L01 - Set Theory. CSci/Math May 2015

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "L01 - Set Theory. CSci/Math May 2015"

Transcription

1 L01 - Set Theory CSci/Math May / 10

2 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 of A are also in B) A B means A is not a subset of B (at least one element of A is not in B) A B means A B and A B 2 / 10

3 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 of A are also in B) A B means A is not a subset of B (at least one element of A is not in B) A B means A B and A B Example 1 A = {2, 3, 4, {5}} Which of the following are elements of A? Which are subsets? (a) 2 (b) {2} (c) 5 (d) {5} (e) {2, 3, 4} (f) {2, 3, 4, {5}} (g) {{5}} (h) {{{5}}} 2 / 10

4 Set Builder Notation A = {expression : rule} = {expression rule} Example 2 A = { 1 n n N} B = { 1 n n N, 1 n 5} C = {x x B, x < 1} D = {2, 4, 6, 8, 10, 12} E = {1, 3, 5, 7, 9,...} List the elements in A, B, and C. Write D and E in set builder notation. What are C and D? 3 / 10

5 Ordered Pairs and Cartesian Product Example 3 Your society is electing a new executive. 4 / 10

6 Ordered Pairs and Cartesian Product Example 3 Your society is electing a new executive. The ordered pair (x, y) indicates that x is president and y is vice-president. Is (Alice, Bob)=(Bob, Alice)? 4 / 10

7 Ordered Pairs and Cartesian Product Example 3 Your society is electing a new executive. The ordered pair (x, y) indicates that x is president and y is vice-president. Is (Alice, Bob)=(Bob, Alice)? Let A = {people running for president} and B = {people running for vice-president}. What does A B mean? 4 / 10

8 Ordered Pairs and Cartesian Product Example 3 Your society is electing a new executive. The ordered pair (x, y) indicates that x is president and y is vice-president. Is (Alice, Bob)=(Bob, Alice)? Let A = {people running for president} and B = {people running for vice-president}. What does A B mean? If A = {Alice, Chris, Dean} and B = {Bob, Eve, Fiona, Georgie}, what is A B, and what does this number mean? 4 / 10

9 Ordered Pairs and Cartesian Product Example 3 Your society is electing a new executive. The ordered pair (x, y) indicates that x is president and y is vice-president. Is (Alice, Bob)=(Bob, Alice)? Let A = {people running for president} and B = {people running for vice-president}. What does A B mean? If A = {Alice, Chris, Dean} and B = {Bob, Eve, Fiona, Georgie}, what is A B, and what does this number mean? If A = {candidates for president}, B = {candidates for vice-president}, C = {candidates for secretary}, D = {candidates for treasurer}, E = {candidates for DSU rep}, what is A B C D E? 4 / 10

10 Union, Intersection, Difference, Complement A B means the union of A and B, i.e. the set consisting of all element in A and B A B means the intersection of A and B, i.e. the set consisting of all elements which are both in A and B A \ B = A B means the difference of A and B, i.e. the set of elements in A which are not in B A c = A means the complement of A, i.e. U \ A where U is the universal set 5 / 10

11 Union, Intersection, Difference, Complement Example 4 Let A = {a, b, c, d, e}, B = {a, d, e}, C = {a, b, c, f, g, h}, and U = {a, b, c, d, e, f, g, h, i, j}. Find (1) A B (2) A B (3) A C (4) A C (5) A B (6) B A (7) A C (8) C (A B) (9) A (10) C (11) A B (12) A C 6 / 10

12 Union, Intersection, Difference, Complement Example 5 A = {2n n Z} {3m m Z} (1) Which of the following are elements of A? (a) 4 (b) 9 (c) 6 (d) 90 (e) 15 (2) Which of the following are subsets of A? (a) {6, 12, 18} (b) {6, 9, 12, 15} (c) { 6, 90, 12} 7 / 10

13 Union, Intersection, Difference, Complement Example 6 A (B C) (1) Draw the Venn Diagram. (2) Can you find another way using only unions and intersections to describe this set? 8 / 10

14 Union, Intersection, Difference, Complement Example 6 A (B C) (1) Draw the Venn Diagram. (2) Can you find another way using only unions and intersections to describe this set? This is one half of the distributive laws. They are A (B C) = (A B) (A C), A (B C) = (A B) (A C). 8 / 10

15 Union, Intersection, Difference, Complement Example 7 (A B) (1) Draw the Venn Diagram. (2) Can you find another way using only complements, unions and intersections to describe this set? 9 / 10

16 Union, Intersection, Difference, Complement Example 7 (A B) (1) Draw the Venn Diagram. (2) Can you find another way using only complements, unions and intersections to describe this set? This is one half of DeMorgan s laws. They are (A B) = A B, (A B) = A B. 9 / 10

17 Indexed Sets Example 8 U = {all living people} A i = {x x was born on the ith day of the month} B j = {x x was born during the jth month of the year} C k = {x x was born on the kth day of the week} Describe in ( words: 3 ) (a) B 1 C k k=1 (c) C 6 (e) 1 j 4 B j 5 (b) (d) ( 15 B j j=1 i=12 1 k 5 C k A i ) 1 j 4 B j 10 / 10

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

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,

More information

The Language of Mathematics

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,

More information

Clicker Question. Theorems/Proofs and Computational Problems/Algorithms MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES

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:

More information

Sets and set operations

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

More information

CmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B

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

More information

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 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,

More information

LESSON SUMMARY. Set Operations and Venn Diagrams

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

More information

2.1 Symbols and Terminology

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:

More information

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

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

More information

CHAPTER 2. Set, Whole Numbers, and Numeration

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

More information

Automata and Formal Languages

Automata and Formal Languages Automata and Formal Languages Winter 2009-2010 Yacov Hel-Or 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,

More information

4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.

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,

More information

+ Section 6.2 and 6.3

+ 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

More information

Complement. If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A.

Complement. If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A. Complement If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A. For example, if A is the event UNC wins at least 5 football games, then A c is the

More information

Set Theory: Shading Venn Diagrams

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).

More information

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

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

More information

2.1 Sets, power sets. Cartesian Products.

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

More information

Discrete Mathematics

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

More information

Applications of Methods of Proof

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

More information

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. 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

More information

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 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

More information

EXAM. Exam #3. Math 1430, Spring 2002. April 21, 2001 ANSWERS

EXAM. Exam #3. Math 1430, Spring 2002. April 21, 2001 ANSWERS EXAM Exam #3 Math 1430, Spring 2002 April 21, 2001 ANSWERS i 60 pts. Problem 1. A city has two newspapers, the Gazette and the Journal. In a survey of 1, 200 residents, 500 read the Journal, 700 read the

More information

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

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

More information

Sections 2.1, 2.2 and 2.4

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

More information

Discrete Mathematics Set Operations

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

More information

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 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.

More information

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 - 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

More information

Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

More information

Statistics 100A Homework 1 Solutions

Statistics 100A Homework 1 Solutions Chapter 1 tatistics 100A Homework 1 olutions Ryan Rosario 1. (a) How many different 7-place license plates are possible if the first 2 places are for letters and the other 5 for numbers? The first two

More information

Homework 20: Compound Probability

Homework 20: Compound Probability Homework 20: Compound Probability Definition The probability of an event is defined to be the ratio of times that you expect the event to occur after many trials: number of equally likely outcomes resulting

More information

WOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology

WOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology First Name: Family Name: Student Number: Class/Tutorial: WOLLONGONG COLLEGE AUSTRALIA A College of the University of Wollongong Diploma in Information Technology Final Examination Spring Session 2008 WUCT121

More information

CONSTITUTION. Education Deans: Independent Colleges and Universities of Texas. Article I Name. Article II Objectives. Article III Membership

CONSTITUTION. Education Deans: Independent Colleges and Universities of Texas. Article I Name. Article II Objectives. Article III Membership CONSTITUTION Education Deans: Independent Colleges and Universities of Texas Article I Name The name of the Organization shall be Education Deans: Independent Colleges and Universities of Texas (EDICUT)

More information

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 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

More information

Chapter 2: Systems of Linear Equations and Matrices:

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,

More information

Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

More information

JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson

JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3

More information

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 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.

More information

Bylaws of the Clemson University Post-Doctoral Association

Bylaws of the Clemson University Post-Doctoral Association Bylaws of the Clemson University Post-Doctoral Association Incorporated on MISSION 1. The mission of the Clemson University Post-doctoral Association (CUPDA or Association) is to improve and support the

More information

Basic Probability Concepts

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

More information

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

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

More information

2.1 The Algebra of Sets

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

More information

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

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,

More information

Definition 14 A set is an unordered collection of elements or objects.

Definition 14 A set is an unordered collection of elements or objects. Chapter 4 Set Theory Definition 14 A set is an unordered collection of elements or objects. Primitive Notation EXAMPLE {1, 2, 3} is a set containing 3 elements: 1, 2, and 3. EXAMPLE {1, 2, 3} = {3, 2,

More information

A Little Set Theory (Never Hurt Anybody)

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

More information

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 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

More information

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

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

More information

SETS AND FUNCTIONS, MATH 215 FALL 2015 (WHYTE)

SETS AND FUNCTIONS, MATH 215 FALL 2015 (WHYTE) SETS AND FUNCTIONS, MATH 215 FALL 2015 (WHYTE) 1. Intro to Sets After some work with numbers, we want to talk about sets. For our purposes, sets are just collections of objects. These objects can be anything

More information

Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

More information

THE LANGUAGE OF SETS AND SET NOTATION

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

More information

Question 31 38, worth 5 pts each for a complete solution, (TOTAL 40 pts) (Formulas, work

Question 31 38, worth 5 pts each for a complete solution, (TOTAL 40 pts) (Formulas, work Exam Wk 6 Name Questions 1 30 are worth 2 pts each for a complete solution. (TOTAL 60 pts) (Formulas, work, or detailed explanation required.) Question 31 38, worth 5 pts each for a complete solution,

More information

Basic Concepts of Set Theory, Functions and Relations

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

More information

7 Relations and Functions

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

More information

Automata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi

Automata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi Automata Theory Automata theory is the study of abstract computing devices. A. M. Turing studied an abstract machine that had all the capabilities of today s computers. Turing s goal was to describe the

More information

All of mathematics can be described with sets. This becomes more and

All of mathematics can be described with sets. This becomes more and CHAPTER 1 Sets All of mathematics can be described with sets. This becomes more and more apparent the deeper into mathematics you go. It will be apparent in most of your upper level courses, and certainly

More information

Benchmark Test : Algebra 1

Benchmark Test : Algebra 1 1 Benchmark: MA.91.A.3.3 If a ar b r, what is the value of a in terms of b and r? A b + r 1 + r B 1 + b r + b C 1 + b r D b r 1 Benchmark: MA.91.A.3.1 Simplify: 1 g(5 3) 4g 13 4 F 11 4 g 16 G g 1 H 15

More information

SHAW UNIVERSITY College of Arts and Sciences Department of Natural Sciences and Mathematics

SHAW UNIVERSITY College of Arts and Sciences Department of Natural Sciences and Mathematics SHAW UNIVERSITY College of Arts and Sciences Department of Natural Sciences and Mathematics MAT 112 (3 credit hours) SUMMER 2009 (PRE: MAT 111) General Mathematics II (Online) Instructor: Simon Ugwuoke,

More information

Introduction to Proofs

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

More information

ELECTION COMMITTEE Revised June 2015

ELECTION COMMITTEE Revised June 2015 PURPOSE ELECTION COMMITTEE Revised June 2015 Secure nominations from NCSSSA Members for elected positions of President, First Vice President, Vice President Designate, Secretary, and Treasurer (Officers).

More information

Factorizations: Searching for Factor Strings

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

More information

Access The Mathematics of Internet Search Engines

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

More information

Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

More information

GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,

More information

Teori Himpunan. Bagian III

Teori Himpunan. Bagian III Teori Himpunan Bagian III Teori Himpunan Himpunan: Kumpulan objek (konkrit atau abstrak) ) yang mempunyai syarat tertentu dan jelas, bisanya dinyatakan dengan huruf besar. a A a A a anggota dari A a bukan

More information

MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

More information

General responsibilities. Build the Power of the Union

General responsibilities. Build the Power of the Union POSITION President JOB DESCRIPTION Build the Power of the Union Specific Activities Chair AFT Seattle Eboard Meetings Support Board members in carrying out their duties Communicate regularly with faculty

More information

South Carolina College- and Career-Ready (SCCCR) Probability and Statistics

South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR)

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.

2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are

More information

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION 1 WHAT IS STATISTICS? Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics

More information

MATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!

MATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing! MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics

More information

ACTIVITY: Identifying Common Multiples

ACTIVITY: Identifying Common Multiples 1.6 Least Common Multiple of two numbers? How can you find the least common multiple 1 ACTIVITY: Identifying Common Work with a partner. Using the first several multiples of each number, copy and complete

More information

Discrete Maths. Philippa Gardner. These lecture notes are based on previous notes by Iain Phillips.

Discrete Maths. Philippa Gardner. These lecture notes are based on previous notes by Iain Phillips. Discrete Maths Philippa Gardner These lecture notes are based on previous notes by Iain Phillips. This short course introduces some basic concepts in discrete mathematics: sets, relations, and functions.

More information

1 Construction of CCA-secure encryption

1 Construction of CCA-secure encryption CSCI 5440: Cryptography Lecture 5 The Chinese University of Hong Kong 10 October 2012 1 Construction of -secure encryption We now show how the MAC can be applied to obtain a -secure encryption scheme.

More information

Greatest Common Factors and Least Common Multiples with Venn Diagrams

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

More information

Parity-based inference control for multi-dimensional range sum queries

Parity-based inference control for multi-dimensional range sum queries Journal of Computer Security 15 (2007) 417 445 417 IOS Press Parity-based inference control for multi-dimensional range sum queries Lingyu Wang a,, Yingjiu Li b, Sushil Jajodia c and Duminda Wijesekera

More information

Ottawa Chapter of Advocis, The Financial Advisors Association of Canada Request for Nominations

Ottawa Chapter of Advocis, The Financial Advisors Association of Canada Request for Nominations Ottawa Chapter of Advocis, The Financial Advisors Association of Canada Request for Nominations The nominating committee of Ottawa Chapter of Advocis is asking for your assistance in identifying individuals

More information

Mathematics for Algorithm and System Analysis

Mathematics for Algorithm and System Analysis Mathematics for Algorithm and System Analysis for students of computer and computational science Edward A. Bender S. Gill Williamson c Edward A. Bender & S. Gill Williamson 2005. All rights reserved. Preface

More information

Basic Set Theory. Chapter Set Theory. can be written: A set is a Many that allows itself to be thought of as a One.

Basic Set Theory. Chapter Set Theory. can be written: A set is a Many that allows itself to be thought of as a One. Chapter Basic Set Theory A set is a Many that allows itself to be thought of as a One. - Georg Cantor This chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical

More information

Notes on Probability. Peter J. Cameron

Notes on Probability. Peter J. Cameron Notes on Probability Peter J. Cameron ii Preface Here are the course lecture notes for the course MAS108, Probability I, at Queen Mary, University of London, taken by most Mathematics students and some

More information

KAPPA DELTA PI AN INTERNATIONAL HONOR SOCIETY IN EDUCATION COLLEGE OF EDUCATION FLORIDA ATLANTIC UNIVERISTY. Bylaws for Rho Omega Chapter

KAPPA DELTA PI AN INTERNATIONAL HONOR SOCIETY IN EDUCATION COLLEGE OF EDUCATION FLORIDA ATLANTIC UNIVERISTY. Bylaws for Rho Omega Chapter KAPPA DELTA PI AN INTERNATIONAL HONOR SOCIETY IN EDUCATION COLLEGE OF EDUCATION FLORIDA ATLANTIC UNIVERISTY Bylaws for Rho Omega Chapter Article I. THE CHAPTER SECTION 1: SECTION 2: SECTION 3: SECTION

More information

Statistics 100A Homework 2 Solutions

Statistics 100A Homework 2 Solutions Statistics Homework Solutions Ryan Rosario Chapter 9. retail establishment accepts either the merican Express or the VIS credit card. total of percent of its customers carry an merican Express card, 6

More information

TOPIC I INTRODUCTION AND SET THEORY

TOPIC I INTRODUCTION AND SET THEORY [1] Introduction TOPIC I INTRODUCTION AND SET THEORY Economics Vs. Mathematical Economics. "Income positively affects consumption. Consumption level can never be negative. The marginal propensity consume

More information

K-Kids. K-Kids Club Structure. K-Kids organizational structure. Sergeant of Arms

K-Kids. K-Kids Club Structure. K-Kids organizational structure. Sergeant of Arms K-Kids K-Kids organizational structure K-Kids Club Structure Faculty Advisor President Kiwanis Advisor Vice President Parent Advisor Secretary Treasurer Board of Directors (1 member from each grade level)

More information

Lecture #11 Relational Database Systems KTH ROYAL INSTITUTE OF TECHNOLOGY

Lecture #11 Relational Database Systems KTH ROYAL INSTITUTE OF TECHNOLOGY Lecture #11 Relational Database Systems KTH ROYAL INSTITUTE OF TECHNOLOGY Contents Storing data Relational Database Systems Entity Relationship diagrams Normalisation of ER diagrams Tuple Relational Calculus

More information

Chapter 9 Joining Data from Multiple Tables. Oracle 10g: SQL

Chapter 9 Joining Data from Multiple Tables. Oracle 10g: SQL Chapter 9 Joining Data from Multiple Tables Oracle 10g: SQL Objectives Identify a Cartesian join Create an equality join using the WHERE clause Create an equality join using the JOIN keyword Create a non-equality

More information

Modelling Software Requirements Exercise on comparing two methods in an empirical study. BLUE 2 nd session Experiment package mss- U

Modelling Software Requirements Exercise on comparing two methods in an empirical study. BLUE 2 nd session Experiment package mss- U Modelling Software Requirements Exercise on comparing two methods in an empirical study. BLUE 2 nd session Experiment package mss- U Surname: Color: BLUE Name: Application: mss Treatment: UML Use Cases

More information

Notes on counting finite sets

Notes on counting finite sets Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................

More information

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

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

More information

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

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

More information

half-line the set of all points on a line on a given side of a given point of the line

half-line the set of all points on a line on a given side of a given point of the line Geometry Week 3 Sec 2.1 to 2.4 Definition: section 2.1 half-line the set of all points on a line on a given side of a given point of the line notation: is the half-line that contains all points on the

More information

1 the relational data model The relational model of data was introduced by E.F. Codd (1970). 1.1 relational model concepts The relational model repres

1 the relational data model The relational model of data was introduced by E.F. Codd (1970). 1.1 relational model concepts The relational model repres database systems formal definitions of the relational data model [03] s. yurttaοs 1 1 the relational data model The relational model of data was introduced by E.F. Codd (1970). 1.1 relational model concepts

More information

Counting principle, permutations, combinations, probabilities

Counting principle, permutations, combinations, probabilities Counting Methods Counting principle, permutations, combinations, probabilities Part 1: The Fundamental Counting Principle The Fundamental Counting Principle is the idea that if we have a ways of doing

More information

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) 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

More information

Methods Used for Counting

Methods Used for Counting COUNTING METHODS From our preliminary work in probability, we often found ourselves wondering how many different scenarios there were in a given situation. In the beginning of that chapter, we merely tried

More information

5. Duties and Responsibilities of the Executive Board

5. Duties and Responsibilities of the Executive Board 5. Duties and Responsibilities of the Executive Board 5.1 General Expectations of Executive Board Members It is expected that Executive Board members will: a. Attend all Executive Board meetings and stay

More information

Intersection of 3 Planes

Intersection of 3 Planes Intersection of 3 Planes With a partner draw diagrams to represent the six cases studied yesterday. Case 1: Three distinct parallel planes 1 Intersection of 3 Planes With a partner draw diagrams to represent

More information

Basic concepts in probability. Sue Gordon

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

More information

SPECIFIC DUTIES AND RESPONSIBILITIES OF THE FBLA STATE PRESIDENT

SPECIFIC DUTIES AND RESPONSIBILITIES OF THE FBLA STATE PRESIDENT FBLA STATE PRESIDENT In addition to the general responsibilities of all FBLA State Officers, the State President 1. Preside at meetings of the FBLA State Executive Council, the State Leadership Conference,

More information

2 Elementary probability

2 Elementary probability 2 Elementary probability This first chapter devoted to probability theory contains the basic definitions and concepts in this field, without the formalism of measure theory. However, the range of problems

More information

Introduction to Probability

Introduction to Probability Introduction to Probability Math 30530, Section 01 Fall 2012 Homework 1 Solutions 1. A box contains four candy bars: two Mars bars, a Snickers and a Kit-Kat. I randomly draw a bar from the box and eat

More information