# Chapter 1. Sigma-Algebras. 1.1 Definition

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 also in F (c) if B 1, B 2,... is a countable collection of sets in F then their union n=1 B n Sometimes we will just write sigma-algebra instead of sigma-algebra of subsets of X. There are two extreme examples of sigma-algebras: the collection {, X} is a sigma-algebra of subsets of X the set P(X) of all subsets of X is a sigma-algebra Any sigma-algebra F of subsets of X lies between these two extremes: {, X} F P(X) An atom of F is a set A F such that the only subsets of A which are also in F are the empty set and A itself. 1

2 2 CHAPTER 1. SIGMA-ALGEBRAS A partition of X is a collection of disjoint subsets of X whose union is all of X. For simplicity, consider a partition consisting of a finite number of sets A 1,..., A N. Thus A i A j = and A 1 A N = X Then the collect F consisting of all unions of the sets A j forms a σ algebra. Here are a few simple observations: Proposition 1 Let F be a sigma-algebra of subsets of X. (i) X F (ii) If A 1,..., A n F then A 1 A n F (iii) If A 1,..., A n F then A 1 A n F (iv) If A 1, A 2,... is a countable collection of sets in F then n=1 A n F (v) If A, B F then A B F. Proof Since F and it follows that X F. For (ii) we have X = c A 1 A n = A 1 A n F Then (iii) follows by complementation: A 1 A n = (A c 1 A c n) c which is in F because each A c i F and, by (i), F is closed under finite unions. Similarly, (iv) follows by taking complements : n=1 A n = [ n=1 Ac n ]c which belongs to F because F is closed under complements and countable unions. Finally, A B = A B c is in F, because A, B c F. QED

3 1.2. GENERATED SIGMA-ALGEBRA σ(b) Generated Sigma-algebra σ(b) Let X be a set and B a non-empty collection of subsets of X. The smallest σ algebra containing all the sets of B is denoted σ(b) and is called the sigma-algebra generated by the collection B. The term smallest here means that any sigma-algebra containing the sets of B would have to contain all the sets of σ(b) as well. We need to check that such a smalled sigma-algebra exists. To this end observe first the following fact: If G is any non-empty collection of sigma-algebras of subsets of X then the intersection G is also a sigma-algebra of subsets of X. Here G = {A X A F for every F G} consists of all sets A which belong to each sigma-algebra F of G. The verification of this statement is left as an (easy) exercise. Given a collection B of subsets of X, let G B be the collection of all sigmaalgebras containing all the sets of B. Note that and so G B is not empty. Then P(X) G B G B is a sigma-algebra, contains all the sets of B, and is minimal among such sigma-algebras. Minimality here means that if F is a sigma-algebra such that B F then G B F Thus G B is the sigma-algebra generated by B: σ(b) = G B If B is itself a sigma-algebra then of course σ(b) = B.

4 4 CHAPTER 1. SIGMA-ALGEBRAS 1.3 The Dynkin π λ Theorem Let X be a set. A collection P of subsets of X is a π-system if (π) P is closed under finite intersections: if A, B P then A B P Note that by the usual induction argument, this condition implies that if A 1,..., A n are a finite number of sets in P then their intersection A 1 A n is also in P. A collection L of subsets of X is called a λ system if (λ1) L contains the empty set (λ2) L is closed under complements: if A L then A c L (λ3) L is closed under countable disjoint union: if A 1, A 2,... L and A i A j = for every i j, then n=1 A n L Unlike a σ algebra, the notions of π system and λ system are not in themselves fundamental. Their significance is contained in the following theorem which will be of great use later in proving uniqueness of measures: Theorem 1 The Dynkin π λ theorem If P is a π system and L a λ system of subsets of X then σ(p ) L, i.e. the sigma-algebra generated by P is contained in L. The proof of this result is long but can be broken up into simple little pieces. As a first step, we have Lemma 1 A λ system is closed under proper differences, i.e. if A, B L, where L is a λ system, and A B then the difference B A is also in L. Proof. It is best to draw a little diagram illustrating the fact that A B. From this you can see that B A is the complement of the set A B c, and the latter, being the disjoint union of A L and B c L, is in L; thus B A L. More formally, B A = B A c = (B c A) c

5 1.3. THE DYNKIN π λ THEOREM 5 is in L because it is the complement of the set B c A which is in L because it is the union of two disjoint sets A and B c both of which are in L. QED The next step is more substantial: Lemma 2 A family which is both a π system and a λ system is a σ algebra. Proof. Let S be a collection of subsets of X which is both a π system and a λ system. To prove that S is a σ algebra it will be enough to show that S is closed under countable unions (not just disjoint countable unions). Let A 1, A 2,... S. We have to show that their union n=1a n is in S. The trick (and it is a very useful trick) is to rewrite n=1 A n as a countable union of disjoint sets: n=1a n = n=1b n where B 1 = A 1 and, for n 1, B n = A n (A 1 A 2 A n 1 ) = A n A c 1 A c 2 A c n (1.1) Thus B n consists of all elements of A n which do not appear in any previous A i. It is clear that the sets B 1, B 2,... are mutually disjoint. Since S is a λ system, each complement A c i is in S, and since S is a π system it follows then that B n, which is a finite intersection of sets in S, is also in S. QED As further preparation for the proof of the main theorem let us make one more observation, though its significance will only become clear later: Lemma 3 Suppose L is a λ system of subsets of X. For any set A L, let S A be the set of all B X for which A B L. Then S A is a λ system. Proof. First note that S A, because A = L. It is also clear that S A is closed under countable disjoint unions. The last thing we have to show is that S A is closed under complements. To this end, let B S A and observe that A B c = A B = A (A B) The utility in writing the difference A B as the proper difference A (A B) lies in the fact that A B A and we can appeal to Lemma 1, along with the facts that A and A B are both in L, to conclude that A (A B) is in L. QED

6 6 CHAPTER 1. SIGMA-ALGEBRAS Now we return to the proof of the main theorem. As before, P is a π system and L a λ system, with P L. Our objective is to show that the sigma-algebra σ(p ) generated by P is contained in L. The strategy will be to produce a sigma-algebra which lies between P and L, i.e. contains P and is contained in L. This will imply that σ(p ), which is the smallest sigma algebra containing P, is contained in L. We look at l(p ), the intersection of all λ systems containing P. Clearing l(p ) is itself also a λ system and contains P, and is thus the minimal λ system containing P. This means that any λ system which contains P must also contain l(p ). The objective will be to show that the λ system l(p ) is also a π system. This would imply that l(p ) is a sigma-algebra. It contains P and, being the minimal λ system containing P, is a subset of L. This would provide our sigma-algebra lying between P and L. So the last piece of the argument is: Lemma 4 l(p ) is a π system. The proof of this uses a bootstrap argument which is often useful in measure theory. We start with a set A P and show that A B is in l(p ) for every B in l(p ); then we turn around and use this to show that if A and B are in l(p ) then so is their intersection. Proof. Let A P, and let S A be the set of all sets B X for which A B is in l(p ). We have already proven that S A is a λ system. Moreover, it is clear that every element of P is in S A. Thus S A is a λ system with P S A. Therefore, l(p ) S A. Which means that we have proven that for any A P and any B l(p ) the intersection A B is in l(p ). So now consider a B l(p ), and look at S B. The preceding paragraph proves that P S B. On the other hand, by Lemma 3, S B is a λ system. Therefore, l(p ) S B. Which means: for any A l(p ), the intersection A B is in l(p ). Thus, l(p ) is a π system. QED Putting all of the strands together, we have: Proof of Dynkin s theorem. We have proven that the λ system l(p ) is also a π system, and is therefore a σ algebra. On the other hand, we also know that P l(p ) L

7 1.3. THE DYNKIN π λ THEOREM 7 because l(p ) is the intersection of all λ systems containing P, and L is just one λ system containing P. Thus we have produced a sigma-algebra l(p ) lying between P and L. Therefore, P σ(p ) l(p ) L since σ(p ) is the intersection of all sigma-algebras which contain P. QED There are several other similar results which can substitute for the Dynkin π λ theorem. The best known alternative is the monotone class lemma, but we shall not go into this.

### Extension of measure

1 Extension of measure Sayan Mukherjee Dynkin s π λ theorem We will soon need to define probability measures on infinite and possible uncountable sets, like the power set of the naturals. This is hard.

### 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

### 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

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

### Our goal first will be to define a product measure on A 1 A 2.

1. Tensor product of measures and Fubini theorem. Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Recall that the new σ -algebra A 1 A 2 with the unit element is the σ -algebra generated by the

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE CHRISTOPHER HEIL 1.4.1 Introduction We will expand on Section 1.4 of Folland s text, which covers abstract outer measures also called exterior measures).

### 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

### 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

### Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.

### Elements of probability theory

2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### Chapter 1 - σ-algebras

Page 1 of 17 PROBABILITY 3 - Lecture Notes Chapter 1 - σ-algebras Let Ω be a set of outcomes. We denote by P(Ω) its power set, i.e. the collection of all the subsets of Ω. If the cardinality Ω of Ω is

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### 6.4 The Infinitely Many Alleles Model

6.4. THE INFINITELY MANY ALLELES MODEL 93 NE µ [F (X N 1 ) F (µ)] = + 1 i

### Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

### The Lebesgue Measure and Integral

The Lebesgue Measure and Integral Mike Klaas April 12, 2003 Introduction The Riemann integral, defined as the limit of upper and lower sums, is the first example of an integral, and still holds the honour

### P (A) = lim P (A) = N(A)/N,

1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS 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 the identity, such that (i) the

### Full and Complete Binary Trees

Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### DISINTEGRATION OF MEASURES

DISINTEGRTION OF MESURES BEN HES Definition 1. Let (, M, λ), (, N, µ) be sigma-finite measure spaces and let T : be a measurable map. (T, µ)-disintegration is a collection {λ y } y of measures on M such

### 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)

### CmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B

CmSc 175 Discrete Mathematics Lesson 10: SETS Sets: finite, infinite, : empty set, U : universal set Describing a set: Enumeration = {a, b, c} Predicates = {x P(x)} Recursive definition, e.g. sequences

### Finite and discrete probability distributions

8 Finite and discrete probability distributions To understand the algorithmic aspects of number theory and algebra, and applications such as cryptography, a firm grasp of the basics of probability theory

### 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

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### 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

### 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed

### Chapter 3: The basic concepts of probability

Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording

### Fundamentele Informatica II

Fundamentele Informatica II Answer to selected exercises 1 John C Martin: Introduction to Languages and the Theory of Computation M.M. Bonsangue (and J. Kleijn) Fall 2011 Let L be a language. It is clear

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### WEAK DOMINANCE: A MYSTERY CRACKED

WEAK DOMINANCE: A MYSTERY CRACKED JOHN HILLAS AND DOV SAMET Abstract. What strategy profiles can be played when it is common knowledge that weakly dominated strategies are not played? A comparison to the

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

### 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

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### 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

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### 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,

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### 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

### Determination of the normalization level of database schemas through equivalence classes of attributes

Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,

### THE ROOMMATES PROBLEM DISCUSSED

THE ROOMMATES PROBLEM DISCUSSED NATHAN SCHULZ Abstract. The stable roommates problem as originally posed by Gale and Shapley [1] in 1962 involves a single set of even cardinality 2n, each member of which

### MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.

MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P

### Lebesgue Measure on R n

8 CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### CHAPTER 2. Set, Whole Numbers, and Numeration

CHAPTER 2 Set, Whole Numbers, and Numeration 2.1. Sets as a Basis for Whole Numbers A set is a collection of objects, called the elements or members of the set. Three common ways to define sets: (1) A

### x < y iff x < y, or x and y are incomparable and x χ(x,y) < y χ(x,y).

12. Large cardinals The study, or use, of large cardinals is one of the most active areas of research in set theory currently. There are many provably different kinds of large cardinals whose descriptions

### Deterministic Finite Automata

1 Deterministic Finite Automata Definition: A deterministic finite automaton (DFA) consists of 1. a finite set of states (often denoted Q) 2. a finite set Σ of symbols (alphabet) 3. a transition function

### Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

### Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

### 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

### 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.............................

### 4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.

Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,

### THIN GROUPS OF FRACTIONS

Contemporary Mathematics THIN GROUPS OF FRACTIONS Patrick DEHORNOY Abstract. A number of properties of spherical Artin Tits groups extend to Garside groups, defined as the groups of fractions of monoids

### 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

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### Theory of Computation

Theory of Computation For Computer Science & Information Technology By www.thegateacademy.com Syllabus Syllabus for Theory of Computation Regular Expressions and Finite Automata, Context-Free Grammar s

### 5. Linear algebra I: dimension

5. Linear algebra I: dimension 5.1 Some simple results 5.2 Bases and dimension 5.3 Homomorphisms and dimension 1. Some simple results Several observations should be made. Once stated explicitly, the proofs

### Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.

Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a

### Lecture Notes on Measure Theory and Functional Analysis

Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents

### Atomic Cournotian Traders May Be Walrasian

Atomic Cournotian Traders May Be Walrasian Giulio Codognato, Sayantan Ghosal, Simone Tonin September 2014 Abstract In a bilateral oligopoly, with large traders, represented as atoms, and small traders,

### Linear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:

Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),

### Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from

### 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

### TECHNIQUES FOR OPTIMIZING THE RELATIONSHIP BETWEEN DATA STORAGE SPACE AND DATA RETRIEVAL TIME FOR LARGE DATABASES

Techniques For Optimizing The Relationship Between Data Storage Space And Data Retrieval Time For Large Databases TECHNIQUES FOR OPTIMIZING THE RELATIONSHIP BETWEEN DATA STORAGE SPACE AND DATA RETRIEVAL

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

### Structural properties of oracle classes

Structural properties of oracle classes Stanislav Živný Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK Abstract Denote by C the class of oracles relative

### On Winning Conditions of High Borel Complexity in Pushdown Games

Fundamenta Informaticae (2005) 1 22 1 IOS Press On Winning Conditions of High Borel Complexity in Pushdown Games Olivier Finkel Equipe de Logique Mathématique U.F.R. de Mathématiques, Université Paris

### Open and Closed Sets

Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

### 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

### Basic Probability Concepts

page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes

### 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,

### Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

### 4 Domain Relational Calculus

4 Domain Relational Calculus We now present two relational calculi that we will compare to RA. First, what is the difference between an algebra and a calculus? The usual story is that the algebra RA is

### 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

### 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......................................

### On strong fairness in UNITY

On strong fairness in UNITY H.P.Gumm, D.Zhukov Fachbereich Mathematik und Informatik Philipps Universität Marburg {gumm,shukov}@mathematik.uni-marburg.de Abstract. In [6] Tsay and Bagrodia present a correct

### 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,

### The Inverse of a Square Matrix

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich

MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 12 May 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014. Prerequisites

### 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

### Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm

MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following

### Clicker Question. Theorems/Proofs and Computational Problems/Algorithms MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES

MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Tuesday, 1/21/14 General course Information Sets Reading: [J] 1.1 Optional: [H] 1.1-1.7 Exercises: Do before next class; not to hand in [J] pp. 12-14:

### Notes: Chapter 2 Section 2.2: Proof by Induction

Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case - S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### Automata and Formal Languages

Automata and Formal Languages Winter 2009-2010 Yacov Hel-Or 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,

### R u t c o r Research R e p o r t. Boolean functions with a simple certificate for CNF complexity. Ondřej Čepeka Petr Kučera b Petr Savický c

R u t c o r Research R e p o r t Boolean functions with a simple certificate for CNF complexity Ondřej Čepeka Petr Kučera b Petr Savický c RRR 2-2010, January 24, 2010 RUTCOR Rutgers Center for Operations

### 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

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for