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Introduction to Mathematical Proofs∗

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Introduction to Mathematical Proofs∗ Introduction to Mathematical Proofs∗ Attila M´at´e Brooklyn College of the City University of New York March 12, 2020 Contents Contents 1 1 Introduction 5 1.1 Readingproofs................................... ...... 5 1.2 Learningproofs .................................. ...... 5 2 Interchange of quantifiers 6 3 More on logic 7 3.1 Negation ........................................ .... 8 3.2 Tautologies..................................... ...... 8 3.2.1 Tautologies showing the equivalence of two formulas . ............. 9 3.3 Equivalenceversusbiconditional . ............. 10 3.4 Openstatements.................................. ...... 11 3.5 Terms,free,andboundvariables. ........... 11 3.5.1 Substitutability. ..... 11 3.6 Negatingquantifiedstatements. ........... 12 3.7 Restrictedquantifiers . ......... 13 3.8 Firstorderlogic ................................. ....... 14 3.9 Reading ......................................... 14 3.10 Homework ....................................... 14 4 Sets 14 4.1 Theemptyset..................................... 15 4.2 Relationsbetweensets ............................ ........ 15 4.3 Setoperations ................................... ...... 15 4.4 Venndiagrams.................................... ..... 17 4.5 Thepowersetofaset............................... ...... 17 4.6 Reading ......................................... 17 4.7 Homework ........................................ 17 ∗Written for the course Mathematics 2001 (Transition to Advanced Mathematics) at Brooklyn College of CUNY. 1 5 Prime factorization 18 5.1 TwoproofsofEuclid’slemma . ........ 18 5.2 Uniqueprimefactorization . .......... 19 6 What is a proof? 19 6.1 Formalproofs .................................... ..... 19 6.1.1 Symbolforprovability . ..... 20 6.2 Computersdoingformalproofs . ......... 20 6.3 Proofsinmathematicalessays . .......... 21 6.4 Canonelearnhowtofindproofs? . ........ 21 7 Various patterns of mathematical proofs 21 7.1 Findingaproofversuspresentingit . ............ 21 7.2 Usinganintermediateassumption . ........... 22 7.3 Directproof ..................................... ..... 22 7.4 Proofbycases.................................... ..... 22 7.5 Indirectproof ................................... ...... 22 7.6 Induction:steppingup ............................ ........ 23 7.7 Induction: trueiftrueforsmallerintegers . ............... 23 7.7.1 Induction: failure at smallest integers . ........... 23 7.8 Second order justification of mathematical induction . ................. 24 8 Simple examples of proofs 24 8.1 Trivialsituations ............................... ........ 24 8.1.1 Vacuousassertions . 24 8.1.2 Unnecessaryassumptions . ..... 25 8.2 Simpledirectproofs.............................. ........ 25 8.2.1 Bruteforceversusbeauty . ..... 26 8.3 Proofbycontrapositive . ......... 26 8.4 Proofbycases.................................... ..... 27 8.5 Problemsinvolvingrealnumbers. ........... 27 8.6 Equalityofsets .................................. ...... 28 8.7 Reading ......................................... 28 8.8 Homework ........................................ 28 9 Counter examples and indirect proofs 28 9.1 Counterexamples ................................. ...... 28 9.1.1 Analgebramistake.............................. 28 9.1.2 Euler’spowersumconjecture . ...... 29 9.2 Indirectproofs.................................. ....... 29 9.2.1 Anumberthatisneverasquare . 29 9.2.2 Anonexistenttriangle . ..... 29 9.2.3 Numbersnotinageometricprogression . ........ 30 9.2.4 TheSylvester–Gallaitheorem . ....... 30 9.3 Reading ......................................... 31 9.4 Homework ........................................ 31 10 Mathematical induction 31 10.1 Recursivedefinition . ......... 31 2 10.2 Reading ........................................ 32 10.3 Homework ....................................... 32 11 Equivalence relations 32 11.1 Cartesianproducts .............................. ........ 32 11.2 Relations...................................... ...... 33 11.3 Equivalencerelations . .......... 33 11.4 Reading ........................................ 34 11.5 Homework ....................................... 34 12 Divisiblity 34 12.1 Pythagoreantriples . ......... 37 12.2 The simplest case of Fermat’s Last Theorem . ............. 37 12.3 Reading ........................................ 39 12.4 Homework ....................................... 39 13 Congruences 39 13.1 Cancelationlemma ............................... ....... 39 13.2 Addingandmultiplyingcongruences . ............ 40 13.3 Fermat’slittletheorem . .......... 40 13.4 Residueclasses................................. ........ 40 13.4.1 Compatibility with addition and multiplication . .............. 41 13.4.2 Operationsonresidueclasses . ........ 41 13.5 Reading ........................................ 42 13.6 Homework ....................................... 42 14 Irrationality of square roots 42 14.1 The traditional proof of the irrationality of √2....................... 42 14.2 Newerproofsofirrationality . ............ 43 14.3 Reading ........................................ 44 14.4 Homework ....................................... 44 15 Incommensurablitity: the golden ratio 44 15.1 Theintercepttheorem . ........ 44 15.2 Commensurability............................... ........ 45 15.3 Theregularpentagon ............................. ........ 45 15.4 Incommensurability . ......... 47 16 Continued fractions 48 16.1 The irrationality of the golden ratio revisited . .................. 49 16.2 The case of √2 ........................................ 50 16.3 Othersquareroots ............................... ....... 50 16.4 The irrationality of π ..................................... 51 17 The greatest common divisor 52 17.1 Reading ........................................ 54 17.2 Homework ....................................... 54 18 Functions 54 3 18.1 Domain,range,inverse . ......... 54 18.2 Functionfromasetintoanother. ........... 55 18.3 Composition .................................... ...... 55 18.3.1 Domainofacomposition . 57 18.4 Inversefunction ................................ ........ 57 18.5 Composition ofafunction andits inverse . .............. 57 18.6 Inverseofacomposition . ......... 58 18.7 Restriction.................................... ....... 59 18.8 Injective, bijective, and surjective functions . .................... 59 18.9 Reading ........................................ 59 18.10Homework ...................................... ..... 59 19 Cardinalities 60 19.1 The Cantor-Schr¨oder-Bernstein Theorem . ............... 63 19.2 Reading ........................................ 66 19.3 Homework ....................................... 66 20 Paradoxes in mathematics 66 20.1 Cantor’sparadox ................................ ....... 66 20.2 Realclasses .................................... ...... 66 20.3 Russell’sparadox ............................... ........ 67 20.4 Epimenidesparadox. .. ........ 67 20.5 Berry’sparadox ................................. ....... 67 20.6 Axiomaticsettheory ............................. ........ 68 20.7 Theaxiomofreplacement . ........ 69 20.8 Otheraxiomsystemsofsettheory. ........... 69 20.9 Hilbert’s programandincompleteness . .............. 69 20.10Thecontinuumhypothesis . .......... 70 20.11 Independence of the continuum hypothesis . ............... 70 20.12Computers..................................... ...... 70 21 The Axiom of Completeness of the real numbers 71 22 Sequences and limits 74 22.1 Sequencesandsubsequences . .......... 74 22.2 Limits ......................................... 74 22.2.1 A subsequence of a convergent subsequence is convergent ........... 76 22.3 Limitrules..................................... ...... 76 23 Supremum and limits 78 23.1 Closedandopensets .............................. ....... 78 23.2 Uses of the Axiom of Replacement and of the Axiom of Choice.............. 79 23.3 Moreonsupremumandlimits . ........ 80 24 Limits of functions 81 24.1 Theprecisedefinition . ......... 81 24.2 Whenalimitsisnotunique . ........ 82 24.2.1 Clusterpointsofaset . ..... 82 24.3 Uniquenessoflimits. ......... 83 4 24.4 Limitrules..................................... ...... 83 References 85 1 Introduction These notes are primarily about proofs, and not the mathematical subjects discussed. The first question about proofs that arises immediately is, can I learn how to do (find, create) proofs. The answer to this question is a clear no. Sometimes centuries elapse before a proof eventually found. Aristotle already claimed that the diameter and the circumference of the circle are incommensu- rable,1.1 In other words, he conjectured (guessed) that π is irrational. Yet the first proof of this was only given by Lambert in 1761. Another famous example is Fermat’s Last Theorem,1.2 which, after being first stated without a proof, had to wait more than 350 years for a proof. Many proofs are much easier to find, but finding them is still a challenge. Often one needs to have ideas as to how to proceed that do not seem to have anything to do with the assertion to be proved. 1.1 Reading proofs Reading proofs is usually much easier than finding then. In reading a proof, all you need to to is to check the correctness of each of the steps performed. This may still be difficult, primarily because of the technical knowledge required, but often also because of the lack of details provided by the author: in a proof, steps are often omitted with the expectation that the reader can fill in the details. The other problem with reading proofs is often an unusual
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