# CHAPTER 2. Set, Whole Numbers, and Numeration

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 verbal description. The set of all sophomore female students in this class This is a clear criterion for judging set membership. (2) Listing (within braces): {orange, pink, green} (3) Set-builder notation: Note. (1) If S = {2, 4, 6, 8, 10}, {x x is an even whole number} = {2, 4, 6, 8,... } 4 2 S (4 is an element of S) and 7 /2 S (7 is not an element of S). (2) ; (preferably) or { } denotes the empty set (or null set), the set with no members. 11

2 12 2. SET, WHOLE NUMBERS, AND NUMERATION Two sets A and B are equal (A = B) if they have precisely the same elements. Note. (1) We write A 6= B if A and B are not equal. (2) Elements are usually listed only once in a set, so {a, a, b, c} = {a, b, c}. (3) The order of listing is immaterial, so (4) {;} 6= ; (or {{ }} 6= { }). Why? {a, b, c} = {c, a, b}. Definition. A 1 1 correspondence between two sets A and B is a pairing of the elements of A with the elements of B so that each element of A corresponds with exactly one element of B, and vice-versa. We write A B and say A and B are equivalent or matching sets. Example. Definition. (1) Set A is a subset of set B, written A B, if and only if every element of A is also an element of B. Example. {1, 2, 4} {1, 2, 3, 4, 5}, but {1, 2, 3} 6 {1, 2, 4} Note. For every set A, A A and ; A (Why?) (2) Set A is a proper subset of set B, written A B, if A B and A 6= B (i.e., B has an element not in A). Example. {1, 2, 4} {1, 2, 3, 4, 5}, but {1, 2, 3} 6 {1, 2, 3}

3 2.1. SETS AS A BASIS FOR WHOLE NUMBERS 13 Note. A = B if and only if A B and B A. Definition. A set is finite if it is empty or can be put in 1 1 correspondence with a set of the form {1, 2, 3,..., n} where n is a counting number. A set is infinite if it is not finite. Note. A set is infinite if and only if it is equivalent to a proper subset of itself. Example. {1, 2, 3, 4,... } l l l l {2, 4, 6, 8,... } What are some implications here? Is every infinite set equivalent to the set of counting numbers? Venn diagrams are used to show relationships between sets. We use a rectangle called the universal set U to represent all elements considered in a discussion or context and circles for other sets. A B This is a proper subset since x /2 A.

4 14 2. SET, WHOLE NUMBERS, AND NUMERATION Two sets are disjoint if they have no elements in common. Operations on sets: (1) The union of two sets A and B, denoted A [ B, is the set containing all elements belonging to A or to B (or to both). A [ B (2) The intersection of two sets A and B, denoted A \ B, is the set of all elements common to both A and B. A \ B

5 2.1. SETS AS A BASIS FOR WHOLE NUMBERS 15 (3) The complement of a set A, denoted A, is the set of all elements in the universe U not in A. A (4) The set di erence (or relative complement) of set B from set A, denoted A B, is the set of all elements of A that are not in B. A B (5) The Cartesian product of set A with set B, denoted A B and read A cross B, is the set of all ordered pairs (a, b) where a 2 A and b 2 B. Note. A [ B = {x x 2 A or x 2 B} A \ B = {x x 2 A and x 2 B} A = {x x 2 U and x /2 A} A B = {x x 2 A and x /2 B} A B = {(a, b) x 2 A and x 2 B}

6 16 2. SET, WHOLE NUMBERS, AND NUMERATION Problem (Page 56 # 25). Let A = {50, 55, 60, 65, 70, 75, 80}, B = {50, 60, 70, 80}, C = {60, 70, 80}, D = {55, 65}. (a) A [ (B \ C) = A [ {60, 70, 80} = A (b) (A [ B) \ C = A \ C = C (c)(a \ C) [ (C \ D) = C [ ; = C (d)(a \ C) \ (C [ D) = C [ {55, 60, 65, 70, 80} = C (e)(b C) \ A = {50} {50} \ A = {50} (f)(a D) \ (B C) = {50, 60, 70, 75, 80} \ {50} = {50}

7 2.1. SETS AS A BASIS FOR WHOLE NUMBERS 17 Problem (Page 56 # 39). A university professor asked his class of 42 students when they had studied for his class the previous weekend. Their responses were as follows: 9 had studied on Friday 18 had studied on Saturday 30 had studied on Sunday 3 had studied on both Friday and Saturday 10 had studied on both Saturday and Sunday 6 had studied on both Friday and Sunday 2 had studied on Friday, Satuday, and Sunday Assuming that all 42 students responded and answered honestly, answer the following questions. (a) How many students studied on Sunday but not on either Friday or Saturday? (b) How many students did all of their studying one one day? (c) How many of the students did not study at all for this class last weekend? Strategy 7 Draw a diagram. Here we use a Venn diagram (A = Friday, B = Saturday, C = Sunday). 2 had studied on Friday, Satuday, and Sunday 2

8 18 2. SET, WHOLE NUMBERS, AND NUMERATION 3 had studied on both Friday and Saturday 10 had studied on both Saturday and Sunday 6 had studied on both Friday and Sunday had studied on Friday 18 had studied on Saturday 30 had studied on Sunday (a) How many students studied on Sunday but not on either Friday or Saturday? 16 (b) How many students did all of their studying one one day? 25 (c) How many of the students did not study at all for this class last weekend? 2

9 2.2. WHOLE NUMBERS AND NUMERATION Whole numbers and numeration A number is an idea or abstraction that represents a quantity. A numeral is the symbol used to represent a number. There are three common uses of numbers: (1) A cardinal number describes how many elements are in a finite set. is the set of whole numbers. W = {0, 1, 2, 3,... } (2) The ordinal numbers deal with ordering: first, second, third, etc. There are three ways of ordering whole numbers: (a) the usual counting chant: 3 < 6 since 3 comes before 6 in the chant. (b) using 1-1 correspondence: We let n(a) represent the number of elements in a set A and n(b) represent the number of elements in a set B. Let a = n(a) and b = n(b). Then a < b or b > a if A is equivalent to a proper subset of B. Example.

10 20 2. SET, WHOLE NUMBERS, AND NUMERATION (c) using the whole number line. 4 < 8 since 4 is to the left of 8 on the whole number line. (3) Identification numbers are used to name things, such as telephone numbers, social security numbers, and CBU 899 numbers. Numeration systems use various numerals to represent numbers. (1) The tally numeration system uses single strokes for numbers and can be improved by grouping. For 37: or We see that grouping helps considerably. What are some advantages and disadvantages of this system? (2) The Egyption numeration system uses the following numerals: This is an additive system since the values of the individual numerals are added together to form numbers. Note. The order in which the numerals are written for a number is immaterial.

11 2.2. WHOLE NUMBERS AND NUMERATION 21 (3) The Roman numeration system uses the following basic numerals: It is a positional system in that the position of a numeral a ects the value being represented. For example, IX is di erent from XI. It is also a subtractive system in that a lower numeral to the left of a numeral means subtraction rather than addition: IV means 5-1 or 4. Here are some common pairings: Finally, this is also a multiplicative system in that XV means or 15,000. Example. VMCMXLXIV means V M CM XL IV = = 6944.

12 22 2. SET, WHOLE NUMBERS, AND NUMERATION (4) The Babylonian numeration system uses the following two symbols: This system uses place value, where symbols represent di erent values depending on the place in which they were written. Place value is based on 60. But consider the following: Does this represent = 74 or (60) + 14 = 3614? To help in cases like this, the placeholder was introduced to indicate a vacant place. (What number seems to be missing in this and all previous systems?) With this new symbol, means = 74, while means (60) + 14 = 3614.

13 2.2. WHOLE NUMBERS AND NUMERATION 23 (5) The Mayan numeration system was a vertical place value system and had a symbol for zero. The three symbols used are Here are some sample simple numbers: This last number is di erent in that it occupies two levels, the zero for ones and the dot for one twenty. Remember, this is a vertical system. This system also had a varying base: (1 7200) + (6 360) + (0 20) + (3 1) = = 9363 Express the following in our numeration system:

14 24 2. SET, WHOLE NUMBERS, AND NUMERATION The answers in our numeration system: The following table summarizes the attributes of the numeration systems we have studied: 2.3. The Hindu-Arabic System This is our system. Features: (1) 10 symbols can be used to represent all whole numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. (1A) Suppose we only used 5 symbols 0, 1, 2, 3, 4. (2) Grouping by 10 s ours is a base 10 system. (a) Bundles of sticks model each group of 10 sticks is bundled with a rubber band.

15 2.3. THE HINDU-ARABIC SYSTEM 25 Ten bundles of ten are bundled to make 100, etc. (b) Base ten pieces: blocks, 4 flats, 2 longs, 7 units = 3(1000) + 4(100) + 2(10) + 7(1). (2A) Grouping by fives base 5. Write this as 32five or 325 and read as three two base 5. Some block representations of base five numbers follow.

16 26 2. SET, WHOLE NUMBERS, AND NUMERATION The first 10 base five numerals follow: Adding 15 to a numeral follows in the block model. (3) We have place value, which implies the system is positional. (3A) The same for base five. (4) The system is additive and multiplicative. A numerals s expanded form or expanded notation expresses a numeral as the sum of its digits times their respective place values. 75, 234 = 7(10, 000) + 5(1000) + 2(100) + 3(10) + 4(1) (4A) The same for base 5. 23, 2345 = 2(10, 0005) + 3(10005) + 2(1005) + 3(105) + 4(15) = 2(625) + 3(125) + 2(25) + 3(5) + 4(1) = This serves as an example of converting base five to base ten.

17 2.3. THE HINDU-ARABIC SYSTEM 27 Example (Converting from base ten to base five). Convert 7326 to base five. 5 0 = 1, 5 1 = 5, 5 2 = 25, 5 3 = 125, 5 4 = 625, 5 5 = 3125, 5 6 = 15625,... Example. Convert 7326 to base = = 1, 7 1 = 7, 7 2 = 49, 7 3 = 343, 7 4 = 2401, 7 5 = 16807, = Each Hindu-Arabic numeral has an associated name. Note. The word and does not appear in any name.

### The Mathematics Driving License for Computer Science- CS10410

The Mathematics Driving License for Computer Science- CS10410 Venn Diagram, Union, Intersection, Difference, Complement, Disjoint, Subset and Power Set Nitin Naik Department of Computer Science Venn-Euler

### Math 117 Chapter 7 Sets and Probability

Math 117 Chapter 7 and Probability Flathead Valley Community College Page 1 of 15 1. A set is a well-defined collection of specific objects. Each item in the set is called an element or a member. Curly

### Numeration and the Whole Numbers If you attend a student raffle, you might hear the following announcement when the entry forms are drawn

5 Numeration Systems Numeration and the Whole Numbers If you attend a student raffle, you might hear the following announcement when the entry forms are drawn The student with identification number 50768-973

### 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,

Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish

### Sets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.

Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, complex numbers, C integers, All students in

### 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,

### 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

### 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

### 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

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

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

### 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

### 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

### 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

### 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

### SETS. Chapter Overview

Chapter 1 SETS 1.1 Overview This chapter deals with the concept of a set, operations on sets.concept of sets will be useful in studying the relations and functions. 1.1.1 Set and their representations

### 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,

### 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:

### A set is an unordered collection of objects.

Section 2.1 Sets A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to contain

### 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

### 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:

### 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.

### The set consisting of all natural numbers that are in A and are in B is the set f1; 3; 5g;

Chapter 5 Set Theory 5.1 Sets and Operations on Sets Preview Activity 1 (Set Operations) Before beginning this section, it would be a good idea to review sets and set notation, including the roster method

### 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

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

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

### Notes. Sets. Notes. Introduction II. Notes. Definition. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry.

Sets Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.6 1.7 of Rosen cse235@cse.unl.edu Introduction

### 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

### 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.

### Automata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi

Automata Theory Automata theory is the study of abstract computing devices. A. M. Turing studied an abstract machine that had all the capabilities of today s computers. Turing s goal was to describe the

### 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

### 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

### 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

### Sets (DSLC 1.2) Intuitively, a set is a collection of things, called its elements, ormembers. To say that x is an element of S, wewrite

Sets (DSLC 1.2) Intuitively, a set is a collection of things, called its elements, ormembers. To say that x is an element of S, wewrite x 2 S. Other ways of saying this: x belongs to S, S contains x, x

### 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

### 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.

### 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

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

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

### Sets, Venn Diagrams & Counting

MT 142 College Mathematics Sets, Venn Diagrams & Counting Module SC Terri Miller revised January 5, 2011 What is a set? Sets set is a collection of objects. The objects in the set are called elements of

### 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

### Assigning Probabilities

What is a Probability? Probabilities are numbers between 0 and 1 that indicate the likelihood of an event. Generally, the statement that the probability of hitting a target- that is being fired at- is

### + 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

### 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.

### 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

### 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

Set (mathematics) From Wikipedia, the free encyclopedia A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right. Sets are one of the most fundamental

### 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

### Pythagorean Theorem. Inquiry Based Unit Plan

Pythagorean Theorem Inquiry Based Unit Plan By: Renee Carey Grade: 8 Time: 5 days Tools: Geoboards, Calculators, Computers (Geometer s Sketchpad), Overhead projector, Pythagorean squares and triangle manipulatives,

### Solving and Graphing Inequalities

Algebra I Pd Basic Inequalities 3A What is the answer to the following questions? Solving and Graphing Inequalities We know 4 is greater than 3, so is 5, so is 6, so is 7 and 3.1 works also. So does 3.01

### 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

### 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).

### 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.

### 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

### 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

### 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

### Name Date. Goal: Understand and represent the intersection and union of two sets.

F Math 12 3.3 Intersection and Union of Two Sets p. 162 Name Date Goal: Understand and represent the intersection and union of two sets. A. intersection: The set of elements that are common to two or more

### Greatest Common Factors and Least Common Multiples with Venn Diagrams

Greatest Common Factors and Least Common Multiples with Venn Diagrams Stephanie Kolitsch and Louis Kolitsch The University of Tennessee at Martin Martin, TN 38238 Abstract: In this article the authors

### 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,

### 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

### Chapter 1 Section 5: Equations and Inequalities involving Absolute Value

Introduction The concept of absolute value is very strongly connected to the concept of distance. The absolute value of a number is that number s distance from 0 on the number line. Since distance is always

### Discrete mathematics is the study of techniques, ideas and modes

CHAPTER 1 Discrete Systems Discrete mathematics is the study of techniques, ideas and modes of reasoning that are indispensable in applied disciplines such as computer science or information technology.

### Common sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility.

Lecture 6: Income and Substitution E ects c 2009 Je rey A. Miron Outline 1. Introduction 2. The Substitution E ect 3. The Income E ect 4. The Sign of the Substitution E ect 5. The Total Change in Demand

### Algorithm set of steps used to solve a mathematical computation. Area The number of square units that covers a shape or figure

Fifth Grade CCSS Math Vocabulary Word List *Terms with an asterisk are meant for teacher knowledge only students need to learn the concept but not necessarily the term. Addend Any number being added Algorithm

### Lecture 2 : Basics of Probability Theory

Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different

### GRADE 7 MATH TEACHING GUIDE

GRDE 7 MTH Lesson 2.1: Union and Intersection of Sets Pre-requisite Concepts: Whole Numbers, definition of sets, Venn diagrams Objectives: In this lesson, you are expected to: 1. Describe and define a.

### MATH 112 Section 2.3: Numeration

MATH 112 Section 2.3: Numeration Prof. Jonathan Duncan Walla Walla College Fall Quarter, 2006 Outline 1 Numbers vs. Numerals 2 Early Numeration Systems Tally Marks Egyptian Numeration System Roman Numeration

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

### 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

### axiomatic vs naïve set theory

ED40 Discrete Structures in Computer Science 1: Sets Jörn W. Janneck, Dept. of Computer Science, Lund University axiomatic vs naïve set theory s Zermelo-Fraenkel Set Theory w/choice (ZFC) extensionality

### 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

### 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

### 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?

### Classical Sets and Fuzzy Sets Classical Sets Operation on Classical Sets Properties of Classical (Crisp) Sets Mapping of Classical Sets to Functions

Classical Sets and Fuzzy Sets Classical Sets Operation on Classical Sets Properties of Classical (Crisp) Sets Mapping of Classical Sets to Functions Fuzzy Sets Notation Convention for Fuzzy Sets Fuzzy

### Algebra I Notes Relations and Functions Unit 03a

OBJECTIVES: F.IF.A.1 Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element

### 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

### The Basics of Counting. Niloufar Shafiei

The Basics of Counting Niloufar Shafiei Counting applications Counting has many applications in computer science and mathematics. For example, Counting the number of operations used by an algorithm to

### 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

### Notes 2 for Honors Probability and Statistics

Notes 2 for Honors Probability and Statistics Ernie Croot August 24, 2010 1 Examples of σ-algebras and Probability Measures So far, the only examples of σ-algebras we have seen are ones where the sample

### 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,

### The Human Binary Calculator: Investigating Positional Notation

The Human Binary Calculator: Investigating Positional Notation Subject: Mathematics/Computer Science Topic: Numbers Grade Level: 8-12 Time: 40-60 min Pre/Post Show Math Activity Introduction: In Show Math,

### Set Theory Basic Concepts and Definitions

Set Theory Basic Concepts and Definitions The Importance of Set Theory One striking feature of humans is their inherent need and ability to group objects according to specific criteria. Our prehistoric

### TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

### Prairie-Hills Elementary School District 144 Kindergarten ~ MATH Curriculum Map

Quarter 1 Prairie-Hills Elementary School District 144 Kindergarten ~ MATH Curriculum Map Domain(s): Counting and Cardinality Geometry Cluster(s): Identify and describe shapes (squares, circles, triangles,

### Course Syllabus. MATH 1350-Mathematics for Teachers I. Revision Date: 8/15/2016

Course Syllabus MATH 1350-Mathematics for Teachers I Revision Date: 8/15/2016 Catalog Description: This course is intended to build or reinforce a foundation in fundamental mathematics concepts and skills.

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

### Section 4.1 Inequalities & Applications. Inequalities. Equations. 3x + 7 = 13 y = 7 3x + 2y = 6. 3x + 7 < 13 y > 7 3x + 2y 6. Symbols: < > 4.

Section 4.1 Inequalities & Applications Equations 3x + 7 = 13 y = 7 3x + 2y = 6 Inequalities 3x + 7 < 13 y > 7 3x + 2y 6 Symbols: < > 4.1 1 Overview of Linear Inequalities 4.1 Study Inequalities with One

### Sets and Logic. Chapter Sets

Chapter 2 Sets and Logic This chapter introduces sets. In it we study the structure on subsets of a set, operations on subsets, the relations of inclusion and equality on sets, and the close connection

### 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,

### 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

### LEARNING OBJECTIVES FOR THIS CHAPTER

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

### Basics of Probability

Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.

### 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

### 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

### Probability - Part I. Definition : A random experiment is an experiment or a process for which the outcome cannot be predicted with certainty.

Probability - Part I Definition : A random experiment is an experiment or a process for which the outcome cannot be predicted with certainty. Definition : The sample space (denoted S) of a random experiment

### 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

### Applied Liberal Arts Mathematics MAT-105-TE

Applied Liberal Arts Mathematics MAT-105-TE This TECEP tests a broad-based overview of mathematics intended for non-math majors and emphasizes problem-solving modeled on real-life applications. Topics

### Section 1.1 Systems of Numeration and Additive Systems of Numeration

Chapter 1 Counting and Measuring Section 1.1 Systems of Numeration and Additive Systems of Numeration Counting The language of mathematics began when human civilization began to develop words and later

### Mathematics Grade 5. Prepublication Version, April 2013 California Department of Education 32

Mathematics In, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions