Skip to main content\(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\R}{\mathbb R}
\newcommand{\Q}{\mathbb Q}
\newcommand{\mynot}{\neg}
\newcommand{\card}{\mathbf{card}}
\newcommand{\x}{:}
\newcommand{\Mod}[1]{\ (\mathrm{mod}\ #1)}
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Appendix C Answers and Hints for Selected Exercises
1 Introduction to Writing Proofs in Mathematics
1.1 Statements and Conditional Statements
Exercises
1.
1.a
1.b
1.c
1.d
1.e
1.f
1.g
1.h
1.i
1.j
1.k
2.
2.a
2.b
2.c
2.d
2.e
3.
3.a
3.b
3.c
3.d
4.
4.a
4.b
6.
6.a
6.b
6.c
9.
9.a
9.b
9.c
9.d
9.e
9.f
9.g
1.2 Constructing Direct Proofs
Exercises
1.
1.a
2.
2.c
3.
3.a
3.b
4.
4.a
5.
5.b
6.
6.a
6.d
9.
9.a
9.c
10.
10.a
2 Logical Reasoning
2.1 Statements and Logical Operators
Exercises
1.
2.
2.a
2.b
2.c
4.
4.c
5.
5.a
5.b
5.c
5.d
5.e
7.
9.
9.c
9.d
11.
11.a
11.b
11.c
11.d
2.2 Logically Equivalent Statements
Exercises
1.
1.a
1.b
1.c
1.d
2.
3.
3.a
3.b
3.c
3.d
3.e
7.
7.a
7.b
10.
10.c
10.d
10.f
11.
11.d
2.3 Open Sentences and Sets
Exercises
1.
1.a
1.b
1.c
1.d
1.e
1.f
2.
2.b
3.
3.b
3.c
4.
4.a
4.b
5.
5.a
5.c
5.e
2.4 Quantifiers and Negations
Exercises
1.
1.a
1.b
1.c
2.
2.a
2.b
2.f
3.
3.a
3.c
3.e
3.h
4.
4.a
4.e
5.
6.
6.a
6.b
6.c
6.d
6.e
10.
10.a
3 Constructing and Writing Proofs in Mathematics
3.1 Direct Proofs
Exercises
1.
1.a
1.b
1.c
2.
2.a
2.d
3.
3.b
3.d
3.e
3.f
3.g
4.
4.a
4.b
5.
8.
8.a
8.b
11.
11.a
11.b
12.
12.b
15.
15.c
3.2 More Methods of Proof
Exercises
1.
1.a
1.b
1.c
1.d
2.
2.a
3.
3.a
3.b
4.
4.a
4.b
4.c
6.
6.a
6.b
6.c
8.
8.c
9.
10.
15.
16.
17.
17.b
18.
3.3 Proof by Contradiction
Exercises
1.
1.a
2.
2.a
2.b
2.d
3.
3.a
3.b
5.
5.a
5.b
6.
6.a
6.b
11.
11.a
12.
13.
13.a
14.
3.4 Using Cases in Proofs
Exercises
1.
2.
3.
4.
5.
5.a
5.c
6.
6.a
8.
9.
9.b
10.
10.a
11.
11.a
12.
12.b
3.5 The Division Algorithm and Congruence
Exercises
2.
2.a
2.b
3.
4.
4.a
4.b
5.
5.a
5.b
5.c
6.
7.
7.a
7.b
8.
8.a
9.
10.
10.b
10.c
11.
11.a
3.6 Review of Proof Methods
Exercises
2.
2.c
5.
7.
4 Mathematical Induction
4.1 The Principle of Mathematical Induction
Exercises
1.
1.a
1.b
1.c
1.d
2.
2.a
2.b
3.
3.a
6.
6.b
6.c
7.
7.e
7.f
8.
8.a
13.
14.
4.2 Other Forms of Mathematical Induction
Exercises
1.
1.a
2.
5.
8.
12.
16.
16.a
16.b
4.3 Induction and Recursion
Exercises
1.
2.
2.a
2.c
2.f
6.
8.
9.
9.a
9.b
12.
12.a
12.b
13.
13.a
13.b
14.
14.a
16.
16.b
18.
18.a
5 Set Theory
5.1 Sets and Operations on Sets
Exercises
1.
1.a
1.b
1.c
1.d
1.e
2.
2.a
2.b
3.
3.a
3.b
3.c
3.d
3.e
3.f
3.g
3.h
3.i
3.j
3.k
3.l
3.m
3.n
4.
5.
5.a
5.b
5.c
5.d
6.
6.a
7.
7.a
7.b
7.c
7.d
7.e
7.f
7.g
7.h
7.i
7.j
7.k
7.l
7.m
7.n
9.
9.b
10.
10.a
5.2 Proving Set Relationships
Exercises
1.
1.a
1.b
3.
3.a
3.b
3.c
5.
5.a
5.c
7.
7.a
7.b
7.e
10.
12.
12.a
15.
15.a
15.b
5.3 Properties of Set Operations
Exercises
1.
1.a
1.c
2.
4.
4.a
4.c
6.
6.a
9.
9.a
5.4 Cartesian Products
Exercises
1.
1.a
1.b
1.c
1.d
1.e
1.f
1.g
1.h
3.
4.
5.5 Indexed Families of Sets
Exercises
1.
1.a
1.d
2.
2.a
2.c
2.d
2.f
3.
3.a
3.b
4.
4.a
5.
5.a
8.
8.a
12.
12.a
6 Functions
6.1 Introduction to Functions
Exercises
1.
1.a
1.b
1.d
3.
3.a
3.b
3.c
4.
4.b
4.c
4.d
5.
5.b
5.d
6. The number of divisors function.
6.a
6.b
6.c
6.d
6.e
6.f
6.g
7.
7.a
7.b
7.c
6.2 More about Functions
Exercises
1.
1.a
1.b
1.c
3.
3.a
3.b
3.c
3.d
4.
4.a
4.d
5.
5.a
5.c
6.
7.
7.a
7.b
8.
8.a
8.c
9.
9.a
6.3 Injections, Surjections, and Bijections
Exercises
2.
2.a
2.c
3.
3.a
3.b
3.h
4.
4.a
4.b
7.
9.
9.a
9.b
6.4 Composition of Functions
Exercises
2.
3.
3.a
3.b
3.c
3.d
4.
4.a
5.
5.a
6.
7.
7.a
7.b
7.f
6.5 Inverse Functions
Exercises
2.
2.b
2.d
3.
3.a
3.b
3.c
4.
6.
6.a
6.b
7.
7.a
7.b
6.6 Functions Acting on Sets
Exercises
1.
1.a
1.d
1.f
1.h
2.
2.b
2.d
2.e
2.f
3.
3.a
3.b
4.
4.a
5.
6.
9.
9.a
9.b
7 Equivalence Relations
7.1 Relations
Exercises
1.
1.a
1.b
1.c
2.
2.a
2.b
2.c
2.d
3.
3.a
3.b
4.
4.a
4.b
4.c
4.d
6.
6.a
6.b
6.c
6.d
9.
9.a
9.b
7.2 Equivalence Relations
Exercises
1.
3.
4.
6.
6.a
6.b
10.
10.a
15.
15.c
7.3 Equivalence Classes
Exercises
1.
2.
3.
4.
5.
5.a
6.
6.a
9.
9.a
9.c
7.4 Modular Arithmetic
Exercises
1.
1.a
1.b
2.
2.a
2.e
2.g
3.
3.a
3.b
5.
5.a
17.
17.a
19.
8 Topics in Number Theory
8.1 The Greatest Common Divisor
Exercises
1.
1.a
1.b
1.c
1.d
2.
2.a
4.
4.a
4.b
5.
5.a
5.b
5.e
6.
6.a
7.
7.a
7.b
7.c
8.2 Prime Numbers and Prime Factorizations
Exercises
1.
1.a
1.b
2.
3.
4.
4.a
4.b
7.
7.a
7.b
7.c
9.
11.
16.
16.b
17.
18.
8.3 Linear Diophantine Equations
Exercises
8.3.3.
8.3.3.a
8.3.3.b
8.3.3.c
8.3.3.d
8.3.4.
8.3.5.
8.3.6.
8.3.6.a
8.3.6.b
8.3.6.c
9 Finite and Infinite Sets
9.1 Finite Sets
Exercises
2.
3.
4.
5.
5.a
5.b
7.
7.a
8.
8.a
10.
9.2 Countable Sets
Exercises
1.
1.a
1.b
1.c
1.d
2.
2.a
2.e
2.f
3.
5.
6.
7.
8.
9.
9.a
9.b
11.
12.
9.3 Uncountable Sets
Exercises
1.
1.a
1.b
2.
3.
4.