Report CopyRight/DMCA Form For : Proofs And Mathematical Reasoning University Of Birmingham
1 Introduction 6,2 Mathematical language and symbols 6. 2 1 Mathematics is a language 6,2 2 Greek alphabet 6. 2 3 Symbols 6,2 4 Words in mathematics 7,3 What is a proof 9. 3 1 Writer versus reader 9,3 2 Methods of proofs 9. 3 3 Implications and if and only if statements 10,4 Direct proof 11. 4 1 Description of method 11,4 2 Hard parts 11,4 3 Examples 11. 4 4 Fallacious proofs 15,4 5 Counterexamples 16,5 Proof by cases 17. 5 1 Method 17,5 2 Hard parts 17,5 3 Examples of proof by cases 17. 6 Mathematical Induction 19,6 1 Method 19,6 2 Versions of induction 19. 6 3 Hard parts 20,6 4 Examples of mathematical induction 20. 7 Contradiction 26,7 1 Method 26,7 2 Hard parts 26. 7 3 Examples of proof by contradiction 26,8 Contrapositive 29. 8 1 Method 29,8 2 Hard parts 29,8 3 Examples 29, 9 1 What common mistakes do students make when trying to present the proofs 31. 9 2 What are the reasons for mistakes 32,9 3 Advice to students for writing good proofs 32. 9 4 Friendly reminder 32,c University of Birmingham 2014. 10 Sets 34,10 1 Basics 34,10 2 Subsets and power sets 34. 10 3 Cardinality and equality 35,10 4 Common sets of numbers 36. 10 5 How to describe a set 37,10 6 More on cardinality 37. 10 7 Operations on sets 38,10 8 Theorems 39,11 Functions 41. 11 1 Image and preimage 41,11 2 Composition of the functions 42. 11 3 Special functions 42,11 4 Injectivity surjectivity bijectivity 43. 11 5 Inverse function 44,11 6 Even and odd functions 44. 11 7 Exercises 45,12 Appendix 47,c University of Birmingham 2014. Talk to any group of lecturers about how their students handle proof and reasoning when. presenting 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 is. hardly a surprise that this is a regular comment The project that Agata Stefanowicz worked on. at the University of Birmingham over the summer of 2014 had as its aim clarifying and codifying. views of staff on these matters and then using these as the basis of an introduction to the basic. methods 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 logical. mathematics Through a judicious selection of examples and techniques students are presented. with instructive examples and straightforward advice on how to improve the way they produce. and present good mathematics An added feature that further enhances the written text is the. use of linked videos files that offer the reader the experience of live mathematics developed by. an expert And Chapter 9 that looks at common mistakes that are made when students present. proofs 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 within. the pages that follow But this will be doubly true if they engage with the problems by trying. them as they go through this guide,Michael Grove Joe Kyle. September 2014,c University of Birmingham 2014,Acknowledgements. I would like to say a big thank you to the Mathematics Support Centre team for the opportunity. to work on an interesting project and for the help and advice from the very first day Special. gratitude goes to Dr Joe Kyle for his detailed comments on my work and tips on creating the. document Thank you also to Michael Grove for his cheerful supervision fruitful brainstorming. conversations and many ideas on improving the document I cannot forget to mention Dr Simon. Goodwin and Dr Corneliu Hoffman thank you for your time and friendly advice The document. would not be the same without help from the lecturers at the University of Birmingham who took. part 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. c University of Birmingham 2014,1 Introduction, From the first day at university you will hear mention of writing Mathematics in a good style and. using proper English You will probably start wondering what is the whole deal with words when. you just wanted to work with numbers If on top of this scary welcome talk you get a number of. definitions and theorems thrown at you in your first week where most of them include strange notions. that you cannot completely make sense of do not worry It is important to notice how big difference. there is between mathematics at school and at the university Before the start of the course many of. us visualise really hard differential equations long calculations and x long digit numbers Most of us. will be struck seeing theorems like a 0 0 Now while it is obvious to everybody mathematicians. are 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 and. the methods of proofs A number of examples will be given which should be a good resource for further. study and an extra exercise in constructing your own arguments We will start with introducing the. mathematical language and symbols before moving onto the serious matter of writing the mathematical. proofs Each theorem is followed by the notes which are the thoughts on the topic intended to give. a deeper idea of the statement You will find that some proofs are missing the steps and the purple. notes will hopefully guide you to complete the proof yourself If stuck you can watch the videos which. should explain the argument step by step Most of the theorems presented some easier and others. more complicated are discussed in first year of the mathematics course The last two chapters give. the basics of sets and functions as well as present plenty of examples for the reader s practice. 2 Mathematical language and symbols,2 1 Mathematics is a language. Mathematics at school gives us good basics in a country where mathematical language is spoken. after GCSEs and A Levels we would be able to introduce ourselves buy a train ticket or order a pizza. To have a fluent conversation however a lot of work still needs to be done. Mathematics at university is going to surprise you First you will need to learn the language to. be able to communicate clearly with others This section will provide the grammar notes i e the. commonly used symbols and notation so that you can start writing your mathematical statements in. a good style And like with any other foreign language practice makes perfect so take advantage. of any extra exercises which over time will make you fluent in a mathematical world. 2 2 Greek alphabet, Greek alphabet upper and lower cases and the names of the letters. 2 3 Symbols, Writing proofs is much more efficient if you get used to the simple symbols that save us writing long. sentences very useful during fast paced lectures Below you will find the basic list with the symbols. on the left and their meaning on the right hand side which should be a good start to exploring further. mathematics Note that these are useful shorthands when you need to note the ideas down quickly In. general though when writing your own proofs your lecturers will advise you to use words instead of. the fancy notation especially at the beginning until you are totally comfortable with the statements. if then When reading mathematical books you will notice that the word implies appears. more often than the symbol,c University of Birmingham 2014. A alpha N nu,gamma O omicron,E epsilon P rho,Z zeta sigma. H eta T tau,theta Y upsilon,I iota phi,K kappa X chi. lambda psi,M mu omega,Table 1 Greek letters,Quantifiers. universal quantifier for all,existential quantifier there exists. Symbols in set theory,intersection,or proper subset. composition of functions, Common symbols used when writing proofs and definitions. if and only if,is defined as,is equivalent to,or such that. E or contradiction,or end of proof,2 4 Words in mathematics. Many symbols presented above are useful tools in writing mathematical statements but nothing. more than a convenient shorthand You must always remember that a good proof should also include. words As mentioned at the beginning of the paper correct English or any other language in which. you are literate is as important as the symbols and numbers when writing mathematics Since it is. important to present proofs clearly it is good to add the explanation of what is happening at each. step using full sentences The whole page with just numbers and symbols without a single word will. nearly always be an example of a bad proof, Tea or coffee Mathematical language though using mentioned earlier correct English differs. slightly from our everyday communication The classic example is a joke about a mathematician. c University of Birmingham 2014, who asked whether they would like a tea or coffee answers simply yes This is because or in. mathematics is inclusive so A or B is a set of things where each of them must be either in A or in B. In another words elements of A or B are both those in A and those in B On the other hand when. considering a set A and B then each of its elements must be both in A and B. Exercise 2 1 Question There are 3 spoons 4 forks and 4 knives on the table What fraction of the. utensils are forks OR knives, Answer Forks or knives means that we consider both of these sets We have 4 of each so there. are 8 together Therefore we have that forks or knives constitute to 11 of all the utensils. If we were asked what fraction of the utensils are forks and knives then the answer would be 0. since no utensil is both fork and knive, Please refer to section 10 where the operations on sets are explained in detail The notions or. and and are illustrated on the Venn diagrams which should help to understand them better. c University of Birmingham 2014,3 What is a proof, The search for a mathematical proof is the search for a. knowledge which is more absolute than the knowledge accu. mulated by any other discipline,Simon Singh, A proof is a sequence of logical statements one implying another which gives an explanation of why. a given statement is true Previously established theorems may be used to deduce the new ones one. may also refer to axioms which are the starting points rules accepted by everyone Mathematical. proof is absolute which means that once a theorem is proved it is proved for ever Until proven. though the statement is never accepted as a true one. Writing proofs is the essence of mathematics studies You will notice very quickly that from day. one at university lecturers will be very thorough with their explanations Every word will be defined. notations clearly presented and each theorem proved We learn how to construct logical arguments and. what a good proof looks like It is not easy though and requires practice therefore it is always tempting. for students to learn theorems and apply them leaving proofs behind This is a really bad habit and. does not pay off during final examinations instead go through the proofs given in lectures and. textbooks understand them and ask for help whenever you are stuck There are a number of methods. which can be used to prove statements some of which will be presented in the next sections Hard. and tiring at the beginning constructing proofs gives a lot of satisfaction when the end is reached. successfully,3 1 Writer versus reader, Kevin Houston in his book 2 gives an idea to think of a proof like a small battle between the. reader and the writer At the beginning of mathematics studies you will often be the reader learning. the proofs given by your lecturers or found in textbooks You should then take the active attitude. which means working through the given proof with pen and paper Reading proofs is not easy and may. get boring if you just try to read it like a novel comfortable on your sofa with the half concentration. level Probably the most important part is to question everything what the writer is telling you. Treat it as the argument between yourself and the author of the proof and ask them why at each. step of their reasoning, When it comes to writing your own proof the final version should be clear and have no gaps in. understanding Here a good idea is to think about someone else as the person who would question. each of the steps you present The argument should flow and have enough explanations so that the. reader will find the answer to every why they might ask. 3 2 Methods of proofs, There are many techniques that can be used to prove the statements It is often not obvious at. the beginning which one to use although with a bit of practice we may be able to give an educated. guess and hopefully reach the required conclusion It is important to notice that there is no one ideal. proof a theorem can be established using different techniques and none of them will be better or. worse as long as they are all valid For example in Proofs from the book we may find six different. proofs of the infinity of primes one of which is presented in section 7 Go ahead and master the. techniques you might discover the passion for pure mathematics.