HTML Links for Discrete Math Textbook
All content copyright D.E. Ensley and J.W. Crawley

Chapter 1 Puzzles, Patterns, and Mathematical Language

Section 1.1 First Examples

 Exploring the Josephus problem: Josephus

 Drawing without lifting pencil: Draw this

 The (4 by 4) grid game (see Section 7.5 for more variations): Grid game

Section 1.2 Number Puzzles and Sequences

 Identify the parameters for a recursive sequence: Sequence self test

 Use sigma (summation) notation: Sigma notation

 Four problems that practice calculating recursive sequences and comparing them to closed formulas for n = 1, 2, 3, 4, then some random values.

1. a₁ = 2, a_n = a_n−1 + 2n, and closed formula n² + n:  Recursive sequences #1
2. a₁ = 5, a_n = a_n−1 + (2n−1), and closed formula n² + 4:  Recursive sequences #2
3. a₁ = 5, a_n = a_n−1 + (n+4), and closed formula (n² + 9n) / 2:  Recursive sequences #3
4. a₁ = 3, a_n = a_n−1 + (n+4), and closed formula (n² + 9n − 4) / 2:  Recursive sequences #4

 Four problems that practice calculating summations and comparing them to closed formulas for n = 1, 2, 3, 4, then some random values.

1. 2 + 4 + 6 + ... + 2n and closed formula n² + n:  Summation calculations #1
2. 1 + 2 + 3 + ... + n and closed formula (n² + n) / 2 :  Summation calculations #2
3. 5 + 7 + 9 + ... + (2n+3) and closed formula n² + 4n:  Summation calculations #3
4. 1 + 4 + 7 + ... + (3n−2) and closed formula (3n² − n) / 2:  Summation calculations #4

Section 1.3 Truth-tellers, Liars, and Propositional Logic

 Fill in a truth table for two or three variables: Truth tables

 Use a truth table to see if two expressions are logically equivalent: Equivalent expressions

Section 1.4 Predicates

 Identify which elements of a domain satisfy a predicate: Predicates and domains

 Identify which elements of a domain satisfy the negation of a predicate: Negations of predicates

 Quantified statements, part 1: Identify true statements

 Quantified statements, part 2: Supply an appropriate domain

Section 1.5 Implications

 Identify which elements of a domain make an implication false: Negations of predicates with implications

 Same exercise, but phrased as finding all counterexamples to a quantified implication: Counterexamples

 Fill in a truth table for a proposition containing an implication: Truth tables with implications

 Applications of truth tables: Truth table applications

Chapter 2 A Primer of Mathematical Writing

Section 2.1 Mathematical Writing

 Provide counterexamples: Counterexamples

 Fill in the blanks: Proofs about numbers

 Tracing proofs: Proof reader

 Scrambled proofs, two versions:
  Use up and down arrows to correctly order a scrambled proof: Scrambled proofs
  Use drag and drop to correctly order a scrambled proof: Scrambled proofs

Section 2.2 Proofs About Numbers

 Provide counterexamples: Counterexamples

 Scrambled proofs, two versions:
  Use up and down arrows to correctly order a scrambled proof: Scrambled proofs
  Use drag and drop to correctly order a scrambled proof: Scrambled proofs

 Tracing proofs: Proof reader

 Practice with proof by cases: Proof by cases

Section 2.3 Mathematical Induction

Section 2.3, Part 1. Induction for recursively defined sequences.

 These four exercises build on the exercises in Section 1.2 (checking if a recursive definition and a closed formula give the same sequence). The student is told that equality has been verified for n from 1 to some particular number, and is asked to check the next value of n.

1. a₁ = 2, a_n = a_n−1 + 2n, and closed formula n² + n:  Recursive sequences #1
2. a₁ = 5, a_n = a_n−1 + (2n−1), and closed formula n² + 4:  Recursive sequences #2
3. a₁ = 5, a_n = a_n−1 + (n+4), and closed formula (n² + 9n) / 2:  Recursive sequences #3
4. a₁ = 3, a_n = a_n−1 + (n+4), and closed formula (n² + 9n − 4) / 2:  Recursive sequences #4

 For the same four problems, the student is told that n from 1 to m−1 has been checked, and is asked to check the next value of n. (That is, the student does the "induction step.")

1. a₁ = 2, a_n = a_n−1 + 2n, and closed formula n² + n:  Induction for recursive sequences #1
2. a₁ = 5, a_n = a_n−1 + (2n−1), and closed formula n² + 4:  Induction for recursive sequences #2
3. a₁ = 5, a_n = a_n−1 + (n+4), and closed formula (n² + 9n) / 2:  Induction for recursive sequences #3
4. a₁ = 3, a_n = a_n−1 + (n+4), and closed formula (n² + 9n − 4) / 2:  Induction for recursive sequences #4

Section 2.3, part 2. Induction for summations

 These four exercises build on the exercises in Section 1.2 (checking if a summation and a closed formula give the same sequence). The student is told that equality has been verified for n from 1 to some particular number, and is asked to check the next value of n.

1. 2 + 4 + 6 + ... + 2n and closed formula n² + n:  Summation calculations #1
2. 1 + 2 + 3 + ... + n and closed formula (n² + n) / 2 :  Summation calculations #2
3. 5 + 7 + 9 + ... + (2n+3) and closed formula n² + 4n:  Summation calculations #3
4. 1 + 4 + 7 + ... + (3n−2) and closed formula (3n² − n) / 2:  Summation calculations #4

 For the same four problems, given that n from 1 to m−1 has been checked, the student is asked to check the next value of n. (That is, the student does the "induction step.")

1. 2 + 4 + 6 + ... + 2n and closed formula n² + n:  Induction for summations #1
2. 1 + 2 + 3 + ... + n and closed formula (n² + n) / 2 :  Induction for summations #2
3. 5 + 7 + 9 + ... + (2n+3) and closed formula n² + 4n:  Induction for summations #3
4. 1 + 4 + 7 + ... + (3n−2) and closed formula (3n² − n) / 2:  Induction for summations #4

Section 2.3, part 3. Scrambled induction proofs

 Scrambled proofs, two versions:
  Use up and down arrows to correctly order a scrambled proof: Scrambled proofs
  Use drag and drop to correctly order a scrambled proof: Scrambled proofs

Section 2.4. More About Induction

Induction for divisibility proofs

 Fill in the blanks: Divisibility proofs

 These four exercises provide a framework for using induction for various divisibility proofs.

1. For the sequence a₁ = 1, a_n = a_n−1 + 2n, show that a_n is always odd:  Divisibility proof #1
2. For the sequence a₁ = 7, a_n = 3a_n−1 + 6n, show that a_n is always of the form 4k+3:  Divisibility proof #2
3. For the sequence a₁ = 1, a₂ = 2, a_n = 2a_n−1 + a_n−2, show that a_n + n is always even:  Divisibility proof #3
4. Show that n² + n is always even:  Divisibility proof #4

Section 2.5 Contradiction and the Pigeonhole Principle

 Start a proof by contradiction: Proof by contradiction

 Fill in the blanks: Proof by contradiction

 Scrambled proofs, two versions:
  Use up and down arrows to correctly order a scrambled proof: Scrambled proofs
  Use drag and drop to correctly order a scrambled proof: Scrambled proofs

 The Pigeonhole Principle at work: Pigeonhole Principle

Section 2.6 Excursion: Representations of Numbers

 Base conversions:
  Part 1 - convert base 10 to/from other bases: Base Conversion #1
  Part 2 - convert between bases 2, 8, and 16: Base Conversion #2

Chapter 3 Sets and Boolean Algebra

Section 3.1 Set Definitions and Operations

 Set notation: Set notation

 Set operations - calculate union, intersection, difference: Set operations

 Provide counterexamples: Counterexamples

 Two set Venn diagrams: Venn diagrams

 Three set Venn diagrams: Venn diagrams

Section 3.2 More Operations on Sets

 Provide counterexamples: Counterexamples

Section 3.3 Proving Set Properties

 Fill in the blanks: 'Elementwise' proofs for sets

 Set proofs using set properties, part 1 - choose justifications for steps: Justifications

 Set proofs using set properties, part 2 - scrambled proofs, two versions:
  Use up and down arrows to correctly order a scrambled proof: Scrambled proofs
  Use drag and drop to correctly order a scrambled proof: Scrambled proofs

 Set proofs using set properties, part 3 - scrambled proofs, and choose justifications, two versions:
  Use up and down arrows to correctly order a scrambled proof: Scrambled proofs with justifications
  Use drag and drop to correctly order a scrambled proof: Scrambled proofs with justifications

Section 3.4 Boolean Algebra

 Boolean algebra proofs, part 1 - choose justifications for steps: Justifications

 Boolean algebra proofs, part 2 - scrambled proofs, two versions:
  Use up and down arrows to correctly order a scrambled proof: Scrambled proofs
  Use drag and drop to correctly order a scrambled proof: Scrambled proofs

 Boolean algebra proofs, part 3 - scrambled proofs, and choose justifications, two versions:
  Use up and down arrows to correctly order a scrambled proof: Scrambled proofs with justifications
  Use drag and drop to correctly order a scrambled proof: Scrambled proofs with justifications

Section 3.5 Excursion: Logic Circuits

 Truth tables for logical expressions: Truth tables

Chapter 4 Functions and Relations

Section 4.1 Definitions, Diagrams, and Inverses

 Two set arrow diagrams for functions: Two set arrow diagrams - functions

 Two set arrow diagrams for relations between sets X and Y: Two set arrow diagrams - relations

 One set arrow diagrams for relations on a set S: One set arrow diagrams - relations
  Same as the previous, but using a two set arrow diagram: Two set arrow diagrams

 Fill in the blanks: Inverse functions

Section 4.2 The Composition Operation

 Calculate composition of two functions (using two-set arrow diagrams): Function composition
  Same, but using tables: Function composition
  Same, but using one-set arrow diagrams: Function composition

 More function composition problems (using arrow diagrams): More function composition
  Same, but using tables: More function composition

 Calculate composition of two relations (using two-set arrow diagrams): Relation composition (version 1)

 Calculate composition of two relations (using one-set arrow diagrams): Relation composition (version 2)

 Calculate R ∘ R and R ∘ (R ∘ R): Relation composition (version 3)

Section 4.3 Properties of Functions and Set Cardinality

 Provide arrow diagrams and identify function properties (1-1? onto?): Function properties

 Give examples of function with various function properties (1-1? onto?): Create examples

 Provide counterexamples: Counterexamples

 Fill in the blanks: Proofs about function properties

 Scrambled proofs, two versions:
  Use up and down arrows to correctly order a scrambled proof: Scrambled proofs
  Use drag and drop to correctly order a scrambled proof: Scrambled proofs

Section 4.4 Properties of Relations

 Provide counterexamples for relation properties (reflexive, irreflexive, antisymmetric, transitive)
  Version 1 - relation presented as set of ordered pairs: Counterexamples
  Version 2 - relation presented as arrow diagram: Counterexamples
  Version 3 - relation on Z presented as property of integers: Counterexamples

 Scrambled proofs, two versions:
  Use up and down arrows to correctly order a scrambled proof: Scrambled proofs
  Use drag and drop to correctly order a scrambled proof: Scrambled proofs

Section 4.5 Equivalence Relations

 Provide counterexamples for relation properties (reflexive, symmetric, transitive):
  Version 1 - relation presented as set of ordered pairs: Counterexamples
  Version 2 - relation presented as arrow diagram: Counterexamples
  Version 3 - relation on Z presented as property of integers: Counterexamples

 Same, but also include irreflexive and antisymmetric properties:
  Version 1 - relation presented as set of ordered pairs: Counterexamples
  Version 2 - relation presented as arrow diagram: Counterexamples
  Version 3 - relation on Z presented as property of integers: Counterexamples

Chapter 5 Combinatorics

Section 5.1 Introduction

 Count outcomes for dice: Dice counts

 One-to-one correspondences (3 different correspondences)
  Coin toss sequences and subsets: Correspondence 1
  Certain sums and binary sequences: Correspondence 2
  Bags of fruit and subsets: Correspondence 3

Section 5.2 Basic Rules for Counting

 Four versions of practice problems.

1. Student enters the numerical answer.:  Version 1
2. Allows "sum of products" input, such as 18∗17+20∗19.  Version 2
3. Allows "sum of products" and also P(n,r) notation.  Version 3
4. Allows "sum of products" and also nPr notation.  Version 4

Section 5.3 Combinations and the Binomial Theorem

 Two versions of practice problems. Both allow sum of products input.

1. Allows C(n,r) notation.  Version 1
2. Allows nCr notation.  Version 2

Section 5.4 Binary Sequences

 Two versions of practice problems. Both allow sum of products input.

1. Allows P(n,r) and C(n,r) notation.  Version 1
2. Allows nPr and nCr notation.  Version 2

Chapter 6 Probability

Section 6.1 Introduction

 Simulate looking for duplicate birthdays in groups: Birthdays

 Calculate probabilities for dice: Dice probabilities

 Simulate a simple dice game: Dice game

 Two versions of practice problems, where the answer is entered as a numerator divided by a denominator. Both the numerator and denominator can be entered as a sum of products.

1. Allows P(n,r) and C(n,r) notation.  Version 1
2. Allows nPr and nCr notation.  Version 2

Section 6.2 Sum and Product Rules for Probability

 Practice with the sum and product rule (and the complement rule):  Sum and product rule
   NOTE. The student can enter expressions such as 3∗1/16+1/3−1/2∗1/2, that is "sums of products" where each factor is either a number or a fraction. As the app detects the division symbol, it automatically displays the input as a fraction.

Section 6.3 Probability in Games of Chance

 Two versions of practice problems. Each version allows input such as (3/4)^3(0.7)^5. The app displays the fractions in fraction form and displays the ^3, etc., in exponential form.

1. Allows P(n,r) and C(n,r) notation.  Version 1
2. Allows nPr and nCr notation.  Version 2

 Simulate Bernoulli trials: Bernoulli trials

 Simulate a best of 3 series: Best of 3 series

 Simulate a best of 5 series: Best of 5 series

 Simulate a best of 7 series: Best of 7 series

 Simulate a best of 7 series with home field advantage: Home field advantage

Section 6.4 Expected Value in Games of Chance

 Simulate a "best of" series where you set the series length and probabilities:
   Series simulation with user input, version 1
   Series simulation with user input, version 2

Section 6.5 Recursion Revisited

 Simulate a "win by 2" game, similar to a tennis game tied at "deuce": Tennis simulation

 Simulate the "Hank and Ted" game:": Hank and Ted Game

Section 6.6 Excursion: Matrices and Markov Chains

 Markov chain matrix calculator: Matrix calculator

Chapter 7 Graphs and Trees

Section 7.1 Graph Theory

 Find Euler trails or Euler circuits: Eulerian graphs

 Create Euler circuits by adding edges to a graph: Eulerize a graph

Section 7.2 Proofs About Graphs and Trees

 Fill in the blanks: Proofs about graphs

Section 7.3 Isomorphism and Planarity

 Planar graphs:
   Part 1 (exploration for complete and complete bipartite graphs): Planar graphs, part 1
   Part 2 (practice with a variety of graphs): Planar graphs, part 2

 Isomorphic graphs:
   Part 1 (exploration): Isomorphic graphs, part 1
   Part 2 (using drag/drop to find the isomorphism): Isomorphic graphs, part 2

Section 7.4 Connections to Matrices and Relations

 Fill in the adjacency matrix for a relation:
   Version 1 (relation given as ordered pairs)
   Version 2 (relation given as graph)

 Convert adjacency matrix for a relation to a graph: From matrix to graph

 Closure of relations:
   Closure, version 1 (fixed set of 5 problems)
   Closure, version 2 (randomly generated problems)

 Provide counterexamples for relation properties (repeat Section 4.4, 4.5 but adds adjacency matrix version)
  Version 1 - relation presented as set of ordered pairs: Counterexamples
  Version 2 - relation presented as arrow diagram: Counterexamples
  Version 3 - relation presented as adjacency matrix: Counterexamples

Section 7.5 Graphs in Puzzles and Games

 Two versions of the nim game.
   Computer plays randomly: Nim version 1
   Computer plays smartly: Nim version 2

 Two more versions of the nim game. These versions display the current game "status" (nim-sum).
   Computer plays randomly: Nim version 3
   Computer plays smartly: Nim version 4

 Several versions of the grid game.
   Same as in Section 1.1: Grid game version 1
   Same as version 1, but with different grid sizes: Grid game version 2
   Same as version 2, but computer doesn't use the symmetry strategy: Grid game version 3
   Computer plays randomly until only a few remaining cells: Grid game version 4
   Versions 2, 3, and 4 again, but showing the "nim sum" of the rows and columns:
    Grid game version 2 alternate
    Grid game version 3 alternate
    Grid game version 4 alternate

Section 7.7 Excursion: Hamiltonian Cycles and the TSP

 Find Hamiltonian cycles: Hamiltonian cycles