicated, are discussed in first year of the mathematics course. The last two chapters givethe basics of sets and functions as well as present plenty of examples for the readers practice.

    2 Mathematical language and symbols

    2.1 Mathematics is a language

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