DOCSLIB.ORG

Axioms, Definitions, and Proofs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Proofs Definitions (An Alternative to Discrete Mathematics) G. Viglino Ramapo College of New Jersey January 2018 CONTENTS Chapter 1 A LOGICAL BEGINNING 1.1 Propositions 1 1.2 Quantifiers 14 1.3 Methods of Proof 23 1.4 Principle of Mathematical Induction 33 1.5 The Fundamental Theorem of Arithmetic 43 Chapter 2 A TOUCH OF SET THEORY 2.1 Basic Definitions and Examples 53 2.2 Functions 63 2.3 Infinite Counting 77 2.4 Equivalence Relations 88 2.5 What is a Set? 99 Chapter 3 A TOUCH OF ANALYSIS 3.1 The Real Number System 111 3.2 Sequences 123 3.3 Metric Space Structure of 135 3.4 Continuity 147 Chapter 4 A TOUCH OF TOPOLOGY 4.1 Metric Spaces 159 4.2 Topological Spaces 171 4.3 Continuous Functions and Homeomorphisms 185 4.4 Product and Quotient Spaces 196 Chapter 5 A TOUCH OF GROUP THEORY 5.1 Definition and Examples 207 5.2 Elementary Properties of Groups 220 5.3 Subgroups 228 5.4 Automorphisms and Isomorphisms 237 Appendix A: Solutions to CYU-boxes Appendix B: Answers to Selected Exercises PREFACE The underlying goal of this text is to help you develop an appreciation for the essential roles axioms and definitions play throughout mathematics. We offer a number of paths toward that goal: a Logic path (Chapter 1); a Set Theory path (Chapter 2); an Analysis path (Chapter 3); a Topology path (Chapter 4); and an Algebra path (Chapter 5). You can embark on any of the last three independent paths upon comple- tion of the first two: pter 3 Cha alysis h of An . A Touc y Ch 2 or A A To apter er he uch 4 pt T to of T a et u C opolo h S ch gy C of ha ch of p ou A te Chapter 1 T b r A st 5 .A Logical Beginning. ra c t A lg eb ra A one-semester course may include one, two, or even all three of the last three chapters, in total or in part, depending on the preparation of the students, and the ambition of the instructor. For our part, we have made every effort to assist you in the journey you are about to take. We did our very best to write a readable book, without compromising mathematical integrity. Along the way, you will encounter numerous Check Your Understanding boxes designed to challenge your understanding of each newly-introduced concept. Detailed solutions to each of the Check Your Understanding problems appear in Appendix A, but you should only turn to that appendix after making a valiant effort to solve the given prob- lem on your own, or with others. In the words of Desecrates: We never understand a thing so well, and make it our own, when we learn it from another, as when we have discovered it for ourselves. There are also plenty of exercises at the end of each section, covering the gamut of diffi- culty, providing you with ample opportunities to further hone your mathematical skills. Exercises that play a role in future developments are marked with a sharp (#). I wish to thank my colleague, Professor Marion Berger, for her numerous comments and suggestions throughout the development of the text. Her generous participation is deeply appreciated. I am also grateful to Professor Maxim Goldberg-Rugalev for his invaluable input. 1.1 Propositions 1 1 CHAPTER 1 A Logical Beginning Mathematics strives, as best it can, to transpose our physical universe into a non-physical form—a thought-universe, as it were, where defini- tions are the physical objects, and where the reasoning process rules. At some point, one has to ask whether we humans are creating mathemat- ics or whether, through some kind of wonderful mental telescope, we are capable of discovering a meager portion of the mathematical uni- verse which, in all of its glory, may very well be totally independent of us the spectators. Whichever; but one thing appears to be clear: the nature of mathematics, as we observe it or create it, appears to be inspired by what we perceive to be intuitive logic. §1. PROPOSITIONS We begin by admitting that “True” and “False” are undefined con- cepts, and that the word “sentence” is also undefined. Nonetheless: DEFINITION 1.1 A proposition (or statement) is a sentence PROPOSITION that is True or False, but not both. Fine, we have a definition, but its meaning rests on undefined words! We feel your frustration. Still, mathematics may very well be the closest thing to perfection we have. So, let’s swallow our pride and proceed. We can all agree that 53+8= is a True proposition, and that 53+9= is a False proposition. But how about this sentence: THE SUM OF ANY TWO ODD INTEGERS IS EVEN. First off, in order to understand the sentence one has to know all of its words, including: sum, integer, odd, and even. Only then can one hope to determine if the sentence is a proposition; and, if it is, to establish whether it is True or False. All in good time. For now, let us consider two other sentences: HE IS AN AMERICAN CITIZEN. This sentence cannot be evaluated to be True or False without first knowing exactly who “He” is. It is a variable proposition (more formally called a predicate), involving the variable “He.” Once He is specified, then we have a proposition which is either True or False. THE TWO OF US ARE RELATED. Even if we know precisely who the “two of us” are, we still may not have a proposition. Why not? Because the word “related” is just too vague. 2 Chapter 1 A Logical Beginning COMPOUND PROPOSITIONS The two numbers 3 and 5 can be put together in several ways to arrive at other numbers: 35+ , 35– , 35 , and so on. Similarly, two propo- sitions, p and q, can be put together to arrive at other propositions, called compound propositions. One way, is to “and-them:” In truth table form: DEFINITION 1.2 Let p, q be propositions. The conjunction of p q pq p and q p and q (or simply: p and q), written pq , T T T pq is that proposition which is True if both p and T F F q are True, and is False otherwise. F T F For example, the proposition: F F F 75 and 35+8= is True. (since both 75 and 35+8= are True) The proposition: 75 and 35+9= is False. (since 75 and 35+9= are not both True) Another way to put two propositions together is to “or-them:” In truth table form: DEFINITION 1.3 Let p, q be propositions. The disjunction of p p q pq and q (or simply: p or q), written pq , is T T T p or q T F T pq that proposition which is False when both p F T T and q are False, and is True otherwise. F F F For example, the proposition: 75 or 35+8= is True, as is the Note that p or q is True when both p and q are True. As such, it proposition 75 or 35+9= . The proposition 75 or 35+9= is said to be the inclusive-or. The exclusive-or is typically used in is False, since neither 75 nor 35+9= is True. conversation, as in: Do you want coffee or tea (one or the other, but not both) CHECK YOUR UNDERSTANDING 1.1 Let p be the proposition 75= , and q be the proposition 35+8= . (a) True (b) False (a) Is pq True or False? (b) Is pq True or False? The above and/or operators, which act on two given propositions, are said to be binary operators. The following operator is a unary operator in that it deals with only one given proposition: In truth table form: DEFINITION 1.4 Let p be a proposition. The negation of p, p p NOT P read “not p,” and written p , is that proposi- T F tion which is False if p is True, and True if p F T is False. For example, the proposition: ~7 5 is True, since the proposition 75 is False. The proposition: ~3+8 5= is False, since the proposition 35+8= is True. 1.1 Propositions 3 CHECK YOUR UNDERSTANDING 1.2 Indicate if the proposition p is True of False, given that (a) p is the proposition 53 . (a) False (b) False (b) p is the proposition q , where q is the proposition 35= . The negation operator “~” takes precedence (is performed prior) to both the “ ” and the “ ” operators. For example, the proposition: 35 35+9= is True. [since 35 is True] As is the case with algebraic expressions, the order of operations in a logical expression can be overridden by means of parentheses. Brackets”[ ]”play the same role as parentheses, and are For example, the proposition: used to bracket an expres- sion which itself contains 53 35+8= is False parentheses. [since 53 35+8= is True ] CHECK YOUR UNDERSTANDING 1.3 Assume that p and q are True propositions, and that s is a False prop- osition. Determine if the given compound proposition is True or False. (a) pq (b) pq (c) sq (d) sq True: d, f, g, h, j (e) pq (f) pq (g) sq (h) sq False: a, b, c, e, i, k (i) ps qs (j) ps qs (k) ps qs CONDITIONAL STATEMENT Consider the sentence: IF IT IS MONDAY, THEN JOHN WILL CALL HIS MOTHER.
Recommended publications
  • A Guide to Topology

    A Guide to Topology

    i i “topguide” — 2010/12/8 — 17:36 — page i — #1 i i A Guide to Topology i i i i i i “topguide” — 2011/2/15 — 16:42 — page ii — #2 i i c 2009 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2009929077 Print Edition ISBN 978-0-88385-346-7 Electronic Edition ISBN 978-0-88385-917-9 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “topguide” — 2010/12/8 — 17:36 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY MAA Guides # 4 A Guide to Topology Steven G. Krantz Washington University, St. Louis ® Published and Distributed by The Mathematical Association of America i i i i i i “topguide” — 2010/12/8 — 17:36 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Paul Zorn, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Patricia B. Humphrey Virginia E. Knight Mark A. Peterson Jonathan Rogness Thomas Q. Sibley Joe Alyn Stickles i i i i i i “topguide” — 2010/12/8 — 17:36 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City Uni- versity of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successfulexpositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition.
  • The Symbol of Infinity Represented by the Art

    The Symbol of Infinity Represented by the Art

    ering & ine M g a n n E Tamir, Ind Eng Manage 2014, 3:3 a l g a i e r m t s DOI: 10.4172/2169-0316.1000e124 e u n d t n I Industrial Engineering & Management ISSN: 2169-0316 Editorial Open Access Artistic Demonstrations by Euclidean Geometry: Possible in 2D but Impossible in 3D Abraham Tamir* Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel Infinity is a concept that has different meanings in mathematics, hole. The right hand side of Figure 2 entitled “The False Mirror” is philosophy, cosmology and everyday language. However the common the artwork of the Belgian surrealist artist Rene Magritte. A giant eye to all meanings is that infinity is something that its content is higher is formed as a frame of a blue sky with clouds. The pupil of the eye than everything else or a process that will never reach its end. The creates a dead centre in a sharp colour contrast to the white and blue of mathematical symbol of infinity is demonstrated in the different the sky, and also with a contrast of form–the hard outline of the pupil artworks that are the major subject of this article. The most accepted against the soft curves and natural form of the clouds. Surprisingly, the definition of infinity is “a quantity greater than any assignable quantity combination of the original artwork on the right with its mirror image of the same kind”. Other definitions are as follows. In geometry it creates the symbol of infinity.
  • 6.5 the Recursion Theorem

    6.5 the Recursion Theorem

    6.5. THE RECURSION THEOREM 417 6.5 The Recursion Theorem The recursion Theorem, due to Kleene, is a fundamental result in recursion theory. Theorem 6.5.1 (Recursion Theorem, Version 1 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. For every total recursive function f, there is some n such that ϕn = ϕf(n). The recursion Theorem can be strengthened as follows. Theorem 6.5.2 (Recursion Theorem, Version 2 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. There is a total recursive function h such that for all x ∈ N,ifϕx is total, then ϕϕx(h(x)) = ϕh(x). 418 CHAPTER 6. ELEMENTARY RECURSIVE FUNCTION THEORY A third version of the recursion Theorem is given below. Theorem 6.5.3 (Recursion Theorem, Version 3 ) For all n ≥ 1, there is a total recursive function h of n +1 arguments, such that for all x ∈ N,ifϕx is a total recursive function of n +1arguments, then ϕϕx(h(x,x1,...,xn),x1,...,xn) = ϕh(x,x1,...,xn), for all x1,...,xn ∈ N. As a first application of the recursion theorem, we can show that there is an index n such that ϕn is the constant function with output n. Loosely speaking, ϕn prints its own name. Let f be the recursive function such that f(x, y)=x for all x, y ∈ N. 6.5. THE RECURSION THEOREM 419 By the s-m-n Theorem, there is a recursive function g such that ϕg(x)(y)=f(x, y)=x for all x, y ∈ N.
  • Basic Structures: Sets, Functions, Sequences, and Sums 2-2

    Basic Structures: Sets, Functions, Sequences, and Sums 2-2

    CHAPTER Basic Structures: Sets, Functions, 2 Sequences, and Sums 2.1 Sets uch of discrete mathematics is devoted to the study of discrete structures, used to represent discrete objects. Many important discrete structures are built using sets, which 2.2 Set Operations M are collections of objects. Among the discrete structures built from sets are combinations, 2.3 Functions unordered collections of objects used extensively in counting; relations, sets of ordered pairs that represent relationships between objects; graphs, sets of vertices and edges that connect 2.4 Sequences and vertices; and finite state machines, used to model computing machines. These are some of the Summations topics we will study in later chapters. The concept of a function is extremely important in discrete mathematics. A function assigns to each element of a set exactly one element of a set. Functions play important roles throughout discrete mathematics. They are used to represent the computational complexity of algorithms, to study the size of sets, to count objects, and in a myriad of other ways. Useful structures such as sequences and strings are special types of functions. In this chapter, we will introduce the notion of a sequence, which represents ordered lists of elements. We will introduce some important types of sequences, and we will address the problem of identifying a pattern for the terms of a sequence from its first few terms. Using the notion of a sequence, we will define what it means for a set to be countable, namely, that we can list all the elements of the set in a sequence.
  • April 22 7.1 Recursion Theorem

    April 22 7.1 Recursion Theorem

    CSE 431 Theory of Computation Spring 2014 Lecture 7: April 22 Lecturer: James R. Lee Scribe: Eric Lei Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. An interesting question about Turing machines is whether they can reproduce themselves. A Turing machine cannot be be defined in terms of itself, but can it still somehow print its own source code? The answer to this question is yes, as we will see in the recursion theorem. Afterward we will see some applications of this result. 7.1 Recursion Theorem Our end goal is to have a Turing machine that prints its own source code and operates on it. Last lecture we proved the existence of a Turing machine called SELF that ignores its input and prints its source code. We construct a similar proof for the recursion theorem. We will also need the following lemma proved last lecture. ∗ ∗ Lemma 7.1 There exists a computable function q :Σ ! Σ such that q(w) = hPwi, where Pw is a Turing machine that prints w and hats. Theorem 7.2 (Recursion theorem) Let T be a Turing machine that computes a function t :Σ∗ × Σ∗ ! Σ∗. There exists a Turing machine R that computes a function r :Σ∗ ! Σ∗, where for every w, r(w) = t(hRi; w): The theorem says that for an arbitrary computable function t, there is a Turing machine R that computes t on hRi and some input. Proof: We construct a Turing Machine R in three parts, A, B, and T , where T is given by the statement of the theorem.
  • Section 2.1: Proof Techniques

    Section 2.1: Proof Techniques

    Section 2.1: Proof Techniques January 25, 2021 Abstract Sometimes we see patterns in nature and wonder if they hold in general: in such situations we are demonstrating the appli- cation of inductive reasoning to propose a conjecture, which may become a theorem which we attempt to prove via deduc- tive reasoning. From our work in Chapter 1, we conceive of a theorem as an argument of the form P → Q, whose validity we seek to demonstrate. Example: A student was doing a proof and suddenly specu- lated “Couldn’t we just say (A → (B → C)) ∧ B → (A → C)?” Can she? It’s a theorem – either we prove it, or we provide a counterexample. This section outlines a variety of proof techniques, including direct proofs, proofs by contraposition, proofs by contradiction, proofs by exhaustion, and proofs by dumb luck or genius! You have already seen each of these in Chapter 1 (with the exception of “dumb luck or genius”, perhaps). 1 Theorems and Informal Proofs The theorem-forming process is one in which we • make observations about nature, about a system under study, etc.; • discover patterns which appear to hold in general; • state the rule; and then • attempt to prove it (or disprove it). This process is formalized in the following definitions: • inductive reasoning - drawing a conclusion based on experi- ence, which one might state as a conjecture or theorem; but al- mostalwaysas If(hypotheses)then(conclusion). • deductive reasoning - application of a logic system to investi- gate a proposed conclusion based on hypotheses (hence proving, disproving, or, failing either, holding in limbo the conclusion).
  • Domain and Range of a Function

    Domain and Range of a Function

    4.1 Domain and Range of a Function How can you fi nd the domain and range of STATES a function? STANDARDS MA.8.A.1.1 MA.8.A.1.5 1 ACTIVITY: The Domain and Range of a Function Work with a partner. The table shows the number of adult and child tickets sold for a school concert. Input Number of Adult Tickets, x 01234 Output Number of Child Tickets, y 86420 The variables x and y are related by the linear equation 4x + 2y = 16. a. Write the equation in function form by solving for y. b. The domain of a function is the set of all input values. Find the domain of the function. Domain = Why is x = 5 not in the domain of the function? 1 Why is x = — not in the domain of the function? 2 c. The range of a function is the set of all output values. Find the range of the function. Range = d. Functions can be described in many ways. ● by an equation ● by an input-output table y 9 ● in words 8 ● by a graph 7 6 ● as a set of ordered pairs 5 4 Use the graph to write the function 3 as a set of ordered pairs. 2 1 , , ( , ) ( , ) 0 09321 45 876 x ( , ) , ( , ) , ( , ) 148 Chapter 4 Functions 2 ACTIVITY: Finding Domains and Ranges Work with a partner. ● Copy and complete each input-output table. ● Find the domain and range of the function represented by the table. 1 a. y = −3x + 4 b. y = — x − 6 2 x −2 −10 1 2 x 01234 y y c.
  • Adaptive Lower Bound for Testing Monotonicity on the Line

    Adaptive Lower Bound for Testing Monotonicity on the Line

    Adaptive Lower Bound for Testing Monotonicity on the Line Aleksandrs Belovs∗ Abstract In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. This is an active area of research. In this paper we study query complexity of ε-testing monotonicity of a function f :[n] → [r]. All our lower bounds are for adaptive two-sided testers. log r • We prove a nearly tight lower bound for this problem in terms of r. TheboundisΩ log log r when ε =1/2. No previous satisfactory lower bound in terms of r was known. • We completely characterise query complexity of this problem in terms of n for smaller values of ε. The complexity is Θ ε−1 log(εn) . Apart from giving the lower bound, this improves on the best known upper bound. Finally, we give an alternative proof of the Ω(ε−1d log n−ε−1 log ε−1) lower bound for testing monotonicity on the hypergrid [n]d due to Chakrabarty and Seshadhri (RANDOM’13). 1 Introduction The framework of property testing was formulated by Rubinfeld and Sudan [19] and Goldreich et al. [16]. A property testing problem is specified by a property P, which is a class of functions mapping some finite set D into some finite set R, and proximity parameter ε, which is a real number between 0 and 1. An ε-tester is a bounded-error randomised query algorithm which, given oracle access to a function f : D → R, distinguishes between the case when f belongs to P and the case when f is ε-far from P.
  • MTH 304: General Topology Semester 2, 2017-2018

    MTH 304: General Topology Semester 2, 2017-2018

    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
  • Finding an Interpretation Under Which Axiom 2 of Aristotelian Syllogistic Is

    Finding an Interpretation Under Which Axiom 2 of Aristotelian Syllogistic Is

    Finding an interpretation under which Axiom 2 of Aristotelian syllogistic is False and the other axioms True (ignoring Axiom 3, which is derivable from the other axioms). This will require a domain of at least two objects. In the domain {0,1} there are four ordered pairs: <0,0>, <0,1>, <1,0>, and <1,1>. Note first that Axiom 6 demands that any member of the domain not in the set assigned to E must be in the set assigned to I. Thus if the set {<0,0>} is assigned to E, then the set {<0,1>, <1,0>, <1,1>} must be assigned to I. Similarly for Axiom 7. Note secondly that Axiom 3 demands that <m,n> be a member of the set assigned to E if <n,m> is. Thus if <1,0> is a member of the set assigned to E, than <0,1> must also be a member. Similarly Axiom 5 demands that <m,n> be a member of the set assigned to I if <n,m> is a member of the set assigned to A (but not conversely). Thus if <1,0> is a member of the set assigned to A, than <0,1> must be a member of the set assigned to I. The problem now is to make Axiom 2 False without falsifying Axiom 1 as well. The solution requires some experimentation. Here is one interpretation (call it I) under which all the Axioms except Axiom 2 are True: Domain: {0,1} A: {<1,0>} E: {<0,0>, <1,1>} I: {<0,1>, <1,0>} O: {<0,0>, <0,1>, <1,1>} It’s easy to see that Axioms 4 through 7 are True under I.
  • Convergence of Functions: Equi-Semicontinuity

    Convergence of Functions: Equi-Semicontinuity

    transactions of the american mathematical society Volume 276, Number 1, March 1983 CONVERGENCEOF FUNCTIONS: EQUI-SEMICONTINUITY by szymon dolecki, gabriella salinetti and roger j.-b. wets Abstract. We study the relationship between various types of convergence for extended real-valued functionals engendered by the associated convergence of their epigraphs; pointwise convergence being treated as a special case. A condition of equi-semicontinuity is introduced and shown to be necessary and sufficient to allow the passage from one type of convergence to another. A number of compactness criteria are obtained for families of semicontinuous functions; in the process we give a new derivation of the Arzelá-Ascoli Theorem. Given a space X, by Rx we denote the space of all functions defined on X and with values in R, the extended reals. We are interested in the relationship between various notions of convergence in Rx, in particular, but not exclusively, between pointwise convergence and that induced by the convergence of the epigraphs. We extend and refine the results of De Giorgi and Franzoni [1975] (collection of "equi-Lipschitzian" functions with respect to pseudonorms) and of Salinetti and Wets [1977] (sequences of convex functions on reflexive Banach spaces). The range of applicability of the results is substantially enlarged, in particular the removal of the convexity, reflexivity and the norm dependence assumptions is significant in many applications; for illustrative purposes an example is worked out further on. The work in this area was motivated by: the search for " valid" approximations to extremal statistical problems, variational inequalities and difficult optimization problems, cf. the above mentioned articles.
  • The Axiom of Choice and Its Implications

    The Axiom of Choice and Its Implications

    THE AXIOM OF CHOICE AND ITS IMPLICATIONS KEVIN BARNUM Abstract. In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial. Contents 1. Introduction 1 2. The Axiom of Choice and Its Equivalents 1 2.1. The Axiom of Choice and its Well-known Equivalents 1 2.2. Some Other Less Well-known Equivalents of the Axiom of Choice 3 3. Applications of the Axiom of Choice 5 3.1. Equivalence Between The Axiom of Choice and the Claim that Every Vector Space has a Basis 5 3.2. Some More Applications of the Axiom of Choice 6 4. Controversial Results 10 Acknowledgments 11 References 11 1. Introduction The Axiom of Choice states that for any family of nonempty disjoint sets, there exists a set that consists of exactly one element from each element of the family. It seems strange at first that such an innocuous sounding idea can be so powerful and controversial, but it certainly is both. To understand why, we will start by looking at some statements that are equivalent to the axiom of choice. Many of these equivalences are very useful, and we devote much time to one, namely, that every vector space has a basis. We go on from there to see a few more applications of the Axiom of Choice and its equivalents, and finish by looking at some of the reasons why the Axiom of Choice is so controversial.