Classical Analysis I

Size: px
Start display at page:

Download "Classical Analysis I"

Transcription

1 Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if x is not an element of A, we write x A. If every element of a set A is also an element of B, then A is subset of B. This is denoted as A B. Two sets A and B are equal, denoted A = B, if A and B have the same elements. Equivalently, A = B if A B and B A. If A is a set and P is a property, then {x P(x)} is the subset of X consisting of all elements x of X such that P(x) is true. The empty set has no elements; it has the property that it is a subset of any set. Given two sets A and B we define: The union A B is the set The intersection A B is the set A B = {x x A or x B}. A B = {x x A and x B}. The relative complement A \ B is the set A \ B = {x x A and x B}. Sets A and B are called disjoint if A B =. When it is understood that all sets under considerations are subsets of a fixed set X, then the complement A c of the set A X is defined by A c = X \ A = {x X x A}. The set of all subsets of a given set X is called the power set and is denoted by P(X). 1

2 The concept of union and intersection of two sets extends to unions and intersections of arbitrary families of sets. By a family of sets we mean a nonempty set F whose elements are sets themselves. If F is a family of sets, then A = {x x A for some A F} A F A F A = {x x A for all A F}. Proposition 1.1 (de Morgan s laws). Let {A i i J} be the family of ssubsets of X. Then ( i I A ) c i = i I Ac i. ( i I A ) c i = i I Ac i. If a A and b B, then (a,b) is called an ordered pair. Two ordered pairs (a,b) and (a,b ) are equal if a = a and b = b. The (Cartesian) product A B of A and B is the set of all ordered pairs (a,b) where a A and b B. The product A B = if and only if A = or B =. In general, A B B A. The product of three sets A,B and C is defines as A B C = (A B) C. Proceeding inductively one defines the product of n by A 1 A n = (A 1 A n 1 ) A n. For a A 1 A n we write (a 1,...,a n ) instead of ( ((a 1,a 2 ),a 3 ),,a n ). We call a k the kth component of a. 1.1 Relations Let X and Y be two sets. A relation R from X to Y is a subset R X Y. If X = Y, then R is said to be a relation on X. If x X and y Y, we write xry if (x,y) R Equivalence relation A relation on X is called an equivalence relation if it satisfies the following conditions: (a) (reflexivity) For all x X, x x. 2

3 (b) (symmetry ) If x y, then y x. (c) (transitivity) If x y and y z, then x y. Example 1.2. On the set Z = N N define the relation: (m,n) (j,k) if and only if m + k = n + j. This is an equivalence relation. Indeed, it is obvious that is reflexive and symmetric. If (m,n) (j,k) and (j,k) (r,s), then m + k = n + j and j + s = k + r. Then m + k + j + s = n + j + k + r from which we conclude that m + s = n + r. This means that (m,n) (r,s). Example 1.3. If is an equivalence relation on X, then the equivalence class of x as the set [x] = {y A x y}. The quotient set of A and is the set of all equivalence classes of A with respect to, that is, the set {[x] x A}. The quotient set is denoted by A/. We have (1) Let x,y A. If x y, then [x] = [y] and, if x y, then [x] [y] = (2) x A [x] = A Functions Given two sets X and Y, a relation f from X to Y is called a function, denoted by f : X Y, if for every x X, there is exactly one y Y such that (a,b) f. In other words, if (x,y) f and (x,y ) f, then y = y. This definition of a function identifies a function with its graph. We think of f as a rule of assigning, to the element x X, the element y Y for which (x,y) f. If (x,y) f, then y is the value of f at x, and we write y = f(x). The set X is called the domain of the function and the set Y is called the codomain. the set {f(x) x X} is called the range of f, denoted by im(f), or the image of f, denoted by R(f). If f : X Y is a function, A X and B Y, then f(a) = {f(a) a A} is called the image of A under f and the set f 1 (B) = {x X f(x) C} 3

4 is called the preimage of B under f. Given two functions f : X Y and g : Y Z, we define the composition g f of f and g by g f : X Z x g(f(x)). If X Y, then f is surjective if im(f) = Y, injective if f(x) = f(y) implies x = y for all x,y X, and bijective if f is both injective and surjective. Proposition 1.4. Let f : X Y. Then f is bijective if and only if there is a function g : Y X satisfying g f = id X and f g = id Y. Proof. = Since f is bijective, for every y Y, there is x X such that f(x) = y. Since f is injective, this x is uniquely determined. This defines a function g : Y X with the desired properties. = From f g = id Y follows that f is surjective. If x,y X and f(x) = f(y), then x = g(f(x)) = g(f(y)) = y showing that f is injective. Let f : X Y bijective. Then the inverse function f 1 of f is the unique function f 1 : Y X such that f f 1 = id Y and f 1 f = id X. Proposition 1.5. Let f : X Y and g : Y Z be bijective. Then g f : X Z is bijective and (g f) 1 = f 1 g 1. Proposition 1.6. The following hold for the function f : X Y. If A B X, then f(a) f(b). If A i X for every i J, then f ( i J A ) i i J f(a i). If A i X for every i J, then f ( i J A ) i i J f(a i). If A X, then f(a c ) f(x) \ f(a). If B i Y for every i J, f 1( i I B ) i = i I f 1 (B i ). If B i Y for every i J, f 1( i I B ) i = i I f 1 (B i ). If B Y, f 1 (B c ) = [f 1 (B)] c. 4

5 1.1.3 Order relations A relation < is called a partial ordering on X if it satisfies: (1) For all x X, x x. (2) (transitivity) If x < y and y < z, then x < y. If this holds, then the pair (X,<) is called a partially ordered set. If, in addition, the partial order < satisfies: (4) trichotomy) For every x,y X, either x < y or y < x or x = y, then then is called total order on X and (X,<) is a totally ordered set. Example 1.7. Let (P(X), ) be the power set of X. For A,B P(X), define A < B if and only if A B and A B. Hence < is a proper inclusion of subsets of X. Then < is a partial order on P(X) and (P(X),<) is a partially ordered set. We also write x y to mean either x < y or x = y. Let (X,<) be a partially ordered set and A be a nonempty subset of X. An element x X is called an upper boundof A if a x for all a A, It is called a lower bound if x a for all a A. The subset A bounded above if it has an upper bound, bounded below if it has a lower bound, and bounded if it is both bounded above and below. An element m X is the maximum of A, written m = max(a), if m is an upper bound of A and m A. An element m X is the minimum of A, written m = min(a), if m is a lower bound of A and m A. Let A be a nonempty subset of a partially ordered subset of X which is bounded above. If the set of all upper bounds of A has a minimum, then this element is called the least upper bound of A or supremum of A and is written sup(a). Similarly, if A is bounded below and the set of all lower bounds has a maximum, then this element is called the greatest lower bound of A or infimum and is written inf(a). 5

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

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.

More information

f(x) is a singleton set for all x A. If f is a function and f(x) = {y}, we normally write

f(x) is a singleton set for all x A. If f is a function and f(x) = {y}, we normally write Math 525 Chapter 1 Stuff If A and B are sets, then A B = {(x,y) x A, y B} denotes the product set. If S A B, then S is called a relation from A to B or a relation between A and B. If B = A, S A A is called

More information

INTRODUCTORY SET THEORY

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,

More information

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

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.

More information

Equivalence relations

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

More information

A Problem With The Rational Numbers

A Problem With The Rational Numbers Solvability of Equations Solvability of Equations 1. In fields, linear equations ax + b = 0 are solvable. Solvability of Equations 1. In fields, linear equations ax + b = 0 are solvable. 2. Quadratic equations

More information

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

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

More information

Sets, Relations and Functions

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

More information

Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year , First Semester Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

More information

TOPOLOGICAL PROOFS OF THE EXTREME AND INTERMEDIATE VALUE THEOREMS. Contents

TOPOLOGICAL PROOFS OF THE EXTREME AND INTERMEDIATE VALUE THEOREMS. Contents TOPOLOGICAL PROOFS OF THE EXTREME AND INTERMEDIATE VALUE THEOREMS JAMES MURPHY Abstract. In this paper, I will present some elementary definitions in Topology. In particular, I will explain topological

More information

Lecture 16 : Relations and Functions DRAFT

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

More information

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

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

More information

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

More information

RELATIONS AND FUNCTIONS

RELATIONS AND FUNCTIONS Chapter 1 RELATIONS AND FUNCTIONS There is no permanent place in the world for ugly mathematics.... It may be very hard to define mathematical beauty but that is just as true of beauty of any kind, we

More information

Course 421: Algebraic Topology Section 1: Topological Spaces

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...............

More information

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

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

More information

2.3. Relations. Arrow diagrams. Venn diagrams and arrows can be used for representing

2.3. Relations. Arrow diagrams. Venn diagrams and arrows can be used for representing 2.3. RELATIONS 32 2.3. Relations 2.3.1. Relations. Assume that we have a set of men M and a set of women W, some of whom are married. We want to express which men in M are married to which women in W.

More information

Cartesian Products and Relations

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

More information

Sets and Cardinality Notes for C. F. Miller

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

More information

Class XII: Math Chapter: Relations and Functions. Concepts and Formulae. Key Concepts

Class XII: Math Chapter: Relations and Functions. Concepts and Formulae. Key Concepts Class XII: Math Chapter: Relations and Functions Concepts and Formulae Key Concepts 1. A relation R between two non empty sets A and B is a subset of their Cartesian Product A B. If A B then relation R

More information

RELATIONS AND FUNCTIONS

RELATIONS AND FUNCTIONS Chapter 1 RELATIONS AND FUNCTIONS 1.1 Overview 1.1.1 Relation A relation R from a non-empty set A to a non empty set B is a subset of the Cartesian product A B. The set of all first elements of the ordered

More information

LECTURE NOTES ON RELATIONS AND FUNCTIONS

LECTURE NOTES ON RELATIONS AND FUNCTIONS LECTURE NOTES ON RELATIONS AND FUNCTIONS PETE L. CLARK Contents 1. Relations 1 1.1. The idea of a relation 1 1.2. The formal definition of a relation 2 1.3. Basic terminology and further examples 2 1.4.

More information

Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008

Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008 Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008 David R. Wilkins Copyright c David R. Wilkins 1989 2008 Contents 1 Basic Theorems of Complex Analysis 1 1.1 The Complex Plane........................

More information

Sets and set operations: cont. Functions.

Sets and set operations: cont. Functions. CS 441 Discrete Mathematics for CS Lecture 8 Sets and set operations: cont. Functions. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Set Definition: set is a (unordered) collection of objects.

More information

Structure of Measurable Sets

Structure of Measurable Sets Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations

More information

SETS, RELATIONS, AND FUNCTIONS

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

More information

Week 5: Binary Relations

Week 5: Binary Relations 1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all

More information

PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

More information

3. Equivalence Relations. Discussion

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,

More information

Introducing Functions

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

More information

vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

More information

Chapter Prove or disprove: A (B C) = (A B) (A C). Ans: True, since

Chapter Prove or disprove: A (B C) = (A B) (A C). Ans: True, since Chapter 2 1. Prove or disprove: A (B C) = (A B) (A C)., since A ( B C) = A B C = A ( B C) = ( A B) ( A C) = ( A B) ( A C). 2. Prove that A B= A B by giving a containment proof (that is, prove that the

More information

Discrete Mathematics. Hans Cuypers. October 11, 2007

Discrete Mathematics. Hans Cuypers. October 11, 2007 Hans Cuypers October 11, 2007 1 Contents 1. Relations 4 1.1. Binary relations................................ 4 1.2. Equivalence relations............................. 6 1.3. Relations and Directed Graphs.......................

More information

Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

More information

Review for Final Exam

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 information

This chapter describes set theory, a mathematical theory that underlies all of modern mathematics.

This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.

More information

POWER SETS AND RELATIONS

POWER SETS AND RELATIONS POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

More information

Set theory as a foundation for mathematics

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

More information

Introduction to Topology

Introduction to Topology Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

More information

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

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/

More information

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

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

More information

MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS

MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS ALLAN YASHINSKI Abstract. We explore the notion of well-definedness when defining functions whose domain is

More information

Methoδos Primers, Vol. 1

Methoδos Primers, Vol. 1 Methoδos Primers, Vol. 1 The aim of the Methoδos Primers series is to make available concise introductions to topics in Methodology, Evaluation, Psychometrics, Statistics, Data Analysis at an affordable

More information

Chapter 1. Informal introdution to the axioms of ZF.

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

More information

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 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

More information

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

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

More information

Foundations of Mathematics I Set Theory (only a draft)

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

More information

Introduction to Relations

Introduction to Relations CHAPTER 7 Introduction to Relations 1. Relations and Their Properties 1.1. Definition of a Relation. Definition: A binary relation from a set A to a set B is a subset R A B. If (a, b) R we say a is related

More information

Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

More information

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

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

More information

Math 3000 Running Glossary

Math 3000 Running Glossary Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (

More information

Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook. John Rognes

Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook. John Rognes Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook John Rognes November 29th 2010 Contents Introduction v 1 Set Theory and Logic 1 1.1 ( 1) Fundamental Concepts..............................

More information

Geometric Transformations

Geometric Transformations Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004

NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004 NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................

More information

MA651 Topology. Lecture 6. Separation Axioms.

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

More information

Equivalence Relations

Equivalence Relations Equivalence Relations Definition An equivalence relation on a set S, is a relation on S which is reflexive, symmetric and transitive. Examples: Let S = Z and define R = {(x,y) x and y have the same parity}

More information

Logic & Discrete Math in Software Engineering (CAS 701) Dr. Borzoo Bonakdarpour

Logic & Discrete Math in Software Engineering (CAS 701) Dr. Borzoo Bonakdarpour Logic & Discrete Math in Software Engineering (CAS 701) Background Dr. Borzoo Bonakdarpour Department of Computing and Software McMaster University Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS

More information

Problem Set 1 Solutions Math 109

Problem Set 1 Solutions Math 109 Problem Set 1 Solutions Math 109 Exercise 1.6 Show that a regular tetrahedron has a total of twenty-four symmetries if reflections and products of reflections are allowed. Identify a symmetry which is

More information

SETS AND FUNCTIONS, MATH 215 FALL 2015 (WHYTE)

SETS AND FUNCTIONS, MATH 215 FALL 2015 (WHYTE) SETS AND FUNCTIONS, MATH 215 FALL 2015 (WHYTE) 1. Intro to Sets After some work with numbers, we want to talk about sets. For our purposes, sets are just collections of objects. These objects can be anything

More information

INTRODUCTION TO TOPOLOGY

INTRODUCTION TO TOPOLOGY INTRODUCTION TO TOPOLOGY ALEX KÜRONYA In preparation January 24, 2010 Contents 1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology

More information

Limits and convergence.

Limits and convergence. Chapter 2 Limits and convergence. 2.1 Limit points of a set of real numbers 2.1.1 Limit points of a set. DEFINITION: A point x R is a limit point of a set E R if for all ε > 0 the set (x ε,x + ε) E is

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

Completeness I. Chapter Rational Numbers

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

More information

CHAPTER 3. Mapping Concepts and Mapping Problems for. Scalar Valued Functions of a Scalar Variable

CHAPTER 3. Mapping Concepts and Mapping Problems for. Scalar Valued Functions of a Scalar Variable A SERIES OF CLASS NOTES TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS REMEDIAL CLASS NOTES A COLLECTION OF HANDOUTS FOR REMEDIATION IN FUNDAMENTAL CONCEPTS

More information

Assignment 7; Due Friday, November 11

Assignment 7; Due Friday, November 11 Assignment 7; Due Friday, November 9.8 a The set Q is not connected because we can write it as a union of two nonempty disjoint open sets, for instance U = (, 2) and V = ( 2, ). The connected subsets are

More information

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

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

More information

Basic Set Theory. Chapter Set Theory. can be written: A set is a Many that allows itself to be thought of as a One.

Basic Set Theory. Chapter Set Theory. can be written: A set is a Many that allows itself to be thought of as a One. Chapter Basic Set Theory A set is a Many that allows itself to be thought of as a One. - Georg Cantor This chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical

More information

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

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

More information

0 ( x) 2 = ( x)( x) = (( 1)x)(( 1)x) = ((( 1)x))( 1))x = ((( 1)(x( 1)))x = ((( 1)( 1))x)x = (1x)x = xx = x 2.

0 ( x) 2 = ( x)( x) = (( 1)x)(( 1)x) = ((( 1)x))( 1))x = ((( 1)(x( 1)))x = ((( 1)( 1))x)x = (1x)x = xx = x 2. SOLUTION SET FOR THE HOMEWORK PROBLEMS Page 5. Problem 8. Prove that if x and y are real numbers, then xy x + y. Proof. First we prove that if x is a real number, then x 0. The product of two positive

More information

18.312: Algebraic Combinatorics Lionel Levine. Lecture 8

18.312: Algebraic Combinatorics Lionel Levine. Lecture 8 18.312: Algebraic Combinatorics Lionel Levine Lecture date: March 1, 2011 Lecture 8 Notes by: Christopher Policastro Remark: In the last lecture, someone asked whether all posets could be constructed from

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

2.1.1 Examples of Sets and their Elements

2.1.1 Examples of Sets and their Elements Chapter 2 Set Theory 2.1 Sets The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define

More information

A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

A set is a Many that allows itself to be thought of as a One. (Georg Cantor) Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

More information

Basic Category Theory for Models of Syntax (Preliminary Version)

Basic Category Theory for Models of Syntax (Preliminary Version) Basic Category Theory for Models of Syntax (Preliminary Version) R. L. Crole ( ) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K. Abstract. These notes

More information

Chapter 1. Logic and Proof

Chapter 1. Logic and Proof Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known

More information

Notes on counting finite sets

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......................................

More information

Introduction to Mathematical Economics

Introduction to Mathematical Economics Lecture Notes on Introduction to Mathematical Economics Walter Bossert Département de Sciences Economiques Université de Montréal CP 6128, succursale Centre-ville Montréal QC H3C 3J7 Canada walterbossert@umontrealca

More information

An Introduction to Real Analysis. John K. Hunter. Department of Mathematics, University of California at Davis

An Introduction to Real Analysis. John K. Hunter. Department of Mathematics, University of California at Davis An Introduction to Real Analysis John K. Hunter Department of Mathematics, University of California at Davis Abstract. These are some notes on introductory real analysis. They cover the properties of the

More information

2.3 Bounds of sets of real numbers

2.3 Bounds of sets of real numbers 2.3 Bounds of sets of real numbers 2.3.1 Upper bounds of a set; the least upper bound (supremum) Consider S a set of real numbers. S is called bounded above if there is a number M so that any x S is less

More information

CHAPTER 5: MODULAR ARITHMETIC

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

More information

Continuous functions

Continuous functions CHAPTER 3 Continuous functions In this chapter I will always denote a non-empty subset of R. This includes more general sets, but the most common examples of I are intervals. 3.1. The ǫ-δ definition of

More information

Section 6.4 Closures of Relations

Section 6.4 Closures of Relations Section 6.4 Closures of Relations Definition: The closure of a relation R with respect to property P is the relation obtained by adding the minimum number of ordered pairs to R to obtain property P. In

More information

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

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

More information

Chapter 1. Sigma-Algebras. 1.1 Definition

Chapter 1. Sigma-Algebras. 1.1 Definition Chapter 1 Sigma-Algebras 1.1 Definition Consider a set X. A σ algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) F (b) if B F then its complement B c is

More information

Fixed Point Theorems in Topology and Geometry

Fixed Point Theorems in Topology and Geometry Fixed Point Theorems in Topology and Geometry A Senior Thesis Submitted to the Department of Mathematics In Partial Fulfillment of the Requirements for the Departmental Honors Baccalaureate By Morgan Schreffler

More information

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

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

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective

More information

Revision of ring theory

Revision of ring theory CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic

More information

Solutions to Homework Problems from Chapter 3

Solutions to Homework Problems from Chapter 3 Solutions to Homework Problems from Chapter 3 31 311 The following subsets of Z (with ordinary addition and multiplication satisfy all but one of the axioms for a ring In each case, which axiom fails (a

More information

Math 112 Solutions for Problem Set 2 Spring, 2013 Professor Hopkins

Math 112 Solutions for Problem Set 2 Spring, 2013 Professor Hopkins Math 112 Solutions for Problem Set 2 Spring, 2013 Professor Hopkins 1. (Rudin, Ch 1, #6). Fix b > 1. (a) If m,n,p,q are integers, n > 0, q > 0, and r = m/n = p/q, prove that (b m ) 1/n = (b p ) 1/q. Hence

More information

Chapter 10. Abstract algebra

Chapter 10. Abstract algebra Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

More information

Lecture 17 : Equivalence and Order Relations DRAFT

Lecture 17 : Equivalence and Order Relations DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

More information

Metric Spaces. Chapter 1

Metric Spaces. Chapter 1 Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

More information

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

More information

Notes on Discrete Mathematics. Miguel A. Lerma

Notes on Discrete Mathematics. Miguel A. Lerma Notes on Discrete Mathematics Miguel A. Lerma Contents Introduction 5 Chapter 1. Logic, Proofs 6 1.1. Propositions 6 1.2. Predicates, Quantifiers 11 1.3. Proofs 13 Chapter 2. Sets, Functions, Relations

More information

Compactness in metric spaces

Compactness in metric spaces MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a, b] of the real line, and more generally the

More information

!"#$%&'&()*+&,-(.%"/01.*"(,,,,,,,.",2*(&3%,456&7%3

!#$%&'&()*+&,-(.%/01.*(,,,,,,,.,2*(&3%,456&7%3 "#$%&'&()*+&,-(.%"/01.*"(,,,,,,,.",2*(&3%,456&7%3 849:,-,;,?49,42@?

More information

Define the set of rational numbers to be the set of equivalence classes under #.

Define the set of rational numbers to be the set of equivalence classes under #. Rational Numbers There are four standard arithmetic operations: addition, subtraction, multiplication, and division. Just as we took differences of natural numbers to represent integers, here the essence

More information