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Contents

I Introduction

1 Syllabus

1.1 Discrete Mathematics I

1.2 Goals

1.3 Instructor: Jason Riedy

1.4 Text

1.5 Grading

1.6 On homework

1.7 Submitting homework

2 Syllabus schedule

II Notes for chapters 1, 2, and 3

3 Notes for 18 August

3.1 Syllabus and class mechanics

3.2 Introductions

3.3 Inductive and deductive reasoning

3.4 Inductive

3.5 Deductive

4 Notes for 20 August

4.1 Review: Inductive and deductive reasoning

4.2 Inductive reasoning on sequences

4.3 A tool for sequences: successive differences

4.4 Successive differences are not useful for everything.

4.5 An application where successive differences work, amazingly.

4.6 Next time: Problem solving techniques.

4.7 Homework

5 Notes for 22 August

5.1 The problem solving section is important enough for a full class

5.2 Review successive differences: a tool for inductive reasoning on sequences

5.3 Moving from a table to a formula

5.4 Starting point

5.5 The plan for deriving a formula

5.6 The derivation

5.7 Homework

6 Solutions for first week’s assignments

6.1 Notes on received homeworks

6.2 Exercises for Section 1.1

6.3 Explain the “trick” of Section 1.1’s example

6.4 Exercises for Section 1.2

7 Notes for 25 August

7.1 Problem solving principles

7.2 Making a lists and tables

7.3 Searching by guessing

7.4 Understanding dependencies, or ”working backward”

7.5 Next time: more techniques

7.6 Homework

8 Notes for 27 August

8.1 Review: Pólya’s problem-solving principles

8.2 Effective trial and error by bisection

8.3 Simpler sub-problems for finding patterns

8.4 Other sources for tactics and examples

8.5 Next time: Reading graphs and charts

8.6 Homework

9 Notes for 29 August

9.1 Review: Pólya’s problem-solving principles

9.2 Notes on the homework

9.3 Reading graphs: delayed until Monday (or later)

9.4 Homework

10 Solutions for second week’s assignments

10.1 Exercises for Section 1.3

10.2 Making change

10.3 Writing out problems

10.4 Computing with numbers

11 Notes for reading graphs

11.1 Reading graphs

11.2 Creating a graphical depiction of data

11.3 Graph galleries and resources

12 Homework for reading graphs

12.1 Homework

13 Notes for the third week: set theory

13.1 Language of set theory

13.2 Basic definitions

13.3 Translating sets into (and from) English

13.4 Relations

13.5 Translating relations into (and from) English

13.6 Consequences of the set relation definitions

13.7 Visualizing two or three sets: Venn diagrams

13.8 Operations

13.9 Translating operations into English

13.10 Special operations

13.11 Cardinality and the power set

14 Homework for the third week: set theory

14.1 Homework

15 Solutions for third week’s assignments

15.1 Section 1.4, problem 54

15.2 Section 2.1

15.3 Section 2.2

15.4 Section 2.3

III Notes for chapters 4, 5, and 6

16 Notes for the fourth week: symbolic logic

16.1 Language of logic

16.2 Symbolic logic

16.3 Logical operators and truth tables

16.4 Properties of logical operators

16.5 Truth tables and logical expressions

16.6 Conditionals

16.7 Quantifiers

16.8 Logical deduction: Delayed until after the test

17 Homework for the fourth week: symbolic logic

17.1 Homework

18 Solutions for fourth week’s assignments

18.1 Section 3.1

18.2 Section 3.2

18.3 Section 3.3

18.4 Section 3.4

18.5 Section 3.1 again

18.6 Negating statements

18.7 Function from truth table

19 Notes for the fifth week: review

19.1 Review

19.2 Inductive and deductive reasoning

19.3 Problem solving

19.4 Sequences

19.5 Set theory

19.6 Symbolic logic

20 First exam and solutions

21 Notes for the sixth week: numbers and computing

21.1 Positional Numbers

21.2 Converting Between Bases

21.3 Operating on Numbers

21.4 Computing with Circuits

22 Homework for the sixth week: numbers and computing

22.1 Homework

23 Solutions for sixth week’s assignments

23.1 Section 4.1, problems 35 and 36

23.2 Section 4.2

23.3 Section 4.3

23.4 Positional form

23.5 Operations

24 Notes for the seventh week: primes, factorization, and modular arithmetic

24.1 Divisibility

24.2 Primes

24.3 Factorization

24.4 Modular Arithmetic

24.5 Divisibility Rules

25 Homework for the seventh week: primes, factorization, and modular arithmetic

25.1 Homework

26 Solutions for seventh week’s assignments

26.1 Section 5.1 (prime numbers)

26.2 Section 5.1 (factorization)

26.3 Section 5.4 (modular arithmetic)

26.4 Section 5.1 (divisibility rules)

27 Notes for the eighth week: GCD, LCM, ax + by = c

27.1 Modular arithmetic

27.2 Divisibility rules

27.3 Greatest common divisor

27.4 Least common multiple

27.5 Euclidean GCD algorithm

27.6 Linear Diophantine equations : Likely delayed

28 Homework for the eighth week: GCD, LCM, ax + by = c

28.1 Homework

29 Solutions for eighth week’s assignments

29.1 Exercises 5.3

29.2 Computing GCDs

29.3 Computing LCMs

30 Notes for the ninth week: ax + by = c and fractions

30.1 Linear Diophantine equations

30.2 Into real numbers

30.3 Rational numbers

30.4 Review of rational arithmetic

30.5 Complex fractions

31 Homework for the ninth week: ax + by = c and fractions

31.1 Homework

32 Solutions for ninth week’s assignments

32.1 Linear Diophantine equations

32.2 Exercises 6.3

33 Notes for the tenth week: Irrationals and decimals

33.1 Real numbers

33.2 Exponents and roots

33.3 Decimal expansions and percentages

33.4 Fixed and floating-point arithmetic

34 Homework for the tenth week: Irrationals and decimals

34.1 Homework

35 Solutions for tenth week’s assignments

35.1 Exercises 6.4

35.2 Exercises 6.3

35.3 Exercises 6.5

35.4 Rounding and floating-point

36 Second exam and solutions

IV Notes for chapters 7 and 8

37 Notes for the twelfth week

37.1 Covered So Far

37.2 What Will Be Covered

37.3 An Algebraic Example

37.4 The Example’s Graphical Side

37.5 Definitions

37.6 Algebraic Rules for Transformations Between Equivalent Equations

37.7 Transformation Examples

37.8 Manipulating Formulæ by Transformations

38 Homework for the twelfth week

38.1 Homework

39 Solutions for twelfth week’s assignments

39.1 Exercises for 7.1

39.2 Exercises for 7.2

40 Homework for the thirteenth week

40.1 Homework

41 Solutions for the thirteenth week’s assignments

41.1 Exercises for 8.2

41.2 Exercises for 8.3

41.3 Exercises for 8.7

41.4 Exercises for 8.8

42 Homework for the fourteenth week

42.1 Homework

43 Solutions for the fourteenth week’s assignments

43.1 Exercises for 7.3

43.2 Exercises for 7.4

43.3 Exercises for 7.5

43.4 Exercises for 7.7

43.5 Exercises for 8.1

43.6 Exercises for 8.3

43.7 Exercises for 8.6

43.8 Exercises for 8.7

43.9 Exercises for 8.8

44 Third exam, due 1 December

45 Third exam solutions

46 Final exam

V Resources

47 Math Lab

48 On-line

48.1 General mathematics education resources

48.2 Useful software and applications

I Introduction

1 Syllabus

1.1 Discrete Mathematics I

1.2 Goals

1.3 Instructor: Jason Riedy

1.4 Text

1.5 Grading

1.6 On homework

1.7 Submitting homework

2 Syllabus schedule

II Notes for chapters 1, 2, and 3

3 Notes for 18 August

3.1 Syllabus and class mechanics

3.2 Introductions

3.3 Inductive and deductive reasoning

3.4 Inductive

3.5 Deductive

4 Notes for 20 August

4.1 Review: Inductive and deductive reasoning

4.2 Inductive reasoning on sequences

4.3 A tool for sequences: successive differences

4.4 Successive differences are not useful for everything.

4.5 An application where successive differences work, amazingly.

4.6 Next time: Problem solving techniques.

4.7 Homework

5 Notes for 22 August

5.1 The problem solving section is important enough for a full class

5.2 Review successive differences: a tool for inductive reasoning on sequences

5.3 Moving from a table to a formula

5.4 Starting point

5.5 The plan for deriving a formula

5.6 The derivation

5.7 Homework

6 Solutions for first week’s assignments

6.1 Notes on received homeworks

6.2 Exercises for Section 1.1

6.3 Explain the “trick” of Section 1.1’s example

6.4 Exercises for Section 1.2

7 Notes for 25 August

7.1 Problem solving principles

7.2 Making a lists and tables

7.3 Searching by guessing

7.4 Understanding dependencies, or ”working backward”

7.5 Next time: more techniques

7.6 Homework

8 Notes for 27 August

8.1 Review: Pólya’s problem-solving principles

8.2 Effective trial and error by bisection

8.3 Simpler sub-problems for finding patterns

8.4 Other sources for tactics and examples

8.5 Next time: Reading graphs and charts

8.6 Homework

9 Notes for 29 August

9.1 Review: Pólya’s problem-solving principles

9.2 Notes on the homework

9.3 Reading graphs: delayed until Monday (or later)

9.4 Homework

10 Solutions for second week’s assignments

10.1 Exercises for Section 1.3

10.2 Making change

10.3 Writing out problems

10.4 Computing with numbers

11 Notes for reading graphs

11.1 Reading graphs

11.2 Creating a graphical depiction of data

11.3 Graph galleries and resources

12 Homework for reading graphs

12.1 Homework

13 Notes for the third week: set theory

13.1 Language of set theory

13.2 Basic definitions

13.3 Translating sets into (and from) English

13.4 Relations

13.5 Translating relations into (and from) English

13.6 Consequences of the set relation definitions

13.7 Visualizing two or three sets: Venn diagrams

13.8 Operations

13.9 Translating operations into English

13.10 Special operations

13.11 Cardinality and the power set

14 Homework for the third week: set theory

14.1 Homework

15 Solutions for third week’s assignments

15.1 Section 1.4, problem 54

15.2 Section 2.1

15.3 Section 2.2

15.4 Section 2.3

III Notes for chapters 4, 5, and 6

16 Notes for the fourth week: symbolic logic

16.1 Language of logic

16.2 Symbolic logic

16.3 Logical operators and truth tables

16.4 Properties of logical operators

16.5 Truth tables and logical expressions

16.6 Conditionals

16.7 Quantifiers

16.8 Logical deduction: Delayed until after the test

17 Homework for the fourth week: symbolic logic

17.1 Homework

18 Solutions for fourth week’s assignments

18.1 Section 3.1

18.2 Section 3.2

18.3 Section 3.3

18.4 Section 3.4

18.5 Section 3.1 again

18.6 Negating statements

18.7 Function from truth table

19 Notes for the fifth week: review

19.1 Review

19.2 Inductive and deductive reasoning

19.3 Problem solving

19.4 Sequences

19.5 Set theory

19.6 Symbolic logic

20 First exam and solutions

21 Notes for the sixth week: numbers and computing

21.1 Positional Numbers

21.2 Converting Between Bases

21.3 Operating on Numbers

21.4 Computing with Circuits

22 Homework for the sixth week: numbers and computing

22.1 Homework

23 Solutions for sixth week’s assignments

23.1 Section 4.1, problems 35 and 36

23.2 Section 4.2

23.3 Section 4.3

23.4 Positional form

23.5 Operations

24 Notes for the seventh week: primes, factorization, and modular arithmetic

24.1 Divisibility

24.2 Primes

24.3 Factorization

24.4 Modular Arithmetic

24.5 Divisibility Rules

25 Homework for the seventh week: primes, factorization, and modular arithmetic

25.1 Homework

26 Solutions for seventh week’s assignments

26.1 Section 5.1 (prime numbers)

26.2 Section 5.1 (factorization)

26.3 Section 5.4 (modular arithmetic)

26.4 Section 5.1 (divisibility rules)

27 Notes for the eighth week: GCD, LCM, ax + by = c

27.1 Modular arithmetic

27.2 Divisibility rules

27.3 Greatest common divisor

27.4 Least common multiple

27.5 Euclidean GCD algorithm

27.6 Linear Diophantine equations : Likely delayed

28 Homework for the eighth week: GCD, LCM, ax + by = c

28.1 Homework

29 Solutions for eighth week’s assignments

29.1 Exercises 5.3

29.2 Computing GCDs

29.3 Computing LCMs

30 Notes for the ninth week: ax + by = c and fractions

30.1 Linear Diophantine equations

30.2 Into real numbers

30.3 Rational numbers

30.4 Review of rational arithmetic

30.5 Complex fractions

31 Homework for the ninth week: ax + by = c and fractions

31.1 Homework

32 Solutions for ninth week’s assignments

32.1 Linear Diophantine equations

32.2 Exercises 6.3

33 Notes for the tenth week: Irrationals and decimals

33.1 Real numbers

33.2 Exponents and roots

33.3 Decimal expansions and percentages

33.4 Fixed and floating-point arithmetic

34 Homework for the tenth week: Irrationals and decimals

34.1 Homework

35 Solutions for tenth week’s assignments

35.1 Exercises 6.4

35.2 Exercises 6.3

35.3 Exercises 6.5

35.4 Rounding and floating-point

36 Second exam and solutions

IV Notes for chapters 7 and 8

37 Notes for the twelfth week

37.1 Covered So Far

37.2 What Will Be Covered

37.3 An Algebraic Example

37.4 The Example’s Graphical Side

37.5 Definitions

37.6 Algebraic Rules for Transformations Between Equivalent Equations

37.7 Transformation Examples

37.8 Manipulating Formulæ by Transformations

38 Homework for the twelfth week

38.1 Homework

39 Solutions for twelfth week’s assignments

39.1 Exercises for 7.1

39.2 Exercises for 7.2

40 Homework for the thirteenth week

40.1 Homework

41 Solutions for the thirteenth week’s assignments

41.1 Exercises for 8.2

41.2 Exercises for 8.3

41.3 Exercises for 8.7

41.4 Exercises for 8.8

42 Homework for the fourteenth week

42.1 Homework

43 Solutions for the fourteenth week’s assignments

43.1 Exercises for 7.3

43.2 Exercises for 7.4

43.3 Exercises for 7.5

43.4 Exercises for 7.7

43.5 Exercises for 8.1

43.6 Exercises for 8.3

43.7 Exercises for 8.6

43.8 Exercises for 8.7

43.9 Exercises for 8.8

44 Third exam, due 1 December

45 Third exam solutions

46 Final exam

V Resources

47 Math Lab

48 On-line

48.1 General mathematics education resources

48.2 Useful software and applications

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