Answers for the Progress Checks

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Appendix B Answers for the Progress Checks

1 Introduction to Writing Proofs in Mathematics
1.1 Statements and Conditional Statements
Statements

Progress Check 1.1. Statements.

1.1.a

1.1.b

1.1.c

1.1.d

1.1.e

1.1.f

1.1.g

1.1.h

1.1.i

Techniques of Exploration

Progress Check 1.2. Explorations.

1.2.a

1.2.b

1.2.c

1.2.d

Conditional Statements

Progress Check 1.5. Explorations with Conditional Statements.

1.5.a

1.5.a.i

1.5.a.ii

1.5.a.iii

1.5.b

Further Remarks about Conditional Statements

Progress Check 1.6. Working with a Conditional Statement.

1.6.a

1.6.b

1.6.c

1.6.d

Closure Properties of Number Systems

Progress Check 1.8.

1.8.a

1.8.b

1.8.c

1.2 Constructing Direct Proofs
Writing Guidelines for Mathematics Proofs

Progress Check 1.11. Proving Propositions.

1.11.a

1.11.b

1.11.c

Some Comments about Constructing Direct Proofs

Progress Check 1.12. Exploring a Proposition.

Progress Check 1.13. Constructing and Writing a Proof.

2 Logical Reasoning
2.1 Statements and Logical Operators
Other Forms of Conditional Statements

Progress Check 2.3. The “Only If” Statement.

2.3.a

2.3.b

2.3.c

2.3.d

Constructing Truth Tables

Progress Check 2.5. Constructing Truth Tables.

2.5.d

Tautologies and Contradictions

Progress Check 2.7. Tautologies and Contradictions.

2.7.c

2.2 Logically Equivalent Statements
Another Method of Establishing Logical Equivalencies

Progress Check 2.11.

2.11.a

2.11.b

2.3 Open Sentences and Sets
Some Set Notation

Progress Check 2.14. Set Notation.

2.14.a

2.14.b

Variables and Open Sentences

Progress Check 2.16.

2.16.a

2.16.a.i

2.16.a.ii

2.16.b

2.16.b.i

2.16.b.ii

Set Builder Notation

Progress Check 2.18. Working with Truth Sets.

2.18.a

2.18.b

2.18.c

Progress Check 2.20. Set Builder Notation.

2.20.b

2.4 Quantifiers and Negations
Negations of Quantified Statements

Progress Check 2.26. Negating Quantified Statements.

2.26.a

2.26.b

2.26.c

2.26.d

2.26.e

Counterexamples and Negations of Conditional Statements

Progress Check 2.27. Using Counterexamples.

2.27.a

2.27.b

Quantifiers in Definitions

Progress Check 2.28. Multiples of Three.

2.28.a

2.28.d

2.28.e

Statements with More than One Quantifier

Progress Check 2.29. Negating a Statement with Two Quantifiers.

3 Constructing and Writing Proofs in Mathematics
3.1 Direct Proofs
Writing Guidelines for Equation Numbers

Progress Check 3.2. A Property of Divisors.

3.2.b

3.2.c

Using Counterexamples

Progress Check 3.3. Using a Counterexample.

Congruence

Progress Check 3.4. Congruence Modulo 8.

3.4.a

3.4.b

3.4.c

3.4.d

Progress Check 3.6. Proving Proposition 3.5.

3.6.a

3.6.b

3.6.c

3.6.d

3.6.e

3.2 More Methods of Proof
Using Other Logical Equivalencies

Progress Check 3.9. Using Another Logical Equivalency.

3.9.a

3.9.b

3.9.c

3.3 Proof by Contradiction
Writing Guidelines: Keep the Reader Informed

Progress Check 3.19. Starting a Proof by Contradiction.

3.19.a

3.19.b

3.19.c

3.19.d

Important Note

Progress Check 3.20. Exploration and a Proof by Contradiction.

3.20.a

3.20.b

3.20.c

Proving that Something Does Not Exist

Progress Check 3.22.

3.22.a

3.22.b

3.4 Using Cases in Proofs
Writing Guidelines for a Proof Using Cases

Progress Check 3.26. Using Cases: \(\boldsymbol{n}\) Is Even or \(\boldsymbol{n}\) Is Odd.

Absolute Value

Progress Check 3.29.

3.29.a

3.29.b

3.29.b.i

3.29.b.ii

3.29.b.iii

3.29.b.iv

3.5 The Division Algorithm and Congruence
The Division Algorithm

Progress Check 3.32. Using the Division Algorithm.

3.32.a

3.32.a.i

3.32.a.ii

3.32.b

3.32.b.i

3.32.b.ii

3.32.b.iii

3.32.b.iv

3.32.b.v

3.32.b.vi

Properties of Congruence

Progress Check 3.35. Proving Item 1 of Theorem 3.34.

Using Cases Based on Congruence Modulo \(\boldsymbol{n}\)

Progress Check 3.40. Using Properties of Congruence.

4 Mathematical Induction
4.1 The Principle of Mathematical Induction
Inductive Sets

Progress Check 4.1. Inductive Sets.

4.1.a

4.1.b

4.1.c

4.1.d

4.1.e

4.1.f

4.1.g

4.1.h

Summation Notation

Progress Check 4.3. An Example of a Proof by Induction.

4.3.b

Some Comments about Mathematical Induction

Progress Check 4.6. Proof of Proposition 4.5.

4.6.a

4.6.b

4.6.c

4.2 Other Forms of Mathematical Induction
Using the Extended Principle of Mathematical Induction

Progress Check 4.10. Formulating Conjectures.

4.10.a

4.10.b

4.10.c

Using the Second Principle of Mathematical Induction

Progress Check 4.12. Using the Second Principle of Induction.

4.12.b

4.12.d

4.3 Induction and Recursion
The Fibonacci Numbers

Progress Check 4.15. Every Third Fibonacci Number Is Even.

5 Set Theory
5.1 Sets and Operations on Sets
Set Equality, Subsets, and Proper Subsets

Progress Check 5.5. Using Set Notation.

5.5.a

5.5.b

5.5.c

5.5.d

5.5.e

5.5.f

5.5.g

5.5.h

5.5.i

5.5.j

More about Venn Diagrams

Progress Check 5.8. Using Venn Diagrams.

5.8.a

5.8.a.i

5.8.a.ii

5.8.a.iii

5.8.a.iv

5.8.b

5.8.c

5.2 Proving Set Relationships
The Choose-an-Element Method

Progress Check 5.13. Subsets and Set Equality.

5.13.a

5.13.b

Progress Check 5.14. Using the Choose-an-Element Method.

5.14.b

Proving Set Equality

Progress Check 5.17. Set Equality.

Disjoint Sets

Progress Check 5.21. Proving Two Sets Are Disjoint.

5.3 Properties of Set Operations
Proof of One of the Commutative Laws in Theorem 5.24

Progress Check 5.25. Exploring a Distributive Property.

5.25.a

5.25.b

Important Properties of Set Complements

Progress Check 5.27.

Progress Check 5.28.

5.4 Cartesian Products
Cartesian Products

Progress Check 5.30. Relationships between Cartesian Products.

5.30.a

5.30.a.i

5.30.a.ii

5.30.a.iii

5.30.a.iv

5.30.a.v

5.30.a.vi

5.30.a.vii

5.30.a.viii

5.30.a.ix

5.30.a.x

5.30.b

The Cartesian Plane

Progress Check 5.32. Cartesian Products of Intervals.

5.32.a

5.32.a.i

5.32.a.ii

5.32.a.iii

5.32.a.iv

5.32.a.v

5.32.a.vi

5.32.a.vii

5.32.a.viii

5.32.a.ix

5.32.a.x

5.32.b

5.5 Indexed Families of Sets
The Union and Intersection of an Indexed Family of Sets

Progress Check 5.35. An Infinite Family of Sets.

5.35.a

5.35.b

5.35.c

5.35.d

5.35.e

5.35.f

Progress Check 5.36. Indexed Families of Sets.

5.36.a

5.36.b

5.36.c

Progress Check 5.38. A Continuation of Example 5.37.

5.38.a

5.38.b

5.38.c

5.38.d

Pairwise Disjoint Families of Sets

Progress Check 5.41. Disjoint Families of Sets.

5.41.c

6 Functions
6.1 Introduction to Functions
The Definition of a Function

Progress Check 6.1. Images and Preimages.

6.1.a

6.1.b

6.1.c

6.1.d

6.1.e

6.1.f

The Codomain and Range of a Function

Progress Check 6.2. Codomain and Range.

6.2.a

6.2.a.i

6.2.a.ii

6.2.a.iii

6.2.b

6.2.b.i

6.2.b.ii

6.2.b.iii

The Graph of a Real Function

Progress Check 6.4. Using the Graph of a Real Function.

6.4.a

6.4.b

6.4.c

Arrow Diagrams

Progress Check 6.7. Working with Arrow Diagrams.

6.7.b

6.2 More about Functions
Functions Involving Congruences

Progress Check 6.10. Functions Defined by Congruences.

6.10.a

6.10.b

Equality of Functions

Progress Check 6.11. Equality of Functions.

Mathematical Processes as Functions

Progress Check 6.12. Average of a Finite Set of Numbers.

6.12.a

6.12.b

6.12.c

6.12.d

Sequences as Functions

Progress Check 6.13. Sequences.

6.13.a

6.13.b

6.13.c

Functions of Two Variables

Progress Check 6.14. Working with a Function of Two Variables.

6.14.a

6.14.b

6.14.c

6.3 Injections, Surjections, and Bijections
Injections

Progress Check 6.16. Working with the Definition of an Injection.

6.16.e

Surjections

Progress Check 6.17. Working with the Definition of a Surjection.

6.17.d

The Importance of the Domain and Codomain

Progress Check 6.21. The Importance of the Domain and Codomain.

Working with a Function of Two Variables

Progress Check 6.22. A Function of Two Variables.

6.22.a

6.22.b

6.22.c

6.4 Composition of Functions
Composition and Arrow Diagrams

Progress Check 6.26. The Composition of Two Functions.

Decomposing Functions

Progress Check 6.27. Decomposing Functions.

6.27.a

6.27.b

6.27.c

6.27.d

Theorems about Composite Functions

Progress Check 6.28. Compositions of Injections and Surjections.

6.28.a

6.28.c

6.5 Inverse Functions
The Ordered Pair Representation of a Function

Progress Check 6.32. Sets of Ordered Pairs that Are Not Functions.

6.32.a

6.32.b

The Inverse of a Function

Progress Check 6.33. Exploring the Inverse of a Function.

6.33.b

6.33.c

6.33.c.i

6.33.c.ii

6.33.c.iii

6.33.e

6.6 Functions Acting on Sets
Functions Acting on Sets

Progress Check 6.42. Beginning Activity 1 Revisited.

6.42.a

6.42.b

6.42.c

6.42.d

Set Operations and Functions Acting on Sets

Progress Check 6.44. Set Operations and Functions Acting on Sets.

6.44.a

6.44.b

6.44.c

6.44.c.i

6.44.c.ii

6.44.c.iii

6.44.c.iv

6.44.d

6.44.e

7 Equivalence Relations
7.1 Relations
Introduction to Relations

Progress Check 7.2.

7.2.a

7.2.a.i

7.2.a.ii

7.2.a.iii

7.2.a.iv

7.2.b

7.2.b.i

7.2.b.ii

7.2.b.iii

7.2.b.iii.A

7.2.b.iii.B

Notation for Relations

Progress Check 7.4. The Divides Relation.

7.4.a

7.4.b

7.4.b.i

7.4.b.ii

7.4.b.iii

Functions as Relations

Progress Check 7.5. A Set of Ordered Pairs.

7.5.a

7.5.b

7.5.b.i

7.5.b.ii

7.5.b.iii

7.5.b.iv

7.5.c

Visual Representations of Relations

Progress Check 7.8. The Directed Graph of a Relation.

7.2 Equivalence Relations
Directed Graphs and Properties of Relations

Progress Check 7.11. Properties of Relations.

Definition of an Equivalence Relation

Progress Check 7.13. A Relation that Is an Equivalence Relation.

Examples of Other Equivalence Relations

Progress Check 7.15. Another Equivalence Relation.

7.3 Equivalence Classes
The Definition of an Equivalence Class

Progress Check 7.16. Equivalence Classes from Beginning Activity 1.

Congruence Modulo \(\boldsymbol{n}\) and Congruence Classes

Progress Check 7.17. Congruence Modulo 4.

Properties of Equivalence Classes

Progress Check 7.19. Equivalence Classes.

7.19.a

7.19.b

7.19.c

7.4 Modular Arithmetic
The Integers Modulo \(\boldsymbol{n}\)

Progress Check 7.25. Modular Arithmetic in \(\boldsymbol{\Z_2}\text{,}\) \(\boldsymbol{\Z_5}\text{,}\) and \(\boldsymbol{\Z_6}\).

7.25.a

7.25.c

7.25.d

7.25.e

7.25.e.i

7.25.e.ii

8 Topics in Number Theory
8.1 The Greatest Common Divisor
The Greatest Common Divisor

Progress Check 8.4. Illustrations of Lemma 8.3.

8.4.a

8.4.b

8.4.c

The Euclidean Algorithm

Progress Check 8.6.

8.6.a

8.6.b

Writing \(\boldsymbol{\gcd (a, b)}\) in Terms of \(a\) and \(b\)

Progress Check 8.9. Writing the gcd as a Linear Combination.

8.9.a

8.9.b

8.2 Prime Numbers and Prime Factorizations
Relatively Prime Integers

Progress Check 8.12. Relatively Prime Integers.

8.12.a

8.12.b

8.12.c

Progress Check 8.15. Completing the Proof of Theorem 8.14.

8.3 Linear Diophantine Equations

Progress Check 8.22. An Example of a Linear Diophantine Equation.

8.22.b

Progress Check 8.23. Revisiting Beginning Activity 2.

Progress Check 8.26. Linear Diophantine Equations.

8.26.a

8.26.b

9 Finite and Infinite Sets
9.1 Finite Sets
Equivalent Sets

Progress Check 9.2. Examples of Equivalent Sets.

9.2.a

9.2.b

9.2.c

9.2 Countable Sets
Infinite Sets

Progress Check 9.12. Examples of Infinite Sets.

9.12.a

9.12.b

9.12.c

Countably Infinite Sets

Progress Check 9.13. Examples of Countably Infinite Sets.

9.13.a

9.13.b

9.13.c

9.3 Uncountable Sets
Uncountable Subsets of \(\boldsymbol{\mathbb{R}}\)

Progress Check 9.28. Dodge Ball and Cantor's Diagonal Argument.

Progress Check 9.30. Proof of Theorem 9.29.

9.30.a

9.30.b