# 4.1. Sets. Introduction. Prerequisites. Learning Outcomes. Learning Style

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 ets 4.1 Introduction If we can identify a property which is common to several objects, it is often useful to group them together. uch a grouping is called a set. Engineers for example, may wish to study all components of a production run which fail to meet some specified tolerance. Mathematicians may lookat sets of numbers with particular properties, for example, the set of all even numbers, or the set of all numbers greater than zero. In this blockwe introduce some terminology that is commonly used to describe sets, and practice using set notation. This notation will be particularly useful when we come to study probability in later blocks. Prerequisites efore starting this lockyou should... Learning Outcomes fter completing this lockyou should be able to... understand what is meant by a set use set notation understand the meaning of the concepts of the intersection and union of two sets understand what is meant by the complement of a set be able to use Venn diagrams to illustrate sets. 1 Learning tyle To achieve what is expected of you... allocate sufficient study time briefly revise the prerequisite material attempt every guided exercise and most of the other exercises

2 1. ets set is any collection of objects. Here, the word object is used in its most general sense: an object may be a diode, an aircraft, a number, or a letter, for example. set is often described by listing the collection of objects - these are the members or elements of the set. We usually write this list of elements in curly brackets, and denote the full set by a capital letter. For example, C = {the resistors produced in a factory on a particular day} D = {on, off} E = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} The elements of set C, above, are the resistors produced in a factory on a particular day. These could be listed individually but as the number is large it is not practical to do this. et D lists the two possible states of a simple switch, and the elements of set E are the digits used in the decimal system. ometimes we can describe a set in words. For example, is the set all odd numbers. Clearly all the elements of this set cannot be listed. imilarly, is the set of binary digits viz = {0, 1}. has only two elements. set with a finite number of elements is called a finite set.,c,d and E are finite sets. The set has an infinite number of elements and so is not a finite set. It is known as an infinite set. Two sets are equal if they contain exactly the same elements. For example, the sets {9, 10, 14} and {10, 14, 9} are equal since the order in which elements are written is unimportant. Note also that repeated elements are ignored. The set {2, 3, 3, 3, 5, 5} is equal to the set {2, 3, 5}. ubsets ometimes one set is contained completely within another set. For example if X = {2, 3, 4, 5, 6} and Y = {2, 3, 6} then all the elements of Y are also elements of X. We say that Y is a subset of X and write Y X. Example Given = {0, 1, 2, 3}, = {0, 1, 2, 3, 4, 5, 6} and C = {0, 1}, state which sets are subsets. olution is a subset of, that is C is a subset of, that is C C is a subset of, that is C. Engineering Mathematics: Open Learning Unit Level 1 2

3 Now do this exercise factory produces cars over a five day period; Monday to Friday. Consider the following sets, (a) = {cars produced from Monday to Friday} (b) = {cars produced from Monday to Thursday} (c) C = {cars produced on Friday} (d) D = {cars produced on Wednesday} (e) E = {cars produced on Wednesday and Thursday} tate which sets are subsets. nswer The symbol. To show that an element belongs to a particular set we use the symbol, (Greekepsilon). This symbol means is a member of or belongs to. The symbol means is not a member of or does not belong to. For example if X = {all even numbers} then we may write 4 X, 6 X, 7 X and 11 X. The empty set and the universal set ometimes a set will contain no elements. For example, suppose we define the set K by K = {all odd numbers which are divisible by 4} ince there are no odd numbers which are divisible by 4, then K has no elements. The set with no elements is called the empty set, and it is denoted by. On the other hand, the set containing all the objects of interest in a particular situation is called the universal set, denoted by. The precise universal set will depend upon the context. If, for example, we are concerned only with whole numbers then will be the set of whole numbers. If we are concerned only with the decimal digits then we take = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. The complement of a set Given a set and a universal set we can define a new set, called the complement of and denoted by. The complement of contains all the elements of the universal set that are not in. 3 Engineering Mathematics: Open Learning Unit Level 1

4 Example Given = {2, 3, 7}, = {0, 1, 2, 3, 4} and = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} state (a) (b) olution (a) The elements of are those which belong to but not to. = {0, 1, 4, 5, 6, 8, 9} (b) = {5, 6, 7, 8, 9} ometimes a set is described in a mathematical way. uppose the set Q contains all numbers which are divisible by 4 and 7. We can write Q = {x : x is divisible by 4 and x is divisible by 7} The symbol : stands for such that. We read the above as Q is the set comprising all elements x, such that x is divisible by 4 and by Venn diagrams ets are often represented pictorially by Venn diagrams (see Figure 1). C D Figure 1 Here,, C, D represent sets. The sets, have no items in common so are drawn as nonintersecting regions whilst the sets C, D have some items in common so are drawn overlapping. In a Venn diagram the universal set is always represented by a rectangle and sets of interest by area regions within this rectangle. Example Represent the sets = {0, 1} and = {0, 1, 2, 3, 4} using a Venn diagram. olution The elements 0 and 1 are in set, represented by the small circle in Figure 2. The large circle represents set and so contains the elements 0,1,2,3 and 4. suitable universal set in this case is the set of all integers. The universal set is shown by the rectangle. Note that. This is shown in the Venn diagram by being completely inside. Engineering Mathematics: Open Learning Unit Level 1 4

5 2 O Figure 2 The set is contained completely within Now do this exercise Given = {0, 1} and = {2, 3, 4} draw Venn diagrams showing (a) and (b) (c) nswer 3. The intersection and union of sets Intersection Given two sets, and, the intersection of and is a set which contains elements that are common both to and. We write to denote the intersection of and. Mathematically we write this as: Key Point = {x : x and x } This says that the intersection contains all the elements x such that x belongs to and also x belongs to. Note that and are identical. The intersection of two sets can be represented by a Venn diagram as shown in Figure 6. Figure 6 The overlapping area represents 5 Engineering Mathematics: Open Learning Unit Level 1

6 Example Given = {3, 4, 5, 6}, = {3, 5, 9, 10, 15} and C = {4, 6, 10} state (a) (b) C and draw a Venn diagram representing these intersections. olution (a) The elements common to both and are 3 and 5. Hence = {3, 5} (b) The only element common to and C is 10. Hence C = {10} C Now do this exercise Given D = {a, b, c} and F = {the entire alphabet} state D F The intersection of three or more sets is possible. nswer Example Given = {0, 1, 2, 3}, = {1, 2, 3, 4, 5} and C = {2, 3, 4, 7, 9} state (a) (b) ( ) C (c) C (d) ( C) olution (a) The elements common to and are 1, 2 and 3 so = {1, 2, 3} (b) We need to consider the sets ( ) and C. is given in (a). The elements common to ( ) and C are 2 and 3. Hence ( ) C = {2, 3} (c) The elements common to and C are 2, 3 and 4 so C = {2, 3, 4} (d) We lookat the sets and ( C). The common elements are 2 and 3. Hence ( C) ={2, 3} Note from (b) and (d) that ( ) C = ( C). The example illustrates a general rule. For any sets, and C it is true that ( ) C = ( C) The position of the brackets is thus unimportant. They are usually omitted and we write C. Engineering Mathematics: Open Learning Unit Level 1 6

7 uppose that sets and have no elements in common. Then their intersection contains no elements and we say that and are disjoint sets. We express this as = Recall that is the empty set. Disjoint sets are represented by separate area regions in the Venn diagram. Union The union of two sets and is a set which contains all the elements of together with all the elements of. We write to denote the union of and. We can describe the set formally by: Key Point = {x : x or x or both} Thus the elements of the set are those quantities x such that x is a member of or a member of or a member of both and. The deeply shaded areas of Figure 7 represents. (a) Figure 7 In Figure 7(a) the sets intersect, whereas in Figure 7(b) the sets have no region in common. We say they are disjoint. (b) Example Given = {0, 1}, = {1, 2, 3} and C = {2, 3, 4, 5} write down (a) (b) C (c) C olution (a) = {0, 1, 2, 3}. (b) C = {0, 1, 2, 3, 4, 5}. (c) C = {1, 2, 3, 4, 5}. Recall that there is no need to repeat elements in a set. unimportant so =. Clearly the order of the union is 7 Engineering Mathematics: Open Learning Unit Level 1

8 Now do this exercise Given = {2, 3, 4, 5, 6}, = {2, 4, 6, 8, 10} and C = {3, 5, 7, 9, 11} state (a) (b) ( ) C (c) (d) ( ) C (e) C nswer More exercises for you to try 1. Given a set, its complement and a universal set state which of the following expressions are true and which are false. (a) = (b) = (c) = (d) = (e) = (f) = (g) = (h) = (i) = (j) = (k) = (l) = 2. Given = {a, b, c, d, e, f}, = {a, c, d, f, h} and C = {e, f, x, y} obtain the sets: (a) (b) C (c) ( C) (d) C ( ) (e) C (f) ( C) 3. List the elements of the following sets: (a) = {x : x is odd and x is greater than 0 and less than 12} (b) = {x : x is even and x is greater than 19 and less than 31} 4. Given = {5, 6, 7, 9}, = {0, 2, 4, 6, 8} and = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} list the elements of each of the following sets: (a) (b) (c) (d) (e) (f) ( ) (g) ( ) (h) ( ) (i) ( ) What do you notice about your answers to (c),(g) and (d),(f)? 5. Given that and are intersecting sets, i.e. are not disjoint, show on a Venn diagran the following sets (a) (b) (c) (d) (e) nswer Engineering Mathematics: Open Learning Unit Level 1 8

9 End of lock Engineering Mathematics: Open Learning Unit Level 1

10 (a) is a subset of, that is,. (b) C is a subset of, that is, C. (c) D is a subset of, that is, D. (d) E is a subset of, that is, E. (e) D is a subset of, that is, D. (f) E is a subset of, that is, E. (g) D is a subset of E, that is, D E. ackto the theory Engineering Mathematics: Open Learning Unit Level 1 10

11 (a) Note that and have no elements in common. This is represented pictorially in the Venn diagram by circles which are totally separate from each other as shown in Figure Figure 3 The sets and have no elements in common. (b) The complement of is the set whose elements do not belong to. The set is shown in Figure 4. ' Figure 4 The complement of contains elements which are not in. (c) The set,, is shown in Figure 5. ' Figure 5 Everything outside of (but within ) is. ackto the theory 11 Engineering Mathematics: Open Learning Unit Level 1

12 The elements common to D and F are a, b and c, and so D F = {a, b, c} Note that D is a subset of F and so D F = D. ackto the theory Engineering Mathematics: Open Learning Unit Level 1 12

13 (a) = {2, 3, 4, 5, 6, 8, 10} (b) We need to lookat the sets ( ) and C. The elements common to both of these sets are 3 and 5. Hence ( ) C = {3, 5}. (c) = {2, 4, 6} (d) We consider the sets ( ) and C. We form the union of these two sets to obtain ( ) C = {2, 3, 4, 5, 6, 7, 9, 11}. (e) The set formed by the union of all three sets will contain all the elements from all the sets. C = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} ackto the theory 13 Engineering Mathematics: Open Learning Unit Level 1

14 1. a) T, b) F, c) T, d) F, e) F, f) T, g) F, h) F, i) T, j) F, k) F, l) T. 2. a) {a, b, c, d, e, f, h}, b) {f}, c) {a, c, d, e, f}, d) {e, f}, e) {f}, f) {a, c, d, e, f, h}. 3. a) {1, 3, 5, 7, 9, 11}, b) {20, 22, 24, 26, 28, 30}. 4. a) {0, 1, 2, 3, 4, 8}, b) {1, 3, 5, 7, 9}, c) {0, 1, 2, 3, 4, 5, 7, 8, 9}, d) {1, 3}, e) {0, 2, 4, 5, 6, 7, 8, 9}, f) {1, 3}, g) {0, 1, 2, 3, 4, 5, 7, 8, 9}, h) {1, 3, 5, 6, 7, 9}, i) {0, 2, 4, 8}. 5. (a) (b) (c) (d) (e) ackto the theory Engineering Mathematics: Open Learning Unit Level 1 14

### THE LANGUAGE OF SETS AND SET NOTATION

THE LNGGE OF SETS ND SET NOTTION Mathematics is often referred to as a language with its own vocabulary and rules of grammar; one of the basic building blocks of the language of mathematics is the language

### 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

### Sections 2.1, 2.2 and 2.4

SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete

### 2.1 Symbols and Terminology

2.1 Symbols and Terminology Definitions: set is a collection of objects. The objects belonging to the set are called elements, ormembers, oftheset. Sets can be designated in one of three different ways:

### LESSON SUMMARY. Set Operations and Venn Diagrams

LESSON SUMMARY CXC CSEC MATHEMATICS UNIT Three: Set Theory Lesson 4 Set Operations and Venn Diagrams Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volumes 1 and 2. (Some helpful exercises

### 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:

### 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

### The Language of Mathematics

CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,

### 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,

### Announcements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets

CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.3-2.6 Homework 2 due Tuesday Recitation 3 on Friday

### Sets and set operations

CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used

### 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

### What is a set? Sets. Specifying a Set. Notes. The Universal Set. Specifying a Set 10/29/13

What is a set? Sets CS 231 Dianna Xu set is a group of objects People: {lice, ob, Clara} Colors of a rainbow: {red, orange, yellow, green, blue, purple} States in the S: {labama, laska, Virginia, } ll

### Lecture 1. Basic Concepts of Set Theory, Functions and Relations

September 7, 2005 p. 1 Lecture 1. Basic Concepts of Set Theory, Functions and Relations 0. Preliminaries...1 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2

### 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

### not to be republishe NCERT SETS Chapter Introduction 1.2 Sets and their Representations

SETS Chapter 1 In these days of conflict between ancient and modern studies; there must surely be something to be said for a study which did not begin with Pythagoras and will not end with Einstein; but

### 2.1 The Algebra of Sets

Chapter 2 Abstract Algebra 83 part of abstract algebra, sets are fundamental to all areas of mathematics and we need to establish a precise language for sets. We also explore operations on sets and relations

### 1.2 Inductive Reasoning

Page 1 of 6 1.2 Inductive Reasoning Goal Use inductive reasoning to make conjectures. Key Words conjecture inductive reasoning counterexample Scientists and mathematicians look for patterns and try to

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### 3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

### A Little Set Theory (Never Hurt Anybody)

A Little Set Theory (Never Hurt Anybody) Matthew Saltzman Department of Mathematical Sciences Clemson University Draft: August 21, 2013 1 Introduction The fundamental ideas of set theory and the algebra

### 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

### Working with whole numbers

1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and

### Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to

### 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

### 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.

### Basic concepts in probability. Sue Gordon

Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

### 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

### Set Theory: Shading Venn Diagrams

Set Theory: Shading Venn Diagrams Venn diagrams are representations of sets that use pictures. We will work with Venn diagrams involving two sets (two-circle diagrams) and three sets (three-circle diagrams).

### 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

### Set operations and Venn Diagrams. COPYRIGHT 2006 by LAVON B. PAGE

Set operations and Venn Diagrams Set operations and Venn diagrams! = { x x " and x " } This is the intersection of and. # = { x x " or x " } This is the union of and. n element of! belongs to both and,

### 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

### Access The Mathematics of Internet Search Engines

Lesson1 Access The Mathematics of Internet Search Engines You are living in the midst of an ongoing revolution in information processing and telecommunications. Telephones, televisions, and computers are

### 9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes

The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of

### Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

### 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

### 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

### Discrete Mathematics Set Operations

Discrete Mathematics 1-3. Set Operations Introduction to Set Theory A setis a new type of structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.

### Fractions. Cavendish Community Primary School

Fractions Children in the Foundation Stage should be introduced to the concept of halves and quarters through play and practical activities in preparation for calculation at Key Stage One. Y Understand

### 0N1 (MATH19861) Mathematics for Foundation Year. math19861.html. Lecture Notes.

0N1 (MATH19861) Mathematics for Foundation Year http://www.maths.manchester.ac.uk/~avb/ math19861.html Lecture Notes 16 Sep 2016 0N1 Mathematics Course Arrangements 16 Sep 2016 2 Arrangements for the Course

### 7 Relations and Functions

7 Relations and Functions In this section, we introduce the concept of relations and functions. Relations A relation R from a set A to a set B is a set of ordered pairs (a, b), where a is a member of A,

### 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,

### + Section 6.2 and 6.3

Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

### STAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia

STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that

### 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

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.

Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### 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

### Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

### Elements of probability theory

The role of probability theory in statistics We collect data so as to provide evidentiary support for answers we give to our many questions about the world (and in our particular case, about the business

### 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

### 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

### No Solution Equations Let s look at the following equation: 2 +3=2 +7

5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

### ) ( ) Thus, (, 4.5] [ 7, 6) Thus, (, 3) ( 5, ) = (, 6). = ( 5, 3).

152 Sect 10.1 - Compound Inequalities Concept #1 Union and Intersection To understand the Union and Intersection of two sets, let s begin with an example. Let A = {1, 2,,, 5} and B = {2,, 6, 8}. Union

### Session 6 Number Theory

Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

### NUMBER SYSTEMS. William Stallings

NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html

### Check Skills You ll Need. New Vocabulary union intersection disjoint sets. Union of Sets

NY-4 nion and Intersection of Sets Learning Standards for Mathematics..31 Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets). Check Skills You ll Need

### Math 166 - Week in Review #4. A proposition, or statement, is a declarative sentence that can be classified as either true or false, but not both.

Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Week in Review #4 Sections A.1 and A.2 - Propositions, Connectives, and Truth Tables A proposition, or statement, is a declarative sentence that

### Cubes and Cube Roots

CUBES AND CUBE ROOTS 109 Cubes and Cube Roots CHAPTER 7 7.1 Introduction This is a story about one of India s great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy

### 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,

### Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

### ALGORITHM AND FLOW CHART

ALGORITHM AND FLOW CHART 1.1 Introduction 1.2 Problem Solving 1.3 Algorithm 1.3.1 Examples of Algorithm 1.3.2 Properties of an Algorithm 1.4 Flow Chart 1.4.1 Flow Chart Symbols 1.4.2 Some Flowchart Examples

### Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments

### 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,

### Reading 13 : Finite State Automata and Regular Expressions

CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model

### 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal:

Exercises 1 - number representations Questions 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal: (a) 3012 (b) - 435 2. For each of

### Absolute Value Equations and Inequalities

Key Concepts: Compound Inequalities Absolute Value Equations and Inequalities Intersections and unions Suppose that A and B are two sets of numbers. The intersection of A and B is the set of all numbers

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### 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

### 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,

### 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

### 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

### 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.

### Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

### COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

### 14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style

Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

### Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

### 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

### 6.1. The Exponential Function. Introduction. Prerequisites. Learning Outcomes. Learning Style

The Exponential Function 6.1 Introduction In this block we revisit the use of exponents. We consider how the expression a x is defined when a is a positive number and x is irrational. Previously we have

### 1 / Basic Structures: Sets, Functions, Sequences, and Sums - definition of a set, and the use of the intuitive notion that any property whatever there

C H A P T E R Basic Structures: Sets, Functions, Sequences, and Sums.1 Sets. Set Operations.3 Functions.4 Sequences and Summations Much of discrete mathematics is devoted to the study of discrete structures,

### C H A P T E R Regular Expressions regular expression

7 CHAPTER Regular Expressions Most programmers and other power-users of computer systems have used tools that match text patterns. You may have used a Web search engine with a pattern like travel cancun

### Statistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then

Probability Statistical Inference Question: How often would this method give the correct answer if I used it many times? Answer: Use laws of probability. 1 Example: Tossing a coin If the coin is fair (chance

### Pigeonhole Principle Solutions

Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such

### 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

### Basic Concepts of Set Theory, Functions and Relations

March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2 1.3. Identity and cardinality...3

### Discrete Mathematics

Discrete Mathematics Chih-Wei Yi Dept. of Computer Science National Chiao Tung University March 16, 2009 2.1 Sets 2.1 Sets 2.1 Sets Basic Notations for Sets For sets, we ll use variables S, T, U,. We can

### Introduction to Proofs

Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas are

### Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

### Basic Set Theory. 1. Motivation. Fido Sue. Fred Aristotle Bob. LX 502 - Semantics I September 11, 2008

Basic Set Theory LX 502 - Semantics I September 11, 2008 1. Motivation When you start reading these notes, the first thing you should be asking yourselves is What is Set Theory and why is it relevant?

### (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems.

3130CIT: Theory of Computation Turing machines and undecidability (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems. An undecidable problem

### Unit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12

Unit 3: Algebra Date Topic Page (s) Algebra Terminology Variables and Algebra Tiles 3 5 Like Terms 6 8 Adding/Subtracting Polynomials 9 1 Expanding Polynomials 13 15 Introduction to Equations 16 17 One

### Financial Mathematics

Financial Mathematics For the next few weeks we will study the mathematics of finance. Apart from basic arithmetic, financial mathematics is probably the most practical math you will learn. practical in

### Inf1A: Deterministic Finite State Machines

Lecture 2 InfA: Deterministic Finite State Machines 2. Introduction In this lecture we will focus a little more on deterministic Finite State Machines. Also, we will be mostly concerned with Finite State