## Proofs And Mathematical Reasoning University Of Birmingham-Free PDF

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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.