# 13 Infinite Sets Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385

Save this PDF as:

Size: px
Start display at page:

Download "13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385"

## Transcription

1 mcs-ftl 2010/9/8 0:40 page 379 # Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite set does have an infinite number of elements, but it turns out that not all infinite sets have the same size some are bigger than others! And, understanding infinity is not as easy as you might think. Some of the toughest questions in mathematics involve infinite sets. Why should you care? Indeed, isn t computer science only about finite sets? Not exactly. For example, we deal with the set of natural numbers N all the time and it is an infinite set. In fact, that is why we have induction: to reason about predicates over N. Infinite sets are also important in Part IV of the text when we talk about random variables over potentially infinite sample spaces. So sit back and open your mind for a few moments while we take a very brief look at infinity Injections, Surjections, and Bijections We know from Theorem that if there is an injection or surjection between two finite sets, then we can say something about the relative sizes of the two sets. The same is true for infinite sets. In fact, relations are the primary tool for determining the relative size of infinite sets. Definition Given any two sets A and B, we say that A surj B iff there is a surjection from A to B, A inj B iff there is an injection from A to B, A bij B iff there is a bijection between A and B, and A strict B iff there is a surjection from A to B but there is no bijection from B to A. Restating Theorem with this new terminology, we have: Theorem For any pair of finite sets A and B, jaj jbj iff A surj B; jaj jbj iff A inj B; jaj D jbj iff A bij B; jaj > jbj iff A strict B: 1

2 mcs-ftl 2010/9/8 0:40 page 380 #386 Chapter 13 Infinite Sets Theorem suggests a way to generalize size comparisons to infinite sets; namely, we can think of the relation surj as an at least as big relation between sets, even if they are infinite. Similarly, the relation bij can be regarded as a same size relation between (possibly infinite) sets, and strict can be thought of as a strictly bigger relation between sets. Note that we haven t, and won t, define what the size of an infinite set is. The definition of infinite sizes is cumbersome and technical, and we can get by just fine without it. All we need are the as big as and same size relations, surj and bij, between sets. But there s something else to watch out for. We ve referred to surj as an as big as relation and bij as a same size relation on sets. Most of the as big as and same size properties of surj and bij on finite sets do carry over to infinite sets, but some important ones don t as we re about to show. So you have to be careful: don t assume that surj has any particular as big as property on infinite sets until it s been proved. Let s begin with some familiar properties of the as big as and same size relations on finite sets that do carry over exactly to infinite sets: Theorem For any sets, A, B, and C, 1. A surj B and B surj C IMPLIES A surj C. 2. A bij B and B bij C IMPLIES A bij C. 3. A bij B IMPLIES B bij A. Parts 1 and 2 of Theorem follow immediately from the fact that compositions of surjections are surjections, and likewise for bijections. Part 3 follows from the fact that the inverse of a bijection is a bijection. We ll leave a proof of these facts to the problems. Another familiar property of finite sets carries over to infinite sets, but this time it s not so obvious: Theorem (Schröder-Bernstein). For any pair of sets A and B, if A surj B and B surj A, then A bij B. The Schröder-Bernstein Theorem says that if A is at least as big as B and, conversely, B is at least as big as A, then A is the same size as B. Phrased this way, you might be tempted to take this theorem for granted, but that would be a mistake. For infinite sets A and B, the Schröder-Bernstein Theorem is actually pretty technical. Just because there is a surjective function f W A! B which need not be a bijection and a surjective function g W B! A which also need not 2

3 mcs-ftl 2010/9/8 0:40 page 381 # Countable Sets be a bijection it s not at all clear that there must be a bijection h W A! B. The challenge is to construct h from parts of both f and g. We ll leave the actual construction to the problems Infinity Is Different A basic property of finite sets that does not carry over to infinite sets is that adding something new makes a set bigger. That is, if A is a finite set and b A, then ja [ fbgj D jaj C 1, and so A and A [ fbg are not the same size. But if A is infinite, then these two sets are the same size! Theorem Let A be a set and b A. Then A is infinite iff A bij A [ fbg. Proof. Since A is not the same size as A [ fbg when A is finite, we only have to show that A [ fbg is the same size as A when A is infinite. That is, we have to find a bijection between A [ fbg and A when A is infinite. Since A is infinite, it certainly has at least one element; call it a 0. Since A is infinite, it has at least two elements, and one of them must not be equal to a 0 ; call this new element a 1. Since A is infinite, it has at least three elements, one of which must not equal a 0 or a 1 ; call this new element a 2. Continuing in this way, we conclude that there is an infinite sequence a 0, a 1, a 2,..., a n,..., of different elements of A. Now it s easy to define a bijection f W A [ fbg! A: f.b/ WWD a 0 ; f.a n / WWD a nc1 for n 2 N; f.a/ WWD a for a 2 A fb; a 0 ; a 1 ; : : : g: 13.2 Countable Sets Definitions A set C is countable iff its elements can be listed in order, that is, the distinct elements in C are precisely c 0 ; c 1 ; : : : ; c n ; : : : : This means that if we defined a function f on the nonnegative integers by the rule that f.i/ WWD c i, then f would be a bijection from N to C. More formally, Definition A set C is countably infinite iff N bij C. A set is countable iff it is finite or countably infinite. 3

4 mcs-ftl 2010/9/8 0:40 page 382 #388 Chapter 13 Infinite Sets Discrete mathematics is often defined as the mathematics of countable sets and so it is probably worth spending a little time understanding what it means to be countable and why countable sets are so special. For example, a small modification of the proof of Theorem shows that countably infinite sets are the smallest infinite sets; namely, if A is any infinite set, then A surj N Unions Since adding one new element to an infinite set doesn t change its size, it s obvious that neither will adding any finite number of elements. It s a common mistake to think that this proves that you can throw in countably infinitely many new elements just because it s ok to do something any finite number of times doesn t make it ok to do it an infinite number of times. For example, suppose that you have two countably infinite sets A D fa 0 ; a 1 ; a 2 ; : : : g and B D fb 0 ; b 1 ; b 2 ; : : : g. You might try to show that A[B is countable by making the following list for A [ B: a 0 ; a 1 ; a 2 ; : : : ; b 0 ; b 1 ; b 2 ; : : : (13.1) But this is not a valid argument because Equation 13.1 is not a list. The key property required for listing the elements in a countable set is that for any element in the set, you can determine its finite index in the list. For example, a i shows up in position i in Equation 13.1, but there is no index in the supposed list for any of the b i. Hence, Equation 13.1 is not a valid list for the purposes of showing that A [ B is countable when A is infinite. Equation 13.1 is only useful when A is finite. It turns out you really can add a countably infinite number of new elements to a countable set and still wind up with just a countably infinite set, but another argument is needed to prove this. Theorem If A and B are countable sets, then so is A [ B. Proof. Suppose the list of distinct elements of A is a 0, a 1,..., and the list of B is b 0, b 1,.... Then a valid way to list all the elements of A [ B is a 0 ; b 0 ; a 1 ; b 1 ; : : : ; a n ; b n ; : : : : (13.2) Of course this list will contain duplicates if A and B have elements in common, but then deleting all but the first occurrence of each element in Equation 13.2 leaves a list of all the distinct elements of A and B. Note that the list in Equation 13.2 does not have the same defect as the purported list in Equation 13.1, since every item in A [ B has a finite index in the list created in Theorem

5 mcs-ftl 2010/9/8 0:40 page 383 # Countable Sets b 0 b 1 b 2 b 3 : : : a 0 c 0 c 1 c 4 c 9 a 1 c 3 c 2 c 5 c 10 a 2 c 8 c 7 c 6 c 11 a 3 c 15 c 14 c 13 c 12 : : :: Figure 13.1 A listing of the elements of C D A B where A D fa 0 ; a 1 ; a 2 ; : : : g and B D fb 0 ; b 1 ; b 2 ; : : : g are countably infinite sets. For example, c 5 D.a 1 ; b 2 / Cross Products Somewhat surprisingly, cross products of countable sets are also countable. At first, you might be tempted to think that infinity times infinity (whatever that means) somehow results in a larger infinity, but this is not the case. Theorem The cross product of two countable sets is countable. Proof. Let A and B be any pair of countable sets. To show that C D A B is also countable, we need to find a listing of the elements f.a; b/ j a 2 A; b 2 B g: There are many such listings. One is shown in Figure 13.1 for the case when A and B are both infinite sets. In this listing,.a i ; b j / is the kth element in the list for C where a i is the ith element in A, b j is the j th element in B, and k D max.i; j / 2 C i C max.i j; 0/: The task of finding a listing when one or both of A and B are finite is left to the problems at the end of the chapter Q Is Countable Theorem also has a surprising Corollary; namely that the set of rational numbers is countable. Corollary The set of rational numbers Q is countable. 5

6 mcs-ftl 2010/9/8 0:40 page 384 #390 Chapter 13 Infinite Sets Proof. Since ZZ is countable by Theorem , it suffices to find a surjection f from Z Z to Q. This is easy to to since ( a=b if b 0 f.a; b/ D 0 if b D 0 is one such surjection. At this point, you may be thinking that every set is countable. That is not the case. In fact, as we will shortly see, there are many infinite sets that are uncountable, including the set of real numbers R Power Sets Are Strictly Bigger It turns out that the ideas behind Russell s Paradox, which caused so much trouble for the early efforts to formulate Set Theory, also lead to a correct and astonishing fact discovered by Georg Cantor in the late nineteenth century: infinite sets are not all the same size. Theorem For any set A, the power set P.A/ is strictly bigger than A. Proof. First of all, P.A/ is as big as A: for example, the partial function f W P.A/! A where f.fag/ WWD a for a 2 A is a surjection. To show that P.A/ is strictly bigger than A, we have to show that if g is a function from A to P.A/, then g is not a surjection. So, mimicking Russell s Paradox, define A g WWD f a 2 A j a g.a/ g: A g is a well-defined subset of A, which means it is a member of P.A/. But A g can t be in the range of g, because if it were, we would have A g D g.a 0 / for some a 0 2 A. So by definition of A g, a 2 g.a 0 / iff a 2 A g iff a g.a/ for all a 2 A. Now letting a D a 0 yields the contradiction a 0 2 g.a 0 / iff a 0 g.a 0 /: So g is not a surjection, because there is an element in the power set of A, namely the set A g, that is not in the range of g. 6

7 mcs-ftl 2010/9/8 0:40 page 385 # Power Sets Are Strictly Bigger R Is Uncountable To prove that the set of real numbers is uncountable, we will show that there is a surjection from R to P.N/ and then apply Theorem to P.N/. Lemma R surj P.N/. Proof. Let A N be any subset of the natural numbers. Since N is countable, this means that A is countable and thus that A D fa 0 ; a 1 ; a 2 ; : : : g. For each i 0, define bin.a i / to be the binary representation of a i. Let x A be the real number using only digits 0, 1, 2 as follows: x A WWD 0:2 bin.a 0 /2 bin.a 1 /2 bin.a 2 /2 : : : (13.3) We can then define a surjection f W R! P.N/ as follows: ( A if x D x A for some A 2 N; f.x/ D 0 otherwise: Hence R surj P.N/. Corollary R is uncountable. Proof. By contradiction. Assume R is countable. Then N surj R. By Lemma , R surj P.N/. Hence N surj P.N/. This contradicts Theorem for the case when A D N. So the set of rational numbers and the set of natural numbers have the same size, but the set of real numbers is strictly larger. In fact, R bij P.N /, but we won t prove that here. Is there anything bigger? Even Larger Infinities There are lots of different sizes of infinite sets. For example, starting with the infinite set N of nonnegative integers, we can build the infinite sequence of sets N; P.N/; P.P.N//; P.P.P.N///; : : : By Theorem , each of these sets is strictly bigger than all the preceding ones. But that s not all, the union of all the sets in the sequence is strictly bigger than each set in the sequence. In this way, you can keep going, building still bigger infinities. 7

8 mcs-ftl 2010/9/8 0:40 page 386 #392 Chapter 13 Infinite Sets The Continuum Hypothesis Georg Cantor was the mathematician who first developed the theory of infinite sizes (because he thought he needed it in his study of Fourier series). Cantor raised the question whether there is a set whose size is strictly between the smallest infinite set, N, and P.N/. He guessed not: Cantor s Continuum Hypothesis. There is no set A such that P.N/ is strictly bigger than A and A is strictly bigger than N. The Continuum Hypothesis remains an open problem a century later. Its difficulty arises from one of the deepest results in modern Set Theory discovered in part by Gödel in the 1930s and Paul Cohen in the 1960s namely, the ZFC axioms are not sufficient to settle the Continuum Hypothesis: there are two collections of sets, each obeying the laws of ZFC, and in one collection, the Continuum Hypothesis is true, and in the other, it is false. So settling the Continuum Hypothesis requires a new understanding of what sets should be to arrive at persuasive new axioms that extend ZFC and are strong enough to determine the truth of the Continuum Hypothesis one way or the other Infinities in Computer Science If the romance of different size infinities and continuum hypotheses doesn t appeal to you, not knowing about them is not going to lower your professional abilities as a computer scientist. These abstract issues about infinite sets rarely come up in mainstream mathematics, and they don t come up at all in computer science, where the focus is generally on countable, and often just finite, sets. In practice, only logicians and set theorists have to worry about collections that are too big to be sets. In fact, at the end of the 19th century, even the general mathematical community doubted the relevance of what they called Cantor s paradise of unfamiliar sets of arbitrary infinite size. That said, it is worth noting that the proof of Theorem gives the simplest form of what is known as a diagonal argument. Diagonal arguments are used to prove many fundamental results about the limitations of computation, such as the undecidability of the Halting Problem for programs and the inherent, unavoidable inefficiency (exponential time or worse) of procedures for other computational problems. So computer scientists do need to study diagonal arguments in order to understand the logical limits of computation. Ad a well-educated computer scientist will be comfortable dealing with countable sets, finite as well as infinite. 8

9 MIT OpenCourseWare J / J Mathematics for Computer Science Fall 2010 For information about citing these materials or our Terms of Use, visit:

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### If f is a 1-1 correspondence between A and B then it has an inverse, and f 1 isa 1-1 correspondence between B and A.

Chapter 5 Cardinality of sets 51 1-1 Correspondences A 1-1 correspondence between sets A and B is another name for a function f : A B that is 1-1 and onto If f is a 1-1 correspondence between A and B,

### Georg Cantor (1845-1918):

Georg Cantor (845-98): The man who tamed infinity lecture by Eric Schechter Associate Professor of Mathematics Vanderbilt University http://www.math.vanderbilt.edu/ schectex/ In papers of 873 and 874,

### This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 20

CS 70 Discrete Mathematics and Probability Theory Fall 009 Satish Rao, David Tse Note 0 Infinity and Countability Consider a function (or mapping) f that maps elements of a set A (called the domain of

### In mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann)

Chapter 1 Sets and Functions We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895) In mathematics

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### Sets and Subsets. Countable and Uncountable

Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There

### Finite and Infinite Sets

Chapter 9 Finite and Infinite Sets 9. Finite Sets Preview Activity (Equivalent Sets, Part ). Let A and B be sets and let f be a function from A to B..f W A! B/. Carefully complete each of the following

### Mathematics for Computer Science

Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math

### Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

### Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

### It is not immediately obvious that this should even give an integer. Since 1 < 1 5

Math 163 - Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some

### Introducing Functions

Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1

### Math/CSE 1019: Discrete Mathematics for Computer Science Fall Suprakash Datta

Math/CSE 1019: Discrete Mathematics for Computer Science Fall 2011 Suprakash Datta datta@cse.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cse.yorku.ca/course/1019 1

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### We give a basic overview of the mathematical background required for this course.

1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

### Math 320 Course Notes. Chapter 7: Countable and Uncountable Sets

Math 320 Course Notes Magnhild Lien Chapter 7: Countable and Uncountable Sets Hilbert s Motel: Imagine a motel with in nitely many rooms numbered 1; 2; 3; 4 ; n; : One evening an "in nite" bus full with

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### TREES IN SET THEORY SPENCER UNGER

TREES IN SET THEORY SPENCER UNGER 1. Introduction Although I was unaware of it while writing the first two talks, my talks in the graduate student seminar have formed a coherent series. This talk can be

### Introduction to mathematical arguments

Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. Math

### This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.

3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### Set theory as a foundation for mathematics

Set theory as a foundation for mathematics Waffle Mathcamp 2011 In school we are taught about numbers, but we never learn what numbers really are. We learn rules of arithmetic, but we never learn why these

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### POSITIVE INTEGERS, INTEGERS AND RATIONAL NUMBERS OBTAINED FROM THE AXIOMS OF THE REAL NUMBER SYSTEM

MAT 1011 TECHNICAL ENGLISH I 03.11.2016 Dokuz Eylül University Faculty of Science Department of Mathematics Instructor: Engin Mermut Course assistant: Zübeyir Türkoğlu web: http://kisi.deu.edu.tr/engin.mermut/

### CHAPTER 3. Sequences. 1. Basic Properties

CHAPTER 3 Sequences We begin our study of analysis with sequences. There are several reasons for starting here. First, sequences are the simplest way to introduce limits, the central idea of calculus.

### p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)

.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### S(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.

MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.

### SETS, RELATIONS, AND FUNCTIONS

September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four

### CHAPTER 1. Basic Ideas

CHPTER 1 asic Ideas In the end, all mathematics can be boiled down to logic and set theory. ecause of this, any careful presentation of fundamental mathematical ideas is inevitably couched in the language

### Chapter 1. Informal introdution to the axioms of ZF.

Chapter 1. Informal introdution to the axioms of ZF. 1.1. Extension. Our conception of sets comes from set of objects that we know well such as N, Q and R, and subsets we can form from these determined

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### For the purposes of this course, the natural numbers are the positive integers. We denote by N, the set of natural numbers.

Lecture 1: Induction and the Natural numbers Math 1a is a somewhat unusual course. It is a proof-based treatment of Calculus, for all of you who have already demonstrated a strong grounding in Calculus

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### 5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1

MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing

### Sets, Relations and Functions

Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu ugust 26, 2014 These notes provide a very brief background in discrete

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### Introduction Russell s Paradox Basic Set Theory Operations on Sets. 6. Sets. Terence Sim

6. Sets Terence Sim 6.1. Introduction A set is a Many that allows itself to be thought of as a One. Georg Cantor Reading Section 6.1 6.3 of Epp. Section 3.1 3.4 of Campbell. Familiar concepts Sets can

### (Refer Slide Time: 1:41)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

### A brief history of type theory (or, more honestly, implementing mathematical objects as sets)

A brief history of type theory (or, more honestly, implementing mathematical objects as sets) Randall Holmes, Boise State University November 8, 2012 [notes revised later; this is the day the talk was

### Section 2.6 Cantor s Theorem and the ZFC Axioms

1 Section 2.6 and the ZFC Axioms Purpose of Section: To state and prove Cantor theorem, which guarantees the existence of infinities larger than the cardinality of the continuum. We also state and discuss

### Probability Foundations for Electrical Engineers Prof. Krishna Jagannathan Department of Electrical Engineering Indian Institute of Technology, Madras

Probability Foundations for Electrical Engineers Prof. Krishna Jagannathan Department of Electrical Engineering Indian Institute of Technology, Madras Lecture - 9 Borel Sets and Lebesgue Measure -1 (Refer

### WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

### Set theory as a foundation for mathematics

V I I I : Set theory as a foundation for mathematics This material is basically supplementary, and it was not covered in the course. In the first section we discuss the basic axioms of set theory and the

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### Chapter Three. Functions. In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics.

Chapter Three Functions 3.1 INTRODUCTION In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics. Definition 3.1: Given sets X and Y, a function from X to

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### Foundations of Mathematics I Set Theory (only a draft)

Foundations of Mathematics I Set Theory (only a draft) Ali Nesin Mathematics Department Istanbul Bilgi University Kuştepe Şişli Istanbul Turkey anesin@bilgi.edu.tr February 12, 2004 2 Contents I Naive

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### Completeness I. Chapter Rational Numbers

Chapter 5 Completeness I Completeness is the key property of the real numbers that the rational numbers lack. Before examining this property we explore the rational and irrational numbers, discovering

### MATHS 315 Mathematical Logic

MATHS 315 Mathematical Logic Second Semester, 2006 Contents 2 Formal Statement Logic 1 2.1 Post production systems................................. 1 2.2 The system L.......................................

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### 2.1 Sets, power sets. Cartesian Products.

Lecture 8 2.1 Sets, power sets. Cartesian Products. Set is an unordered collection of objects. - used to group objects together, - often the objects with similar properties This description of a set (without

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### 4. TILING AND DISSECTION 4.2. Scissors Congruence

What is Area? Despite being such a fundamental notion of geometry, the concept of area is very difficult to define. As human beings, we have an intuitive grasp of the idea which suffices for our everyday

### CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS

CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we construct the set of rational numbers Q using equivalence

### LOGICAL INFERENCE & PROOFs. Debdeep Mukhopadhyay Dept of CSE, IIT Madras

LOGICAL INFERENCE & PROOFs Debdeep Mukhopadhyay Dept of CSE, IIT Madras Defn A theorem is a mathematical assertion which can be shown to be true. A proof is an argument which establishes the truth of a

### 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008

MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### Lecture 1: Elementary Number Theory

Lecture 1: Elementary Number Theory The integers are the simplest and most fundamental objects in discrete mathematics. All calculations by computers are based on the arithmetical operations with integers

### Review for Final Exam

Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters

### More Mathematical Induction. October 27, 2016

More Mathematical Induction October 7, 016 In these slides... Review of ordinary induction. Remark about exponential and polynomial growth. Example a second proof that P(A) = A. Strong induction. Least

### 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008

MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

### Discrete Mathematics, Chapter 5: Induction and Recursion

Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-founded

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### A Mathematical Toolkit

A Mathematical Toolkit Philosophy 389 1 Naive Set Theory Cantor, in his 1895 work Contributions to the Founding of the Theory of Transfinite Numbers, gives the following definition of a set: By a set [Menge]

### How many numbers there are?

How many numbers there are? RADEK HONZIK Radek Honzik: Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz Contents 1 What are numbers 2 1.1 Natural

### Recursion Theory in Set Theory

Contemporary Mathematics Recursion Theory in Set Theory Theodore A. Slaman 1. Introduction Our goal is to convince the reader that recursion theoretic knowledge and experience can be successfully applied

### Notes on counting finite sets

Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................

### Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Note 3

CS 70 Discrete Mathematics and Probability Theory Fall 06 Seshia and Walrand Note 3 Mathematical Induction Introduction. In this note, we introduce the proof technique of mathematical induction. Induction

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### Basic Proof Techniques

Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

### Fundamentals of Mathematics Lecture 6: Propositional Logic

Fundamentals of Mathematics Lecture 6: Propositional Logic Guan-Shieng Huang National Chi Nan University, Taiwan Spring, 2008 1 / 39 Connectives Propositional Connectives I 1 Negation: (not A) A A T F

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### Equivalence relations

Equivalence relations A motivating example for equivalence relations is the problem of constructing the rational numbers. A rational number is the same thing as a fraction a/b, a, b Z and b 0, and hence

### Topology and Convergence by: Daniel Glasscock, May 2012

Topology and Convergence by: Daniel Glasscock, May 2012 These notes grew out of a talk I gave at The Ohio State University. The primary reference is [1]. A possible error in the proof of Theorem 1 in [1]

### EULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p.

EULER S THEOREM KEITH CONRAD. Introduction Fermat s little theorem is an important property of integers to a prime modulus. Theorem. (Fermat). For prime p and any a Z such that a 0 mod p, a p mod p. If

### 7 Relations and Partial Orders

mcs-ftl 2010/9/8 0:40 page 213 #219 7 Relations and Partial Orders A relation is a mathematical tool for describing associations between elements of sets. Relations are widely used in computer science,

### Summer High School 2009 Aaron Bertram. 1. Prime Numbers. Let s start with some notation:

Summer High School 2009 Aaron Bertram 1. Prime Numbers. Let s start with some notation: N = {1, 2, 3, 4, 5,...} is the (infinite) set of natural numbers. Z = {..., 3, 2, 1, 0, 1, 2, 3,...} is the set of

### Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence