# Activity Symbol per Hour 2.2 Venn Diagrams and Subsets universe of discourse. universal set. Venn diagrams, F I G U R E 1

Save this PDF as:

Size: px
Start display at page:

Download "Activity Symbol per Hour 2.2 Venn Diagrams and Subsets universe of discourse. universal set. Venn diagrams, F I G U R E 1"

## Transcription

1 U 2.2 A A FIGURE 1 Venn Diagrams and Subsets In every problem there is either a stated or implied universe of discourse. The universe of discourse includes all things under discussion at a given time. For example, in studying reactions to a proposal that a certain campus raise the minimum age of individuals to whom beer may be sold, the universe of discourse might be all the students at the school, the nearby members of the public, the board of trustees of the school, or perhaps all these groups of people. In the mathematical theory of sets, the universe of discourse is called the universal set. The letter U is typically used for the universal set. The universal set might change from problem to problem. In most areas of mathematics, our reasoning can be aided and clarified by utilizing various kinds of drawings and diagrams. In set theory, we commonly use Venn diagrams, developed by the logician John Venn ( ). In these diagrams, the universal set is represented by a rectangle, and other sets of interest within the universal set are depicted by oval regions, or sometimes by circles or other shapes. In the Venn diagram of Figure 1, the entire region bounded by the rectangle represents the universal set U, while the portion bounded by the oval represents set A. (The size of the oval representing A is irrelevant.) The colored region inside U and outside the oval is labeled A (read A prime ). This set, called the complement of A, contains all elements that are contained in U but not contained in A.

2 2.2 Venn Diagrams and Subsets 57 The Complement of a Set For any set A within the universal set U, the complement of A, written A, is the set of elements of U that are not elements of A. That is, A {x x U and x A}. U N M FIGURE 2 EXAMPLE 1 Let U a, b, c, d, e, f, g, h, M a, b, e, f, N b, d, e, g, h. Find each of the following sets. (a) M Set M contains all the elements of set U that are not in set M. Since set M contains the elements a, b, e, and f, these elements will be disqualified from belonging to set M, and consequently set M will contain c, d, g, and h, or M c, d, g, h. (b) N Set N contains all the elements of U that are not in set N, so N a, c, f}. Consider the complement of the universal set, U. The set U is found by selecting all the elements of U that do not belong to U. There are no such elements, so there can be no elements in set U. This means that for any universal set U, U 0. w consider the complement of the empty set, 0. Since 0 x x U and x 0 and set 0 contains no elements, every member of the universal set U satisfies this definition. Therefore for any universal set U, 0 U. Suppose that we are given the universal set U 1, 2, 3, 4, 5, while A 1, 2, 3. Every element of set A is also an element of set U. Because of this, set A is called a subset of set U, written A U. ( A is not a subset of set U would be written A U. ) A Venn diagram showing that set M is a subset of set N is shown in Figure 2. Subset of a Set Set A is a subset of set B if every element of A is also an element of B. In symbols, A B. EXAMPLE 2 Write or in each blank to make a true statement. (a) 3, 4, 5, 6 3, 4, 5, 6, 8 Since every element of 3, 4, 5, 6 is also an element of 3, 4, 5, 6, 8, the first set is a subset of the second, so goes in the blank.

3 58 CHAPTER 2 The Basic Concepts of Set Theory (b) 1, 2, 3 2, 4, 6, 8 The element 1 belongs to 1, 2, 3 but not to 2, 4, 6, 8. Place in the blank. (c) 5, 6, 7, 8 5, 6, 7, 8 Every element of 5, 6, 7, 8 is also an element of 5, 6, 7, 8. Place in the blank. As Example 2(c) suggests, every set is a subset of itself: B B for any set B. The statement of set equality in Section 2.1 can be formally presented using subset terminology. Set Equality (Alternative definition) If A and B are sets, then A B if A B and B A. When studying subsets of a set B, it is common to look at subsets other than set B itself. Suppose that B 5, 6, 7, 8 and A 6, 7. A is a subset of B, but A is not all of B; there is at least one element in B that is not in A. (Actually, in this case there are two such elements, 5 and 8.) In this situation, A is called a proper subset of B. To indicate that A is a proper subset of B, write A B. Proper Subset of a Set Set A is a proper subset of set B if A B and A B. In symbols, A B. (tice the similarity of the subset symbols, and, to the inequality symbols from algebra, and. ) EXAMPLE 3 Decide whether,, or both could be placed in each blank to make a true statement. (a) 5, 6, 7 5, 6, 7, 8 Every element of 5, 6, 7 is contained in 5, 6, 7, 8, so could be placed in the blank. Also, the element 8 belongs to 5, 6, 7, 8 but not to 5, 6, 7, making 5, 6, 7 a proper subset of 5, 6, 7, 8. This means that could also be placed in the blank. (b) a, b, c a, b, c The set a, b, c is a subset of a, b, c. Since the two sets are equal, a, b, c is not a proper subset of a, b, c. Only may be placed in the blank. Set A is a subset of set B if every element of set A is also an element of set B. This definition can be reworded by saying that set A is a subset of set B if there are

4 2.2 Venn Diagrams and Subsets 59 no elements of A that are not also elements of B. This second form of the definition shows that the empty set is a subset of any set, or 0 B for any set B. This is true since it is not possible to find any elements of 0 that are not also in B. (There are no elements in 0.) The empty set 0 is a proper subset of every set except itself: 0 B if B is any set other than 0. Every set (except 0 ) has at least two subsets, 0 and the set itself. EXAMPLE 4 Find all possible subsets of each set. (a) (b) 7, 8 By trial and error, the set {7, 8 has four subsets: 0, 7, 8, 7, 8. a, b, c Here trial and error leads to 8 subsets for a, b, c: 0, a, b, c, a, b, a, c, b, c, a, b, c. In Example 4, the subsets of 7, 8 and the subsets of a, b, c were found by trial and error. An alternative method involves drawing a tree diagram, a systematic way of listing all the subsets of a given set. Figures 3(a) and (b) show tree diagrams for 7, 8 and a, b, c. 7 subset? 8 subset? 4 subsets {7, 8} {7} {8} 0 a subset? b subset? c subset? 8 subsets {a, b, c} {a, b} {a, c} {a} {b, c} {b} {c} 0 (a) (b) FIGURE 3 In Example 4, we determined the number of subsets of a given set by making a list of all such subsets and then counting them. The tree diagram method also produced a list of all possible subsets. In many applications, we don t need to display all the subsets but simply determine how many there are. Furthermore, the trial and error method and the tree diagram method would both involve far too much work if the original set had a very large number of elements. For these reasons, it is desirable to have a formula for the number of subsets. To obtain such a formula, we use inductive reasoning. That is, we observe particular cases to try to discover a general pattern. Begin with the set containing the least number of elements possible the empty set. This set, 0, has only one subset, 0 itself. Next, a set with one element has only

5 60 CHAPTER 2 The Basic Concepts of Set Theory two subsets, itself and 0. These facts, together with those obtained above for sets with two and three elements, are summarized here. Number of elements Number of subsets Powers of , ,048, ,554, ,073,741,824 The small numbers at the upper right are called exponents. Here the base is 2. tice how quickly the values grow; this is the origin of the phrase exponential growth. This chart suggests that as the number of elements of the set increases by one, the number of subsets doubles. This suggests that the number of subsets in each case might be a power of 2. Every number in the second row of the chart is indeed a power of 2. Add this information to the chart. Number of elements Number of subsets This chart shows that the number of elements in each case is the same as the exponent on the 2. Inductive reasoning gives the following generalization. Number of Subsets The number of subsets of a set with n elements is 2 n. Since the value 2 n includes the set itself, we must subtract 1 from this value to obtain the number of proper subsets of a set containing n elements. Number of Proper Subsets The number of proper subsets of a set with n elements is 2 n 1. As shown in the chapter on problem solving, although inductive reasoning is a good way of discovering principles or arriving at a conjecture, it does not provide a proof that the conjecture is true in general. A proof must be provided by other means. The two formulas above are true, by observation, for n 0, 1, 2, or 3. (For a general proof, see Exercise 71 at the end of this section.) EXAMPLE 5 Find the number of subsets and the number of proper subsets of each set. (a) 3, 4, 5, 6, 7 This set has 5 elements and subsets. Of these, are proper subsets. (b) 1, 2,3, 4, 5, 9, 12, 14 This set has 8 elements. There are subsets and 255 proper subsets.

### 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)

### 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

### Glencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 3-3, 5-8 8-4, 8-7 1-6, 4-9

Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 6-8 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,

### Prentice Hall Mathematics: Course 1 2008 Correlated to: Arizona Academic Standards for Mathematics (Grades 6)

PO 1. Express fractions as ratios, comparing two whole numbers (e.g., ¾ is equivalent to 3:4 and 3 to 4). Strand 1: Number Sense and Operations Every student should understand and use all concepts and

### Samples of Allowable Supplemental Aids for STAAR Assessments. Updates from 12/2011

Samples of Allowable Supplemental Aids for STAAR Assessments Updates from 12/2011 All Subjects: Mnemonic Devices A mnemonic device is a learning technique that assists with memory. Only mnemonic devices

### Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)

New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct

### Factorizations: Searching for Factor Strings

" 1 Factorizations: Searching for Factor Strings Some numbers can be written as the product of several different pairs of factors. For example, can be written as 1, 0,, 0, and. It is also possible to write

### Full and Complete Binary Trees

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

### Section 1.5 Exponents, Square Roots, and the Order of Operations

Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares.

### Building a Bridge to Academic Vocabulary in Mathematics

Building a Bridge to Academic Vocabulary in Mathematics AISD Elementary Mathematics Department How Students Develop a Repertoire of Academic English in Mathematics Developed and researched by the AISD

### The Taxman Game. Robert K. Moniot September 5, 2003

The Taxman Game Robert K. Moniot September 5, 2003 1 Introduction Want to know how to beat the taxman? Legally, that is? Read on, and we will explore this cute little mathematical game. The taxman game

### NEW MEXICO Grade 6 MATHEMATICS STANDARDS

PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical

### 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

### Open-Ended Problem-Solving Projections

MATHEMATICS Open-Ended Problem-Solving Projections Organized by TEKS Categories TEKSING TOWARD STAAR 2014 GRADE 7 PROJECTION MASTERS for PROBLEM-SOLVING OVERVIEW The Projection Masters for Problem-Solving

### Time needed. Before the lesson Assessment task:

Formative Assessment Lesson Materials Alpha Version Beads Under the Cloud Mathematical goals This lesson unit is intended to help you assess how well students are able to identify patterns (both linear

### Level 1 - Maths Targets TARGETS. With support, I can show my work using objects or pictures 12. I can order numbers to 10 3

Ma Data Hling: Interpreting Processing representing Ma Shape, space measures: position shape Written Mental method s Operations relationship s between them Fractio ns Number s the Ma1 Using Str Levels

### GRADE 5 SUPPLEMENT. Set A2 Number & Operations: Primes, Composites & Common Factors. Includes. Skills & Concepts

GRADE 5 SUPPLEMENT Set A Number & Operations: Primes, Composites & Common Factors Includes Activity 1: Primes & Common Factors A.1 Activity : Factor Riddles A.5 Independent Worksheet 1: Prime or Composite?

### Indicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities.

3 rd Grade Math Learning Targets Algebra: Indicator 1: Use procedures to transform algebraic expressions. 3.A.1.1. Students are able to explain the relationship between repeated addition and multiplication.

### Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary

Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:

### Assignment 5 - Due Friday March 6

Assignment 5 - Due Friday March 6 (1) Discovering Fibonacci Relationships By experimenting with numerous examples in search of a pattern, determine a simple formula for (F n+1 ) 2 + (F n ) 2 that is, a

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### 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

### 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

### Prentice Hall MyMathLab Algebra 1, 2011

Prentice Hall MyMathLab Algebra 1, 2011 C O R R E L A T E D T O Tennessee Mathematics Standards, 2009-2010 Implementation, Algebra I Tennessee Mathematics Standards 2009-2010 Implementation Algebra I 3102

### Unit 6 Number and Operations in Base Ten: Decimals

Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,

### Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

### 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

### Grade 4 Mathematics Patterns, Relations, and Functions: Lesson 1

Grade 4 Mathematics Patterns, Relations, and Functions: Lesson 1 Read aloud to the students the material that is printed in boldface type inside the boxes. Information in regular type inside the boxes

### 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

### Paper 2 Revision. (compiled in light of the contents of paper1) Higher Tier Edexcel

Paper 2 Revision (compiled in light of the contents of paper1) Higher Tier Edexcel 1 Topic Areas 1. Data Handling 2. Number 3. Shape, Space and Measure 4. Algebra 2 Data Handling Averages Two-way table

### Radius, Diameter, Circumference, π, Geometer s Sketchpad, and You! T. Scott Edge

TMME,Vol.1, no.1,p.9 Radius, Diameter, Circumference, π, Geometer s Sketchpad, and You! T. Scott Edge Introduction I truly believe learning mathematics can be a fun experience for children of all ages.

### Probability Distributions

CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution

### Introduction. Appendix D Mathematical Induction D1

Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to

### Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space

### Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %

Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the

### 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

### Big Ideas in Mathematics

Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards

### Excel -- Creating Charts

Excel -- Creating Charts The saying goes, A picture is worth a thousand words, and so true. Professional looking charts give visual enhancement to your statistics, fiscal reports or presentation. Excel

### Area and Perimeter: The Mysterious Connection TEACHER EDITION

Area and Perimeter: The Mysterious Connection TEACHER EDITION (TC-0) In these problems you will be working on understanding the relationship between area and perimeter. Pay special attention to any patterns

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

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

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

### Introduction to the TI-Nspire CX

Introduction to the TI-Nspire CX Activity Overview: In this activity, you will become familiar with the layout of the TI-Nspire CX. Step 1: Locate the Touchpad. The Touchpad is used to navigate the cursor

### UNIT 2 Braille Lesson Plan 1 Braille

Codes and Ciphers Activity Mathematics Enhancement Programme Lesson Plan 1 Braille 1 Introduction T: What code, designed more than 150 years ago, is still used extensively today? T: The system of raised

### Chapter 2: Systems of Linear Equations and Matrices:

At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,

### Using Patterns of Integer Exponents

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. How can you develop and use the properties of integer exponents? The table below shows powers of

### The Circumference Function

2 Geometry You have permission to make copies of this document for your classroom use only. You may not distribute, copy or otherwise reproduce any part of this document or the lessons contained herein

### Using Algebra Tiles for Adding/Subtracting Integers and to Solve 2-step Equations Grade 7 By Rich Butera

Using Algebra Tiles for Adding/Subtracting Integers and to Solve 2-step Equations Grade 7 By Rich Butera 1 Overall Unit Objective I am currently student teaching Seventh grade at Springville Griffith Middle

### Classifying Lesson 1 Triangles

Classifying Lesson 1 acute angle congruent scalene Classifying VOCABULARY right angle isosceles Venn diagram obtuse angle equilateral You classify many things around you. For example, you might choose

### Developing Conceptual Understanding of Number. Set J: Perimeter and Area

Developing Conceptual Understanding of Number Set J: Perimeter and Area Carole Bilyk cbilyk@gov.mb.ca Wayne Watt wwatt@mts.net Perimeter and Area Vocabulary perimeter area centimetres right angle Notes

### 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,

### Maths Workshop for Parents 2. Fractions and Algebra

Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)

### Scope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B

Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced

### Mathematical Conventions. for the Quantitative Reasoning Measure of the GRE revised General Test

Mathematical Conventions for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview The mathematical symbols and terminology used in the Quantitative Reasoning measure

### Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades

Appendix A Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades To respond correctly to TIMSS test items, students need to be familiar with the mathematics

### Planning Guide. Grade 6 Factors and Multiples. Number Specific Outcome 3

Mathematics Planning Guide Grade 6 Factors and Multiples Number Specific Outcome 3 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg6/html/pg6_factorsmultiples/index.html

### Anchorage School District/Alaska Sr. High Math Performance Standards Algebra

Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,

### Overview. Essential Questions. Grade 8 Mathematics, Quarter 4, Unit 4.3 Finding Volume of Cones, Cylinders, and Spheres

Cylinders, and Spheres Number of instruction days: 6 8 Overview Content to Be Learned Evaluate the cube root of small perfect cubes. Simplify problems using the formulas for the volumes of cones, cylinders,

### Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

### 1.2. Successive Differences

1. An Application of Inductive Reasoning: Number Patterns In the previous section we introduced inductive reasoning, and we showed how it can be applied in predicting what comes next in a list of numbers

### Using Graphic Organizers to Aid Comprehension Grade Two

Ohio Standards Connections Reading Process: Concepts of Print, Comprehension Strategies and Self- Monitoring Strategies Benchmark A Establish a purpose for reading and use a range of reading comprehension

### VISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University

VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS

E XPLORING QUADRILATERALS E 1 Geometry State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Statement of Purpose: The activities in this

### Curriculum Overview YR 9 MATHS. SUPPORT CORE HIGHER Topics Topics Topics Powers of 10 Powers of 10 Significant figures

Curriculum Overview YR 9 MATHS AUTUMN Thursday 28th August- Friday 19th December SUPPORT CORE HIGHER Topics Topics Topics Powers of 10 Powers of 10 Significant figures Rounding Rounding Upper and lower

### 2 SYSTEM DESCRIPTION TECHNIQUES

2 SYSTEM DESCRIPTION TECHNIQUES 2.1 INTRODUCTION Graphical representation of any process is always better and more meaningful than its representation in words. Moreover, it is very difficult to arrange

### Decomposing Numbers (Operations and Algebraic Thinking)

Decomposing Numbers (Operations and Algebraic Thinking) Kindergarten Formative Assessment Lesson Designed and revised by Kentucky Department of Education Mathematics Specialists Field-tested by Kentucky

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### Summary of important mathematical operations and formulas (from first tutorial):

EXCEL Intermediate Tutorial Summary of important mathematical operations and formulas (from first tutorial): Operation Key Addition + Subtraction - Multiplication * Division / Exponential ^ To enter a

### Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School

Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education

### Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014

Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the

### Decision Mathematics D1 Advanced/Advanced Subsidiary. Tuesday 5 June 2007 Afternoon Time: 1 hour 30 minutes

Paper Reference(s) 6689/01 Edexcel GCE Decision Mathematics D1 Advanced/Advanced Subsidiary Tuesday 5 June 2007 Afternoon Time: 1 hour 30 minutes Materials required for examination Nil Items included with

### The right edge of the box is the third quartile, Q 3, which is the median of the data values above the median. Maximum Median

CONDENSED LESSON 2.1 Box Plots In this lesson you will create and interpret box plots for sets of data use the interquartile range (IQR) to identify potential outliers and graph them on a modified box

### Process / Operation Symbols

Flowchart s and Their Meanings Flowchart s Defined By Nicholas Hebb The following is a basic overview, with descriptions and meanings, of the most common flowchart symbols - also commonly called flowchart

### 6 3 The Standard Normal Distribution

290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since

### Decimals and Percentages

Decimals and Percentages Specimen Worksheets for Selected Aspects Paul Harling b recognise the number relationship between coordinates in the first quadrant of related points Key Stage 2 (AT2) on a line

### Guide to Leaving Certificate Mathematics Ordinary Level

Guide to Leaving Certificate Mathematics Ordinary Level Dr. Aoife Jones Paper 1 For the Leaving Cert 013, Paper 1 is divided into three sections. Section A is entitled Concepts and Skills and contains

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence

### Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series

Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them

### Excel 2007: Basics Learning Guide

Excel 2007: Basics Learning Guide Exploring Excel At first glance, the new Excel 2007 interface may seem a bit unsettling, with fat bands called Ribbons replacing cascading text menus and task bars. This

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### G C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.

### 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.

### Tom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.

Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find

### IB Math Research Problem

Vincent Chu Block F IB Math Research Problem The product of all factors of 2000 can be found using several methods. One of the methods I employed in the beginning is a primitive one I wrote a computer

### To Evaluate an Algebraic Expression

1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum

### Algebra 1: Basic Skills Packet Page 1 Name: Integers 1. 54 + 35 2. 18 ( 30) 3. 15 ( 4) 4. 623 432 5. 8 23 6. 882 14

Algebra 1: Basic Skills Packet Page 1 Name: Number Sense: Add, Subtract, Multiply or Divide without a Calculator Integers 1. 54 + 35 2. 18 ( 30) 3. 15 ( 4) 4. 623 432 5. 8 23 6. 882 14 Decimals 7. 43.21

Ohio Standards Connection Patterns, Functions and Algebra Benchmark E Solve open sentences and explain strategies. Indicator 4 Solve open sentences by representing an expression in more than one way using

### Grade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills

Grade 6 Mathematics Assessment Eligible Texas Essential Knowledge and Skills STAAR Grade 6 Mathematics Assessment Mathematical Process Standards These student expectations will not be listed under a separate

### Polynomial and Rational Functions

Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

### 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.

### Problem of the Month: Double Down

Problem of the Month: Double Down The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core

### University of Southern California Marshall Information Services

University of Southern California Marshall Information Services Determine Breakeven Price Using Excel - Using Goal Seek, Data Tables, Vlookup & Charts This guide covers how to determine breakeven price

### c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.

Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions

### 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