Introduction to Mathematical Reasoning
Purpose of Course showclose
The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems. Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations. The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting. The most important aspect of this course is that you will learn what it means to prove a mathematical proposition. We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions. The environments we use are propositions and predicates, finite sets and relations, integers, fractions and rational numbers, and infinite sets. Each topic in this course is standard except the first one, puzzles. There are several reasons for including puzzles. First and foremost, a challenging puzzle can be a microcosm of mathematical development. A great puzzle is like a laboratory for proving propositions. The puzzler initially feels the tension that comes from not knowing how to start just as the mathematician feels when first investigating a topic or trying to solve a problem. The mathematician “plays” with the topic or problem, developing conjectures which he/she then tests in some special cases. Similarly, the puzzler “plays” with the puzzle. Sometimes the conjectures turn out to be provable, but often they do not, and the mathematician goes back to playing. At some stage, the puzzler (mathematician) develops sufficient sense of the structure and only then can he begin to build the solution (prove the theorem). This multistep process is perfectly mirrored in solving the KenKen problems this course presents. Some aspects of the solutions motivate ideas you will encounter later in the course. For example, modular congruence is a standard topic in number theory, and it is also useful in solving some KenKen problems. Another reason for including puzzles is to foster creativity.
Optionally, you can read this New York Times article by Benedict Carney, “Tracing the Spark of Creative ProblemSolving”, to get the whole story; note that this may require you to sign up for the New York Times online for free.
Course Information showclose
Course Designer: Professor Harold B. Reiter
Primary Resources: This course is comprised of a range of different free, online materials. However, the course makes primary use of the following resources:
 YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s Lectures
 Old Dominion University: Shunichi Toida’s Discrete Mathematics Web Course Material
 Subunit 1.2 Activity
 Subunit 1.3 Activity
 Subsubunit 1.4.1 Activity
 Subsubunit 1.4.2 Activity
 Unit 1 Assessments
 Subsubunit 3.4.5.1 Activity
 Subsubunit 3.4.5.2 Activity
 Subsubunit 3.7.1 Activity
 The Final Exam
In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.
Time Commitment: This course should take you a total of 112.25 hours to complete. Each unit includes a time advisory that lists the amount of time you are expected to spend on each subunit. These should help you plan your time accordingly. It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself. For example, Unit 1 should take approximately 31.5 hours to complete. Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 4 hours) on Monday night; subunit 1.2, which is optional (a total of 4 hours) on Tuesday night; subunits 1.3 and 1.4 (a total of 4.25 hours) on Wednesday night; etc.
Tips/Suggestions: As you read, take careful notes on a separate sheet of paper. Mark down any important equations, formulas, definitions, and proofs that stand out to you. These notes will be useful to review as you study for the Final Exam.
This course features a number of Khan Academy™ videos. Khan Academy™ has a library of over 3,000 videos covering a range of topics (math, physics, chemistry, finance, history and more), plus over 300 practice exercises. All Khan Academy™ materials are available for free at www.khanacademy.org.

Learning Outcomes showclose
 Read and dissect proofs of elementary propositions related to discrete mathematical objects such as integers, finite sets, graphs and relations, and functions.
 Translate verbal statements into symbolic ones by using the elements of mathematical logic.
 Determine when a proposed mathematical argument is logically correct.
 Determine when a compound sentence is a tautology, a contradiction, or a contingency.
 Translate riddles and other brainteasers into the language of predicates and propositions.
 Solve problems related to place value, divisors, and remainders.
 Use modular arithmetic to solve various equations, including quadratic equations in Z_{6}, Z_{7}, Z_{11} and Diophantine equations.
 Prove and use the salient characteristics of the rational, irrational, and real number systems to verify properties of various number systems.
 Use mathematical induction to construct proofs of propositions about sets of positive integers.
 Classify relations as being reflexive, symmetric, antisymmetric, transitive, a partial ordering, a total ordering, or an equivalence relation.
 Determine if a relation is a function, and if so, whether or not it is a bijection.
 Manipulate finite and infinite sets by using functions and set operations.
 Determine if a set is finite, countable, or uncountable.
 Use the properties of countable and uncountable sets in various situations.
 Recognize some standard countable and uncountable sets.
 Determine and effectively use an appropriate counting tool to find the number of objects in a finite set.
Course Requirements showclose
√ Have access to a computer.
√ Have continuous broadband Internet access.
√ Have the ability/permission to install plugins or software (e.g. Adobe Reader or Flash).
√ Have the ability to download and save files and documents to a computer.
√ Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).
√ Have competency in the English language.
√ Have read the Saylor Student Handbook.
Unit Outline show close
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Unit 1: Logic
In this unit, you will begin by considering various puzzles, including KenKen and Sudoku. You will learn the importance of tenacity in approaching mathematical problems including puzzles and brain teasers. You will also learn why giving names to mathematical ideas will enable you to think more effectively about concepts that are built upon several ideas. Then, you will learn that propositions are (English) sentences whose truth value can be established. You will see examples of selfreferencing sentences which are not propositions. You will learn how to combine propositions to build compound ones and then how to determine the truth value of a compound proposition in terms of its component propositions. Then, you will learn about predicates, which are functions from a collection of objects to a collection of propositions, and how to quantify predicates. Finally, you will study several methods of proof including proof by contradiction, proof by complete enumeration, etc.
Unit 1 Time Advisory show close
Unit 1 Learning Outcomes show close

1.1 Sudoku
 Reading: Wikipedia’s “Sudoku”Link: Wikipedia’s “Sudoku” (PDF)
Instructions: Please click on the link above and read the entire article. Try not to get sidetracked looking at variations. Pay special attention to the growth of the number of Latin squares as the size increases. Note that if you want to look ahead at the type of problem you will be asked to solve, check the file “Logic.pdf” at the end of Unit 2.
Reading this article should take approximately 30 minutes.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0. You can find the original Wikipedia version of this article here.  Reading: Mathematical Circles Topics: Tom Davis’ “Sudoku”
Link: Mathematical Circles Topics: Tom Davis’ “Sudoku” (PDF)
Instructions: Please click on the link above. Scroll down the webpage to “Games,” and select the “Sudoku” link to download the PDF. Read Tom Davis’ paper, paying special attention to the way he names the cells and to his development of language. Next, if you have not done Sudoku puzzles before, Web Sudoku and Daily Sudoku and are two popular sites. Do one or two before moving on to KenKen. Please note that this reading also covers the topics outlined in subsubunits 2.1.2 and 2.1.3.
This will take you about 3 hours if you have not done Sudoku before, and about 2 hours if you have.
Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.
 Reading: Wikipedia’s “Sudoku”

1.1.1 What Is a Latin Square?
 Reading: Wikipedia’s “Latin Square”Link: Wikipedia’s “Latin Square” (PDF)
Instructions: Please click on the link above and read the entire article on Latin Squares.
Reading this article should take approximately 30 minutes.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0. You can find the original Wikipedia version of this article here.
 Reading: Wikipedia’s “Latin Square”

1.1.2 Building a Language around Sudoku
Note: This topic is covered by the Davis reading assigned below subunit 1.1. In particular, please focus on the introduction, especially the bulleted list of terminology on page 2 of the PDF.

1.1.3 XWing and Unique Candidates Strategies
Note: This topic is covered by the Davis reading assigned below subunit 1.1. In particular, please focus on Section 6, “XWings and Swordfish,” and Section 11, “Unique Solution Constraints.”

1.2 KenKen
 Activity: The New York Times’ “KenKen Puzzles”
Link: The New York Times’ “KenKen Puzzles” (HTML)
Instructions: Please keep in mind that this activity is optional. After reading Harold et al.’s paper, click on the link above to access KenKen puzzles, and attempt to complete one of these puzzles. Note that you can choose the level of difficulty (easier, medium, and harder). After a few practices, challenge yourself to attempt a KenKen puzzle that is at the next level of difficulty. Do not allow yourself to get addicted!
You should dedicate no more than 1 hour to practicing KenKen puzzles.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Optional Reading: The Saylor Foundation: Harold Reiter’s “Introduction to Mathematical Reasoning”
Link: The Saylor Foundation: Harold Reiter’s “Introduction to Mathematical Reasoning” (PDF)
Instructions: This article is optional. If you have an interest in solving KenKen problems, then you will find this section interesting. Otherwise, omitting it will not hinder your understanding of subsequent material. Should you choose to work through this section, please click on the link above, and read the paper by Harold Reiter, et al. for an introduction to KenKen. Complete the exercises in the PDF. Please note that this reading covers the topics outlined in subsubunits 1.2.1 through 1.2.3.
Reading this article and completing the exercises should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Activity: The New York Times’ “KenKen Puzzles”

1.2.1 Parity
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 3 of “Using KenKen to Build Reasoning Skills” to learn how to use parity in KenKen puzzles.

1.2.2 Counting
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 4 of “Using KenKen to Build Reasoning Skills” to learn how to use counting.

1.2.3 Stacked Cages
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 5 of “Using KenKen to Build Reasoning Skills” to learn how to use the idea of stacked cages.

1.2.4 XWing Strategy
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 6 of “Using KenKen to Build Reasoning Skills” to learn how to use the Xwing strategy.

1.2.5 Pair Analysis
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 6 of “Using KenKen to Build Reasoning Skills” to learn how to use the idea of stacked cages.

1.2.6 Parallel and Orthogonal Cages
Note: This topic is covered by the reading assigned below subunit 1.2. Read Section 7 of “Using KenKen to Build Reasoning Skills” to learn how to use parallel and orthogonal cages.

1.2.7 Unique Candidates
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 8 of “Using KenKen to Build Reasoning Skills” to learn how to use the unique candidate rule.

1.2.8 Modular Arithmetic
Note: This topic is covered by the reading assigned below subunit 1.2. Please read Section 9 of “Using KenKen to Build Reasoning Skills,” even though you have not yet studied modular arithmetic. When you get to this part of the course, you will be asked to come back and take another look at this section.

1.3 SET
 Activity: Set Enterprises’ “Daily Puzzle”
Link: Set Enterprises’ “Daily Puzzle” (HTML)
Instructions: Please note that this activity is optional. If you choose to work through this activity, please click on the link above, read the game rules by clicking on the “daily puzzle rules” link, and play a bit.
You should dedicate no more than 1 hour to exploring SET.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Activity: Set Enterprises’ “Daily Puzzle”

1.4 Other Brain Teasers
 Web Media: Khan Academy’s “Brain Teasers”Link: Khan Academy’s “Brain Teasers” (YouTube)
Instructions: Please click on the link above and pick out a few videos to watch on brain teasers. The puzzle will be introduced to you at the beginning of the video. You should pause the video and attempt to solve the puzzle before viewing the solution. Watch the solutions only if you absolutely cannot solve the puzzle; then, go back and reattempt the problem.
You should spend approximately 1 hour on this site, watching a few of these videos and attempting to solve the problems.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialNoDerivs 3.0 Unported License. It is attributed to the Khan Academy.
 Web Media: Khan Academy’s “Brain Teasers”

1.4.1 Truth Tellers and Liars
 Activity: University of Chicago’s Math Explorer’s Club: Antonio Montalban and Yannet Interian’s “Module on Puzzles”
Link: University of Chicago’s Math Explorer’s Club: Antonio Montalban and Yannet Interian’s “Module on Puzzles” (HTML)
Instructions: Please click on the link above and work on the problems on this webpage: liars and truthtellers puzzles, the Rubik’s cube, knots and graphs, and arithmetic and geometry.
Solving these problems should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Activity: University of Chicago’s Math Explorer’s Club: Antonio Montalban and Yannet Interian’s “Module on Puzzles”

1.4.2 Coin Weighing Puzzles
 Activity: Cut the Knot: Alexander Bogomolny’s “A Fake Among Eight Coins”Link: Cut the Knot: Alexander Bogomolny’s “A Fake Among Eight Coins” (HTML)
Instructions: Problems about finding the counterfeit coin among a large group of otherwise genuine coins are quite abundant. Please click on the link above and attempt to solve the problem on this webpage. Solutions appear at the bottom of the webpage. If this type of logical thinking interests you, attempt to find similar problems to solve with an online search.
You should spend approximately 15 minutes attempting to solve this problem.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Activity: Cut the Knot: Alexander Bogomolny’s “A Fake Among Eight Coins”

1.5 Propositional Logic
 Reading: Hofstra University: Stefan Waner and Steven R. Costenoble’s “Introduction to Logic”Link: Hofstra University: Stefan Waner and Steven R. Costenoble’s “Introduction to Logic” (HTML)
Instructions: Please click on the link above and read the entire webpage. This text will enable you to see the very close connection between propositional logic and naïve set theory, which you will study in Unit 3. Please note that this lecture covers the topics outlined in subsubunits 1.5.1 and 1.5.2 as well as any inclusive subsubunits.
Reading this webpage should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Lecture: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 1: Propositional Logic”
Link: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 1: Propositional Logic” (YouTube)
Instructions: Click on the link above and watch the entire lecture. In particular, focus on the information provided from the 12minute mark until the 18minute mark. In this lecture, you will learn which sentences are propositions. Please note that this lecture covers the topics outlined in subsubunits 1.5.1 and 1.5.2 as well as any inclusive subsubunits.
Watching this video and pausing to take notes should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: Hofstra University: Stefan Waner and Steven R. Costenoble’s “Introduction to Logic”

1.5.1 Compound Proposition
Note: This topic is covered by the lecture and article assigned below subunit 1.1.

1.5.1.1 Logical Connectives, Conjunction, Disjunction, and Negation
 Web Media: YouTube: The University of Aukland Library’s “Boolean Operators” and Philosophical Techne’s “Introduction to Propositional Logic Part I”
Links: YouTube: The University of Aukland Library’s “Boolean Operators” (YouTube) and Philosophical Techne’s “Introduction to Propositional Logic Part I” (YouTube)
Instructions: Please watch both of the videos linked above. This will help you later when you are asked to build proofs of statements about rational numbers and about integers.
Watching these lectures and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.
 Web Media: YouTube: The University of Aukland Library’s “Boolean Operators” and Philosophical Techne’s “Introduction to Propositional Logic Part I”

1.5.1.2 Implication and Other Boolean Connectives
Note: This topic is covered by the video lecture assigned below subunit 1.1. In particular, focus on and review the lecture from the 25minute mark to the 32minute mark for a discussion on implication and other Boolean connectives.

1.5.1.3 Properties of Connectives
 Reading: Wikipedia’s “Logical Connective”
Link: Wikipedia’s “Logical Connective” (PDF)
Instructions: Please click on the link above and read this article, which covers the properties of connectives. While reading, pay special attention to the connection between the Boolean connective and its Venn diagram.
Reading this article and taking notes should take approximately 1 hour.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0. You can find the original Wikipedia version of this article here.
 Reading: Wikipedia’s “Logical Connective”

1.5.2 Truth Tables
 Reading: University of Cincinnati, Blue Ash: Kenneth R. Koehler’s Logic and Set Theory: “Logical Operations and Truth Tables”, “Properties of Logical Operators”, “Arguments”, and “Boolean Algebra”
Link: University of Cincinnati, Blue Ash: Kenneth R. Koehler’s Logic and Set Theory: “Logical Operations and Truth Tables”, (HTML) “Properties of Logical Operators”, (HTML) “Arguments”, (HTML) and “Boolean Algebra” (HTML)
Instructions: Please click on the links above and read these four sections of Koehler’s lectures on logic and set theory. These sections cover the topics outlined below subunit 1.5, including all the subsubunits.
Reading these sections should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Lecture: YouTube: National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 2: Propositional Logic (Contd.)”Link: YouTube: National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 2: Propositional Logic (Contd.)” (YouTube)
Instructions: Please click on the link above and watch the entire video lecture to learn about propositional logic.
Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: University of Cincinnati, Blue Ash: Kenneth R. Koehler’s Logic and Set Theory: “Logical Operations and Truth Tables”, “Properties of Logical Operators”, “Arguments”, and “Boolean Algebra”

1.5.2.1 What Is a Truth Table?
Note: This topic is covered by the Koehler reading assigned below subsubunit 1.5.2. Make sure to review the “Logical Operations and Truth Tables” section for an introduction that helps define truth tables.

1.5.2.2 The Boolean Algebra of Propositions
Note: This topic is covered by the Koehler reading assigned below subsubunit 1.5.2. Make sure to review the section on “Boolean Algebra.”

1.5.2.3 Tautologies, Contingencies, and Contradictions
Note: This topic is covered by the Koehler reading assigned below subsubunit 1.5.2. Make sure to read the definitions of tautology and contradictions (terms highlighted in bold) in the opening paragraphs of the “Logical Operations and Truth Tables” section. Please note that a contingency is simply a proposition that is caught between tautology (at the top) and contradiction (at the bottom). In other words, it is a proposition which is true for some values of its components and false for others. For example “if it rains today, it will snow tomorrow” is a contingency, because it can be true or false depending on the truth values of the two component propositions.

1.5.2.4 Logical Equivalence
Note: This topic is covered by the Koehler reading assigned below subsubunit 1.5.2. In particular, focus on the text after the heading “Equivalence” toward the end of the “Logical Operations and Truth Tables” section.

1.6 Predicate Logic
 Lecture: YouTube: National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 3: Predicates and Quantifiers” and “Lecture 4: Predicates and Quantifiers Continued”
Links: YouTube: National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 3: Predicates and Quantifiers” (YouTube) and “Lecture 4: Predicates and Quantifiers Continued” (YouTube)
Instructions: Please click on the links above and watch these lectures in their entirety to learn about predicates and quantifiers. These videos will also cover the topics outlined in subsubunit 1.6.1, including 1.6.1.1 and 1.6.1.2.
Watching these lectures and pausing to take notes should take approximately 3 hours.
Terms of Use: Please respect the copyright and terms of use displayed on these webpages above.
 Lecture: YouTube: National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 3: Predicates and Quantifiers” and “Lecture 4: Predicates and Quantifiers Continued”

1.6.1 Universal and Existential Qualifiers
Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6.

1.6.1.1 Negating Existential and Universal Predicates
Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6. Note that the negation of an existentially quantified predicate is a universally quantified one, and viceversa.

1.6.1.2 The Algebra of Predicates
Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6. The main idea here is that predicates can be manipulated in much the same way as numbers, sets, or propositions as we have seen already in the course.
 Assessment: The Saylor Foundation’s “Logic Homework Set”
Link: The Saylor Foundation’s “Logic Homework Set” (PDF)
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of logical connectives, propositions, negations, quantifiers, truth tables, and counterexamples. When you are done, check your work against those provided in the accompanying solutions file, the Saylor Foundation’s “Logic Homework Set Solutions” (PDF).
This exercise set should take between 1 and 2 hours to complete, depending on your comfort level with the material.
 Assessment: The Saylor Foundation’s “Logic Homework Set”

1.6.2 Building Proofs
Note: This topic is covered by the lecture assigned below subunit 1.6.

1.6.2.1 Modus Ponens and Modus Tollens
 Lecture: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 7: Methods of Proof”
Link: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 7: Methods of Proof” (YouTube)
Instructions: Please click on the link above and watch the entire video lecture.
Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Lecture: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 7: Methods of Proof”

1.6.2.2 Proofs by Contradiction
 Reading: California State University, San Bernardino: Peter Williams’ “Notes on Methods of Proof”
Link: California State University, San Bernardino: Peter Williams’ “Notes on Methods of Proof” (HTML)
Instructions: Please click on the link above and read the following sections: “Introduction”, “Definition and Theorems”, “Disproving Statements”, and “Types of Proofs”. The types of proofs include Direct Proofs, Proof by Contradiction, Existence Proofs, and Uniqueness Proofs. You may stop the reading here; we will cover the sixth one, Mathematical Induction, later in the course.
Reading these sections should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: California State University, San Bernardino: Peter Williams’ “Notes on Methods of Proof”

1.6.2.3 Problem Solving Strategies
 Reading: Old Dominion University: Shunichi Toida’s “Problem Solving”
Link: Old Dominion University: Shunichi Toida’s “Problem Solving” (HTML)
Instructions: Please click on the link above and read through the examples in the article. The problems are not difficult, but they do serve as clear illustrations of the various aspects of entrylevel problem solving.
Reading this article should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: Old Dominion University: Shunichi Toida’s “Problem Solving”

1.6.2.4 Contrapositive and Equivalent Forms
 Reading: Gowers’ Weblog: “Basic Logical Relationships between Statements, Converses, and Contrapositives”
Link: Gowers’ Weblog: “Basic Logical Relationships between Statements, Converses, and Contrapositives” (HTML)
Instructions: Please click on the link above and read this article, paying special attention to the parts of converses and contrapositives.
Reading this article and taking notes should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: Gowers’ Weblog: “Basic Logical Relationships between Statements, Converses, and Contrapositives”

Unit 1 Assessment
 Assessment: Oswego City School District Regents Exam Prep Center: Donna Roberts’ “Logic and Related Conditionals Quiz”
Link: Oswego City School District Regents Exam Prep Center: Donna Roberts’ “Logic and Related Conditionals Quiz” (HTML)
Instructions: Please click on the link above and complete this 10question quiz on logic and related conditionals. Once you choose an answer, a pop up will tell you if you have chosen correctly or incorrectly. You may also click on the drop down menu for an explanation.
Completing this quiz should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: The Saylor Foundation’s “Logic Problems”
Link: The Saylor Foundation’s “Logic Problems” (PDF)
Instructions: Please click on the link above to download the assignment. In order to solidify your problem solving and logical skills, work the eight problems. Then, check your answers against the Saylor Foundation’s “Logic Problems Solutions” (PDF).
Completing this assessment should take approximately 2 hours.
 Assessment: Oswego City School District Regents Exam Prep Center: Donna Roberts’ “Logic and Related Conditionals Quiz”

Unit 2: Sets, Part I
In this unit, you will explore the ideas of what is called ‘naive set theory.’ Contrasted with ‘axiomatic set theory,’ naive set theory assumes that you already have an intuitive understanding of what it means to be a set. You should mainly be concerned with how two or more given sets can be combined to build other sets and how the number of members (i.e. the cardinality) of such sets is related to the cardinality of the given sets.
Unit 2 Time Advisory show close
Unit 2 Learning Outcomes show close
 Lecture: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 10: Sets”
Link: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 10: Sets” (YouTube)
Instructions: Please click on the link above and watch this lecture. This lecture defines sets and will familiarize you with set notation and set language.
Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.  Reading: Old Dominion University: Shunichi Toida’s “Introduction to Set Theory”, (HTML) “Representation of Sets”, (HMTL) “Basics of Sets”, (HTML and Java) and “Mathematical Reasoning”Links: Old Dominion University: Shunichi Toida’s “Introduction to Set Theory”, (HTML) “Representation of Sets”, (HMTL) “Basics of Sets”, (HTML and Java) and “Mathematical Reasoning” (HTML)
Instructions: Please click on the links above and read these webpages in their entirety. These texts discuss the basics of set theory. Note that there are three ways to define a set. The third method, recursion, will come up again later in the course, but this is a great time to learn it.
Reading these webpages should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Lecture: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL): Dr. Kamala Krithivasan’s “Lecture 10: Sets”
 2.1 What Is a Set? Set Builder Notation

2.1.1 The Empty Set, the Universal Set
 Reading: University of California, San Diego: Edward Bender and S. Williamson's Discrete Mathematics: “Arithmetic, Logic and Numbers, Unit SF: Sets and Functions”Link: University of California, San Diego: Edward Bender and S. Williamson's Discrete Mathematics: “Arithmetic, Logic and Numbers, Unit SF: Sets and Functions” (PDF)
Instructions: Please click on the link above to download the PDF. Please read pages SF1 through SF8 of the file for an introduction to sets, set notation, set properties and proofs, and ordering sets.
Reading this article should take approximately 1 hour.
Terms of Use: The linked material above has been reposted by the kind permission of Edward Bender and S. Williamson. Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.  Web Media: YouTube: Kasidej Anchaleenukoon’s “Set Theory”
Link: YouTube: Kasidej Anchaleenukoon’s “Set Theory” (YouTube)
Instructions: Please click the link above and watch this video for an elementary introduction to set theory. This will be useful to you in case you feel uneasy about the reading above.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: University of California, San Diego: Edward Bender and S. Williamson's Discrete Mathematics: “Arithmetic, Logic and Numbers, Unit SF: Sets and Functions”

2.1.2 The Characteristic Function of a Set (Also Called the Indicator Function)
Note: This topic is covered by the Bernstein reading assigned below subsubunit 2.1.1. In particular, make sure to focus on “Definition 2: Characteristic Function,” starting on page 10 of the PDF.

2.1.3 Sets with Sets as Members
 Reading: University of California, San Diego: Edward Bender and S. Williamson’s Discrete Mathematics: “Arithmetic, Logic and Numbers, Unit SF: Sets and Functions”Link: University of California, San Diego: Edward Bender and S. Williamson’s Discrete Mathematics: “Arithmetic, Logic and Numbers, Unit SF: Sets and Functions” (PDF)
Instructions: Please click on the link above to download the PDF. Please read pages SF9 through SF11 to learn about subsets of sets. This text also is useful for learning how to prove various properties of sets. If needed, review pages SF1SF8, which were covered in subsubunit 2.1.1 above. In particular, please focus on example 9.
Reading these sections should take approximately 30 minutes.
Terms of Use: The linked material above has been reposted by the kind permission of Edward Bender and S. Williamson. Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
 Reading: University of California, San Diego: Edward Bender and S. Williamson’s Discrete Mathematics: “Arithmetic, Logic and Numbers, Unit SF: Sets and Functions”

2.2 Building New Sets from Given Sets
 Reading: Old Dominion University: Shunichi Toida’s “Set Operations”
Link: Old Dominion University: Shunichi Toida’s “Set Operations” (HTML and Java)
Instructions: Please click on the link above and read the entire webpage. Then test your understanding by working the four problems at the bottom.
Reading this webpage and completing these problems should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: Old Dominion University: Shunichi Toida’s “Set Operations”

2.2.1 Properties of Union, Intersection, and Complementation
 Reading: Old Dominion University: Shunichi Toida’s “Properties of Set Operation”
Link: Old Dominion University: Shunichi Toida’s “Properties of Set Operation” (HMTL)
Instructions: Please click on the link above and read the entire webpage. It is important that you become aware that sets combine under union and intersection in very much the same ways that numbers combine under addition and multiplication. For example, AUB=BUA is a way to say union is commutative in the same way as x + y = y + x says addition is commutative. One difference, however, is that the properties of addition and multiplication are defined as part of the number system (in our development) whereas the properties of sets under the operations we have defined are provable and hence must be proved.
Reading this webpage should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.  Reading: Simpson College’s Department of Computer Science: Lydia Sinapova’s “Boolean Algebra”
Link: Simpson College’s Department of Computer Science: Lydia Sinapova’s “Boolean Algebra” (PDF)
Instructions: Please click on the link above, scroll down the webpage to week 7, and click on the link for “Boolean Algebra” to download the lecture as a PDF. Please read this entire lecture, paying special attention to the definition of Boolean Algebra and to the isomorphism between the two systems of propositional logic and that of sets. Work the three exercises at the bottom of the PDF and then have a look at the solutions at the end of the document. Note that this reading also covers the topic outlined in subsubunit 2.2.2.
Reading this lecture and completing these exercises should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: Old Dominion University: Shunichi Toida’s “Properties of Set Operation”

2.2.2 The (Boolean) Algebra of Subsets of a Set
 Reading: The University of Western Australia: Greg Gamble’s “Set Theory, Logic, and Boolean Algebra”Link: The University of Western Australia: Greg Gamble’s “Set Theory, Logic, and Boolean Algebra” (PDF)
Instructions: Please click on the first link above and then select the link to the lecture “Set Theory, Logic, and Boolean Algebra” to download the PDF. Please read the entire lecture.
Reading this lecture should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: The University of Western Australia: Greg Gamble’s “Set Theory, Logic, and Boolean Algebra”

2.2.3 Using Characteristic Functions to Prove Properties of Sets
 Reading: Jerusalem College of Technology: Dr. Th. DanaPicard’s “The Characteristic Function of a Set”Link: Jerusalem College of Technology: Dr. Th. DanaPicard’s “The Characteristic Function of a Set” (HTML)
Instructions: Please click on the link above and read the entire webpage. This brief text will show you how to use characteristic functions to prove properties of sets. However, there are other reasons to learn how to do this. You will see later in the course that functions (not just characteristic functions play a critical role in the theory of cardinality (set size).
Reading this webpage should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: Jerusalem College of Technology: Dr. Th. DanaPicard’s “The Characteristic Function of a Set”

2.3 The Cartesian Product of Two or More Sets
 Reading: Jerusalem College of Technology: Dr. Th. DanaPicard’s “The Cartesian Product of Sets”Link: Jerusalem College of Technology: Dr. Th. DanaPicard’s “The Cartesian Product of Sets” (HTML)
Instructions: Please click on the link above and read this webpage for a definition and overview of the Cartesian product of sets.
Reading this webpage should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: Jerusalem College of Technology: Dr. Th. DanaPicard’s “The Cartesian Product of Sets”

2.3.1 The Disjoint Union and Addition
 Web Media: YouTube: American Public University’s “Disjoint Sets”
Link: YouTube: American Public University’s “Disjoint Sets” (YouTube)
Instructions: Please click on the link above and watch this brief lecture on disjoint sets.
Watching this and pausing to take notes should take less than 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: YouTube: American Public University’s “Disjoint Sets”

2.3.2 The Cartesian Product and Multiplication
 Reading: Nikos Drakos and Ross Moore’s Discrete Mathematics: “The Cartesian Product of Sets”Link: Nikos Drakos and Ross Moore’s Discrete Mathematics: “The Cartesian Product of Sets” (HTML)
Instructions: Please click on the link above and read this webpage, paying special attention to the proof of proposition 3.3.3 at the end of the page. There is a nice proof of this using characteristic functions, which you will be asked to produce later in the course.
Reading this webpage should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: Nikos Drakos and Ross Moore’s Discrete Mathematics: “The Cartesian Product of Sets”
 2.4 Counting Finite Sets

2.4.1 The Cardinality of the Power Set of a Set
 Web Media: YouTube: American Public University’s “Equivalent Sets”
Link: YouTube: American Public University’s “Equivalent Sets” (YouTube)
Instructions: Please click on the link above and watch this lecture, which discusses equivalent sets.
Watching this lecture and pausing to take notes should 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: YouTube: American Public University’s “Equivalent Sets”

2.4.2 The formula A + B = A U B + A U B
 Reading: University of Hawaii: G.N. Hile’s “Set Cardinality”
Link: University of Hawaii: G.N. Hile’s “Set Cardinality” (HTML)
Instructions: Please click on the link above and read this webpage, which demonstrates the basic inclusion/exclusion equation outlined in the title of this subunit. The examples on this webpage are especially interesting; pay attention to example 2, which is about playing cards.
Reading this webpage and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: The Saylor Foundation’s “Elementary Set Theory Homework Set”
Link: The Saylor Foundation’s “Elementary Set Theory Homework Set” (PDF)
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of elementary set theory. When you are done, check your work against the answers provided in the accompanying solutions file, the Saylor Foundation’s “Elementary Set Theory Homework Set Solutions” (PDF).
This exercise set should take between 2 and 3 hours to complete, depending on your comfort level with the material.
 Reading: University of Hawaii: G.N. Hile’s “Set Cardinality”

Unit 3: Introduction to Number Theory
This unit is primarily concerned with the set of natural numbers N = {0, 1, 2, 3, . . .}. The axiomatic approach to N will be postponed until the unit on recursion and mathematical induction. This unit will help you understand the multiplication and additive structure of N. This unit begins with integer representation: place value. This fundamental idea enables you to completely understand the algorithms we learned in elementary school for addition, subtraction, multiplication, and division of multidigit integers. The beautiful idea in the Fusing Dots paper will enable you to develop a much deeper understanding of the representation of integers and other real numbers. Then, you will learn about the multiplicative building blocks, the prime numbers. The Fundamental Theorem of Arithmetic guarantees that every positive integer greater than 1 is a prime number or can be written as a product of prime numbers in essentially one way. The Division Algorithm enables you to associate with each ordered pair of nonzero integers – a unique pair of integers – the quotient and the remainder. Another important topic is modular arithmetic. This arithmetic comes from an understanding of how remainders combine with one another under the operations of addition and multiplication. Finally, the unit discusses the Euclidean Algorithm, which provides a method for solving certain equations over the integers. Such equations with integer solutions are sometimes called Diophantine Equations.
Unit 3 Time Advisory show close
Unit 3 Learning Outcomes show close

3.1 Place Value Notation
 Reading: The Saylor Foundation: Harold Reiter’s “Fusing Dots”
Link: The Saylor Foundation: Harold Reiter’s “Fusing Dots” (PDF)
Instructions: Please click on the link above and read this essay, “Fusing Dots,” paying special attention to the exercises at the end. Please note that this reading covers all of the subunits assigned below subunit 3.1. You may find the second half of this reading very difficult. Try to read through Laurie Jarvis’ “Understanding Place Value” first (subsubunit 3.1.12) and then come back to this more challenging paper. You can access the solutions for selected problems here (PDF). Don’t worry about understanding all of the details your first time through the reading. Instead, concentrate on the material in the first five sections of the document, and then attempt to generally understand the subsequent sections on Fusing Dots. The supporting details will become more familiar as you work through the various subunits.
Reading this essay should take approximately 2 hours, and completing the exercises should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: The Saylor Foundation: Harold Reiter’s “Fusing Dots”

3.1.1 Representing Integers in Base b Notation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please read Section 2 of “Fusing Dots.”

3.1.1.1 Base 5 Notation: Doing Arithmetic in Base 5
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, work the problem that involves finding the product of two numbers both given in base 5 notation, without translating to decimal notation.

3.1.1.2 Decimal Notation
 Web Media: WiscOnline: Laurie Jarvis’ “Understanding Place Value”
Link: WiscOnline: Laurie Jarvis’ “Understanding Place Value” (Flash)
Instructions: Please click on the link above and then use the arrow keys to navigate through this slideshow presentation. This information will most likely serve as a review of place value.
Reviewing this presentation should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: WiscOnline: Laurie Jarvis’ “Understanding Place Value”

3.1.1.3 Using Repeated Subtraction
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 3 of “Fusing Dots.”

3.1.1.4 Using Repeated Division
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 4 of “Fusing Dots.”

3.1.2 Representing Rational Numbers in the Unit Interval in Base b Notation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of “Fusing Dots.”

3.1.2.1 Using Repeated Subtraction
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of “Fusing Dots.”

3.1.2.2 Using Repeated Multiplication
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of “Fusing Dots.”

3.1.2.3 Translating between Representations
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of “Fusing Dots.”

3.1.2.4 Representing Repeating Base b Numbers as Quotients
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 5 of “Fusing Dots.”
 3.1.3 Other Interesting Methods of Representation

3.1.3.1 Base phi Notation
Note: The Base number here is the irrational number phi. This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 6 of “Fusing Dots” and see problem 6 at the end of Section 6.

3.1.3.2 Fibonacci Representation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 6 of “Fusing Dots” and see problem 7 at the end of Section 6.

3.1.3.3 Cantor’s Representation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, focus on Section 6 of “Fusing Dots.” Problems 12 through 15 at the end of Section 6 all deal with Cantor’s representation, also known as factorial notation.

3.1.3.4 Base Negative 4 Notation
Note: This topic is covered by the reading assigned below subunit 3.1. In particular, please focus on Section 7 of “Fusing Dots.” Problem 2 at the end of section 10 is devoted to base negative 4 notation and arithmetic.

3.2 Prime Numbers
 Reading: The MacTutor History of Mathematics Archive: J.J. O’Connor and E.F. Robertson’s “Prime Numbers” and The University of Tennessee at Martin’s The Prime Pages: Chris K. Caldwell’s “Euclid’s Proof of the Infinitude of Primes”
Link: The MacTutor History of Mathematics Archive: J.J. O’Connor and E.F. Robertson’s “Prime Numbers” (HTML) and The University of Tennessee at Martin’s The Prime Pages: Chris K. Caldwell’s “Euclid’s Proof of the Infinitude of Primes” (HTML)
Instructions: Please click on the link above to “Prime Numbers” and read the webpage, which includes a good overview of prime numbers and also a list of unsolved problems. Pay special attention to the unsolved problems 1 and 2. Then click on the second link above and read the proof of Euclid.
Reading these webpages should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Reading: Wikipedia’s “Prime Number”Link: Wikipedia’s “Prime Number” (PDF)
Instructions: Please click on the link above and read this entry on prime numbers, which will give you an idea of the connections between number theory and other areas of mathematics.
Reading this article and taking notes should take approximately 2 hours.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0. You can find the original Wikipedia version of this article here.
 Reading: The MacTutor History of Mathematics Archive: J.J. O’Connor and E.F. Robertson’s “Prime Numbers” and The University of Tennessee at Martin’s The Prime Pages: Chris K. Caldwell’s “Euclid’s Proof of the Infinitude of Primes”

3.2.1 An Infinitude of Primes
 Reading: University of Tennessee at Martin: Chris K. Caldwell’s “Euclid’s Proof of the Infinitude of Primes”
Link: University of Tennessee at Martin: Chris K. Caldwell’s “Euclid’s Proof of the Infinitude of Primes” (HTML)
Instructions: Please click on the link above and read the entire webpage on Euclid’s proof of the infinitude of primes. Be sure you understand why the prime P is not already in the list of primes; if necessary, reread this text a few times until you have fully grasped this concept.
Reading this article should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: University of Tennessee at Martin: Chris K. Caldwell’s “Euclid’s Proof of the Infinitude of Primes”

3.2.2 Conjectures about Primes
 Reading: The University of Utah: Peter Alfeld’s “Prime Number Problems”
Link: The University of Utah: Peter Alfeld’s “Prime Number Problems” (HTML)
Instructions: Please click on the link above and read this webpage to get an idea of some of the many unsolved problems about prime numbers.
Reading this text and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The University of Utah: Peter Alfeld’s “Prime Number Problems”

3.2.2.1 The Twin Prime Conjecture
 Reading: Plus Magazine: “Mathematical Mysteries: Twin Primes”
Link: Plus Magazine: “Mathematical Mysteries: Twin Primes” (HTML)
Instructions: Please click on the link above and read this webpage. Take note of the definition of Brun’s constant. Also note that this is related to the Intel’s famous $475 million recall of Pentium chips. Please also feel free to click on the link to “Enumeration to 1e14 of the twin primes and Brun’s constant” link at the end of the webpage to read associated content.
Reading this webpage should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: Plus Magazine: “Mathematical Mysteries: Twin Primes”

3.2.2.2 Goldbach’s Conjecture
 Reading: I Programmer: Mike James’ “Goldbach Conjecture: Closer to Solved?”
Link: I Programmer: Mike James’ “Goldbach Conjecture: Closer to Solved?” (HTML)
Instructions: Please click on the link above and read this brief article to learn a about the Goldbach conjecture. Problems like this are the subject of intense work by mathematicians around the world, and progress is made nearly every year towards solving them.
Reading this article should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: I Programmer: Mike James’ “Goldbach Conjecture: Closer to Solved?”

3.2.2.3 The Riemann Hypothesis
 Reading: Wikipedia’s “Riemann Hypothesis”
Link: Wikipedia’s “Riemann Hypothesis” (PDF)
Instructions: Please click on the link above and read this article for information about Riemann Hypothesis.
Reading this article should take approximately 1 hour.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0. You can find the original Wikipedia version of this article here.
 Reading: Wikipedia’s “Riemann Hypothesis”

3.3 Fundamental Theorem of Arithmetic (FTA)
 Reading: Cut the Knot: Alexander Bogomolny’s “Euclid’s Algorithm” and “GCD and the Fundamental Theorem of Arithmetic”
Link: Cut the Knot: Alexander Bogomolny’s “Euclid’s Algorithm” (HTML) and “GCD and the Fundamental Theorem of Arithmetic” (HTML)
Instructions: Please click on the first link above, scroll down to the “Fundamental Theorem of Arithmetic” heading, and read this brief introductory information. Then, click on the second link above and read the entire webpage for information about the Fundamental Theorem of Arithmetic (FTA). Please note that we are going to postpone the proof of FTA until the end of Unit 4. This reading covers the topics outlined in subsubunits 3.3.1 and 3.3.2.
Reading these articles should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.  Web Media: Khan Academy’s “The Fundamental Theorem of Arithmetic”
Link: Khan Academy’s “The Fundamental Theorem of Arithmetic” (YouTube)
Instructions: Please click on the link above and watch this brief video, which provides an informative, though far less technical, introduction to the Fundamental Theorem of Arithmetic.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialNoDerivs 3.0 Unported License. It is attributed to the Khan Academy.
 Reading: Wikipedia’s “Fundamental Theorem of Arithmetic”
Link: Wikipedia’s “Fundamental Theorem of Arithmetic” (PDF)
Instructions: Please click on the link above and read this entire article on the fundamental theorem of arithmetic. The article may take more time to read than some others. Please note that this reading covers the topics outlined in subsubunits 3.3.1 and 3.3.2.
Reading this article should take approximately 2 hours.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0. You can find the original Wikipedia version of this article here.
 Reading: Cut the Knot: Alexander Bogomolny’s “Euclid’s Algorithm” and “GCD and the Fundamental Theorem of Arithmetic”

3.3.1 A Proof of FTA
Note: This topic is covered by the readings assigned below subunit 3.3. In particular, please focus on Bogomolny’s “Euclid’s Algorithm” reading and section 2 and 3, “Euclid’s Algorithm” and “Alternative Proof”, in the Wikipedia article.

3.3.2 Some Applications of FTA
Note: This topic is covered by the readings assigned below subunit 3.3. In particular, please focus on Section 1 “Applications” of the Wikipedia article. Nearly all the proofs of irrationality of the square root of a composite nonsquare number depend on FTA. Of course, there are also many other applications.

3.3.3 A System that Does Not Have Unique Factorization
 Reading: University of California, Berkeley: Zvezdelina StankovaFrenkel’s “Unique and Nonunique Factorization”Link: University of California, Berkeley: Zvezdelina StankovaFrenkel’s “Unique and Nonunique Factorization” (HTML)
Instructions: Please click on the link above and read this webpage. In particular, focus on the exercise in the reading. Do not be intimidated by the notation in the essay, just read it down to the part on ideals.
Reading this webpage should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: University of California, Berkeley: Zvezdelina StankovaFrenkel’s “Unique and Nonunique Factorization”

3.4 Modular Arithmetic, the Algebra of Remainders
 Web Media: YouTube: TheMathster’s “Modular Arithmetic 1”, “Modular Arithmetic 2”, “Modular Arithmetic 3”, “Modular Arithmetic 4”, “Modular Arithmetic 5”, “Modular Arithmetic 6”, and “Modular Arithmetic 7”Link: YouTube: TheMathster’s “Modular Arithmetic 1”, “Modular Arithmetic 2”, “Modular Arithmetic 3”, “Modular Arithmetic 4”, “Modular Arithmetic 5”, “Modular Arithmetic 6”, and “Modular Arithmetic 7” (YouTube)
Instructions: Please click on the links above and watch these lectures in sequential order. These videos address the concepts outlined in subsubunits 3.4.1 through 3.4.4. Then, if you chose to work through the KenKen material in subunit 1.2, go to section 9 of the paper “Using KenKen to Build Reasoning Skills” from subunit 1.2, and reread the section to recall how to use modular arithmetic as a strategy for KenKen puzzles.
Watching these lectures and rereading this section of “Using KenKen to Build Reasoning Skills” should take approximately 2 hours and 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Web Media: YouTube: TheMathster’s “Modular Arithmetic 1”, “Modular Arithmetic 2”, “Modular Arithmetic 3”, “Modular Arithmetic 4”, “Modular Arithmetic 5”, “Modular Arithmetic 6”, and “Modular Arithmetic 7”

3.4.1 Division by 3, 9, and 11
 Web Media: YouTube: Dr. James Tanton’s “Divisibility by 3 and 9: Why Do They Work?” and “Divisibility by 11”
Link: YouTube: Dr. James Tanton’s “Divisibility by 3 and 9: Why Do They Work?” (YouTube) and “Divisibility by 11” (YouTube)
Instructions: Please click on the first link above and watch this video, which will help you understand the divisibility rules for 3 and 9. Then click on the second link above and watch the entire lecture, which discusses divisibility by 11.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Web Media: YouTube: Dr. James Tanton’s “Divisibility by 3 and 9: Why Do They Work?” and “Divisibility by 11”

3.4.2 Chinese Remainder Theorem
Note: This topic is covered by the video lectures assigned below subunit 3.4. In particular, please focus on “Modular Arithmetic 3” and “Modular Arithmetic 4.”

3.4.3 Building the Field Z?
 Reading: The Saylor Foundation’s “Introduction to Modular Arithmetic, Building the Rings to Z? and Z?”
Link: The Saylor Foundation’s “Introduction to Modular Arithmetic, Building the Rings to Z? and Z?” (PDF)
Instructions: Please click on the link above and read this entire essay, paying special attention to understanding the operations ⊕ and ⊗ (read 'oplus' and 'otimes') in Z? and Z?. Then, work the problems 1 and 4 on Z??. You may check your solutions against the Saylor Foundation’s “Introduction to Modular Arithmetic, Building the Rings to Z? and Z? Solutions” (PDF).
Reading, taking notes, and completing the problems will take approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: The Saylor Foundation’s “Introduction to Modular Arithmetic, Building the Rings to Z? and Z?”

3.4.4 Fundamental Theorems of Modular Arithmetic
Note: This topic is covered by the video lectures assigned below subunit 3.4. In particular, please focus on “Modular Arithmetic 1” and “Modular Arithmetic 2.”
 3.4.5 Square Roots in Modular Arithmetic

3.4.5.1 The Addition of Remainders
 Activity: The Art of Problem Solving’s “2000 AMC 12 Problems”
Link: The Art of Problem Solving’s “2000 AMC 12 Problems” (HTML)
Instructions: Please click on the link above and try to solve the problem before checking the solution. This problem asks: what is the units’ digit of the 2012^{th} Fibonacci number? See if you can work this using your understanding that remainders work perfectly with respect to addition. After you have attempted this problem, review the solution on this webpage.
Completing this assignment should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Activity: The Art of Problem Solving’s “2000 AMC 12 Problems”

3.4.5.2 The Multiplication of Remainders
 Activity: Oracle ThinkQuest: “The Units Digit of Powers of a Number”
Link: Oracle ThinkQuest: “The Units Digit of Powers of a Number” (HTML)
Instructions: By now, the following type of problem should be familiar: what is the units’ digit of the expression 7^2012 X 13^2011? See if you can work this using your understanding that remainders work perfectly with respect to multiplication. In other words, if you know the remainder when N is divided by d, then you can find the remainder when N^3 is divided by d.
The solution to this question is mentioned below, but please only check it after you have attempted the problem. After you have completed this problem, click on the link above, and work to solve the problem on this webpage. After you have attempted the problem, click on the link to see the solution.
Solution: The solution to the initial problem mentioned above is that the remainder when N^3 is divided by d is the same as when the r^3 is divided by d, where r is the remainder when N is divided by d.
Completing this assignment should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Activity: Oracle ThinkQuest: “The Units Digit of Powers of a Number”
 3.5 Functions in Number Theory

3.5.1 The Floor and the Ceiling Functions
 Reading: The University of Western Australia: Greg Gamble’s “The Floor or Integer Part Function” and “Number Theory 1”
Link: The University of Western Australia: Greg Gamble’s “The Floor or Integer Part Function” (PDF) and “Number Theory 1” (PDF)
Instructions: Please click on the first link above and then select the hyperlink to the lecture titled “The Floor or Integer Part Function” to access the PDF. Please read the entire lecture. Then click on the second link, and select the link titled “v.3.0 numberI.pdf” under “1995 Lectures” to access the PDF. Please read the entire lecture.
Reading these lectures and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: The University of Western Australia: Greg Gamble’s “The Floor or Integer Part Function” and “Number Theory 1”

3.5.2 The Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of Two Integers
 Reading: Wikidot’s “Lecture 3: GCD and LCM”
Link: Wikidot’s “Lecture 3: GCD and LCM” (HTML)
Instructions: Please click on the link above and read this webpage, paying special attention to the relationship between the GCD and LCM.
Reading this webpage and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: Wikidot’s “Lecture 3: GCD and LCM”

3.5.3 The SigmaFunction, Summing Divisors
 Reading: Wikipedia’s “Divisor Function”Link: Wikipedia’s “Divisor Function” (PDF)
Instructions: Please click on the link above and read this article about the sum of the divisors of a number.
Reading this article and taking notes should take approximately 30 minutes.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0. You can find the original Wikipedia version of this article here.  Reading: The Saylor Foundation’s “Just the Factors Ma’am” (PDF)
Link: The Saylor Foundation: Harold Reiter’s “Just the Factors, Ma’am” (PDF)
Instructions: Please click on the link above and read this article, paying special attention to Sections 3 and 4, where you will learn about geometry of the divisors of an integer. Complete the problems on the document above, and then check your answers against the Saylor Foundation’s “Just the Factors, Ma’am Solutions” (PDF). The topics outlined for subunit 3.6, including subsubunits 3.6.1 through 3.6.3, are covered by these sections of reading.
Reading this article and taking notes should take about 3 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages linked above.
 Reading: Wikipedia’s “Divisor Function”
 3.6 The Lattice of Divisors of an Integer

3.6.1 Using GCD and LCM to Define a Lattice
Note: This topic is covered by the reading, “Just the Factors Ma’am,” assigned below Subunit 3.5.4. In particular, review Sections 3 and 4.

3.6.2 Properties of the Lattice Dn of Divisors of n
Note: This topic is covered by the reading, “Just the Factors Ma’am,” assigned below Subunit 3.5.4. In particular, review Sections 3 and 4.

3.6.3 Counting the Divisors of a Number
Note: This topic is covered by the reading, “Just the Factors Ma’am,” assigned below Subunit 3.5.4. In particular, review Sections 3 and 4.

3.7 The Euclidean Algorithm
 Reading: Cut the Knot: Alexander Bogomolny’s “Euclid’s Algorithm,” “An Interactive Illustration,” “Euclid’s Game,” “Binary Euclid’s Algorithm,” and “GCD and the Fundamental Theorem of Arithmetic”
Link: Cut the Knot: Alexander Bogomolny’s “Euclid’s Algorithm” (HTML), “An Interactive Illustration” (HTML and Java), “Euclid’s Game” (HTML and Java), “Binary Euclid’s Algorithm” (HTML), and “GCD and the Fundamental Theorem of Arithmetic” (HTML)
Instructions: Please click on the links above and read each of these webpages for information on the Euclidean Algorithm.
Reading these webpages should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages linked above.  Reading: PlanetMath.org: Michael Slone, Kimberly Lloyd, and Chi Woo’s “Proof of the Fundamental Theorem of Arithmetic”
Link: PlanetMath.org: Michael Slone, Kimberly Lloyd, and Chi Woo’s “Proof of the Fundamental Theorem of Arithmetic” (HTML)
Instructions: Please click on the link above and read this proof of the FTA.
Reading this proof should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.  Reading: MIT OpenCourseWare: Dr. Srini Devadas and Dr. Eric Lehman’s “Lecture 6: Number Theory I”
Link: MIT OpenCourseWare: Dr. Srini Devadas and Dr. Eric Lehman’s “Lecture 6: Number Theory I” (PDF)
Instructions: Please click on the link above and then select the “PDF” link next to “Lecture 6: Number Theory I” to download the file. Please read this lecture, which provides an introduction to decanting (see the Die Hard example on pages 57) and the Euclidean algorithm.
Reading this lecture and taking notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.  Reading: The Saylor Foundation: Harold Reiter’s “Decanting”
Link: The Saylor Foundation: Harold Reiter’s “Decanting” (PDF)
Instructions: Please click on the link above to download the PDF version of the text and read this paper. This is an easier version of this technique. Solutions to selected problems can be found here (PDF).
Reading this paper should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Cut the Knot: Alexander Bogomolny’s “Euclid’s Algorithm,” “An Interactive Illustration,” “Euclid’s Game,” “Binary Euclid’s Algorithm,” and “GCD and the Fundamental Theorem of Arithmetic”

3.7.1 Another Look at the Division Algorithm
 Activity: The University of Western Australia: Greg Gamble’s “Number Theory 1”
Link: The University of Western Australia: Greg Gamble’s “Number Theory 1” (PDF)
Instructions: Note that you have already read this essay in subsubunit 3.5.1. Please click on the link above, and select the “v.3.0 numberI.pdf” link under “1995 Lectures” to download the PDF. Review the section on “Division Algorithm” again, and then attempt the 3 sample problems in the lecture.
Reviewing this section and attempting the sample problems should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Activity: The University of Western Australia: Greg Gamble’s “Number Theory 1”

3.7.2 Solving Ax + By = C over the Integers
 Reading: DavData’s “Solving Ax + By = C”
Link: DavData’s “Solving Ax + By = C” (HTML)
Instructions: Please click on the link above and read the brief text on solving Ax + By = C.
Reading this text should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.  Reading: Carnegie Mellon University: Victor Adamchick’s Concepts of Mathematics: “Integer Divisibility”Link: Carnegie Mellon University: Victor Adamchick’s Concepts of Mathematics: “Integer Divisibility” (PDF)
Instructions: Please click on the link above and then click on “Schedule” on the left side of the webpage. Scroll down the webpage to the section “Integer Divisibility,” and select the “Linear Diophantine Equations” link to download Lecture 5 as a PDF. Please read this studentfriendly version of the lecture, which discusses solving an integer divisibility type of equation. You should focus on solving linear Diophantine equations. In particular, you should be able to find a single solution and then generate all solutions from the one you found.
Reading this lecture should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.
 Reading: DavData’s “Solving Ax + By = C”

Unit 4: Rational Numbers
In this unit, you will learn to prove some basic properties of rational numbers. For example, the set of rational numbers is dense in the set of real numbers. That means that strictly between any two real numbers, you can always find a rational number. The distinction between a fraction and a rational number will also be discussed. There is an easy way to tell whether a number given in decimal form is rational: if the digits of the representation regularly repeat in blocks, then the number is rational. If this is the case, you can find a pair of integers whose quotient is the given decimal. The unit discusses the mediant of a pair of rational fractions, and why the mediant does not depend on the values of its components, but instead on the way they are represented.
Unit 4 Time Advisory show close
Unit 4 Learning Outcomes show close
 Reading: The Saylor Foundation: Harold Reiter’s “Fractions”
Link: The Saylor Foundation: Harold Reiter’s “Fractions” (PDF)
Instructions: Please click on the link above and read this article. Pay special attention to the five problems on rational numbers at the beginning of the paper. Problem 10 will enable you to appreciate the different between the value of a number and the numeral used to express it. Pay special attention to Simpson’s Paradox in the paper. Try the practice problems at the end of the reading. After you have attempted these problems, please check the solutions against the Saylor Foundation’s “Fractions Solutions” (PDF).
Please note that this reading covers the topics outlined for subunit 4.1, as well as inclusive subsubunits 4.1.1 through 4.1.3, and subunit 4.2, as well as subsubunits 4.2.1 through 4.2.3.
Reading this article and taking notes should take approximately 3 hours. You should also spend approximately 3 hours working on the problems provided in the text.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: The Saylor Foundation: Harold Reiter’s “Fractions”
 4.1 Fractions and Rational Numbers Are Not the Same

4.1.1 Numbers and Their Numerals
Note: This topic is covered by the reading assigned below the Unit 4 introduction. In particular, please focus on first 2 paragraphs after the text “Fraction versus rational number,” especially where numbers and numerals are in bold font as this will help you understand the relationship between the two.

4.1.2 The Mediant of Two Fractions
Note: This topic is covered by the reading assigned below the Unit 4 introduction. Make sure to work on problem 10 in the essay to help you better understand the mediant of two fractions.

4.1.3 Building New Rational Numbers from Given Ones
Note: This topic is covered by the reading assigned below the Unit 4 introduction. In particular, pay attention to problems 13 under “Rational Numbers.”

4.2 Representing Rational Numbers as Decimals
 Web Media: YouTube: MrCaryMath’s “Rational vs. Irrational Numbers”
Link: YouTube: MrCaryMath’s “Rational vs. Irrational Numbers” (YouTube)
Instructions: Please click on the link above and watch this video to learn about the difference between rational and irrational numbers.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: YouTube: MrCaryMath’s “Rational vs. Irrational Numbers”

4.2.1 Every Rational Number Has a Nice Decimal Representation
Note: This topic is covered by the reading assigned below the Unit 4 introduction. In particular, focus on problem 4 to learn more about decimal representation.

4.2.2 Turning Repeating Decimals into Fractions
 Web Media: YouTube: Examplayer’s “Recurring Decimals to Fractions Part 1 of 2”Link: YouTube: Examplayer’s “Recurring Decimals to Fractions Part 1 of 2” (YouTube)
Instructions: Please click on the link above and watch this video on the conversion of repeating decimals into fractions of the form a/b with a and b integers.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Web Media: YouTube: Examplayer’s “Recurring Decimals to Fractions Part 1 of 2”

4.2.3 Density of Rational Numbers
Note: This topic is covered by the reading assigned below the Unit 4 introduction. An interesting property of the rational numbers is that between any two rational numbers we insert another rational number. This property is called density. We say the rational numbers are dense in the real numbers. The same property holds for irrational numbers. Try proving these propositions. Problem 6 in the essay discusses this property.
This topic is also covered in the reading assigned below subunit 4.3.2. In particular, focus on Section 6, “Density of Rational Numbers.”  4.3 The Existence of Irrational Numbers

4.3.1 ?2, ?3, and ?6 Are All Irrational Numbers
 Web Media: YouTube: University of Missouri – Kansas City’s “Proof: The Square Root of 2 Is Irrational” and djcoopdawg’s “Proof: Square Root of 3 Is Irrational”Link: YouTube: University of Missouri – Kansas City’s “Proof: The Square Root of 2 Is Irrational” (YouTube) and djcoopdawg’s “Proof: Square Root of 3 Is Irrational” (YouTube)
Instructions: Please click on the first link above and watch the video, which shows a proof of the irrationality of the square root of 2. Can you see how to use these ideas to prove that the square root of 3 and of 6 are also irrational?
Next, click on the second link above, and watch the video, which shows a proof that the square root of 3 is irrational. After watching this video, do you think you could prove how the square root of 6 is also irrational?
Watching these videos, pausing to take notes, and answering the questions above should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.
 Web Media: YouTube: University of Missouri – Kansas City’s “Proof: The Square Root of 2 Is Irrational” and djcoopdawg’s “Proof: Square Root of 3 Is Irrational”

4.3.2 Density of Irrational Numbers
 Reading: New York University: Lawrence Tsang’s “Real Numbers”
Link: New York University: Lawrence Tsang’s “Real Numbers” (PDF)
Instructions: Please click on the link above to access Professor Tsang’s webpage. Select the link to “HW1” to download the PDF. Please read pages 7 through 9, from “Density of Rational Numbers” through “Density of Irrational Numbers.” Please note that this reading also covers the topic of Density of Rational Numbers outlined for subsubunit 4.2.3.
Reading this article, taking notes, and reviewing the proofs several times should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: New York University: Lawrence Tsang’s “Real Numbers”

4.3.3 Algebraic versus Transcendental Numbers
 Reading: Library of Halexandria: Dan Sewell Ward’s “Transcendental Numbers”
Link: Library of Halexandria: Dan Sewell Ward’s “Transcendental Numbers” (HTML)
Instructions: Please click on the link above and read the webpage, which describes transcendental numbers. Note that both pi and e are transcendental.
Reading this webpage should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: Library of Halexandria: Dan Sewell Ward’s “Transcendental Numbers”

4.4 The Field of Rational Numbers
 Reading: New York University: Lawrence Tsang’s “Real Numbers”
Link: New York University: Lawrence Tsang’s “Real Numbers” (PDF)
Instructions: Please click on the link above to access Professor Tsang’s webpage. Select the link to “HW1” to download the PDF. Read pages 17 of the text. The first 6 pages discuss the field and order axioms for real numbers. The Completeness Axiom on page 6 is what distinguishes the rational numbers from the real numbers – the latter is COMPLETE, while the former is not. This resource covers the topics for subsubunits 4.4.1 through 4.4.3.
Reading this article and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: New York University: Lawrence Tsang’s “Real Numbers”

4.4.1 Axioms for a Field
Note: This topic is covered by the reading assigned below subunit 4.4. In particular, please focus on Section 4.1 “Field Axioms” on pages 2 and 3.

4.4.2 Proofs of Properties
Note: This topic is covered by the reading assigned below subunit 4.4. For example, look at the proofs of theorems 4.9 and 4.12.

4.4.3 Order and Incompleteness
Note: This topic is covered by the reading assigned below subunit 4.4. In particular, focus on Section 4.2 “Ordering Axiom” and Section 5 “WellOrdering Principle.”

Unit 5: Mathematical Induction
In this unit, you will prove propositions about an infinite set of positive integers. Mathematical induction is a technique used to formulate all such proofs. The term recursion refers to a method of defining sequences of numbers, functions, and other objects. The term mathematical induction refers to a method of proving properties of such recursively defined objects.
Unit 5 Time Advisory show close
Unit 5 Learning Outcomes show close
 5.1 Mathematical Induction Is Equivalent to the WellOrdering Property of N

5.1.1 The Well Ordering Property of N
 Web Media: YouTube: The Mathsters’ “Induction 10” and “Induction 11”
Link: YouTube: The Mathsters’ “Induction 10” (YouTube) and “Induction 11” (YouTube)
Instructions: Please click on the links above and watch these brief videos. These videos provide informative discussions as to why the wellordering principle of the natural numbers implies the principle of mathematical induction.
Watching these videos and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: YouTube: The Mathsters’ “Induction 10” and “Induction 11”

5.1.2 Strong and Weak Induction
 Web Media: YouTube: isallaboutmath’s “Mathematical Induction (Part III)”
Link: YouTube: Is All About Math’s “Mathematical Induction (Part III)” (YouTube)
Instructions: Please click on the link above and watch the brief video, which provides an informative discussion on the principle of mathematical induction and the wellordering principle of the natural numbers. It specifically addresses the notion of strong mathematical induction.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Reading: YouTube: The Mathsters’ “Induction 12”
Link: YouTube: The Mathsters’ “Induction 12” (YouTube)
Instructions: Please click on the link above and watch the brief video, which discusses why the principle of strong mathematical induction implies the wellordering principle of the natural numbers.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: YouTube: isallaboutmath’s “Mathematical Induction (Part III)”
 5.2 Proofs of Summations and Products

5.2.1 Sums and Products
 Web Media: Khan Academy’s “Proof by Induction”
Link: Khan Academy’s “Proof by Induction” (YouTube)
Instructions: Please click on the link above and watch the brief video, which examines how the principle of mathematical induction is used to prove statements involving sums and products of integers.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialNoDerivs 3.0 Unported License. It is attributed to the Khan Academy.
 Web Media: Khan Academy’s “Proof by Induction”

5.2.2 Divisibility
 Web Media: YouTube: David Metzler’s “Mathematical Induction Again (Divisibility Proof)”Link: YouTube: David Metzler’s “Mathematical Induction Again (Divisibility Proof)” (YouTube)Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and watch the brief video, which illustrates how the principle of strong mathematical induction can prove a statement about divisibility of natural numbers.
Watching this video and pausing to take notes should take approximately 15 minutes.
 Web Media: YouTube: David Metzler’s “Mathematical Induction Again (Divisibility Proof)”

5.2.3 Recursively Defined Functions
 Web Media: YouTube: Math Doctor Bob’s “Example of Proof by Induction 4 – Hard Inequality”
Link: YouTube: Math Doctor Bob’s “Example of Proof by Induction 4 – Hard Inequality” (YouTube)
Instructions: Please click on the link above and watch the brief video, which illustrates using the principle of strong mathematical induction to prove a statement about divisibility of natural numbers.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Reading: Old Dominion University: Shunichi Toida’s “Recursive Definition” and “Recursive Definition of Function”
Link: Old Dominion University: Shunichi Toida’s “Recursive Definition” (HTML and Java) and “Recursive Definition of Function” (HTML and Java)
Instructions: Please click on the links above and read both essays. Notice the similarities between using recursion to define sets and using recursion to define functions. Then answer the four questions at the end of the first essay.
In this type of definition, first a collection of elements to be included initially in the set is specified. These elements can be viewed as the seeds of the set being defined. Next, the rules to be used to generate elements of the set from elements already known to be in the set (initially the seeds) are given. These rules provide a method to construct the set, element by element, starting with the seeds. These rules can also be used to test elements for the membership in the set.
Reading these essays, taking notes, and completing the assignment should take about 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.
 Web Media: YouTube: Math Doctor Bob’s “Example of Proof by Induction 4 – Hard Inequality”

Unit 5 Assessment
 Assessment: The Saylor Foundation’s “Mathematical Induction Homework Set”
Link: The Saylor Foundation’s “Mathematical Induction Homework Set” (PDF)
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topic of proofs using mathematical induction. When you are done, check your work against those provided in the accompanying solutions file, The Saylor Foundation’s “Mathematical Induction Homework Set Solutions” (PDF).
This exercise set should take between 1 and 2 hours to complete, depending on your comfort level with the material.
 Assessment: The Saylor Foundation’s “Mathematical Induction Homework Set”

Unit 6: Relations and Functions
In this unit, you will learn about binary relations from a set A to a set B. Some of these relations are functions from A to B. Restricting our attention to relations from a set A to the set A, this unit discusses the properties of reflexivity(R), symmetry(S), antisymmetry(A), and transitivity(T). Relations that satisfy R, S, and T are called equivalence relations, and those satisfying R, A, and T are called partial orderings.
Unit 6 Time Advisory show close
Unit 6 Learning Outcomes show close

6.1 Binary Relations on a Set A
 Assessment: The Saylor Foundation’s “Relations Homework Set”
Link: The Saylor Foundation’s “Relations Homework Set” (PDF)
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the properties of relations and interrelationships among them, as well as specific examples of relations. When you are done, check your work against the answers provided in the accompanying solutions file, The Saylor Foundation’s “Relations Homework Set Solutions” (PDF).
This exercise set should take between 1 and 2 hours to complete, depending on your comfort level with the material.  Web Media: YouTube: MathDoctorBob’s “Binary Relations”
Link: YouTube: MathDoctorBob’s “Binary Relations” (YouTube)
Instructions: Please click on the link above and watch the video. It may be worth spending some time watching this video twice. Note that the lecturer spends some time discussing the definitions of the properties below for subsubunits 6.1.1 through 6.1.5. The examples he provides exhibit several properties. These are the defining properties of an equivalence relation (see subunit 6.4) and Partial Ordering (see subunit 6.5). Note that this resource covers the topics outlined for subsubunits 6.1.1 through 6.1.6.
Watching this video twice and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Assessment: The Saylor Foundation’s “Relations Homework Set”

6.1.1 Symmetry S
Note: This topic is covered by the video lecture assigned below subunit 6.1.

6.1.2 Reflexivity R
Note: This topic is covered by the video lecture assigned below subunit 6.1.

6.1.3 Antisymmetry AS
Note: This topic is covered by the video lecture assigned below subunit 6.1.

6.1.4 Transitivity T
Note: This topic is covered by the video lecture assigned below subunit 6.1.
 6.2 Binary Relations from A to B

6.2.1 Relations that Are Functions
 Web Media: Khan Academy’s “Relations and Functions”
Link: Khan Academy’s “Relations and Functions” (YouTube)
Instructions: Please click on the link above and watch the brief video, which illustrates the notions of relations and functions. This video also provides examples of relations that are functions and some that are not.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialNoDerivs 3.0 Unported License. It is attributed to the Khan Academy.
 Web Media: Khan Academy’s “Relations and Functions”

6.2.2 Injections
 Web Media: Khan Academy’s “Surjective (Onto) and Injective (OnetoOne) Functions,” “Relating Invertibility to Being Onto and OnetoOne,” and “Proof: Invertibility Implies a Unique Solution to f(x) = y”
Link: Khan Academy’s “Surjective (Onto) and Injective (OnetoOne) Functions” (YouTube), “Relating Invertibility to Being Onto and OnetoOne” (YouTube), and “Proof: Invertibility Implies a Unique Solution to f(x) = y” (YouTube)
Instructions: Please click on the first link above to watch the video about injections (1 to 1 functions) and surjections (onto functions). Then click on the second link to watch the video, which will show the relationship between injections and functions that have an inverse. Finally, click on the link and watch the last lecture.
Watching these videos should take approximately 30 minutes.
Terms of Use: These videos are licensed under a Creative Commons AttributionNonCommercialNoDerivs 3.0 Unported License. They are attributed to the Khan Academy.
 Web Media: Khan Academy’s “Surjective (Onto) and Injective (OnetoOne) Functions,” “Relating Invertibility to Being Onto and OnetoOne,” and “Proof: Invertibility Implies a Unique Solution to f(x) = y”

6.2.3 Surjections
Note: This topic is covered by the video assigned below subunit 6.2.2. A surjective function is one for which every element in the codomain is mapped to by an element in the domain. For such functions, the codomain and range are equal.

6.2.4 Bijections
Note: This topic is covered by the video assigned below subunit 6.2.2. A bijection is a function that is both onetoone and onto.
 Reading: The Saylor Foundation’s “Elementary Functions and Equivalence Relations Homework Set”
Link: The Saylor Foundation’s “Elementary Functions and Equivalence Relations Homework Set” (PDF)
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of verifying properties of given functions and equivalence relations, determining if relations are equivalence relations, and commenting on the structure of a relation by using equivalence classes. When you are done, check your work against the solutions provided in the accompanying solutions file, the Saylor Foundation’s “Elementary Functions and Equivalence Relations Homework Set – Solutions” (PDF).
This assessment should take between 2 and 3 hours to complete, depending on your comfort level with the material.
 Reading: The Saylor Foundation’s “Elementary Functions and Equivalence Relations Homework Set”

6.3 Equivalence Relations
 Web Media: YouTube: MathDoctorBob’s “Binary Relations”
Link: YouTube: MathDoctorBob’s “Binary Relations” (YouTube)
Instructions: Please click on the link above and watch the last 10 minutes of this video again. It is especially important that you understand the relationship between an equivalence relation and the partition it induces.
Reviewing this section of the lecture and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Reading: Old Dominion University: Shunichi Toida’s “Equivalence Relations”
Link: Old Dominion University: Shunichi Toida’s “Equivalence Relations” (HTML and Java)
Instructions: Please click on the link above, and read the entire webpage on equivalence relations. Then, answer the four questions at the bottom of the webpage.
Reading and answering the questions should take about 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Web Media: YouTube: MathDoctorBob’s “Binary Relations”

6.4 Partial Orderings
 Web Media: MathVids: ArsDigita’s “Lecture 17: Equivalence Relations and Partial Orders”
Link: MathVids: ArsDigita’s “Lecture 17: Equivalence Relations and Partial Orders” (Flash)
Instructions: Please click on the link above and watch the entire lecture. Note that you may also click on the link for a PDF of lecture notes as well as a problem set that corresponds to the lecture.
Watching this lecture, reviewing the lecture notes, and practicing the problem sets should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: MathVids: ArsDigita’s “Lecture 17: Equivalence Relations and Partial Orders”

Unit 7: Sets, Part II
In this unit, you will study cardinality. One startling realization is that not all infinite sets are the same size. In fact, there are many different size infinite sets. This can be made perfectly understandable to you at this stage of the course. In Unit 7.4.3, section (d)iii, you learned about bijections from set A to set B. If two sets A and B have a bijection between them, they are said to be equinumerous. It turns out that the relation equinumerous is an equivalence relation on the collection of all subsets of the real numbers (in fact on any set of sets). The equivalence classes (the cells) of this relation are called cardinalities.
Unit 7 Time Advisory show close
Unit 7 Learning Outcomes show close
 Web Media: YouTube: Jeremy Lash, Matt Cerny, Michael Leahy, and Sumedha Pramod’s “Cardinality”
Link: YouTube: Jeremy Lash, Matt Cerny, Michael Leahy, and Sumedha Pramod’s “Cardinality” (YouTube)
Instructions: Please click on the link above and watch this video. Make sure you understand how two sets A and B can be equinumerous.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.  Web Media: YouTube: American Public University’s “Equivalent Sets”, “Infinite Sets and Cardinality”, and “Subset and Proper Subset”
Link: YouTube: American Public University’s “Equivalent Sets” (YouTube), “Infinite Sets and Cardinality” (YouTube), and “Subset and Proper Subset” (YouTube)
Instructions: Please click on the links above and watch these lectures to continue your studies on sets.
Watching these lectures and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Web Media: YouTube: Jeremy Lash, Matt Cerny, Michael Leahy, and Sumedha Pramod’s “Cardinality”
 7.1 Cantor Diagonalization Theorem: The Existence of Uncountable Sets of Real Numbers

7.1.1 Proof of the Theorem
 Reading: Bellevue College: “Cantor’s Diagonalization Theorem”
Link: Bellevue College: “Cantor’s Diagonalization Theorem” (HTML)
Instructions: Please click on the link above and study the proof of Cantor’s Theorem. Even though the proof is only one page, this idea is new to you, and therefore is likely to be harder to understand; thus, you should take your time studying this proof carefully.
Studying this proof should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.
 Web Media: YouTube: University of Missouri – Kansas City’s “Proof: There Are More Real Numbers than Natural Numbers” and Minute Physics’ “How to Count Infinity”
Link: YouTube: University of Missouri – Kansas City’s “Proof: There Are More Real Numbers than Natural Numbers” (YouTube) and Minute Physics’ “How to Count Infinity” (YouTube)
Instructions: Please click on the first link above and watch this brief video to supplement the written proof of Cantor’s Theorem. Then click on the second link and watch the brief video about counting finite sets and Cantor’s Diagonalization Theorem.
Watching these videos and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: Bellevue College: “Cantor’s Diagonalization Theorem”

7.1.2 Even the Cantor Set Is Uncountable, the Base3 Connection with the Cantor Ternary Set
 Web Media: YouTube: ProfessorElvisZap’s “The Cantor Set Is Uncountable”
Link: YouTube: ProfessorElvisZap’s “The Cantor Set Is Uncountable” (YouTube)
Instructions: Please click on the link above and watch the video to see a proof that the Cantor middle third set is uncountable.
Watching this lecture and taking notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Web Media: YouTube: ProfessorElvisZap’s “The Cantor Set Is Uncountable”

7.1.3 Other Examples of Uncountable Subsets of R
 Reading: Brown University: Rich Schwartz’s “Countable and Uncountable Sets”
Link: Brown University: Rich Schwartz’s “Countable and Uncountable Sets” (PDF)
Instructions: Please click on the link above and scroll down to “Handout 8.” Select the PDF link to download the file. Read this entire document to learn about countable and uncountable sets. Focus on the several examples of uncountable subsets of R. Please note that this reading also covers topics outlined in subunit 7.2, including subsubunits 7.2.1 and 7.2.2.
Reading this handout and taking notes should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.  Assessment: The Saylor Foundation’s “Functional Properties Homework Set”Link: The Saylor Foundation’s “Functional Properties Homework Set” (PDF)
Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of functional properties involving images and inverse images of sets as well as computing images and inverse images of sets. When you are done, check your work against the solutions provided in the accompanying solutions file, the Saylor Foundation’s “Functional Properties Homework Set – Solutions” (PDF).This assessment should take between 2 and 3 hours to complete, depending on your comfort level with the material.  Assessment: The Saylor Foundation’s “Cardinality Homework Set”Link: The Saylor Foundation’s “Cardinality Homework Set” (PDF)Instructions: Please click on the link above, and complete the following problems, showing all work. These problems cover the topics of recognizing cardinality properties, determining if sets are countable or uncountable, and determining whether two sets are equivalent. When you are done, check your work against the solutions provided in the accompanying solutions file, the Saylor Foundation’s “Cardinality Homework Set – Solutions” (PDF).This assessment should take between 2 and 3 hours to complete, depending on your comfort level with the material.
 Reading: Brown University: Rich Schwartz’s “Countable and Uncountable Sets”

7.2 The Rational Numbers Are Countable
 Web Media: YouTube: University of Missouri – Kansas City’s “Proof: There Are the Same Number of Rational Numbers as Natural Numbers”
Link: YouTube: University of Missouri – Kansas City’s “Proof: There Are the Same Number of Rational Numbers as Natural Numbers” (YouTube)
Instructions: Please click on the link above and watch the video to see a proof that rational numbers are countable.
Watching this lecture and taking notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Web Media: YouTube: University of Missouri – Kansas City’s “Proof: There Are the Same Number of Rational Numbers as Natural Numbers”

7.2.1 The Proof
 Reading: Theorem of the Week: “Theorem 18: The Rational Numbers Are Countable”
Link: Theorem of the Week: “Theorem 18: The Rational Numbers Are Countable” (HTML)
Instructions: Please click on the link above and study the proof on this webpage, which shows that rational numbers are countable. Note that information on this topic is also found in the reading assigned below subsubunit 7.1.3.
Reading this webpage, taking notes, and studying the proof should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: Theorem of the Week: “Theorem 18: The Rational Numbers Are Countable”

7.2.2 The Algebraic Numbers Are Countable
 Reading: Abstract Nonsense: Alex Youcis’ “Algebraic Numbers Are Countable”
Link: Abstract Nonsense: Alex Youcis’ “Algebraic Numbers Are Countable” (HTML)
Instructions: Please click on the link above and study the proof, which demonstrates that algebraic numbers are countable.
Reading this proof should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Reading: The MacTutor History of Mathematics Archive: John O’Connor’s “Infinity and Infinites”
Link: The MacTutor History of Mathematics Archive: John O’Connor’s “Infinity and Infinites” (HTML)
Instructions: Please click on the link above and read only this webpage; there is no need to click on the “next” or “previous” buttons at this time. Most of these topics have already been covered in the videos for Unit 7, so some of this will be a review.
Reading this webpage should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: Abstract Nonsense: Alex Youcis’ “Algebraic Numbers Are Countable”

7.3 Other Bijections
 Reading: Florida State University: Dr. Penelope Kirby’s “Course Notes 4.2: Property of Functions”
Link: Florida State University: Dr. Penelope Kirby’s “Course Notes 4.2: Property of Functions” (PDF)
Instructions: Please click on the link above and select the link to “Course Notes 4.2: Property of Functions” to download the PDF. Please read the paper, paying special attention to examples 2.2.5, 2.2.6, and two functions a^{x} and log_{a}x in the paragraph above example 2.7.1. Please note that this resource also covers the topic outlined in subsubunit 7.3.1.
Reading this paper should take about 1 hour.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Reading: Florida State University: Dr. Penelope Kirby’s “Course Notes 4.2: Property of Functions”

7.3.1 Bijections on Sets of Real Numbers
Note: This topic is covered in Dr. Kirby’s reading assigned below subunit 7.3.

7.3.2 The Cube and the Interval Are Equinumerous
 Reading: The Cube and the Interval Are Equinumerous
The Cube and the Interval Are Equinumerous
The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.
 Reading: The Cube and the Interval Are Equinumerous

Unit 8: Combinatorics
In this unit, you will learn to count. That is, you will learn to classify the objects of a set in such a way that one of several principles applies.
Unit 8 Time Advisory show close
Unit 8 Learning Outcomes show close

8.1 Counting Problems as Sampling Problems with Conditions on the Structure of the Sample
 Lecture: YouTube: AjarnJJ’s “Permutations and Combinations” and Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning: Professor Kamala Krithivasan’s “Lecture 28: Permutations and Combinations”
Link: YouTube: AjarnJJ’s “Permutations and Combinations” (YouTube) and Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning: Professor Kamala Krithivasan’s “Lecture 28: Permutations and Combinations” (YouTube)
Instructions: Please click on the first link above and watch this brief video for an overview of the development of the formulas for the number of permutations and the number of combinations of n objects. For a much more elaborate introduction to counting, click on the second link above, and watch the lecture. This resource covers the topics outlined in subsubunits 8.1.2 and 8.1.4 below.
Watching these lectures and pausing to take notes should take approximately 1 hour and 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.  Reading: The Saylor Foundation: Harold Reiter’s “Counting”
Link: The Saylor Foundation: Harold Reiter’s “Counting” (PDF)
Instructions: Please click on the link above and read the PDF. Attempt problems 120, starting on page 3. Once you have attempted these problems, check your solutions at the Saylor Foundation’s “Counting Solutions” (PDF). Please note that this reading and these exercises cover the topics outlined in subsubunits 8.1.1 through 8.1.4.
Reading this essay and working on these problems should take approximately 5 hours.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Lecture: YouTube: AjarnJJ’s “Permutations and Combinations” and Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning: Professor Kamala Krithivasan’s “Lecture 28: Permutations and Combinations”

8.1.1 Sampling with Replacement, Order Matters
Note: This topic is covered by the reading assigned below subunit 8.1.

8.1.2 Sampling without Replacement, Order Matters
Note: This topic is covered by the reading assigned below subunit 8.1.

8.1.3 Sampling with Replacement, Order Does Not Matter
Note: This topic is covered by the reading assigned below subunit 8.1.

8.1.4 Sampling without Replacement, Order Does Not Matter
Note: This topic is covered by the reading assigned below subunit 8.1.

8.2 The InclusionExclusion Principle
 Web Media: YouTube: coursehack's “Discrete Math: Tutorial 12Inclusion/Exclusion Principle” and “Discrete Math: Tutorial 13: Example 1 Inclusion/Exclusion”
Link: YouTube: coursehack’s “Discrete Math: Tutorial 12Inclusion/Exclusion Principle” (YouTube) and “Discrete Math: Tutorial 13: Example 1 Inclusion/Exclusion” (YouTube)
Instructions: Please click on the first link above and watch the video for an introduction to the principle. Then, follow this up by clicking on the second link and watching an example of the principle. Notice that the problem is about As, Bs, and Cs, not As, Bs, and Os as the teacher describes at the start.
Watching these videos and taking notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.
 Web Media: YouTube: coursehack's “Discrete Math: Tutorial 12Inclusion/Exclusion Principle” and “Discrete Math: Tutorial 13: Example 1 Inclusion/Exclusion”

8.2.1 The Case with Just Two Sets
 Web Media: YouTube: Brian Veitch’s “Formula for the Union of Sets – Two Sets and Three Sets (Part 3 – Set Series)”
Link: YouTube: Brian Veitch’s “Formula for the Union of Sets – Two Sets and Three Sets (Part 3 – Set Series)” (YouTube)
Instructions: Please click on the link above and watch this brief video, which provides an informative illustration of the addition formula for the cardinality of the union of two and three sets.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: YouTube: Brian Veitch’s “Formula for the Union of Sets – Two Sets and Three Sets (Part 3 – Set Series)”

8.2.2 The Proof
 Reading: Wikipedia’s “InclusionExclusion Principle”
Link: Wikipedia’s “InclusionExclusion Principle” (PDF)
Instructions: Please click on the link above and read this article. This resource also covers the topics outlined in subsubunits 8.2.3 and 8.2.4.
Reading this article and taking notes should take approximately 1 hour.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0. You can find the original Wikipedia version of this article here.
 Reading: Wikipedia’s “InclusionExclusion Principle”

8.2.3 Other Examples
 Web Media: YouTube: Course Shack’s “Discrete Math 2: Tutorial 13 – Example 1: Inclusion/Exclusion” and “Discrete Math 2: Tutorial 14 – Example 2: Inclusion/Exclusion”
Link: YouTube: Course Shack’s “Discrete Math 2: Tutorial 13 – Example 1: Inclusion/Exclusion” (YouTube) and “Discrete Math 2: Tutorial 14 – Example 2: Inclusion/Exclusion” (YouTube)
Instructions: Please click on the links above and watch these brief videos, which provide informative illustrations of the use of the inclusionexclusion formula.
Watching this video and pausing to take notes should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: YouTube: Course Shack’s “Discrete Math 2: Tutorial 13 – Example 1: Inclusion/Exclusion” and “Discrete Math 2: Tutorial 14 – Example 2: Inclusion/Exclusion”

8.3 The PigeonHole Principle (PHP)
 Web Media: YouTube: Dr. James Tanton’s “PigeonHole Principle”
Link: YouTube: Dr. James Tanton’s “PigeonHole Principle” (YouTube)
Instructions: Please click on the link above and watch Dr. Tanton’s introduction to the pigeonhole principle. This video also covers the topic assigned below subsubunit 8.3.1.
Watching this lecture and taking notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.
 Web Media: YouTube: Dr. James Tanton’s “PigeonHole Principle”

8.3.1 The Standard Principle
Note: This topic is covered, in part, by the video assigned below subunit 8.3.
 Lecture: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL)’s “Lecture 27 Pigeon Hole Principle”
Link: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL)’s “Lecture 27 Pigeon Hole Principle” (YouTube)
Instructions: Please click on the link above and watch the brief video, which provides a careful introduction to the pigeonhole principle and provides several examples.
Watching this video and pausing to take notes should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Lecture: YouTube: Indian Institute of Technology Madras’ National Programme on Technology Enhanced Learning (NPTEL)’s “Lecture 27 Pigeon Hole Principle”

8.3.2 Using the PHP Idea in Other Settings
 Web Media: YouTube: Mr. T’s Math Videos’ “Basic PigeonHole Principle Problems”
Link: YouTube: Mr. T’s Math Videos’ “Basic PigeonHole Principle Problems” (YouTube)
Instructions: Please click on the link above and watch the video, which provides some very elementary applications of the pigeonhole principle.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Web Media: YouTube: MathXpress’ “Pigeonhole Principle Problem 3 – Divisibility and Modular Arithmetic” and “Pigeonhole Principle Problem 2 – Correct Operations”
Link: YouTube: MathXpress’ “Pigeonhole Principle Problem 3 – Divisibility and Modular Arithmetic” (YouTube) and “Pigeonhole Principle Problem 2 – Correct Operations” (YouTube)
Instructions: Please click on the link above and watch the brief video. The first video provides some an application of the pigeonhole principle to divisibility and modular arithmetic. The second video provides some applications of the pigeonhole principle to operations involving integers.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: YouTube: Mr. T’s Math Videos’ “Basic PigeonHole Principle Problems”

Final Exam
 Final Exam: The Saylor Foundation’s “MA111 Final Exam”
Link: The Saylor Foundation’s “MA111 Final Exam”
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
 Final Exam: The Saylor Foundation’s “MA111 Final Exam”