# A Little Set Theory (Never Hurt Anybody)

Save this PDF as:
Size: px
Start display at page:

## Transcription

1 A Little Set Theory (Never Hurt Anybody) Matthew Saltzman Department of Mathematical Sciences Clemson University Draft: August 21, Introduction The fundamental ideas of set theory and the algebra of sets are probably the most important concepts across all areas of mathematics other than the algebra of real numbers. Many texts give these ideas too little direct attention, and then utilize concepts that they have not adequately reviewed. In this brief tutorial, we will review enough set theory to get the reader through most lower-level college math courses. 2 What is a Set? A set is one of those fundamental mathematical ideas whose nature we understand without direct reference to other mathematical ideas. Quite simply, Definition 1 A set is a collection of distinct objects. The objects can be real, physical things, or abstract, mathematical things. The number of such objects can be finite or infinite. The only important thing is that the objects be distinct, i.e., uniquely identifiable. This idea of distinctness has some subtleties. For example, a collection of identical blue balls in an urn (you meet a lot of these in probability class) is a set because the balls are in principle uniquely identifiable: we could paint a different number on each of them. On the other hand, the list of digits 1, 5, 3, 2, 1, 4 is not a set, because there is no way to distinguish the two occurrences of 1 in the list. (Order doesn t matter when listing the elements of a set.) 1

2 The most basic way to define a set is to list its members between curly braces: {2, 4, 6, 8, 10}. We can also define a set by giving a rule that determines reliably whether an object is a member or not: {x : 0 < x < 1}. The : is read such that. (Some authors use instead.) So this rule is read as: the set of values x such that x is strictly between 0 and 1. We must be able to determine membership or lack of it for every object. Otherwise the rule is not a valid definition of a set. The other ways to define sets are by building them up from more basic sets. We will be discussing rules for doing this below. As is common in mathematics, we can refer to a set by naming it with a letter. For example, S = {x : 0 < x < 1}. Definition 2 An object x is an element or member of a set S, written x S, if x satisfies the rule defining membership in S. We can write x / S if x is not an element of S. Definition 3 The empty set or null set, denoted, is the set containing no elements. An important property of a set is the number of elements it contains. Definition 4 The cardinality of a set S, written S, is the number of elements in the set. Note that this number could be infinite (as in {0, 2, 4, 6,...} or {x : 0 < x < 1}). There are some issues with doing arithmetic with infinities that we will ignore for now that is, when discussing arithmetic on set cardinalities, we will assume the sets are finite. The cardinality of the empty set is zero ( = 0). 3 Subsets and Equality This section describes relations between sets, that is, concepts that we use to compare two sets. Definition 5 Two sets S and T are equal, written S = T, if S and T contain exactly the same elements. 2

3 (This may seem too trivial to even bother writing down, but in mathematics, it s important to agree on even these simple concepts. Everything that comes later is built on them, and we will only understand the more complicated ideas if we clearly understand the simple ideas they are built from.) Definition 6 A set S is a subset of another set T, written S T if every element of S is also an element of T. Note that, by this definition, S T does not exclude the possibility that every element of T is also an element of S. Definition 7 A set S is a proper subset of T if S T and S T. That is, if every element of S is also in T, but some element of T is not in S. A note on notation: Some authors write S T to denote subsets that may not be proper and S T to denote proper subsets. Others use S T to denote general subsets and S T to denote proper subsets. There are other combinations of notation in use as well. Be sure you know which convention the author of your textbook uses, and choose one convention and stick to it for your own writing. Properties If R, S, and T are any sets, the following properties hold: S = S (reflexive) If S = T, then T = S (symmetric). If R = S and S = T then R = T (transitive). S S S (reflexive) If R S and S T, then R T (transitive). S T and T S if and only if S = T. If S T, then S T, and if S T, then S < T. 3

4 4 Intersections and Unions This section and the next discuss operations on sets, that is, ways in which sets can be combined to form new sets. Definition 8 The intersection of two sets S and T, written S T is the set of elements common to both S and T, i.e., This definition reads: S T = {x : x S and x T }. the set of objects x such that x is an element of S and x is an element of T. Example 1 If S = {e 1, e 2, e 3, e 4, e 5, e 6 } and T = {e 0, e 2, e 4, e 6, e 8 } then S T = {e 2, e 4, e 6 }. Example 2 If S = {x : 0 x 1} and T = {x :.5 < x < 2}, then S T = {x :.5 < x 1}. Definition 9 The union of two sets S and T, written S T, is the set of elements that are in either S or T or both, i.e., S T = {x : x S or x T }. Note that when two statements are connected with or, this means that at least one of the statements is true. Example 3 If S = {e 1, e 2, e 3, e 4, e 5, e 6 } and T = {e 0, e 2, e 4, e 6, e 8 } then S T = {e 0, e 1, e 2, e 3, e 4, e 5, e 6, e 8 }. Example 4 if S = {x : 0 x 1} and T = {x :.5 < x < 2}, then S T = {x : 0 x < 2}. Properties We can combine multiple operations to create new sets from other sets. Sometimes, two different ways of combining sets produce the same result. In this case, we can replace one combination with the other in an equation. If S is a set constructed from other sets, we can replace S in an expression with the expression that describes the construction of S. (When making such substitutions, it s helpful to enclose the replacement expression 4

5 in parentheses.) Repeated application of this principle can lead to profound insights about the relationships between certain sets. The rules for replacement regarding unions and intersections are listed below. A little reflection will convince you of the validity of most of them. S T = T S (commutative) 1 S T = T S (commutative) S S T S T S S = S S = S S = S S = S R (S T ) = (R S) T (associative) 2 R (S T ) = (R S) T (associative) R (S T ) = (R S) (R T ) (distributive over ) R (S T ) = (R S) (R T ) (distributive over ) S (S T ) = S (S T ) = S S T if and only if S T = T. If R T and S T, then R S T. If R S and R T, then R S T. If S T then R S R T and R S R T. S T if and only if S or T. If S T, then S and T. S = T if and only if S T S T. The key property of cardinalities of unions and intersections is this: 1 Perform all operations before any comparisons. 2 Perform all operations in parentheses before any others. 5

6 S T = S + T S T. That is, the number of elements in the union of two sets is the sum of the numbers of elements in each set less the number of elements that appear twice in the two lists. Example 5 If S = {e 1, e 2, e 3, e 4, e 5, e 6 } and T = {e 0, e 2, e 4, e 6, e 8 } then S T = {e 0, e 1, e 2, e 3, e 4, e 5, e 6, e 8 } and S T = {e 2, e 4, e 6 }. So S = 6 and T = 5. But S T = 3, since there are three elements on both lists. When we form the union, we combine the two lists of elements, but each duplicate element is only listed once. Thus, S T = = 8. If two sets have no elements in common, this formula becomes somewhat simpler. Definition 10 Two sets S and T are mutually exclusive if S T =. If S and T are mutually exclusive, then S T = 0, so S T = S + T. Example 6 If S = {e 1, e 3, e 5 } and T = {e 0, e 2, e 4, e 6, e 8 } then S T = {e 0, e 1, e 2, e 3, e 4, e 5, e 6, e 8 } and S T =. So S = 3 and T = 5. But S T = 0, since no element appears on both lists. When we form the union, we combine the two lists of elements, and there are no duplicates. Thus, S T = = 8. 5 Set Differences and Complements Another way to construct new sets from old is to remove some elements from a set and consider the elements that are left over. Definition 11 The difference between S and T, written S \ T, is the set of elements in S but not also in T : S \ T = {x : x S and x / T }. Example 7 If S = {e 1, e 2, e 3, e 4, e 5, e 6 } and T = {e 0, e 2, e 4, e 6, e 8 } then S \ T = {e 1, e 3, e 5 }, but T \ S = {e 0, e 8 }. Example 8 If S = {e 1, e 3, e 5 } and T = {e 0, e 2, e 4, e 6, e 8 } then S \ T = S and T \ S = T. 6

7 Example 9 If S = {x : 0 x 1} and T = {x :.5 < x < 2}, then S \ T = {x : 0 x.5} and T \ S = {x : 1 < x 2}. We can see that, in general, S \ T T \ S (set difference is not commutative). Properties Two important properties of set differences are: S \ T = S \ (S T ) S \ T = S S T The first property states that the elements of T that are not in S play no role in the construction of S \ T. The second property states that the number of elements removed from S to produce S \ T is the same as the number of elements that S and T have in common. Definition 12 The symmetric difference between S and T is (S T ) \ (S T ), i.e., the set of elements in S or T, but not both. The cardinality of a symmetric difference satisfies: (S T ) \ (S T ) = S + T 2 S T We pool the member lists of S and T, but we delete both copies of duplicates. Example 10 If S = {e 1, e 2, e 3, e 4, e 5, e 6 } and T = {e 0, e 2, e 4, e 6, e 8 } then (S T )\(S T ) = {e 0, e 1, e 3, e 5, e 8 }, and (S T )\(S T ) = = 5. Example 11 If S = {x : 0 x 1} and T = {x :.5 < x < 2}, then (S T ) \ (S T ) = {x : 0 x.5 or 1 < x 2}. In many set problems, all sets are defined to be subsets of some reference set. This reference set is called the universe. In the remainder of this section, the discussion will refer to a universe U, and we will assume that all other sets R, S, and T are subsets of U. Definition 13 Relative to a universe U, the complement of S, written S, is the set of all elements of the universe not contained in S, i.e., S = {x : x U and x / S} 7

8 Another common notation for the complement of S is S. Complements of set expressions can be written similarly, for example: 3 (S T ) or S T. The important cardinality property of complements is S = U S Example 12 If U = {e 0, e 1,... e 10 } and S = {e 1, e 2, e 3, e 4, e 5, e 6 }, then S = {e 0, e 7, e 8, e 9, e 10 }. Also, U = 11 and S = 6, so S = 11 6 = 5. Example 13 If U = {x : x is a real number} and S = {x : 0 x 1}, then S = {x : x < 0 or x > 1}. The following are important properties of complements and differences. U = and = U S S = U and S S = S U = U and S U = S S = S S = T if and only if T = S. S S if and only if S =. S T if and only if T S. S \ T = S T (S T ) \ (S T ) = (S \ T ) (T \ S) (S T ) = S T (S T ) = S T The last two properties are known as DeMorgan s laws. They provide a useful way to convert arbitrary set expressions into ones that involve only complements and unions or only complements and intersections. 3 Perform complements before performing any of the other set operations. Parentheses are required around set expressions to be complemented. 8

9 6 Conclusion It is important to know the basic definitions and a few of the key properties described above. But don t spend time memorizing all the properties; they are too abstract and easy to confuse. Instead, keep this guide handy while working problems involving sets, and refer to it liberally. But each time you make use of a property, think carefully about its meaning. You ll find that, with practice, you ll be able to easily re-derive the properties you need, when you need them. 9

### 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.

### 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

### 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

### Linear Equations and Inequalities

Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................

### Sample Induction Proofs

Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

### Lecture 16 : Relations and Functions DRAFT

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

### Permutation Groups. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles April 2, 2003

Permutation Groups Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles April 2, 2003 Abstract This paper describes permutations (rearrangements of objects): how to combine them, and how

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### 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

### 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

### 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,

### Row Echelon Form and Reduced Row Echelon Form

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

### 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,

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### 3 Some Integer Functions

3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

### 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

### 10.2 Series and Convergence

10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Math 4310 Handout - Quotient Vector Spaces

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

### Playing with Numbers

PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also

### Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

### 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

### 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

### 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

### 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

### Section 1.3 P 1 = 1 2. = 1 4 2 8. P n = 1 P 3 = Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3,..., = 1 2 4.

Difference Equations to Differential Equations Section. The Sum of a Sequence This section considers the problem of adding together the terms of a sequence. Of course, this is a problem only if more than

### Useful Number Systems

Useful Number Systems Decimal Base = 10 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Binary Base = 2 Digit Set = {0, 1} Octal Base = 8 = 2 3 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7} Hexadecimal Base = 16 = 2

### 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

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### DIVISIBILITY AND GREATEST COMMON DIVISORS

DIVISIBILITY AND GREATEST COMMON DIVISORS KEITH CONRAD 1 Introduction We will begin with a review of divisibility among integers, mostly to set some notation and to indicate its properties Then we will

### Relational Algebra. Basic Operations Algebra of Bags

Relational Algebra Basic Operations Algebra of Bags 1 What is an Algebra Mathematical system consisting of: Operands --- variables or values from which new values can be constructed. Operators --- symbols

### CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

### 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)

### 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

### Division of whole numbers is defined in terms of multiplication using the idea of a missing factor.

32 CHAPTER 1. PLACE VALUE AND MODELS FOR ARITHMETIC 1.6 Division Division of whole numbers is defined in terms of multiplication using the idea of a missing factor. Definition 6.1. Division is defined

### 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

### Linear Algebra: Vectors

A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector

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

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

### SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections

### 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.

### 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

### The continuous and discrete Fourier transforms

FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1

### 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

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### 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

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### 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

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

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### 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

### 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

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

### CSI 333 Lecture 1 Number Systems

CSI 333 Lecture 1 Number Systems 1 1 / 23 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = 2 10 0 + 9 10 1 + 1 10 2 General: Digit sequence : d n 1 d n 2...

### Grade 6 Math Circles. Binary and Beyond

Faculty of Mathematics Waterloo, Ontario N2L 3G1 The Decimal System Grade 6 Math Circles October 15/16, 2013 Binary and Beyond The cool reality is that we learn to count in only one of many possible number

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### To convert an arbitrary power of 2 into its English equivalent, remember the rules of exponential arithmetic:

Binary Numbers In computer science we deal almost exclusively with binary numbers. it will be very helpful to memorize some binary constants and their decimal and English equivalents. By English equivalents

### Toothpick Squares: An Introduction to Formulas

Unit IX Activity 1 Toothpick Squares: An Introduction to Formulas O V E R V I E W Rows of squares are formed with toothpicks. The relationship between the number of squares in a row and the number of toothpicks

### 8 Primes and Modular Arithmetic

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

### NPV Versus IRR. W.L. Silber -1000 0 0 +300 +600 +900. We know that if the cost of capital is 18 percent we reject the project because the NPV

NPV Versus IRR W.L. Silber I. Our favorite project A has the following cash flows: -1 + +6 +9 1 2 We know that if the cost of capital is 18 percent we reject the project because the net present value is

### INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

### Grade 7/8 Math Circles Sequences and Series

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Sequences and Series November 30, 2012 What are sequences? A sequence is an ordered

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### 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

### 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

### Probability. Section 9. Probability. Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space)

Probability Section 9 Probability Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space) In this section we summarise the key issues in the basic probability

### Unified Lecture # 4 Vectors

Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

### 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

### Regular Languages and Finite Automata

Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### HFCC Math Lab Beginning Algebra 13 TRANSLATING ENGLISH INTO ALGEBRA: WORDS, PHRASE, SENTENCES

HFCC Math Lab Beginning Algebra 1 TRANSLATING ENGLISH INTO ALGEBRA: WORDS, PHRASE, SENTENCES Before being able to solve word problems in algebra, you must be able to change words, phrases, and sentences

### What is Linear Programming?

Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Chapter 7: Products and quotients

Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students

Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students Studies show that most students lose about two months of math abilities over the summer when they do not engage in

### 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.

### 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

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### We can express this in decimal notation (in contrast to the underline notation we have been using) as follows: 9081 + 900b + 90c = 9001 + 100c + 10b

In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should

### The Deadly Sins of Algebra

The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.

### Computational Geometry Lab: FEM BASIS FUNCTIONS FOR A TETRAHEDRON

Computational Geometry Lab: FEM BASIS FUNCTIONS FOR A TETRAHEDRON John Burkardt Information Technology Department Virginia Tech http://people.sc.fsu.edu/ jburkardt/presentations/cg lab fem basis tetrahedron.pdf

### Math 115 Spring 2011 Written Homework 5 Solutions

. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

### 1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Combinatorial Proofs

Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A

### Math 115A - Week 1 Textbook sections: 1.1-1.6 Topics covered: What is a vector? What is a vector space? Span, linear dependence, linear independence

Math 115A - Week 1 Textbook sections: 1.1-1.6 Topics covered: What is Linear algebra? Overview of course What is a vector? What is a vector space? Examples of vector spaces Vector subspaces Span, linear

### Computer Science 281 Binary and Hexadecimal Review

Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two

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

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

### Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### https://williamshartunionca.springboardonline.org/ebook/book/27e8f1b87a1c4555a1212b...

of 19 9/2/2014 12:09 PM Answers Teacher Copy Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B Example C #2 Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Students

### Section 1.1 Real Numbers

. Natural numbers (N):. Integer numbers (Z): Section. Real Numbers Types of Real Numbers,, 3, 4,,... 0, ±, ±, ±3, ±4, ±,... REMARK: Any natural number is an integer number, but not any integer number is