DOCSLIB.ORG

Notes for MATH 582

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Notes for MATH 582 Notes for MATH 582 David J. Fern´andez-Bret´on (based on the handwritten notes from January-April 2016, typed in January-April 2017, polished in January-April 2018) Table of Contents Chapter 1 Basic set theory 3 1.1 What is a Set? . 3 1.2 Axiomatizing set theory . 3 1.3 The Language of set theory . 4 1.4 Cantor's set theory . 6 1.5 Patching up Cantor's set theory . 8 1.6 Other ways of dodging Russell's Paradox . 14 1.7 The last few words about the axiomatic method . 15 Chapter 2 Relations, functions, and objects of that ilk 17 2.1 Ordered pairs, relations and functions . 17 2.2 Equivalence relations . 23 2.3 Partial orders . 26 2.4 Well-orders . 28 Chapter 3 The natural numbers, the integers, the rationals, the reals... and more 32 3.1 Peano Systems . 32 3.1.1 Arithmetic in a Peano system . 34 3.1.2 The order relation in a Peano system . 35 3.1.3 There exists one Peano system . 37 3.2 The integers . 38 3.3 The rational numbers . 38 3.4 The real numbers . 39 3.4.1 Dedekind cuts . 39 3.4.2 Cauchy sequences . 39 3.5 Other mathematical objects . 44 Chapter 4 Ordinal numbers 45 4.1 Definition and basic facts . 45 4.2 Classes . 48 4.3 A few words about NBG ............................................... 50 4.4 The Axiom of Replacement . 50 4.5 Ordinal numbers as representatives of well-ordered sets . 53 4.6 Sets and their ranks . 53 4.7 Ordinal arithmetic . 54 Chapter 5 Cardinality 55 5.1 Equipotence, countable and uncountable sets . 55 5.2 A partial order among cardinalities . 57 5.3 The continuum, the continuum hypothesis, the @ sequence . 58 5.4 Cardinal Arithmetic . 60 Chapter 6 The Axiom of Choice 61 6.1 Equivalences of the Axiom of Choice . 61 6.2 Cardinal numbers after the Axiom of Choice (two more equivalences) . 67 6.3 The Banach-Tarski paradox . 69 6.4 Dedekind-finite sets . 71 6.5 Some more cardinal arithmetic . 73 6.6 The canonical well-ordering of Ord × Ord and its consequences . 73 1 6.7 The last few equivalences . 74 Chapter 7 Well-founded relations and the Axiom of Foundation 76 7.1 Well-founded relations . 76 7.2 The Axiom of Foundation . 78 Appendix A Axioms of Set Theory 80 Appendix B Worksheet 1: Constructing Z 81 Appendix C Worksheet 2: Constructing Q 83 Appendix D Worksheet 3: Ordinal arithmetic 85 Appendix E Worksheet 4: Na¨ıve cardinal arithmetic 86 Appendix F Worksheet 5: The Axiom of Choice 88 Appendix G Statements equivalent to the Axiom of Choice 90 2 Chapter 1 Basic set theory Set theory is sometimes considered to be a branch of Mathematical Logic. In particular, it might be relevant not just for mathematics, but for computer science and philosophy as well. It is often said that set theory was born \on that December of 1873 in which Cantor proved that there are uncountably many reals". It is certainly true that Cantor must have been one of the first few people which considered abstract collections of mathematical objects at the level of abstraction that we now do in set theory. Over the course of the years, set theory has blossomed and developed to the point where it now constitutes a whole branch of mathematics on its own. This course is an introduction to this exciting area of mathematics. This course has two main objectives. The first is to convince students that every mathematical object (including: functions, (real and complex) numbers, points in n-dimensional space, polynomials, and virtually everything that you can think of) can be viewed as a set of some sort (some people like to say that \everything is a set", although I consider it to be more accurate if we say that everything can be implemented within set theory). Thus we will spend the first half of this course introducing the commonly accepted axioms of set theory, and explaining how, starting from these axioms, it is possible to implement most everyday mathematical objects by defining appropriate sets that behave, when interpreted in the appropriate way, as we would intuitively expect these objects to behave. The culmination of this will be the construction of the Real Line R. The second objective of this course is to introduce the student to a few basic topics of set theory proper, most notably Ordinal Numbers (and Transfinite Induction and Recursion), Infinite Cardinal Arithmetic (with some level of sophistication, beyond the elementary and simple-minded distinction between countable and uncountable) and some equivalences and consequences of the Axiom of Choice (including applications to other areas of Mathematics). These topics will roughly constitute the second half of the course. Recently, I have started referring to the topics from the first half as the \foundational" part of set theory, whereas the topics from the second half constitute a fragment of the \mathematical" part of set theory. 1.1 What is a Set? Informally, we can think of a set as some collection of objects, called its elements, where the word \collection" is intended in a very abstract way. A set is usually written either explicitly as the list of its elements (e.g. the set f6; 28g), or as a description of what its elements look like (e.g. the set fn n is a perfect number and n ≤ 100g). The special symbol 2 is introduced as a binary relation, stating that the object to the left of the symbol is an element of the set to the right of it: for example, 6 2 f6; 28g. Formally speaking, though, it is not necessary to explicitly write down any definition of what a set is: a set would just be any object belonging to set theory, and all that is therefore relevant is that these objects behave like the axioms of set theory say that they do. In other words, the axioms themselves play, to a certain extent, the r^oleof definitions (in the same sense that, for example, the group axioms constitute the very definition of what a group is, rather than being some \obvious" statements about some predetermined mathematical object). So the really important part, to begin with our study of sets, is to explicitly state what are the axioms of set theory. 1.2 Axiomatizing set theory There are a number of possible axiomatizations of set theory that have been proposed over the course of the years. The main ones among these are: • The axioms of Zermelo-Fraenkel, along with the Axiom of Choice (abbreviated ZFC), • the axioms of von Neumann-Bernays-G¨odel(abbreviated NBG), 3 • the axioms of Morse-Kelley (abbreviated MK), • Quine's New Foundations (abbreviated NF), • the modification of NF that allows objects known as urelements (abbreviated NFU). Mostly for historical reasons, the axiom system that is currently the most widespread and commonly accepted is ZFC, and so these are the axioms that we will use in this course, although I will briefly mention the other axiom systems, and how they differ from ZFC, in appropriate moments throughout the course. So in theory, we should now proceed to state what the ZFC axioms are. However, the list of ZFC axioms consists of seven axioms and two axiom schemas (resulting in infinitely many axioms overall), which might look quite arbitrary at first sight. So in order to motivate these axioms, we will first spend some time working in what I like to call \Cantor's set theory", see what are its advantages and also its disadvantages, and what problems arise from this way of doing set theory. After this, the ZFC axioms should (for the most part) look quite natural. What is Cantor's set theory, then? Cantor used to do set theory based on only two very basic principles. The first one is the following: Axiom 1 (Principle of Extensionality). A set is determined by its elements. This means that, if A and B are two sets that have the exact same elements, then they are actually the same set, symbolised A = B. This principle is central to the concept of a set, as it explicitly expresses that the identity of a set is given only by the objects that are elements of this set, irrespective of any other property (such as the order in which its elements are written, or thought of; or the number of times that we write such elements). For example, it is based on this principle that we can conclude that the following four sets are actually the same: f6; 28g = f28; 6g = f6; 28; 6g = fn n is a perfect number and n ≤ 100g: The second basic principle in Cantor's set theory is the following: Axiom 2 (Principle of Set Creation). Given any property, there exists a set whose elements are exactly those objects that satisfy the given property. Thus, if we denote a property of some object x by P (x), then the Principle of Set Creation ensures the existence of a set A such that x 2 A if and only if P (x). As we said before, this set is denoted by fx P (x)g, and the Principle of Extensionality ensures that it is unique. For example, if the property P (x) is \x is a prime number and x ≤ 30", then the Principle of Set Creation ensures the existence of the set fx P (x)g = f2; 3; 5; 7; 11; 13; 17; 19; 23; 29g: 1.3 The Language of set theory Now, there are two problems with Cantor's set theory, which we will mention in reverse (according to when they were historically pointed out) order.
Load more

Recommended publications
  • I0 and Rank-Into-Rank Axioms
    I0 and rank-into-rank axioms Vincenzo Dimonte∗ July 11, 2017 Abstract Just a survey on I0. Keywords: Large cardinals; Axiom I0; rank-into-rank axioms; elemen- tary embeddings; relative constructibility; embedding lifting; singular car- dinals combinatorics. 2010 Mathematics Subject Classifications: 03E55 (03E05 03E35 03E45) 1 Informal Introduction to the Introduction Ok, that’s it. People who know me also know that it is years that I am ranting about a book about I0, how it is important to write it and publish it, etc. I realized that this particular moment is still right for me to write it, for two reasons: time is scarce, and probably there are still not enough results (or anyway not enough different lines of research). This has the potential of being a very nasty vicious circle: there are not enough results about I0 to write a book, but if nobody writes a book the diffusion of I0 is too limited, and so the number of results does not increase much. To avoid this deadlock, I decided to divulge a first draft of such a hypothetical book, hoping to start a “conversation” with that. It is literally a first draft, so it is full of errors and in a liquid state. For example, I still haven’t decided whether to call it I0 or I0, both notations are used in literature. I feel like it still lacks a vision of the future, a map on where the research could and should going about I0. Many proofs are old but never published, and therefore reconstructed by me, so maybe they are wrong.
    [Show full text]
  • Epistemological Consequences of the Incompleteness Theorems
    Epistemological Consequences of the Incompleteness Theorems Giuseppe Raguní UCAM - Universidad Católica de Murcia, Avenida Jerónimos 135, Guadalupe 30107, Murcia, Spain - [email protected] After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main episte- mological consequences of the first incompleteness Theorem for the two funda- mental arithmetical theories are shown: the non-mechanizability for the truths of the first-order arithmetic and the peculiarities for the model of the second- order arithmetic. Finally, the common epistemological interpretation of the second incompleteness Theorem is corrected, proposing the new Metatheorem of undemonstrability of internal consistency. KEYWORDS: semantics, languages, epistemology, paradoxes, arithmetic, in- completeness, first-order, second-order, consistency. 1 Semantics in the Languages Consider an arbitrary language that, as normally, makes use of a countable1 number of characters. Combining these characters in certain ways, are formed some fundamental strings that we call terms of the language: those collected in a dictionary. When the terms are semantically interpreted, i. e. a certain meaning is assigned to them, we have their distinction in adjectives, nouns, verbs, etc. Then, a proper grammar establishes the rules arXiv:1602.03390v1 [math.GM] 13 Jan 2016 of formation of sentences. While the terms are finite, the combinations of grammatically allowed terms form an infinite-countable amount of possible sentences. In a non-trivial language, the meaning associated to each term, and thus to each ex- pression that contains it, is not always unique. The same sentence can enunciate different things, so representing different propositions.
    [Show full text]
  • Elementary Functions Part 1, Functions Lecture 1.6D, Function Inverses: One-To-One and Onto Functions
    Elementary Functions Part 1, Functions Lecture 1.6d, Function Inverses: One-to-one and onto functions Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 26 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 One-to-one functions* A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output. Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C: Smith (SHSU) Elementary Functions 2013 28 / 33 One-to-one functions* A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25.
    [Show full text]
  • John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected]
    John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected] LOGIC & PHILOSOPHICAL METHODOLOGY Introduction For present purposes “logic” will be understood to mean the subject whose development is described in Kneale & Kneale [1961] and of which a concise history is given in Scholz [1961]. As the terminological discussion at the beginning of the latter reference makes clear, this subject has at different times been known by different names, “analytics” and “organon” and “dialectic”, while inversely the name “logic” has at different times been applied much more broadly and loosely than it will be here. At certain times and in certain places — perhaps especially in Germany from the days of Kant through the days of Hegel — the label has come to be used so very broadly and loosely as to threaten to take in nearly the whole of metaphysics and epistemology. Logic in our sense has often been distinguished from “logic” in other, sometimes unmanageably broad and loose, senses by adding the adjectives “formal” or “deductive”. The scope of the art and science of logic, once one gets beyond elementary logic of the kind covered in introductory textbooks, is indicated by two other standard references, the Handbooks of mathematical and philosophical logic, Barwise [1977] and Gabbay & Guenthner [1983-89], though the latter includes also parts that are identified as applications of logic rather than logic proper. The term “philosophical logic” as currently used, for instance, in the Journal of Philosophical Logic, is a near-synonym for “nonclassical logic”. There is an older use of the term as a near-synonym for “philosophy of language”.
    [Show full text]
  • Monomorphism - Wikipedia, the Free Encyclopedia
    Monomorphism - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Monomorphism Monomorphism From Wikipedia, the free encyclopedia In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : X → Y such that, for all morphisms g1, g2 : Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism. Contents 1 Relation to invertibility 2 Examples 3 Properties 4 Related concepts 5 Terminology 6 See also 7 References Relation to invertibility Left invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G.
    [Show full text]
  • The Strength of Mac Lane Set Theory
    The Strength of Mac Lane Set Theory A. R. D. MATHIAS D´epartement de Math´ematiques et Informatique Universit´e de la R´eunion To Saunders Mac Lane on his ninetieth birthday Abstract AUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and S Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall refer as MAC. His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasizes, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke{Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory Z, and obtain an apparently new proof that Z is not finitely axiomatisable; we study Friedman's strengthening KPP + AC of KP + MAC, and the Forster{Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and KPP ; we show, again using ill-founded models, that KPP + V = L proves the consistency of KPP ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret
    [Show full text]
  • Solutions to Homework Set 5
    MAT320/640, Fall 2020 Renato Ghini Bettiol Solutions to Homework Set 5 1. Decide if each of the statements below is true or false. If it is true, give a complete proof; if it is false, give an explicit counter-example. (Recall that a function f : X ! Y is called surjective, or onto, if every point of Y belongs to its image, that is, if f(X) = Y .) (a) There exists a continuous function f : R n f0g ! R. (b) There exists a continuous surjective function f : R n f0g ! R. (c) There exists a continuous function f : [0; 1] ! R. (d) There exists a continuous surjective function f : [0; 1] ! R. (e) The function f : R ! R, f(x) = x2, is uniformly continuous. (f) The function f : [0; 1] ! R, f(x) = x2, uniformly continuous. Solution: (a) TRUE: For example, any constant function f : R n f0g ! R is continuous; e.g., f(x) = 0 for all x 2 R n f0g. (b) TRUE: For example, consider the function f : R n f0g ! R given by ( x; if x < 0; f(x) = x − 1; if x > 0: Since the domain of f is R n f0g, it is continuous on its domain. Moreover, f is clearly surjective (draw its graph). (c) TRUE: For example, any constant function f : [0; 1] ! R is continuous; e.g., f(x) = 0 for all x 2 [0; 1]. (d) FALSE: Suppose, by contradiction, that such a function f : [0; 1] ! R exists. Since [0; 1] is compact and f : [0; 1] ! R is continuous, its image f([0; 1]) is compact.
    [Show full text]
  • Logic and Proof Release 3.18.4
    Logic and Proof Release 3.18.4 Jeremy Avigad, Robert Y. Lewis, and Floris van Doorn Sep 10, 2021 CONTENTS 1 Introduction 1 1.1 Mathematical Proof ............................................ 1 1.2 Symbolic Logic .............................................. 2 1.3 Interactive Theorem Proving ....................................... 4 1.4 The Semantic Point of View ....................................... 5 1.5 Goals Summarized ............................................ 6 1.6 About this Textbook ........................................... 6 2 Propositional Logic 7 2.1 A Puzzle ................................................. 7 2.2 A Solution ................................................ 7 2.3 Rules of Inference ............................................ 8 2.4 The Language of Propositional Logic ................................... 15 2.5 Exercises ................................................. 16 3 Natural Deduction for Propositional Logic 17 3.1 Derivations in Natural Deduction ..................................... 17 3.2 Examples ................................................. 19 3.3 Forward and Backward Reasoning .................................... 20 3.4 Reasoning by Cases ............................................ 22 3.5 Some Logical Identities .......................................... 23 3.6 Exercises ................................................. 24 4 Propositional Logic in Lean 25 4.1 Expressions for Propositions and Proofs ................................. 25 4.2 More commands ............................................
    [Show full text]
  • A New Characterization of Injective and Surjective Functions and Group Homomorphisms
    A new characterization of injective and surjective functions and group homomorphisms ANDREI TIBERIU PANTEA* Abstract. A model of a function f between two non-empty sets is defined to be a factorization f ¼ i, where ¼ is a surjective function and i is an injective Æ ± function. In this note we shall prove that a function f is injective (respectively surjective) if and only if it has a final (respectively initial) model. A similar result, for groups, is also proven. Keywords: factorization of morphisms, injective/surjective maps, initial/final objects MSC: Primary 03E20; Secondary 20A99 1 Introduction and preliminary remarks In this paper we consider the sets taken into account to be non-empty and groups denoted differently to be disjoint. For basic concepts on sets and functions see [1]. It is a well known fact that any function f : A B can be written as the composition of a surjective function ! followed by an injective function. Indeed, f f 0 id , where f 0 : A Im(f ) is the restriction Æ ± B ! of f to Im(f ) is one such factorization. Moreover, it is an elementary exercise to prove that this factorization is unique up to an isomorphism: i.e. if f i ¼, i : A X , ¼: X B is 1 ¡ Æ ±¢ ! 1 ! another factorization for f , then we easily see that X i ¡ Im(f ) and that ¼ i ¡ f 0. The Æ Æ ± purpose of this note is to study the dual problem. Let f : A B be a function. We shall define ! a model of the function f to be a triple (X ,i,¼) made up of a set X , an injective function i : A X and a surjective function ¼: X B such that f ¼ i, that is, the following diagram ! ! Æ ± is commutative.
    [Show full text]
  • Type Theory and Applications
    Type Theory and Applications Harley Eades [email protected] 1 Introduction There are two major problems growing in two areas. The first is in Computer Science, in particular software engineering. Software is becoming more and more complex, and hence more susceptible to software defects. Software bugs have two critical repercussions: they cost companies lots of money and time to fix, and they have the potential to cause harm. The National Institute of Standards and Technology estimated that software errors cost the United State's economy approximately sixty billion dollars annually, while the Federal Bureau of Investigations estimated in a 2005 report that software bugs cost U.S. companies approximately sixty-seven billion a year [90, 108]. Software bugs have the potential to cause harm. In 2010 there were a approximately a hundred reports made to the National Highway Traffic Safety Administration of potential problems with the braking system of the 2010 Toyota Prius [17]. The problem was that the anti-lock braking system would experience a \short delay" when the brakes where pressed by the driver of the vehicle [106]. This actually caused some crashes. Toyota found that this short delay was the result of a software bug, and was able to repair the the vehicles using a software update [91]. Another incident where substantial harm was caused was in 2002 where two planes collided over Uberlingen¨ in Germany. A cargo plane operated by DHL collided with a passenger flight holding fifty-one passengers. Air-traffic control did not notice the intersecting traffic until less than a minute before the collision occurred.
    [Show full text]
  • METALOGIC METALOGIC an Introduction to the Metatheory of Standard First Order Logic
    METALOGIC METALOGIC An Introduction to the Metatheory of Standard First Order Logic Geoffrey Hunter Senior Lecturer in the Department of Logic and Metaphysics University of St Andrews PALGRA VE MACMILLAN © Geoffrey Hunter 1971 Softcover reprint of the hardcover 1st edition 1971 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. First published 1971 by MACMILLAN AND CO LTD London and Basingstoke Associated companies in New York Toronto Dublin Melbourne Johannesburg and Madras SBN 333 11589 9 (hard cover) 333 11590 2 (paper cover) ISBN 978-0-333-11590-9 ISBN 978-1-349-15428-9 (eBook) DOI 10.1007/978-1-349-15428-9 The Papermac edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent, in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. To my mother and to the memory of my father, Joseph Walter Hunter Contents Preface xi Part One: Introduction: General Notions 1 Formal languages 4 2 Interpretations of formal languages. Model theory 6 3 Deductive apparatuses. Formal systems. Proof theory 7 4 'Syntactic', 'Semantic' 9 5 Metatheory. The metatheory of logic 10 6 Using and mentioning. Object language and metalang- uage. Proofs in a formal system and proofs about a formal system. Theorem and metatheorem 10 7 The notion of effective method in logic and mathematics 13 8 Decidable sets 16 9 1-1 correspondence.
    [Show full text]
  • Proofs and Mathematical Reasoning
    Proofs and Mathematical Reasoning University of Birmingham Author: Supervisors: Agata Stefanowicz Joe Kyle Michael Grove September 2014 c University of Birmingham 2014 Contents 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 Tips 31 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 .
    [Show full text]