# LESSON SUMMARY. Set Operations and Venn Diagrams

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 and page numbers are given throughout the lesson e.g. Ex 1g page 19) INTRODUCTION In this lesson we will be looking at set operations and Venn diagrams. Many real life problems can be described in terms of sets. Therefore these problems can be solved using set theory. The Venn diagram is an important tool in solving these problems. OBJECTIVES At the end of this lesson you will be able to: a) define and describe different sets; b) determine and count the elements in the intersection and union of sets; c) construct and use Venn diagrams to show subsets, complements, and the intersection and union of sets; d) determine the number of elements in named subsets of not more than three intersecting sets, given the number of elements in some of the other subsets; e) solve problems involving the use of Venn diagrams for not more than three sets; 1 NOSTT CXC CSEC Mathematics Lesson Summary: Unit 3: Lesson 4

2 3.1 Defining and describing sets A set is a collection of items that share something in common, e.g. the set of vowels in the English alphabet. Try and think of some other examples of sets. You must be able to use phrases to describe sets. (Ex 1a page 4) Example: A possible description of the set P = {2, 3, 5, 7, 11, 13, 17} is P is the set of prime numbers less than eighteen. You must also be able to list members of a set given a description of the set. (Ex 1a page 3). Example: A = {even numbers less than 13} then A = {2, 4, 6, 8, 10, 12}. The following sets are important. Finite set: a set where you can count or name all the members, e.g. the set of planets in our solar system. Infinite set: it is not possible to name or count all the members, e.g. the set of natural numbers. Null or Empty set, denoted by { } or ø: this set has no members, e.g. the set of people who have swum across the Caribbean Sea. Universal set, denoted U: for any given set the universal set is the set from which all its members came, e.g. a suitable universal set for B = {protractor, ruler, set square, compass, divider} is set of geometrical instruments. Subset: a set is a subset of another set if and only if all its members are also members of the set, e.g A = {1, 3} is a subset of B = {1, 2, 3, 4, 5}. We write A B. Note that each set is a subset of itself and the empty set is a subset of every set. The number of subsets of a given set is given by S =, where is the number of elements in the set. The number of subset that can be formed from the set B is. Try and list them, remember to include the empty set and the set itself. A few has been done for you: {1}, {2}, {3}, {4}, {5}, {1,2}, {1,3},. 2 NOSTT CXC CSEC Mathematics Lesson Summary: Unit 3: Lesson 4

3 Equal sets: these are sets with the same elements, e.g. A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5} then A = B. Equivalent sets: these sets have the same number of elements, e.g. P = {a, b, c, d, e} and B = {1, 2, 3, 4, 5} then P and B are equivalent. ACTIVITY 1 You can get valuable practice by doing a few items from Ex 1a to Ex 1e, page 3 to Set Builder Notation You may use set builder notation to describe a set. Consider the following notation: { : }. This means the set of all such that. Example: Describe the following set using set builder notation. A = {1, 2, 3, 4, 5}. Solution: A = { }, i.e. the set of all such that is greater than 0 but less than or equal to 5 and is a whole number. ACTIVITY 2 Describe the following set using set builder notation. X = { 3, 2, 1, 0, 1, 2, 3, 4} 3.3 Set Operations and Venn Diagrams 3 NOSTT CXC CSEC Mathematics Lesson Summary: Unit 3: Lesson 4

4 A Venn diagram consists of a rectangle that represents the universal set and circles that represent the subsets. The following are some very important subsets represented in Venn diagrams. The following sets will be used to develop the examples throughout. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 2, 4, 6, 8, 10} B = {3, 6, 9} C = {2, 4} D = {1, 3, 5, 7, 9} Complement: the complement of a set A, written as A (A prime) is the set of members (or elements) in the universal set but not in A. U A A' Intersection of two sets: This is a set that contains all the members that belong to both sets, i.e. A B =. The symbol is used for intersection and means a member of. U A B Clearly A B = { 6 }. 4 NOSTT CXC CSEC Mathematics Lesson Summary: Unit 3: Lesson 4

5 Can you identify the set B A? Union of two sets: This is the set of all elements that are in either of the two sets, i.e AUB =. Therefore AUB =. This is represented by the shaded area in the following Venn diagram. Note U is used for union. U A B Subset: C A A C Disjoint Sets: Two sets are disjointed if they have no common members. U A D NOSTT CXC CSEC Mathematics Lesson Summary: Unit 3: Lesson 4

6 ACTIVITY 3 (Ex 1g page 19) 3.4 Number of elements in a given set If A = { 2, 4, 6, 8, 10} then the number of elements in A or n(a) = Solving problems using Venn diagrams You can use Venn diagrams to solve problems that involve finding the number of elements is certain subsets of two intersecting sets. Example: In a class of 35 students, 29 play draughts, 16 play chess and 10 play both draughts and chess. Each student plays either draughts or chess. Find the number of students who play: (a) Draughts only (b) chess only Solution: 10 play both draughts and chess. D C No students in the rectangle, all students play either chess or draughts. 19 play draughts only 6 play chess only 6 NOSTT CXC CSEC Mathematics Lesson Summary: Unit 3: Lesson 4

7 You may use the following formula simple numerical problems. to solve Example: In a group of 60 students, 31 speak French, 23 speak Spanish and 14 speak neither French nor Spanish. Determine the number of students who speak both French and Spanish. Solution: ; 8 students speak both French and Spanish. Let us look at a problem involving three sets. There are 38 students in form 5. All students study science. 16 study Physics, 20 study Chemistry, 24 study Biology, 7 study Physic and Chemistry, 8 study Physics and Biology, 10 study Biology and Chemistry, 3 study all three subjects. (a) Draw a carefully labelled Venn diagram to represent the data. (b) Determine the number of students who study at least two subjects. (c) Evaluate the number of students who study Physics and Chemistry only. (d) State the number of students who study Biology only. 7 NOSTT CXC CSEC Mathematics Lesson Summary: Unit 3: Lesson 4

8 Solution: when drawing a Venn diagram you need not put arrows, they are included here to assist you. a) U B C students who study Biology only students who study Physics and Chemistry only 4 P b) The number of students who study at least two subject are those students in the intersections of the circle = 19 students. c) 4 students studied Physics and Chemistry only. d) 9 students study biology only. ASSESSMENT Ex 1h page 13 Ex 12a page 716 vol. 2 CONCLUSION We have used sets and Venn diagrams to solve simple numerical problems. The next unit deals with Consumer Arithmetic. Many computational skills that are used daily will be developed. 8 NOSTT CXC CSEC Mathematics Lesson Summary: Unit 3: Lesson 4

### 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

### 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

### LESSON SUMMARY. Manipulation of Real Numbers

LESSON SUMMARY CXC CSEC MATHEMATICS UNIT TWO: COMPUTATION Lesson 2 Manipulation of Real Numbers Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volume 1. (Some helpful exercises and page numbers

### 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

### 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:

### LESSON SUMMARY. Measuring Shapes

LESSON SUMMARY CXC CSEC MATHEMATICS UNIT SIX: Measurement Lesson 11 Measuring Shapes Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volume 1 (Some helpful exercises and page numbers are given

### 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

### 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

### 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

### 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

### 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

### 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

### 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:

### 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

### 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,

### 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

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

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

### 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

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

### 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,

### Standards for Logical Reasoning

Standards for Logical Reasoning General Outcome Develop logical reasoning. General Notes: This topic is intended to develop students numerical and logical reasoning skills, which are applicable to many

### 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,

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

### LESSON SUMMARY. Mathematics for Buying, Selling, Borrowing and Investing

LESSON SUMMARY CXC CSEC MATHEMATICS UNIT Four: Consumer Arithmetic Lesson 5 Mathematics for Buying, Selling, Borrowing and Investing Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volume

### 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

### 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.

### 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

### 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

### 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

### 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.

### (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

### 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

### Use the following information to answer the next two questions.

Sample Questions Students who achieve the acceptable standard should be able to answer all the following questions, except for any part of a question labelled SE. Parts labelled SE are appropriate examples

### GRADE 7 MATH LEARNING GUIDE

GRDE 7 MTH Lesson 2.1: Union and Intersection of Sets Time: 1.5 hours Pre-requisite Concepts: Whole Numbers, definition of sets, Venn diagrams bout the Lesson: fter learning some introductory concepts

### 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

### The gross monthly salary of a manager is \$5 875. Calculate her net annual salary after deductions of \$976 were made monthly.

LESSON SUMMARY CXC CSEC MATHEMATICS UNIT FOUR: Consumer Arithmetic Lesson 6 Earning Money, Paying Bills and Taxes and Currency Conversion Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volume

### 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

### 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

### 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

### 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).

### Hawkes Learning Systems: College Algebra

Hawkes Learning Systems: College Algebra Section 1.2: The Arithmetic of Algebraic Expressions Objectives o Components and terminology of algebraic expressions. o The field properties and their use in algebra.

### Math/CSE 1019: Discrete Mathematics for Computer Science Fall Suprakash Datta

Math/CSE 1019: Discrete Mathematics for Computer Science Fall 2011 Suprakash Datta datta@cse.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cse.yorku.ca/course/1019 1

### 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

### 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

### Lines, Segments, Rays, and Angles

Line and Angle Review Thursday, July 11, 2013 10:22 PM Lines, Segments, Rays, and Angles Slide Notes Title Lines, Segment, Ray A line goes on forever, so we use an arrow on each side to indicate that.

### 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

### 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,

### Compound Inequalities. Section 3-6

Compound Inequalities Section 3-6 Goals Goal To solve and graph inequalities containing the word and. To solve and graph inequalities containing the word or. Vocabulary Compound Inequality Interval Notation

### AREA & CIRCUMFERENCE OF CIRCLES

Edexcel GCSE Mathematics (Linear) 1MA0 AREA & CIRCUMFERENCE OF CIRCLES Materials required for examination Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser.

### 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

### MATHEMATICS TEST. Paper 1 calculator not allowed LEVEL 6 TESTS ANSWER BOOKLET. First name. Middle name. Last name. Date of birth Day Month Year

LEVEL 6 TESTS ANSWER BOOKLET Ma MATHEMATICS TEST LEVEL 6 TESTS Paper 1 calculator not allowed First name Middle name Last name Date of birth Day Month Year Please circle one Boy Girl Year group School

### GRADE 7 MATH TEACHING GUIDE

GRADE 7 MATH Lesson 12: SUBSETS OF REAL NUMBERS Time: 1.5 hours Prerequisite Concepts: whole numbers and operations, set of integers, rational numbers, irrational numbers, sets and set operations, Venn

### 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

### 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

### 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

### 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

### SETS AND VENN DIAGRAMS

The Improving Mathematics ducation in Schools (TIMS) Project STS AND VNN DIAGRAMS NUMBR AND ALGBRA Module 1 A guide for teachers - Years 7 8 June 2011 7YARS 8 Sets and Venn diagrams (Number and Algebra

### Determine whether the sequence is arithmetic, geometric, or neither. 18, 11, 4,

Over Lesson 10 1 Determine whether the sequence is arithmetic, geometric, or neither. 18, 11, 4, A. arithmetic B. geometric C. neither Over Lesson 10 1 Determine whether the sequence is arithmetic, geometric,

Mathematics Fourth Grade Use the four operations with whole numbers to solve problems. 3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations,

### contents Unit 1: Sets...1 Unit 2: Rational Numbers...10 Unit 3: Decimals...17 Unit 4: Exponents...20 Unit 5: Square Root of a Positive Number...

contents Using this Teaching Guide... iv Unit 1: Sets...1 Unit 2: Rational Numbers...10 Unit 3: Decimals...17 Unit 4: Exponents...20 Unit 5: Square Root of a Positive Number...25 Unit 6: Direct and Inverse

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

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

### Course: Math 7. engage in problem solving, communicating, reasoning, connecting, and representing

Course: Math 7 Decimals and Integers 1-1 Estimation Strategies. Estimate by rounding, front-end estimation, and compatible numbers. Prentice Hall Textbook - Course 2 7.M.0 ~ Measurement Strand ~ Students

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

### 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

### Arithmetic Series. or The sum of any finite, or limited, number of terms is called a partial sum. are shorthand ways of writing n 1

CONDENSED LESSON 9.1 Arithmetic Series In this lesson you will learn the terminology and notation associated with series discover a formula for the partial sum of an arithmetic series A series is the indicated

### SAT Mathematics. Numbers and Operations. John L. Lehet

SAT Mathematics Review Numbers and Operations John L. Lehet jlehet@mathmaverick.com www.mathmaverick.com Properties of Integers Arithmetic Word Problems Number Lines Squares and Square Roots Fractions

### OPERATIONS AND PROPERTIES

CHAPTER OPERATIONS AND PROPERTIES Jesse is fascinated by number relationships and often tries to find special mathematical properties of the five-digit number displayed on the odometer of his car. Today

### + 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

### VENN DIAGRAMS. Condition #1 Condition #2 A B C. Answer: B and C have reflection symmetry. C and D have at least one pair of parallel sides.

VENN DIAGRAMS VENN DIAGRAMS AND CLASSIFICATIONS A Venn diagram is a tool used to classify objects. It is usually composed of two or more circles that represent different conditions. An item is placed or

### 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

### LESSON 1, ACTIVITY 5: SHORT VOWELS FOR A

LESSON 1, ACTIVITY 5: SHORT VOWELS FOR A CHAPTER 13 LESSON 1, ACTIVITY 5: LONG VOWELS FOR A LESSON 2, ACTIVITY 5: SHORT VOWELS FOR E LESSON 2, ACTIVITY 5: LONG VOWELS FOR E LESSON 3, ACTIVITY 5: SHORT

### 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

Mathematics 0N1 Solutions 1 1. Write the following sets in list form. 1(i) The set of letters in the word banana. {a, b, n}. 1(ii) {x : x 2 + 3x 10 = 0}. 3(iv) C A. True 3(v) B = {e, e, f, c}. True 3(vi)

### 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.

### L01 - Set Theory. CSci/Math May 2015

L01 - Set Theory CSci/Math 2112 06 May 2015 1 / 10 Elements and Subsets of a Set a A means a is an element of the set A a / A means a is not an element of the set A A B means A is a subset of B (all elements

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

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

### 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,

### 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

### Section 3.3 Equivalence Relations

1 Section 3.3 Purpose of Section To introduce the concept of an equivalence relation and show how it subdivides or partitions a set into distinct categories. Introduction Classifying objects and placing

### Georgia Standards of Excellence Curriculum Frameworks. Mathematics. GSE Analytic Geometry. Unit 7: Applications of Probability

Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Analytic Geometry Unit 7: Applications of Probability Unit 7 Applications of Probability Table of Contents OVERVIEW... 3 STANDARDS

### MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions )

MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions ) Last modified: May 1, 2012 Reading Dineen, pages 57-67 Proof

### Grade 7/8 Math Circles Greek Constructions - Solutions October 6/7, 2015

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Greek Constructions - Solutions October 6/7, 2015 Mathematics Without Numbers The

### Sucesiones repaso Klasse 12 [86 marks]

Sucesiones repaso Klasse [8 marks] a. Consider an infinite geometric sequence with u = 40 and r =. (i) Find. u 4 (ii) Find the sum of the infinite sequence. (i) correct approach () e.g. u 4 = (40) u 4

### 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

### Packet: Sets and Venn Diagrams

Packet: Sets and Venn Diagrams Standards covered: *MA.912.D.7.1 Perform set operations such as union and intersection, complement, and cross product. *MA.912.D.7.2 Use Venn diagrams to explore relationships

### 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

### 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

### Coimisiún na Scrúduithe Stáit State Examinations Commission. Junior Certificate Examination 2015 Sample Paper. Mathematics

2015. S34 S Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination 2015 Mathematics Paper 1 Higher Level Time: 2 hours, 30 minutes 300 marks Examination number Running

### Florida Department of Education/Office of Assessment January 2012. Algebra 1 End-of-Course Assessment Achievement Level Descriptions

Florida Department of Education/Office of Assessment January 2012 Algebra 1 End-of-Course Assessment Achievement Level Descriptions Algebra 1 EOC Assessment Reporting Category Functions, Linear Equations,

### John Adair Intermediate 4th Grade Math Learning Targets I Can Statements..

John Adair Intermediate 4th Grade Math Learning Targets I Can Statements.. NUMBERS AND OPERATIONS IN BASE TEN 4.NBT Generalize place value understanding for multi-digit whole numbers. Explain how the base

### 16.1. Sequences and Series. Introduction. Prerequisites. Learning Outcomes. Learning Style

Sequences and Series 16.1 Introduction In this block we develop the ground work for later blocks on infinite series and on power series. We begin with simple sequences of numbers and with finite series

### Time Requirements: 45 minutes for teaching the skill of using a Venn diagram 45 minutes in the computer lab with an Internet connection

Teacher Information Lesson Title: Venn diagram Lesson Description: This lesson will teach students how to compare and contrast two different objects and draw a Venn diagram to communicate their ideas.

### Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities

Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned

### 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.

### TWO WAY TABLES. Edexcel GCSE Mathematics (Linear) 1MA0

Edexcel GCSE Mathematics (Linear) 1MA0 TWO WAY TABLES Materials required for examination Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser. Tracing paper may

### Geometry. Kellenberg Memorial High School

2015-2016 Geometry Kellenberg Memorial High School Undefined Terms and Basic Definitions 1 Click here for Chapter 1 Student Notes Section 1 Undefined Terms 1.1: Undefined Terms (we accept these as true)

### Chapter 5: Probability: What are the Chances? Probability: What Are the Chances? 5.1 Randomness, Probability, and Simulation

Chapter 5: Probability: What are the Chances? Section 5.1 Randomness, Probability, and Simulation The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 5 Probability: What Are

### Section 1: How will you be tested? This section will give you information about the different types of examination papers that are available.

REVISION CHECKLIST for IGCSE Mathematics 0580 A guide for students How to use this guide This guide describes what topics and skills you need to know for your IGCSE Mathematics examination. It will help