# Discrete Math Resources

 COURSE HOME CHAPTER 1 CHAPTER 2 CHAPTER 3 CHAPTER 4 CHAPTER 5 CHAPTER 6 CHAPTER 7

## Flash Applications These applets accompany the textbook, Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns and Games, by Doug Ensley and Winston Crawley, published by John Wiley and Sons. The development of some material on this site was funded by NSF DUE-0230755.

 All of the material linked from this page requires the Flash player, a free plug-in from Adobe that is available for many operating systems and browsers. Instructors interested in customizing some of these applets without doing any Flash program should download this tutorial. Instructors interested in designing their own applets using Flash should check out the MathDL Flash Forum (www.mathflashforum.org).

The resources below are referenced to the items in the textbook to which the activity is related.

### Section 1.1. First examples

Josephus problem (Example 2, Exercise 3)

Draw this! (Example 3, Exercise 11)

Grid game (Practice Problem 5, Exercise 12)

### Section 1.2. Number puzzles and sequences

Sequence self test (Example 5, Exercises 4 and 7)

Graphical sequence self test (Example 5, Exercises 4 and 7)

Recursive sequences (Example 5, Exercises 8 and 9)

Notation for sums (Example 11, Exercises 19 and 20)

Josephus problem again (Example 12, Exercise 25)

### Section 1.3. Truthtellers, liars and propositional logic

Truth tables (Practice Problems 4 and 8; Exercises 11, 12, 16-22)

Logically equivalent statements (Practice Problems 4 and 8; Exercises 11, 12, 16-22)

### Section 1.4. Predicates

Predicates and domains (Practice Problem 1; Exercises 3 and 4)

Negation of predicates (Example 3; Practice Problem 2; Exercise 5)

Quantified statements (Practice Problems 2, 3 and 4; Exercises 6 and 7)

### Section 1.5. Implications

More truth tables (Exercises 4 and 5)

And even more truth tables and even some with subexpressions

Applications of truth tables (Exercises 2 and 7)

Negation of predicates with implications (Exercises 10 and 11)

### Section 2.1. Mathematical writing

Counterexamples (Practice Problem 4, Exercises 2 and 3)

Fill in the blanks (Exercise 5)

ProofReader (Tracing proofs, Example 6, Exercise 6)

Scrambled proofs (Exercises 4, 5, and 11)

### Section 2.2. Proofs about numbers

Counterexamples (Exercise 4)

ProofReader (Practice Problem 4, Exercises 5, 7, 14, and 23)

Scrambled proofs (Practice Problem 1, Exercises 7, 14, and 23)

### Section 2.3. Mathematical induction

Proving closed forms for recursive sequences. (Practice Problems 1 and 5, Exercises 3 and 4)

Proving closed formulas for sums. (Practice Problems 2 and 4, Exercises 8 and 9)

Scrambled induction proofs (More practice reading proofs)

### Section 2.4. More on induction

Divisibility proofs (Example 6, Exercises 3, 4, 5)

Fill in the blanks (Example 6, Exercises 3, 4, 5)

### Section 2.5. Proof by contradiction and the Pigeonhole Principle

Scrambled proofs (Example 1, Exercises 4 and 7)

Fill in the blanks (Exercises 1 and 2)

Pigeonhole principle in action (Example 7, Practice Problem 4, Exercises 32, 33, and 34)

### Section 2.6. Representations of Numbers

Flash Magic Trick (External Link)

Converting Between Bases (Examples 7-10, Exercises 1-8)

Hexadecimal Colors (Exercise 19)

### Section 3.1. Set definitions and operations

Set notation (Practice Problem 3, Exercises 4 and 11)

Set operations (Practice Problem 4, Exercise 1)

Counterexamples (Practice Problem 7, Exercises 13, 17, and 32)

Two-set Venn diagrams (Warm-up)

Three-set Venn diagrams (Practice Problem 6, Exercises 16 and 17)

### Section 3.2. More operations on sets

Counterexamples (Exercises 10 and 11)

### Section 3.3. Proving set properties

Fill in the blanks (Practice Problem 3, Exercises 4 and 5)

Scrambled Proofs (Practice Problems 4 and 5, Exercises 14 and 16)

### Section 3.4: Boolean algebra

Scrambled Proofs (Practice Problem 4, Exercises 3 and 5)

### Section 3.5: Logic circuits

Truth tables revisited (Practice Problems 1 and 3, Exercise 3)

### Section 4.1. Definitions, diagrams and inverses

Two-set arrow diagrams for functions (Exercises 3 and 6)

Two-set arrow diagrams for relations (Practice Problem 4, Exercises 8 and 9)

One-set arrow diagrams for relations (Practice Problem 2, Exercises 10 and 12)

Fill in the blanks (Exercise 16)

### Section 4.2. The composition operation

Function composition (Practice Problem 2, Exercises 6 and 7)

Oracle of Bacon at UVA (http://www.cs.virginia.edu/oracle/) (Exercise 22)

### Section 4.3. Properties of functions

Fill in the blanks (Practice Problems 2 and 3, Exercises 7 and 8)

Scrambled proofs (Exercises 14 - 18)

### Section 4.4. Properties of relations

Scrambled proofs (Practice Problems 3, 4 and 5)

Counterexamples (Exercises 2, 3, 4, 9, and 10)

### Section 5.1. Introduction

Dice Problems (Exercises 5 and 6)

One-to-one correspondence (Practice Problem 4, Exercise 18)

### Section 5.2. Basic rules of counting

Practice problems (Exercises 5, 6-8, 20, 21)

### Section 5.3. Combinations and the Binomial Theorem

Practice problems (Practice Problems 3 and 4, Exercises 15, 16, 23, 27, 31, 32)

### Section 5.4. Binary sequences

Practice problems (Practice Problems 2 and 4, Exercises 1-3, 16-20)

### Section 6.1. Introduction

Birthday problem (Practice Problem 3, Exercises 14-17)

Dice problems (Exercise 3)

Simple dice game (Exercise 19)

### Section 6.2. Sum and product rules for probability

Practice problems (Exercises 2, 3, 5, 7, and 8)

### Section 6.3. Probability in games

Bernoulli trials (Practice Problem 1, Exercises 1-6)

Series simulator (Practice Problem 2, Exercises 20 and 22)

### Section 6.4. Expected value in games

Series simulator (Practice Problem 4, Exercises 16-20)

### Section 6.5. Recursive games

Tennis problem (Exercises 9-12, 15 and 16)

Hank and Ted (Exercises 18 and 19)

### Section 6.6. Markov chains

Markov chain matrix calculator (Exercises 12-18, 24-29)

### Section 7.1. Graph theory

Eulerian Graphs (Practice Problem 6, Exercise 9)

"Eulerizing a graph" means to add a minimal number of edges to make a new graph that has an Euler circuit. Each additional edge can be interpreted as a "pencil lift" in drawing problems or a "repeated edge" in a traveling circuit problem.

General graph tools from Christopher Mawatma's "Petersen" Project at http://www.mathcove.net/petersen/

### Section 7.2. Proofs about graphs and trees

Fill in the blanks proof (Practice Problems 1 and 2, Exercises 2, 14, and 18)

### Section 7.3: Isomorphism and planarity

Graph Isomorphism (Example 1, Practice Problem 1, Exercise 3)

Planar Graphs (Practice Problem 4, Exercises 10 and 12)

### Section 7.4. Connections to matrices and relations

Practice problems (Exercises 18, 21, and 25)

### Section 7.5. Graphs in puzzles and games

Water Puzzle (Exercise 1)

Nim Game (Practice Problem 5, Exercises 13-17)

Grid Game (Exercise 20)

### Section 7.7: Hamiltonian graphs and TSP

Hamiltonian Graphs (Exercise 5 and 15)

Sean Forman's TSP Generator (Resource for comparing with Exercises 21-24)

 For more details, contact me by sending e-mail to deensley@ship.edu or by using the information at the right. Doug Ensley Department of Mathematics Shippensburg University Shippensburg, PA 17257 Phone: (717) 477-1431 Fax: (717) 477-4009